Subgroups — C₂×C₃×#3325 #10679752

Order 768.

2630 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3325	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#482	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#469	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#447	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃×#418	C₂	1	2	yes	maximal, complemented
13	C₂×#3325	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₃×D₄	—	2	1	no	—
15	C₂³×C₃×D₄	C₂²	1	2	yes	—
16	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
17	C₂²×C₃×D₈	C₂²	1	2	yes	complemented
18	C₂²×C₃×M₁₆	C₂²	1	2	yes	complemented
19	C₃×#555	—	2	4	no	—
20	C₃×#555	C₂²	1	8	yes	complemented
21	C₃×D₄²	—	2	4	no	—
22	C₃×D₄²	C₂²	1	8	yes	complemented
23	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃×#127	C₂²	1	1	yes	characteristic
25	C₂×C₃×#124	—	2	2	no	—
26	C₂×C₃×#121	—	2	1	no	—
27	C₂×C₃×#121	—	2	1	no	—
28	C₂×C₃×#121	C₂²	1	2	yes	—
29	C₂×C₃×#121	C₂²	1	2	yes	complemented
30	C₂×C₃×#120	—	2	2	no	—
31	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₃×#117	—	2	1	no	—
35	C₂×C₃×#117	C₂²	1	2	yes	complemented
36	C₃×#486	C₂²	1	8	yes	complemented
37	C₃×#482	C₂²	1	16	yes	complemented
38	C₃×#476	C₂²	1	8	yes	complemented
39	C₃×#469	C₂²	1	16	yes	complemented
40	C₃×#459	C₂²	1	8	yes	complemented
41	C₃×#447	C₂²	1	16	yes	complemented
42	C₃×C₄×D₈	C₂²	1	8	yes	complemented
43	C₃×#436	C₂²	1	8	yes	complemented
44	C₂×C₃×#107	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₃×#106	C₂²	1	2	yes	complemented
46	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
47	C₃×#418	C₂²	1	16	yes	complemented
48	C₂×C₃×#102	C₂²	1	1	yes	characteristic
49	C₂×C₃×#102	C₂²	1	2	yes	—
50	C₂×C₃×#102	C₂²	1	2	yes	complemented
51	C₂×C₃×#102	C₂²	1	2	yes	complemented
52	C₂×C₃×#102	C₂²	1	2	yes	complemented
53	C₂×C₃×#102	C₂²	1	1	yes	characteristic, complemented
54	C₂×C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
55	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
56	C₂×#555	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂×D₄²	C₂×C₃	1	1	yes	characteristic, complemented
58	#3325	C₂×C₃	1	16	yes	direct product factor
59	C₂×#486	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂×#482	C₂×C₃	1	2	yes	complemented
61	C₂×#476	C₂×C₃	1	1	yes	characteristic, complemented
62	C₂×#469	C₂×C₃	1	2	yes	complemented
63	C₂×#459	C₂×C₃	1	1	yes	characteristic, complemented
64	C₂×#447	C₂×C₃	1	2	yes	complemented
65	C₂×C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
66	C₂×#436	C₂×C₃	1	1	yes	characteristic, complemented
67	C₂×#418	C₂×C₃	1	2	yes	complemented
68	C₂⁵×C₃	—	2	2	no	—
69	C₂²×C₃×Q₈	—	2	1	no	—
70	C₂²×C₃×D₄	—	2	8	no	—
71	C₂²×C₃×D₄	—	2	2	no	—
72	C₂²×C₃×D₄	—	2	2	no	—
