Subgroups — C₂×C₃×#3136 #10679217

Order 768.

2262 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#535	C₂	1	1	yes	maximal, characteristic
3	C₃×#3136	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#490	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#489	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#488	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#486	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#418	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#3136	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×#544	—	2	4	no	—
19	C₃×#544	C₂²	1	8	yes	complemented
20	C₃×#535	—	2	4	no	—
21	C₃×#535	C₂²	1	8	yes	—
22	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
23	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
24	C₂×C₃×#127	C₂²	1	2	yes	—
25	C₂×C₃×#125	—	2	1	no	—
26	C₂×C₃×#125	C₂²	1	2	yes	complemented
27	C₂×C₃×#124	C₂²	1	2	yes	—
28	C₂×C₃×C₂²⋊Q₈	—	2	2	no	—
29	C₂×C₃×#121	—	2	1	no	—
30	C₂×C₃×#121	—	2	1	no	—
31	C₂×C₃×#120	—	2	2	no	—
32	C₂×C₃×#117	C₂²	1	1	yes	characteristic
33	C₂×C₃×#117	C₂²	1	1	yes	characteristic
34	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₃×#490	C₂²	1	16	yes	complemented
36	C₃×#489	C₂²	1	16	yes	complemented
37	C₃×#488	C₂²	1	16	yes	complemented
38	C₃×#486	C₂²	1	16	yes	complemented
39	C₃×#418	C₂²	1	8	yes	complemented
40	C₃×#417	C₂²	1	8	yes	complemented
41	C₃×#416	C₂²	1	8	yes	complemented
42	C₂×C₃×#103	C₂²	1	1	yes	characteristic, complemented
43	C₂×C₃×#103	C₂²	1	2	yes	complemented
44	C₂×C₃×#103	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₃×#102	C₂²	1	2	yes	—
46	C₂×C₃×#102	C₂²	1	2	yes	—
47	C₂×C₃×#102	C₂²	1	2	yes	—
48	C₂×C₃×#102	C₂²	1	2	yes	—
49	C₂×C₃×#102	C₂²	1	8	yes	complemented
50	C₂×C₃×#102	C₂²	1	1	yes	characteristic, complemented
51	C₂×C₃×#102	C₂²	1	2	yes	complemented
52	C₂×C₃×#102	C₂²	1	1	yes	characteristic, complemented
53	C₃×#405	C₂²	1	8	yes	complemented
54	C₂×C₃×#97	C₂²	1	2	yes	complemented
55	C₂×C₃×C₄×C₈	C₂²	1	2	yes	complemented
56	C₂×#544	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂×#535	C₂×C₃	1	1	yes	characteristic
58	#3136	C₂×C₃	1	16	yes	direct product factor
59	C₂×#490	C₂×C₃	1	2	yes	complemented
60	C₂×#489	C₂×C₃	1	2	yes	complemented
61	C₂×#488	C₂×C₃	1	2	yes	complemented
62	C₂×#486	C₂×C₃	1	2	yes	complemented
63	C₂×#417	C₂×C₃	1	1	yes	characteristic, complemented
64	C₂×#418	C₂×C₃	1	1	yes	characteristic, complemented
65	C₂×#416	C₂×C₃	1	1	yes	characteristic, complemented
66	C₂²×#102	C₂×C₃	1	1	yes	characteristic, complemented
67	C₂×#405	C₂×C₃	1	1	yes	characteristic, complemented
68	C₂⁵×C₃	—	2	1	no	—
69	C₂×C₃×#41	—	2	2	no	—
70	C₂×C₃×#41	—	2	4	no	—
71	C₂×C₃×#41	—	2	4	no	—
72	C₂×C₃×#41	C₂³	1	4	yes	—
73	C₂²×C₃×Q₈	—	2	2	no	—
74	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
75	C₂²×C₃×D₄	—	2	1	no	—
76	C₂²×C₃×D₄	—	2	1	no	—
77	C₂²×C₃×D₄	—	2	2	no	—
78	C₂²×C₃×D₄	—	2	2	no	—
