Subgroups — C₂×C₃×#3096 #10679119

Order 768.

2362 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#535	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3096	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#481	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#482	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#482	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#480	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×#418	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#3096	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₃×#41	C₂²	1	1	yes	characteristic
15	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
18	C₃×#535	—	2	4	no	—
19	C₃×#535	C₂²	1	8	yes	—
20	C₂×C₃×#127	—	2	1	no	—
21	C₂×C₃×#124	—	2	1	no	—
22	C₂×C₃×#121	C₂²	1	2	yes	—
23	C₂×C₃×#121	C₂²	1	2	yes	—
24	C₂×C₃×#120	—	2	2	no	—
25	C₃×#520	—	2	4	no	—
26	C₃×#520	C₂²	1	8	yes	complemented
27	C₂×C₃×C₄×Q₈	—	2	1	no	—
28	C₂×C₃×C₄×D₄	—	2	1	no	—
29	C₂×C₃×C₄×D₄	—	2	2	no	—
30	C₂×C₃×#117	C₂²	1	1	yes	characteristic
31	C₂×C₃×#117	C₂²	1	1	yes	characteristic
32	C₂×C₃×#117	C₂²	1	2	yes	complemented
33	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
35	C₃×#482	C₂²	1	16	yes	complemented
36	C₃×#482	C₂²	1	16	yes	complemented
37	C₃×#481	C₂²	1	16	yes	complemented
38	C₃×#480	C₂²	1	16	yes	complemented
39	C₃×#428	C₂²	1	8	yes	complemented
40	C₃×#427	C₂²	1	8	yes	complemented
41	C₂×C₃×#107	C₂²	1	1	yes	characteristic, complemented
42	C₂×C₃×#107	C₂²	1	1	yes	characteristic, complemented
43	C₂×C₃×#107	C₂²	1	2	yes	complemented
44	C₂×C₃×#106	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₃×#106	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₃×#106	C₂²	1	2	yes	complemented
47	C₃×#418	C₂²	1	8	yes	complemented
48	C₂×C₃×#102	C₂²	1	2	yes	—
49	C₂×C₃×#102	C₂²	1	2	yes	—
50	C₂×C₃×#102	C₂²	1	2	yes	—
51	C₂×C₃×#102	C₂²	1	2	yes	—
52	C₂×C₃×#102	C₂²	1	8	yes	complemented
53	C₃×#408	C₂²	1	8	yes	complemented
54	C₂×C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
55	C₂×C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
56	C₂×#535	C₂×C₃	1	1	yes	characteristic
57	C₂×#520	C₂×C₃	1	1	yes	characteristic, complemented
58	#3096	C₂×C₃	1	16	yes	direct product factor
59	C₂×#481	C₂×C₃	1	2	yes	complemented
60	C₂×#482	C₂×C₃	1	2	yes	complemented
61	C₂×#482	C₂×C₃	1	2	yes	complemented
62	C₂×#480	C₂×C₃	1	2	yes	complemented
63	C₂×#428	C₂×C₃	1	1	yes	characteristic, complemented
64	C₂×#427	C₂×C₃	1	1	yes	characteristic, complemented
65	C₂×#418	C₂×C₃	1	1	yes	characteristic, complemented