73	C₂²×C₃×D₄	—	2	8	no	—
74	C₂²×C₃×D₄	—	2	1	no	—
75	C₂²×C₃×D₄	—	2	2	no	—
76	C₂²×C₃×D₄	—	2	4	no	—
77	C₂²×C₃×D₄	—	2	1	no	—
78	C₂²×C₃×D₄	—	4	4	no	—
79	C₂²×C₃×D₄	—	4	8	no	—
80	C₂²×C₃×D₄	D₄	1	1	yes	characteristic
81	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
82	C₂²×C₃×D₄	C₂³	1	2	yes	—
83	C₂²×C₃×D₄	C₂³	1	8	yes	—
84	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
85	C₂³×C₃×C₄	—	2	1	no	—
86	C₂³×C₃×C₄	—	2	1	no	—
87	C₂³×C₃×C₄	C₂³	1	2	yes	—
88	C₂×C₃×D₈	—	2	4	no	—
89	C₂×C₃×D₈	—	4	4	no	—
90	C₂×C₃×D₈	C₂³	1	4	yes	—
91	C₂×C₃×D₈	C₂³	1	8	yes	complemented
92	C₂×C₃×M₁₆	—	4	4	no	—
93	C₂×C₃×M₁₆	C₂³	1	8	yes	complemented
94	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
95	C₂²×C₃×C₈	C₂³	1	2	yes	—
96	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
97	C₃×C₄⋊Q₈	—	2	2	no	—
98	C₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
99	C₃×#127	—	2	2	no	—
100	C₃×#127	C₂³	1	4	yes	—
101	C₃×#124	—	2	16	no	—
102	C₃×#121	—	2	8	no	—
103	C₃×#121	—	2	4	no	—
104	C₃×#121	—	2	4	no	—
105	C₃×#121	—	2	8	no	—
106	C₃×#121	C₂³	1	8	yes	—
107	C₃×#121	C₂³	1	8	yes	complemented
108	C₃×#120	—	2	16	no	—
109	C₃×C₄×D₄	—	2	2	no	—
110	C₃×C₄×D₄	—	2	2	no	—
111	C₃×C₄×D₄	D₄	1	4	yes	—
112	C₃×C₄×D₄	C₂³	1	4	yes	—
113	C₃×C₄×D₄	C₂³	1	4	yes	—
114	C₃×C₄×D₄	C₂³	1	4	yes	complemented
115	C₃×#117	—	2	4	no	—
116	C₃×#117	—	2	8	no	—
117	C₃×#117	C₂³	1	8	yes	complemented
118	C₂×C₃×C₄⋊C₄	—	2	1	no	—
119	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
120	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
121	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
122	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
123	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
124	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
125	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
126	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
127	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
128	C₂×C₃×C₂²⋊C₄	D₄	1	2	yes	—
129	C₂×C₃×C₄²	—	2	2	no	—
130	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
131	C₃×#107	C₂³	1	4	yes	complemented
132	C₃×#106	C₂³	1	8	yes	complemented
133	C₃×C₄⋊C₈	C₂³	1	4	yes	complemented
134	C₃×#102	C₂³	1	4	yes	—
135	C₃×#102	C₂³	1	8	yes	—
136	C₃×#102	C₂³	1	8	yes	complemented
137	C₃×#102	C₂³	1	8	yes	complemented
138	C₃×#102	C₂³	1	4	yes	complemented
139	C₃×#102	C₂³	1	8	yes	complemented
140	C₃×C₂²⋊C₈	C₂³	1	8	yes	complemented
141	C₃×C₄×C₈	C₂³	1	4	yes	complemented
142	C₂³×D₄	—	2	1	no	—
143	C₂³×D₄	C₂²×C₃	1	2	yes	—
144	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
145	C₂²×D₈	C₂²×C₃	1	2	yes	complemented
146	C₂²×M₁₆	C₂²×C₃	1	2	yes	complemented
147	#555	—	2	4	no	—
148	#555	C₂²×C₃	1	8	yes	complemented
149	D₄²	—	2	4	no	—
150	D₄²	C₂²×C₃	1	8	yes	complemented
151	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
152	C₂×#127	C₂²×C₃	1	1	yes	characteristic
153	C₂×#124	—	2	2	no	—
154	C₂×#121	—	2	1	no	—
155	C₂×#121	—	2	1	no	—
156	C₂×#121	C₂²×C₃	1	2	yes	—
157	C₂×#121	C₂²×C₃	1	2	yes	complemented
158	C₂×#120	—	2	2	no	—
159	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
160	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
161	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
162	C₂×#117	—	2	1	no	—
163	C₂×#117	C₂²×C₃	1	2	yes	complemented
164	#486	C₂²×C₃	1	8	yes	complemented
165	#482	C₂²×C₃	1	16	yes	complemented
166	#476	C₂²×C₃	1	8	yes	complemented
167	#469	C₂²×C₃	1	16	yes	complemented
168	#459	C₂²×C₃	1	8	yes	complemented
169	#447	C₂²×C₃	1	16	yes	complemented
170	C₄×D₈	C₂²×C₃	1	8	yes	complemented
171	#436	C₂²×C₃	1	8	yes	complemented
172	C₂×#107	C₂²×C₃	1	1	yes	characteristic, complemented
173	C₂×#106	C₂²×C₃	1	2	yes	complemented
174	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
175	#418	C₂²×C₃	1	16	yes	complemented
176	C₂×#102	C₂²×C₃	1	1	yes	characteristic
177	C₂×#102	C₂²×C₃	1	2	yes	—
178	C₂×#102	C₂²×C₃	1	2	yes	complemented
179	C₂×#102	C₂²×C₃	1	2	yes	complemented
180	C₂×#102	C₂²×C₃	1	2	yes	complemented
181	C₂×#102	C₂²×C₃	1	1	yes	characteristic, complemented
182	C₂×C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
183	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
184	C₂⁴×C₃	—	2	1	no	—
185	C₂⁴×C₃	—	2	2	no	—
186	C₂⁴×C₃	—	2	1	no	—
187	C₂⁴×C₃	—	2	8	no	—
188	C₂⁴×C₃	—	4	8	no	—
189	C₂⁴×C₃	—	4	8	no	—
190	C₂⁴×C₃	—	8	4	no	—
191	C₂⁴×C₃	C₂×D₄	1	2	yes	—
192	C₂×C₃×Q₈	—	2	4	no	—
193	C₂×C₃×Q₈	—	8	2	no	—
194	C₂×C₃×D₄	—	2	16	no	—
195	C₂×C₃×D₄	—	2	8	no	—
196	C₂×C₃×D₄	—	2	8	no	—
197	C₂×C₃×D₄	—	2	4	no	—
198	C₂×C₃×D₄	—	2	1	no	—
199	C₂×C₃×D₄	—	2	1	no	—
200	C₂×C₃×D₄	—	2	8	no	—
201	C₂×C₃×D₄	—	2	2	no	—
202	C₂×C₃×D₄	—	2	4	no	—
203	C₂×C₃×D₄	—	2	4	no	—
204	C₂×C₃×D₄	—	2	2	no	—
205	C₂×C₃×D₄	—	2	4	no	—
206	C₂×C₃×D₄	—	2	2	no	—
207	C₂×C₃×D₄	—	2	2	no	—
208	C₂×C₃×D₄	—	2	4	no	—
209	C₂×C₃×D₄	—	4	8	no	—
210	C₂×C₃×D₄	—	4	16	no	—
211	C₂×C₃×D₄	—	4	8	no	—
212	C₂×C₃×D₄	—	4	32	no	—
213	C₂×C₃×D₄	—	4	8	no	—
214	C₂×C₃×D₄	—	4	4	no	—
215	C₂×C₃×D₄	—	8	2	no	—
216	C₂×C₃×D₄	—	8	4	no	—
217	C₂×C₃×D₄	C₂×D₄	1	4	yes	—
218	C₂×C₃×D₄	#41	1	4	yes	—
219	C₂×C₃×D₄	C₂⁴	1	4	yes	—
220	C₂×C₃×D₄	C₂⁴	1	2	yes	—
221	C₂×C₃×D₄	C₂⁴	1	2	yes	—
222	C₂²×C₃×C₄	—	2	4	no	—
223	C₂²×C₃×C₄	—	2	4	no	—
224	C₂²×C₃×C₄	—	2	1	no	—
225	C₂²×C₃×C₄	—	2	2	no	—
226	C₂²×C₃×C₄	—	2	1	no	—
227	C₂²×C₃×C₄	—	2	4	no	—
228	C₂²×C₃×C₄	—	2	4	no	—
229	C₂²×C₃×C₄	—	4	4	no	—
230	C₂²×C₃×C₄	—	4	4	no	—
231	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
232	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