66	C₂²×#102	C₂×C₃	1	1	yes	characteristic, complemented
67	C₂×#408	C₂×C₃	1	1	yes	characteristic, complemented
68	C₂⁵×C₃	—	2	1	no	—
69	C₂×C₃×#41	—	2	4	no	—
70	C₂×C₃×#41	—	2	4	no	—
71	C₂×C₃×#41	D₄	1	4	yes	—
72	C₂×C₃×#41	C₂³	1	4	yes	—
73	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
74	C₂²×C₃×D₄	—	2	2	no	—
75	C₂²×C₃×D₄	—	2	2	no	—
76	C₂²×C₃×D₄	—	2	1	no	—
77	C₂²×C₃×D₄	—	2	2	no	—
78	C₂²×C₃×D₄	—	2	4	no	—
79	C₂²×C₃×D₄	—	2	4	no	—
80	C₂²×C₃×D₄	D₄	1	2	yes	—
81	C₂²×C₃×D₄	D₄	1	1	yes	characteristic
82	C₂²×C₃×D₄	C₂³	1	2	yes	—
83	C₂²×C₃×D₄	C₂³	1	4	yes	—
84	C₂³×C₃×C₄	—	2	1	no	—
85	C₂³×C₃×C₄	—	2	2	no	—
86	C₂³×C₃×C₄	C₂³	1	2	yes	—
87	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
88	C₂×C₃×M₁₆	—	2	4	no	—
89	C₂×C₃×M₁₆	C₂³	1	4	yes	—
90	C₂²×C₃×C₈	—	2	4	no	—
91	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
92	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
93	C₂²×C₃×C₈	C₂³	1	4	yes	—
94	C₂²×C₃×C₈	C₂³	1	2	yes	—
95	C₃×#127	—	2	8	no	—
96	C₃×#124	—	2	8	no	—
97	C₃×#121	—	2	4	no	—
98	C₃×#121	—	2	4	no	—
99	C₃×#121	C₂³	1	8	yes	—
100	C₃×#121	C₂³	1	8	yes	—
101	C₃×#120	—	2	16	no	—
102	C₃×C₄×Q₈	—	2	8	no	—
103	C₃×C₄×D₄	—	2	8	no	—
104	C₃×C₄×D₄	—	2	16	no	—
105	C₃×#117	—	2	2	no	—
106	C₃×#117	—	2	2	no	—
107	C₃×#117	—	2	4	no	—
108	C₃×#117	C₂³	1	4	yes	—
109	C₃×#117	C₂³	1	4	yes	—
110	C₃×#117	C₂³	1	8	yes	complemented
111	C₂×C₃×C₄⋊C₄	—	2	1	no	—
112	C₂×C₃×C₄⋊C₄	—	2	2	no	—
113	C₂×C₃×C₄⋊C₄	—	2	2	no	—
114	C₂×C₃×C₄⋊C₄	—	2	4	no	—
115	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
116	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
117	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
118	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
119	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
120	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
121	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	complemented
122	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
123	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
125	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
126	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
127	C₂×C₃×C₄²	—	2	1	no	—
128	C₂×C₃×C₄²	—	2	1	no	—
129	C₂×C₃×C₄²	—	2	2	no	—
130	C₃×#107	C₂³	1	4	yes	complemented
131	C₃×#107	C₂³	1	4	yes	complemented
132	C₃×#107	C₂³	1	8	yes	complemented
133	C₃×#106	C₂³	1	4	yes	complemented
134	C₃×#106	C₂³	1	4	yes	complemented
135	C₃×#106	C₂³	1	8	yes	complemented
136	C₃×#102	C₂³	1	8	yes	—
137	C₃×#102	C₂³	1	8	yes	—
138	C₃×#102	C₂³	1	8	yes	—
139	C₃×#102	C₂³	1	8	yes	—