233	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
234	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
235	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
236	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
237	C₃×D₈	—	2	8	no	—
238	C₃×D₈	—	4	8	no	—
239	C₃×M₁₆	—	4	8	no	—
240	C₂×C₃×C₈	—	2	2	no	—
241	C₂×C₃×C₈	—	2	2	no	—
242	C₂×C₃×C₈	—	2	4	no	—
243	C₂×C₃×C₈	C₂⁴	1	2	yes	—
244	C₂×C₃×C₈	C₂⁴	1	2	yes	—
245	C₂×C₃×C₈	C₂⁴	1	4	yes	—
246	C₃×C₄⋊C₄	—	2	4	no	—
247	C₃×C₄⋊C₄	—	2	1	no	—
248	C₃×C₄⋊C₄	—	2	2	no	—
249	C₃×C₄⋊C₄	—	2	1	no	—
250	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
251	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
252	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
253	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
254	C₃×C₂²⋊C₄	—	2	8	no	—
255	C₃×C₂²⋊C₄	—	2	8	no	—
256	C₃×C₂²⋊C₄	—	2	4	no	—
257	C₃×C₂²⋊C₄	—	2	4	no	—
258	C₃×C₂²⋊C₄	—	2	8	no	—
259	C₃×C₂²⋊C₄	C₂×D₄	1	8	yes	—
260	C₃×C₄²	—	2	8	no	—
261	C₃×C₄²	C₂×D₄	1	2	yes	—
262	C₃×C₄²	C₂⁴	1	2	yes	—
263	C₂⁵	—	2	2	no	—
264	C₂²×Q₈	—	2	1	no	—
265	C₂²×D₄	—	2	8	no	—
266	C₂²×D₄	—	2	2	no	—
267	C₂²×D₄	—	2	2	no	—
268	C₂²×D₄	—	2	8	no	—
269	C₂²×D₄	—	2	2	no	—
270	C₂²×D₄	—	2	1	no	—
271	C₂²×D₄	—	2	4	no	—
272	C₂²×D₄	—	2	1	no	—
273	C₂²×D₄	—	4	4	no	—
274	C₂²×D₄	—	4	8	no	—
275	C₂²×D₄	C₃×D₄	1	1	yes	characteristic
276	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
277	C₂²×D₄	C₂³×C₃	1	2	yes	—
278	C₂²×D₄	C₂³×C₃	1	8	yes	—
279	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
280	C₂³×C₄	—	2	1	no	—
281	C₂³×C₄	—	2	1	no	—
282	C₂³×C₄	C₂³×C₃	1	2	yes	—
283	C₂×D₈	—	2	4	no	—
284	C₂×D₈	—	4	4	no	—
285	C₂×D₈	C₂³×C₃	1	4	yes	—
286	C₂×D₈	C₂³×C₃	1	8	yes	complemented
287	C₂×M₁₆	—	4	4	no	—
288	C₂×M₁₆	C₂³×C₃	1	8	yes	complemented
289	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₃	1	2	yes	—
291	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
292	C₄⋊Q₈	—	2	2	no	—
293	C₄⋊Q₈	C₂³×C₃	1	4	yes	complemented
294	#127	—	2	2	no	—
295	#127	C₂³×C₃	1	4	yes	—
296	#124	—	2	16	no	—
297	#121	—	2	8	no	—
298	#121	—	2	4	no	—
299	#121	—	2	4	no	—
300	#121	—	2	8	no	—
301	#121	C₂³×C₃	1	8	yes	—
302	#121	C₂³×C₃	1	8	yes	complemented
303	#120	—	2	16	no	—
304	C₄×D₄	—	2	2	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	C₃×D₄	1	4	yes	—
307	C₄×D₄	C₂³×C₃	1	4	yes	—
308	C₄×D₄	C₂³×C₃	1	4	yes	—
309	C₄×D₄	C₂³×C₃	1	4	yes	complemented
310	#117	—	2	4	no	—
311	#117	—	2	8	no	—
312	#117	C₂³×C₃	1	8	yes	complemented
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
316	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	—	2	1	no	—
321	C₂×C₂²⋊C₄	—	2	1	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	C₃×D₄	1	2	yes	—
324	C₂×C₄²	—	2	2	no	—
325	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
326	#107	C₂³×C₃	1	4	yes	complemented
327	#106	C₂³×C₃	1	8	yes	complemented
328	C₄⋊C₈	C₂³×C₃	1	4	yes	complemented
329	#102	C₂³×C₃	1	4	yes	—
330	#102	C₂³×C₃	1	8	yes	—
331	#102	C₂³×C₃	1	8	yes	complemented
332	#102	C₂³×C₃	1	8	yes	complemented
333	#102	C₂³×C₃	1	4	yes	complemented
334	#102	C₂³×C₃	1	8	yes	complemented
335	C₂²⋊C₈	C₂³×C₃	1	8	yes	complemented
336	C₄×C₈	C₂³×C₃	1	4	yes	complemented
337	C₂³×C₃	—	2	4	no	—
338	C₂³×C₃	—	2	4	no	—
339	C₂³×C₃	—	2	4	no	—
340	C₂³×C₃	—	2	4	no	—
341	C₂³×C₃	—	2	2	no	—
342	C₂³×C₃	—	2	2	no	—
343	C₂³×C₃	—	2	4	no	—
344	C₂³×C₃	—	4	16	no	—
345	C₂³×C₃	—	4	16	no	—
346	C₂³×C₃	—	4	2	no	—
347	C₂³×C₃	—	4	2	no	—
348	C₂³×C₃	—	4	4	no	—
349	C₂³×C₃	—	4	2	no	—
350	C₂³×C₃	—	4	2	no	—
351	C₂³×C₃	—	4	4	no	—
352	C₂³×C₃	—	4	2	no	—
353	C₂³×C₃	—	4	4	no	—
354	C₂³×C₃	—	8	8	no	—
355	C₂³×C₃	—	8	32	no	—
356	C₂³×C₃	C₂²×D₄	1	4	yes	—
357	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
358	C₃×Q₈	—	8	4	no	—
359	C₃×D₄	—	2	4	no	—
360	C₃×D₄	—	2	8	no	—
361	C₃×D₄	—	2	2	no	—
362	C₃×D₄	—	2	4	no	—
363	C₃×D₄	—	2	2	no	—
364	C₃×D₄	—	2	2	no	—
365	C₃×D₄	—	2	4	no	—
366	C₃×D₄	—	4	16	no	—
367	C₃×D₄	—	4	16	no	—
368	C₃×D₄	—	4	2	no	—
369	C₃×D₄	—	4	4	no	—
370	C₃×D₄	—	4	8	no	—
371	C₃×D₄	—	8	4	no	—
372	C₃×D₄	—	8	8	no	—
373	C₃×D₄	C₂×#41	1	4	yes	—
374	C₂×C₃×C₄	—	2	4	no	—
375	C₂×C₃×C₄	—	2	2	no	—
376	C₂×C₃×C₄	—	2	8	no	—
377	C₂×C₃×C₄	—	2	4	no	—
378	C₂×C₃×C₄	—	2	4	no	—
379	C₂×C₃×C₄	—	2	2	no	—
380	C₂×C₃×C₄	—	2	4	no	—
381	C₂×C₃×C₄	—	2	2	no	—
382	C₂×C₃×C₄	—	4	2	no	—
383	C₂×C₃×C₄	—	4	2	no	—
384	C₂×C₃×C₄	—	4	4	no	—
385	C₂×C₃×C₄	—	4	8	no	—
386	C₂×C₃×C₄	—	4	8	no	—
387	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
388	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
389	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
390	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
391	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
392	C₂×C₃×C₄	C₂×#41	1	2	yes	—
393	C₂×C₃×C₄	H(5,𝔽₂)	1	2	yes	—
394	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
395	C₃×C₈	—	2	2	no	—
396	C₃×C₈	—	2	2	no	—
397	C₃×C₈	—	2	4	no	—
398	C₂⁴	—	2	1	no	—
399	C₂⁴	—	2	1	no	—
400	C₂⁴	—	2	2	no	—
401	C₂⁴	—	2	8	no	—
402	C₂⁴	—	4	8	no	—
403	C₂⁴	—	4	8	no	—
404	C₂⁴	—	8	4	no	—
405	C₂⁴	C₂×C₃×D₄	1	2	yes	—
406	C₂×Q₈	—	2	4	no	—
407	C₂×Q₈	—	8	2	no	—
408	C₂×D₄	—	2	16	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	16	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	4	32	no	—
427	C₂×D₄	—	4	8	no	—
428	C₂×D₄	—	4	4	no	—
429	C₂×D₄	—	8	2	no	—
430	C₂×D₄	—	8	4	no	—
431	C₂×D₄	C₂×C₃×D₄	1	4	yes	—
432	C₂×D₄	C₃×#41	1	4	yes	—
433	C₂×D₄	C₂⁴×C₃	1	4	yes	—
434	C₂×D₄	C₂⁴×C₃	1	2	yes	—