140	C₃×C₂²⋊C₈	C₂³	1	8	yes	complemented
141	C₃×C₂²⋊C₈	C₂³	1	8	yes	complemented
142	C₂²×#41	C₂²×C₃	1	1	yes	characteristic
143	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
144	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
145	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
146	#535	—	2	4	no	—
147	#535	C₂²×C₃	1	8	yes	—
148	C₂×#127	—	2	1	no	—
149	C₂×#124	—	2	1	no	—
150	C₂×#121	C₂²×C₃	1	2	yes	—
151	C₂×#121	C₂²×C₃	1	2	yes	—
152	C₂×#120	—	2	2	no	—
153	#520	—	2	4	no	—
154	#520	C₂²×C₃	1	8	yes	complemented
155	C₂×C₄×Q₈	—	2	1	no	—
156	C₂×C₄×D₄	—	2	1	no	—
157	C₂×C₄×D₄	—	2	2	no	—
158	C₂×#117	C₂²×C₃	1	1	yes	characteristic
159	C₂×#117	C₂²×C₃	1	1	yes	characteristic
160	C₂×#117	C₂²×C₃	1	2	yes	complemented
161	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
162	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
163	#482	C₂²×C₃	1	16	yes	complemented
164	#482	C₂²×C₃	1	16	yes	complemented
165	#481	C₂²×C₃	1	16	yes	complemented
166	#480	C₂²×C₃	1	16	yes	complemented
167	#428	C₂²×C₃	1	8	yes	complemented
168	#427	C₂²×C₃	1	8	yes	complemented
169	C₂×#107	C₂²×C₃	1	1	yes	characteristic, complemented
170	C₂×#107	C₂²×C₃	1	2	yes	complemented
171	C₂×#107	C₂²×C₃	1	1	yes	characteristic, complemented
172	C₂×#106	C₂²×C₃	1	1	yes	characteristic, complemented
173	C₂×#106	C₂²×C₃	1	2	yes	complemented
174	C₂×#106	C₂²×C₃	1	1	yes	characteristic, complemented
175	#418	C₂²×C₃	1	8	yes	complemented
176	C₂×#102	C₂²×C₃	1	2	yes	—
177	C₂×#102	C₂²×C₃	1	2	yes	—
178	C₂×#102	C₂²×C₃	1	2	yes	—
179	C₂×#102	C₂²×C₃	1	2	yes	—
180	C₂×#102	C₂²×C₃	1	8	yes	complemented
181	#408	C₂²×C₃	1	8	yes	complemented
182	C₂×C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
183	C₂×C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
184	C₂⁴×C₃	—	2	2	no	—
185	C₂⁴×C₃	—	2	4	no	—
186	C₂⁴×C₃	—	4	4	no	—
187	C₂⁴×C₃	—	8	2	no	—
188	C₂⁴×C₃	—	8	2	no	—
189	C₂⁴×C₃	C₂×D₄	1	2	yes	—
190	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
191	C₃×#41	—	2	16	no	—
192	C₃×#41	—	2	8	no	—
193	C₃×#41	—	2	8	no	—
194	C₂×C₃×Q₈	—	2	4	no	—
195	C₂×C₃×Q₈	C₂×D₄	1	4	yes	—
196	C₂×C₃×D₄	—	2	8	no	—
197	C₂×C₃×D₄	—	2	8	no	—
198	C₂×C₃×D₄	—	2	8	no	—
199	C₂×C₃×D₄	—	2	4	no	—
200	C₂×C₃×D₄	—	2	2	no	—
201	C₂×C₃×D₄	—	2	4	no	—
202	C₂×C₃×D₄	—	2	4	no	—
203	C₂×C₃×D₄	—	2	4	no	—
204	C₂×C₃×D₄	—	2	8	no	—
205	C₂×C₃×D₄	—	2	8	no	—
206	C₂×C₃×D₄	—	4	4	no	—
207	C₂×C₃×D₄	—	4	8	no	—
208	C₂×C₃×D₄	—	8	4	no	—
209	C₂×C₃×D₄	—	8	4	no	—
210	C₂×C₃×D₄	C₂×D₄	1	8	yes	—
211	C₂×C₃×D₄	C₂×D₄	1	4	yes	—
212	C₂×C₃×D₄	C₂⁴	1	4	yes	—
213	C₂²×C₃×C₄	—	2	4	no	—
214	C₂²×C₃×C₄	—	2	4	no	—