435	C₂×D₄	C₂⁴×C₃	1	2	yes	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	4	4	no	—
444	C₂²×C₄	—	4	4	no	—
445	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
446	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
447	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
448	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
451	D₈	—	2	8	no	—
452	D₈	—	4	8	no	—
453	M₁₆	—	4	8	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	4	no	—
457	C₂×C₈	C₂⁴×C₃	1	2	yes	—
458	C₂×C₈	C₂⁴×C₃	1	2	yes	—
459	C₂×C₈	C₂⁴×C₃	1	4	yes	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
465	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
466	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
467	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	C₂×C₃×D₄	1	8	yes	—
474	C₄²	—	2	8	no	—
475	C₄²	C₂×C₃×D₄	1	2	yes	—
476	C₄²	C₂⁴×C₃	1	2	yes	—
477	C₂²×C₃	—	2	4	no	—
478	C₂²×C₃	—	2	4	no	—
479	C₂²×C₃	—	2	4	no	—
480	C₂²×C₃	—	2	4	no	—
481	C₂²×C₃	—	4	2	no	—
482	C₂²×C₃	—	4	8	no	—
483	C₂²×C₃	—	4	8	no	—
484	C₂²×C₃	—	4	2	no	—
485	C₂²×C₃	—	4	4	no	—
486	C₂²×C₃	—	4	2	no	—
487	C₂²×C₃	—	4	2	no	—
488	C₂²×C₃	—	4	4	no	—
489	C₂²×C₃	—	4	2	no	—
490	C₂²×C₃	—	4	4	no	—
491	C₂²×C₃	—	8	32	no	—
492	C₂²×C₃	—	8	4	no	—
493	C₂²×C₃	—	8	8	no	—
494	C₂²×C₃	#546	1	2	yes	—
495	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
496	C₂²×C₃	#576	1	2	yes	—
497	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
498	C₃×C₄	—	2	4	no	—
499	C₃×C₄	—	2	2	no	—
500	C₃×C₄	—	4	2	no	—
501	C₃×C₄	—	4	2	no	—
502	C₃×C₄	—	4	4	no	—
503	C₃×C₄	#546	1	2	yes	—
504	C₃×C₄	C₂×AGL(1,ℤ/8)	1	4	yes	—
505	C₃×C₄	C₂²×#41	1	1	yes	characteristic
506	C₃×C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
507	C₂³	—	2	4	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	4	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	2	2	no	—
513	C₂³	—	2	4	no	—
514	C₂³	—	4	16	no	—
515	C₂³	—	4	16	no	—
516	C₂³	—	4	2	no	—
517	C₂³	—	4	2	no	—
518	C₂³	—	4	2	no	—
519	C₂³	—	4	4	no	—
520	C₂³	—	4	2	no	—
521	C₂³	—	4	4	no	—
522	C₂³	—	4	4	no	—
523	C₂³	—	4	2	no	—
524	C₂³	—	8	8	no	—
525	C₂³	—	8	32	no	—
526	C₂³	C₂²×C₃×D₄	1	4	yes	—
527	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
528	Q₈	—	8	4	no	—
529	D₄	—	2	8	no	—
530	D₄	—	2	2	no	—
531	D₄	—	2	4	no	—
532	D₄	—	2	2	no	—
533	D₄	—	2	2	no	—
534	D₄	—	2	4	no	—
535	D₄	—	2	4	no	—
536	D₄	—	4	16	no	—
537	D₄	—	4	16	no	—
538	D₄	—	4	8	no	—
539	D₄	—	4	4	no	—
540	D₄	—	4	2	no	—
541	D₄	—	8	4	no	—
542	D₄	—	8	8	no	—
543	D₄	C₂×C₃×#41	1	4	yes	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	8	no	—
556	C₂×C₄	—	4	8	no	—
557	C₂×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
558	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
559	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