215	C₂²×C₃×C₄	—	2	1	no	—
216	C₂²×C₃×C₄	—	2	1	no	—
217	C₂²×C₃×C₄	—	2	1	no	—
218	C₂²×C₃×C₄	—	2	1	no	—
219	C₂²×C₃×C₄	—	2	2	no	—
220	C₂²×C₃×C₄	—	2	4	no	—
221	C₂²×C₃×C₄	—	2	8	no	—
222	C₂²×C₃×C₄	—	8	2	no	—
223	C₂²×C₃×C₄	—	8	4	no	—
224	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
225	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
226	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
227	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
228	C₂²×C₃×C₄	#41	1	4	yes	—
229	C₂²×C₃×C₄	#41	1	4	yes	—
230	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
231	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
232	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
233	C₃×M₁₆	—	2	8	no	—
234	C₂×C₃×C₈	—	2	2	no	—
235	C₂×C₃×C₈	—	2	2	no	—
236	C₂×C₃×C₈	—	2	8	no	—
237	C₂×C₃×C₈	—	2	4	no	—
238	C₂×C₃×C₈	C₂⁴	1	2	yes	—
239	C₂×C₃×C₈	C₂⁴	1	4	yes	—
240	C₂×C₃×C₈	C₂⁴	1	2	yes	—
241	C₃×C₄⋊C₄	—	2	2	no	—
242	C₃×C₄⋊C₄	—	2	1	no	—
243	C₃×C₄⋊C₄	—	2	4	no	—
244	C₃×C₄⋊C₄	—	2	8	no	—
245	C₃×C₄⋊C₄	—	2	1	no	—
246	C₃×C₄⋊C₄	—	2	1	no	—
247	C₃×C₄⋊C₄	—	2	1	no	—
248	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
249	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
250	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
251	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
252	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
253	C₃×C₂²⋊C₄	—	2	4	no	—
254	C₃×C₂²⋊C₄	—	2	4	no	—
255	C₃×C₂²⋊C₄	—	2	8	no	—
256	C₃×C₂²⋊C₄	—	2	8	no	—
257	C₃×C₂²⋊C₄	—	2	8	no	—
258	C₃×C₄²	—	2	4	no	—
259	C₃×C₄²	—	2	4	no	—
260	C₃×C₄²	—	2	8	no	—
261	C₂⁵	—	2	1	no	—
262	C₂×#41	—	2	4	no	—
263	C₂×#41	—	2	4	no	—
264	C₂×#41	C₃×D₄	1	4	yes	—
265	C₂×#41	C₂³×C₃	1	4	yes	—
266	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
267	C₂²×D₄	—	2	2	no	—
268	C₂²×D₄	—	2	2	no	—
269	C₂²×D₄	—	2	1	no	—
270	C₂²×D₄	—	2	2	no	—
271	C₂²×D₄	—	2	4	no	—
272	C₂²×D₄	—	2	4	no	—
273	C₂²×D₄	C₃×D₄	1	2	yes	—
274	C₂²×D₄	C₃×D₄	1	1	yes	characteristic
275	C₂²×D₄	C₂³×C₃	1	2	yes	—
276	C₂²×D₄	C₂³×C₃	1	4	yes	—
277	C₂³×C₄	—	2	1	no	—
278	C₂³×C₄	—	2	2	no	—
279	C₂³×C₄	C₂³×C₃	1	2	yes	—
280	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
281	C₂×M₁₆	—	2	4	no	—
282	C₂×M₁₆	C₂³×C₃	1	4	yes	—
283	C₂²×C₈	—	2	4	no	—
284	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₃	1	4	yes	—
287	C₂²×C₈	C₂³×C₃	1	2	yes	—
288	#127	—	2	8	no	—
289	#124	—	2	8	no	—
290	#121	—	2	4	no	—
291	#121	—	2	4	no	—
292	#121	C₂³×C₃	1	8	yes	—
293	#121	C₂³×C₃	1	8	yes	—
294	#120	—	2	16	no	—
295	C₄×Q₈	—	2	8	no	—
296	C₄×D₄	—	2	8	no	—
297	C₄×D₄	—	2	16	no	—