560	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
561	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
562	C₂×C₄	C₂×C₃×#41	1	2	yes	—
563	C₂×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
564	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
565	C₈	—	2	2	no	—
566	C₈	—	2	2	no	—
567	C₈	—	2	4	no	—
568	C₂×C₃	—	4	4	no	—
569	C₂×C₃	—	4	4	no	—
570	C₂×C₃	—	8	2	no	—
571	C₂×C₃	—	8	4	no	—
572	C₂×C₃	#3325	1	4	yes	direct product factor
573	C₂×C₃	C₂×#546	1	1	yes	characteristic
574	C₂×C₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
575	C₂×C₃	C₂×#576	1	1	yes	characteristic
576	C₂²	—	2	4	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	2	4	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	8	no	—
583	C₂²	—	4	8	no	—
584	C₂²	—	4	4	no	—
585	C₂²	—	4	2	no	—
586	C₂²	—	4	2	no	—
587	C₂²	—	4	4	no	—
588	C₂²	—	4	2	no	—
589	C₂²	—	4	4	no	—
590	C₂²	—	8	32	no	—
591	C₂²	—	8	8	no	—
592	C₂²	—	8	4	no	—
593	C₂²	C₃×#546	1	2	yes	—
594	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
595	C₂²	C₃×#576	1	2	yes	—
596	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
597	C₄	—	2	4	no	—
598	C₄	—	2	2	no	—
599	C₄	—	4	2	no	—
600	C₄	—	4	2	no	—
601	C₄	—	4	4	no	—
602	C₄	C₃×#546	1	2	yes	—
603	C₄	C₂×C₃×AGL(1,ℤ/8)	1	4	yes	—
604	C₄	C₂²×C₃×#41	1	1	yes	characteristic
605	C₄	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
606	C₃	C₂×#3325	1	1	yes	characteristic, direct product factor, Sylow
607	C₂	—	4	4	no	—
608	C₂	—	4	4	no	—
609	C₂	—	8	4	no	—
610	C₂	—	8	2	no	—
611	C₂	C₃×#3325	1	4	yes	direct product factor
612	C₂	C₂×C₃×#546	1	1	yes	characteristic
613	C₂	C₂²×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
614	C₂	C₂×C₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄×D₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×#555	C₂	semidirect
2. C₂×C₃×D₄²	C₂	semidirect
3. C₃×#3325	C₂	direct (N × H)
4. C₂×C₃×#486	C₂	semidirect
5. C₂×C₃×#482	C₂	semidirect
6. C₂×C₃×#476	C₂	semidirect
7. C₂×C₃×#469	C₂	semidirect
8. C₂×C₃×#459	C₂	semidirect
9. C₂×C₃×#447	C₂	semidirect
10. C₂×C₃×C₄×D₈	C₂	semidirect
11. C₂×C₃×#436	C₂	semidirect
12. C₂×C₃×#418	C₂	semidirect
13. C₂×#3325	C₃	direct (N × H)
17. C₂²×C₃×D₈	C₂²	semidirect
18. C₂²×C₃×M₁₆	C₂²	semidirect
20. C₃×#555	C₂²	semidirect
22. C₃×D₄²	C₂²	semidirect
23. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
29. C₂×C₃×#121	C₂²	semidirect
33. C₂×C₃×C₄×D₄	C₂²	semidirect
35. C₂×C₃×#117	C₂²	semidirect
36. C₃×#486	C₂²	semidirect
37. C₃×#482	C₂²	semidirect
38. C₃×#476	C₂²	semidirect
39. C₃×#469	C₂²	semidirect
40. C₃×#459	C₂²	semidirect
41. C₃×#447	C₂²	semidirect
42. C₃×C₄×D₈	C₂²	semidirect
43. C₃×#436	C₂²	semidirect
44. C₂×C₃×#107	C₂²	semidirect
45. C₂×C₃×#106	C₂²	semidirect