298	#117	—	2	2	no	—
299	#117	—	2	2	no	—
300	#117	—	2	4	no	—
301	#117	C₂³×C₃	1	4	yes	—
302	#117	C₂³×C₃	1	4	yes	—
303	#117	C₂³×C₃	1	8	yes	complemented
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	—	2	4	no	—
308	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
310	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	complemented
315	C₂×C₂²⋊C₄	—	2	1	no	—
316	C₂×C₂²⋊C₄	—	2	1	no	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	—	2	1	no	—
322	C₂×C₄²	—	2	2	no	—
323	#107	C₂³×C₃	1	4	yes	complemented
324	#107	C₂³×C₃	1	4	yes	complemented
325	#107	C₂³×C₃	1	8	yes	complemented
326	#106	C₂³×C₃	1	4	yes	complemented
327	#106	C₂³×C₃	1	4	yes	complemented
328	#106	C₂³×C₃	1	8	yes	complemented
329	#102	C₂³×C₃	1	8	yes	—
330	#102	C₂³×C₃	1	8	yes	—
331	#102	C₂³×C₃	1	8	yes	—
332	#102	C₂³×C₃	1	8	yes	—
333	C₂²⋊C₈	C₂³×C₃	1	8	yes	complemented
334	C₂²⋊C₈	C₂³×C₃	1	8	yes	complemented
335	C₂³×C₃	—	2	2	no	—
336	C₂³×C₃	—	2	2	no	—
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	—	4	8	no	—
342	C₂³×C₃	—	4	4	no	—
343	C₂³×C₃	—	4	4	no	—
344	C₂³×C₃	—	4	4	no	—
345	C₂³×C₃	—	8	4	no	—
346	C₂³×C₃	—	8	4	no	—
347	C₂³×C₃	—	8	16	no	—
348	C₂³×C₃	#123	1	4	yes	—
349	C₂³×C₃	C₂²×D₄	1	4	yes	—
350	C₂³×C₃	C₂²×D₄	1	2	yes	—
351	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
352	C₃×Q₈	—	2	8	no	—
353	C₃×D₄	—	2	16	no	—
354	C₃×D₄	—	2	8	no	—
355	C₃×D₄	—	2	4	no	—
356	C₃×D₄	—	2	4	no	—
357	C₃×D₄	—	4	4	no	—
358	C₃×D₄	—	4	8	no	—
359	C₃×D₄	—	8	8	no	—
360	C₃×D₄	—	8	8	no	—
361	C₂×C₃×C₄	—	2	4	no	—
362	C₂×C₃×C₄	—	2	8	no	—
363	C₂×C₃×C₄	—	2	4	no	—
364	C₂×C₃×C₄	—	2	4	no	—
365	C₂×C₃×C₄	—	2	2	no	—
366	C₂×C₃×C₄	—	2	2	no	—
367	C₂×C₃×C₄	—	2	2	no	—
368	C₂×C₃×C₄	—	2	2	no	—
369	C₂×C₃×C₄	—	2	4	no	—
370	C₂×C₃×C₄	—	4	2	no	—
371	C₂×C₃×C₄	—	4	4	no	—
372	C₂×C₃×C₄	—	4	2	no	—
373	C₂×C₃×C₄	—	4	2	no	—
374	C₂×C₃×C₄	—	4	2	no	—
375	C₂×C₃×C₄	—	8	4	no	—
376	C₂×C₃×C₄	—	8	8	no	—
377	C₂×C₃×C₄	#123	1	4	yes	—
378	C₂×C₃×C₄	#123	1	8	yes	—
379	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
380	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
381	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
382	C₂×C₃×C₄	C₂×#41	1	2	yes	—
383	C₂×C₃×C₄	C₂×#41	1	2	yes	—
384	C₂×C₃×C₄	C₂×#41	1	4	yes	—
385	C₂×C₃×C₄	C₂×#41	1	4	yes	—
386	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
387	C₃×C₈	—	2	2	no	—
388	C₃×C₈	—	2	2	no	—
389	C₃×C₈	—	2	4	no	—
390	C₂⁴	—	2	2	no	—
391	C₂⁴	—	2	4	no	—
392	C₂⁴	—	4	4	no	—
393	C₂⁴	—	8	2	no	—
394	C₂⁴	—	8	2	no	—
395	C₂⁴	C₂×C₃×D₄	1	2	yes	—
396	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