46. C₂×C₃×C₄⋊C₈	C₂²	semidirect
47. C₃×#418	C₂²	semidirect
50. C₂×C₃×#102	C₂²	semidirect
51. C₂×C₃×#102	C₂²	semidirect
52. C₂×C₃×#102	C₂²	semidirect
53. C₂×C₃×#102	C₂²	semidirect
54. C₂×C₃×C₂²⋊C₈	C₂²	semidirect
55. C₂×C₃×C₄×C₈	C₂²	semidirect
56. C₂×#555	C₂×C₃	semidirect
57. C₂×D₄²	C₂×C₃	semidirect
58. #3325	C₂×C₃	direct (N × H)
59. C₂×#486	C₂×C₃	semidirect
60. C₂×#482	C₂×C₃	semidirect
61. C₂×#476	C₂×C₃	semidirect
62. C₂×#469	C₂×C₃	semidirect
63. C₂×#459	C₂×C₃	semidirect
64. C₂×#447	C₂×C₃	semidirect
65. C₂×C₄×D₈	C₂×C₃	semidirect
66. C₂×#436	C₂×C₃	semidirect
67. C₂×#418	C₂×C₃	semidirect
91. C₂×C₃×D₈	C₂³	semidirect
93. C₂×C₃×M₁₆	C₂³	semidirect
98. C₃×C₄⋊Q₈	C₂³	semidirect
107. C₃×#121	C₂³	semidirect
114. C₃×C₄×D₄	C₂³	semidirect
117. C₃×#117	C₂³	semidirect
131. C₃×#107	C₂³	semidirect
132. C₃×#106	C₂³	semidirect
133. C₃×C₄⋊C₈	C₂³	semidirect
136. C₃×#102	C₂³	semidirect
137. C₃×#102	C₂³	semidirect
138. C₃×#102	C₂³	semidirect
139. C₃×#102	C₂³	semidirect
140. C₃×C₂²⋊C₈	C₂³	semidirect
141. C₃×C₄×C₈	C₂³	semidirect
145. C₂²×D₈	C₂²×C₃	semidirect
146. C₂²×M₁₆	C₂²×C₃	semidirect
148. #555	C₂²×C₃	semidirect
150. D₄²	C₂²×C₃	semidirect
151. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
157. C₂×#121	C₂²×C₃	semidirect
161. C₂×C₄×D₄	C₂²×C₃	semidirect
163. C₂×#117	C₂²×C₃	semidirect
164. #486	C₂²×C₃	semidirect
165. #482	C₂²×C₃	semidirect
166. #476	C₂²×C₃	semidirect
167. #469	C₂²×C₃	semidirect
168. #459	C₂²×C₃	semidirect
169. #447	C₂²×C₃	semidirect
170. C₄×D₈	C₂²×C₃	semidirect
171. #436	C₂²×C₃	semidirect
172. C₂×#107	C₂²×C₃	semidirect
173. C₂×#106	C₂²×C₃	semidirect
174. C₂×C₄⋊C₈	C₂²×C₃	semidirect
175. #418	C₂²×C₃	semidirect
178. C₂×#102	C₂²×C₃	semidirect
179. C₂×#102	C₂²×C₃	semidirect
180. C₂×#102	C₂²×C₃	semidirect
181. C₂×#102	C₂²×C₃	semidirect
182. C₂×C₂²⋊C₈	C₂²×C₃	semidirect
183. C₂×C₄×C₈	C₂²×C₃	semidirect
286. C₂×D₈	C₂³×C₃	semidirect
288. C₂×M₁₆	C₂³×C₃	semidirect
293. C₄⋊Q₈	C₂³×C₃	semidirect
302. #121	C₂³×C₃	semidirect
309. C₄×D₄	C₂³×C₃	semidirect
312. #117	C₂³×C₃	semidirect
326. #107	C₂³×C₃	semidirect
327. #106	C₂³×C₃	semidirect
328. C₄⋊C₈	C₂³×C₃	semidirect
331. #102	C₂³×C₃	semidirect
332. #102	C₂³×C₃	semidirect
333. #102	C₂³×C₃	semidirect
334. #102	C₂³×C₃	semidirect
335. C₂²⋊C₈	C₂³×C₃	semidirect
336. C₄×C₈	C₂³×C₃	semidirect
572. C₂×C₃	#3325	direct (N × H)
606. C₃	C₂×#3325	direct (N × H)
611. C₂	C₃×#3325	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#555	2
C₂×C₃×D₄²	2
C₃×#3325	2
C₂×C₃×#486	2
C₂×C₃×#482	2
C₂×C₃×#476	2
C₂×C₃×#469	2
C₂×C₃×#459	2
C₂×C₃×#447	2
C₂×C₃×C₄×D₈	2
C₂×C₃×#436	2
C₂×C₃×#418	2
C₂×#3325	3

Subgroups — C2×C3×#3325 #10679752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#3325 #10679752