397	#41	—	2	16	no	—
398	#41	—	2	8	no	—
399	#41	—	2	8	no	—
400	C₂×Q₈	—	2	4	no	—
401	C₂×Q₈	C₂×C₃×D₄	1	4	yes	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	8	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	4	4	no	—
413	C₂×D₄	—	4	8	no	—
414	C₂×D₄	—	8	4	no	—
415	C₂×D₄	—	8	4	no	—
416	C₂×D₄	C₂×C₃×D₄	1	8	yes	—
417	C₂×D₄	C₂×C₃×D₄	1	4	yes	—
418	C₂×D₄	C₂⁴×C₃	1	4	yes	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	8	2	no	—
429	C₂²×C₄	—	8	4	no	—
430	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
431	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
432	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
433	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
434	C₂²×C₄	C₃×#41	1	4	yes	—
435	C₂²×C₄	C₃×#41	1	4	yes	—
436	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
437	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
438	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
439	M₁₆	—	2	8	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	2	8	no	—
443	C₂×C₈	—	2	4	no	—
444	C₂×C₈	C₂⁴×C₃	1	2	yes	—
445	C₂×C₈	C₂⁴×C₃	1	4	yes	—
446	C₂×C₈	C₂⁴×C₃	1	2	yes	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
455	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
456	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
457	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
458	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	8	no	—
463	C₂²⋊C₄	—	2	8	no	—
464	C₄²	—	2	4	no	—
465	C₄²	—	2	4	no	—
466	C₄²	—	2	8	no	—
467	C₂²×C₃	—	2	8	no	—
468	C₂²×C₃	—	2	2	no	—
469	C₂²×C₃	—	2	2	no	—
470	C₂²×C₃	—	2	4	no	—
471	C₂²×C₃	—	2	4	no	—
472	C₂²×C₃	—	2	4	no	—
473	C₂²×C₃	—	4	8	no	—
474	C₂²×C₃	—	4	4	no	—
475	C₂²×C₃	—	4	4	no	—
476	C₂²×C₃	—	4	4	no	—
477	C₂²×C₃	—	8	16	no	—
478	C₂²×C₃	—	8	8	no	—
479	C₂²×C₃	C₂×#123	1	4	yes	—
480	C₂²×C₃	C₂×#123	1	2	yes	—
481	C₂²×C₃	#576	1	2	yes	—
482	C₂²×C₃	#577	1	2	yes	—
483	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
484	C₃×C₄	—	2	4	no	—
485	C₃×C₄	—	4	2	no	—
486	C₃×C₄	—	4	4	no	—
487	C₃×C₄	—	4	2	no	—
488	C₃×C₄	—	4	2	no	—
489	C₃×C₄	—	4	2	no	—
490	C₃×C₄	C₂×#123	1	2	yes	—
491	C₃×C₄	C₂×#123	1	4	yes	—
492	C₃×C₄	C₂²×#41	1	1	yes	characteristic
493	C₃×C₄	C₂²×#41	1	1	yes	characteristic
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	4	no	—
497	C₂³	—	2	4	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	4	4	no	—
501	C₂³	—	4	4	no	—
502	C₂³	—	4	8	no	—
503	C₂³	—	4	4	no	—
504	C₂³	—	8	4	no	—
505	C₂³	—	8	4	no	—
506	C₂³	—	8	16	no	—
507	C₂³	C₃×#123	1	4	yes	—
508	C₂³	C₂²×C₃×D₄	1	4	yes	—
509	C₂³	C₂²×C₃×D₄	1	2	yes	—
510	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
511	Q₈	—	2	8	no	—
512	D₄	—	2	16	no	—
513	D₄	—	2	4	no	—
514	D₄	—	2	4	no	—
515	D₄	—	2	8	no	—
516	D₄	—	4	4	no	—
517	D₄	—	4	8	no	—
518	D₄	—	8	8	no	—
519	D₄	—	8	8	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	8	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	4	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	8	4	no	—
535	C₂×C₄	—	8	8	no	—
536	C₂×C₄	C₃×#123	1	4	yes	—
537	C₂×C₄	C₃×#123	1	8	yes	—
538	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
539	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
540	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
541	C₂×C₄	C₂×C₃×#41	1	2	yes	—
542	C₂×C₄	C₂×C₃×#41	1	2	yes	—
543	C₂×C₄	C₂×C₃×#41	1	4	yes	—
544	C₂×C₄	C₂×C₃×#41	1	4	yes	—
545	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
546	C₈	—	2	2	no	—
547	C₈	—	2	2	no	—
548	C₈	—	2	4	no	—
549	C₂×C₃	—	2	4	no	—
550	C₂×C₃	—	4	4	no	—
551	C₂×C₃	—	8	4	no	—
552	C₂×C₃	#3096	1	4	yes	direct product factor
553	C₂×C₃	C₂²×#123	1	1	yes	characteristic
554	C₂×C₃	C₂×#576	1	1	yes	characteristic
555	C₂×C₃	C₂×#577	1	1	yes	characteristic
556	C₂²	—	2	2	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	8	no	—
559	C₂²	—	2	4	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	4	8	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	8	16	no	—
567	C₂²	—	8	8	no	—
568	C₂²	C₂×C₃×#123	1	4	yes	—
569	C₂²	C₂×C₃×#123	1	2	yes	—
570	C₂²	C₃×#576	1	2	yes	—
571	C₂²	C₃×#577	1	2	yes	—
572	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
573	C₄	—	2	4	no	—
574	C₄	—	4	2	no	—
575	C₄	—	4	4	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	2	no	—
578	C₄	—	4	2	no	—
579	C₄	C₂×C₃×#123	1	2	yes	—
580	C₄	C₂×C₃×#123	1	4	yes	—
581	C₄	C₂²×C₃×#41	1	1	yes	characteristic
582	C₄	C₂²×C₃×#41	1	1	yes	characteristic
583	C₃	C₂×#3096	1	1	yes	characteristic, direct product factor, Sylow
584	C₂	—	2	4	no	—
585	C₂	—	4	4	no	—
586	C₂	—	8	4	no	—
587	C₂	C₃×#3096	1	4	yes	direct product factor
588	C₂	C₂²×C₃×#123	1	1	yes	characteristic
589	C₂	C₂×C₃×#576	1	1	yes	characteristic
590	C₂	C₂×C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂×C₃×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#520	C₂	semidirect
3. C₃×#3096	C₂	direct (N × H)
4. C₂×C₃×#481	C₂	semidirect
5. C₂×C₃×#482	C₂	semidirect
6. C₂×C₃×#482	C₂	semidirect
7. C₂×C₃×#480	C₂	semidirect
8. C₂×C₃×#428	C₂	semidirect
9. C₂×C₃×#427	C₂	semidirect
10. C₂×C₃×#418	C₂	semidirect
11. C₂²×C₃×#102	C₂	semidirect
12. C₂×C₃×#408	C₂	semidirect
13. C₂×#3096	C₃	direct (N × H)
26. C₃×#520	C₂²	semidirect
32. C₂×C₃×#117	C₂²	semidirect
34. C₂²×C₃×C₄⋊C₄	C₂²	semidirect
35. C₃×#482	C₂²	semidirect
36. C₃×#482	C₂²	semidirect
37. C₃×#481	C₂²	semidirect
38. C₃×#480	C₂²	semidirect
39. C₃×#428	C₂²	semidirect
40. C₃×#427	C₂²	semidirect
41. C₂×C₃×#107	C₂²	semidirect
42. C₂×C₃×#107	C₂²	semidirect
43. C₂×C₃×#107	C₂²	semidirect
44. C₂×C₃×#106	C₂²	semidirect
45. C₂×C₃×#106	C₂²	semidirect
46. C₂×C₃×#106	C₂²	semidirect
47. C₃×#418	C₂²	semidirect
52. C₂×C₃×#102	C₂²	semidirect
53. C₃×#408	C₂²	semidirect
54. C₂×C₃×C₂²⋊C₈	C₂²	semidirect
55. C₂×C₃×C₂²⋊C₈	C₂²	semidirect
57. C₂×#520	C₂×C₃	semidirect
58. #3096	C₂×C₃	direct (N × H)
59. C₂×#481	C₂×C₃	semidirect
60. C₂×#482	C₂×C₃	semidirect
61. C₂×#482	C₂×C₃	semidirect
62. C₂×#480	C₂×C₃	semidirect
63. C₂×#428	C₂×C₃	semidirect
64. C₂×#427	C₂×C₃	semidirect
65. C₂×#418	C₂×C₃	semidirect
66. C₂²×#102	C₂×C₃	semidirect
67. C₂×#408	C₂×C₃	semidirect
110. C₃×#117	C₂³	semidirect
121. C₂×C₃×C₄⋊C₄	C₂³	semidirect
130. C₃×#107	C₂³	semidirect
131. C₃×#107	C₂³	semidirect
132. C₃×#107	C₂³	semidirect
133. C₃×#106	C₂³	semidirect
134. C₃×#106	C₂³	semidirect
135. C₃×#106	C₂³	semidirect
140. C₃×C₂²⋊C₈	C₂³	semidirect
141. C₃×C₂²⋊C₈	C₂³	semidirect
154. #520	C₂²×C₃	semidirect
160. C₂×#117	C₂²×C₃	semidirect
162. C₂²×C₄⋊C₄	C₂²×C₃	semidirect
163. #482	C₂²×C₃	semidirect
164. #482	C₂²×C₃	semidirect
165. #481	C₂²×C₃	semidirect
166. #480	C₂²×C₃	semidirect
167. #428	C₂²×C₃	semidirect
168. #427	C₂²×C₃	semidirect
169. C₂×#107	C₂²×C₃	semidirect
170. C₂×#107	C₂²×C₃	semidirect
171. C₂×#107	C₂²×C₃	semidirect
172. C₂×#106	C₂²×C₃	semidirect
173. C₂×#106	C₂²×C₃	semidirect
174. C₂×#106	C₂²×C₃	semidirect
175. #418	C₂²×C₃	semidirect
180. C₂×#102	C₂²×C₃	semidirect
181. #408	C₂²×C₃	semidirect
182. C₂×C₂²⋊C₈	C₂²×C₃	semidirect
183. C₂×C₂²⋊C₈	C₂²×C₃	semidirect
303. #117	C₂³×C₃	semidirect
314. C₂×C₄⋊C₄	C₂³×C₃	semidirect
323. #107	C₂³×C₃	semidirect
324. #107	C₂³×C₃	semidirect
325. #107	C₂³×C₃	semidirect
326. #106	C₂³×C₃	semidirect
327. #106	C₂³×C₃	semidirect
328. #106	C₂³×C₃	semidirect
333. C₂²⋊C₈	C₂³×C₃	semidirect
334. C₂²⋊C₈	C₂³×C₃	semidirect
552. C₂×C₃	#3096	direct (N × H)
583. C₃	C₂×#3096	direct (N × H)
587. C₂	C₃×#3096	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×#535	2
C₂×C₃×#520	2
C₃×#3096	2
C₂×C₃×#481	2
C₂×C₃×#482	2
C₂×C₃×#480	2
C₂×C₃×#428	2
C₂×C₃×#427	2
C₂×C₃×#418	2
C₂²×C₃×#102	2
C₂×C₃×#408	2
C₂×#3096	3

Subgroups — C2×C3×#3096 #10679119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#3096 #10679119