79	C₂²×C₃×D₄	—	2	2	no	—
80	C₂²×C₃×D₄	—	2	4	no	—
81	C₂²×C₃×D₄	—	2	4	no	—
82	C₂²×C₃×D₄	C₂³	1	2	yes	—
83	C₂²×C₃×D₄	C₂³	1	4	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	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×C₄⋊Q₈	—	2	8	no	—
96	C₃×C₄⋊Q₈	—	2	4	no	—
97	C₃×C₄⋊Q₈	C₂³	1	8	yes	complemented
98	C₃×#127	—	2	4	no	—
99	C₃×#127	C₂³	1	8	yes	—
100	C₃×#125	—	2	8	no	—
101	C₃×#125	—	2	4	no	—
102	C₃×#125	C₂³	1	8	yes	complemented
103	C₃×#124	—	2	4	no	—
104	C₃×#124	C₂³	1	8	yes	—
105	C₃×C₂²⋊Q₈	—	2	16	no	—
106	C₃×#121	—	2	8	no	—
107	C₃×#121	—	2	8	no	—
108	C₃×#120	—	2	16	no	—
109	C₃×#117	—	2	2	no	—
110	C₃×#117	D₄	1	4	yes	—
111	C₃×#117	C₂³	1	4	yes	—
112	C₃×#117	C₂³	1	4	yes	—
113	C₂×C₃×C₄⋊C₄	—	2	1	no	—
114	C₂×C₃×C₄⋊C₄	—	2	2	no	—
115	C₂×C₃×C₄⋊C₄	—	2	2	no	—
116	C₂×C₃×C₄⋊C₄	—	2	2	no	—
117	C₂×C₃×C₄⋊C₄	—	2	2	no	—
118	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
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×C₄⋊C₄	C₂³	1	4	yes	—
124	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
126	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
127	C₂×C₃×C₂²⋊C₄	D₄	1	2	yes	—
128	C₂×C₃×C₄²	—	2	1	no	—
129	C₂×C₃×C₄²	C₂³	1	2	yes	—
130	C₃×#103	C₂³	1	4	yes	complemented
131	C₃×#103	C₂³	1	4	yes	complemented
132	C₃×#103	C₂³	1	8	yes	complemented
133	C₃×#102	C₂³	1	8	yes	—
134	C₃×#102	C₂³	1	8	yes	—
135	C₃×#102	C₂³	1	8	yes	—
136	C₃×#102	C₂³	1	8	yes	—
137	C₃×#102	C₂³	1	4	yes	complemented
138	C₃×#102	C₂³	1	4	yes	complemented
139	C₃×#102	C₂³	1	8	yes	complemented
140	C₃×#97	C₂³	1	8	yes	complemented
141	C₃×C₄×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	#544	—	2	4	no	—
147	#544	C₂²×C₃	1	8	yes	complemented
148	#535	—	2	4	no	—
149	#535	C₂²×C₃	1	8	yes	—
150	C₂×C₄⋊Q₈	—	2	1	no	—
151	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
152	C₂×#127	C₂²×C₃	1	2	yes	—
153	C₂×#125	—	2	1	no	—
154	C₂×#125	C₂²×C₃	1	2	yes	complemented
155	C₂×#124	C₂²×C₃	1	2	yes	—
156	C₂×C₂²⋊Q₈	—	2	2	no	—
157	C₂×#121	—	2	1	no	—
158	C₂×#121	—	2	1	no	—
159	C₂×#120	—	2	2	no	—
160	C₂×#117	C₂²×C₃	1	1	yes	characteristic
161	C₂×#117	C₂²×C₃	1	1	yes	characteristic
162	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
163	#490	C₂²×C₃	1	16	yes	complemented
164	#489	C₂²×C₃	1	16	yes	complemented
165	#488	C₂²×C₃	1	16	yes	complemented
166	#486	C₂²×C₃	1	16	yes	complemented
167	#418	C₂²×C₃	1	8	yes	complemented
168	#417	C₂²×C₃	1	8	yes	complemented
169	#416	C₂²×C₃	1	8	yes	complemented
170	C₂×#103	C₂²×C₃	1	1	yes	characteristic, complemented
171	C₂×#103	C₂²×C₃	1	2	yes	complemented
172	C₂×#103	C₂²×C₃	1	1	yes	characteristic, complemented
173	C₂×#102	C₂²×C₃	1	2	yes	—
174	C₂×#102	C₂²×C₃	1	2	yes	—
175	C₂×#102	C₂²×C₃	1	2	yes	—
176	C₂×#102	C₂²×C₃	1	2	yes	—
177	C₂×#102	C₂²×C₃	1	8	yes	complemented
178	C₂×#102	C₂²×C₃	1	1	yes	characteristic, complemented
179	C₂×#102	C₂²×C₃	1	2	yes	complemented
180	C₂×#102	C₂²×C₃	1	1	yes	characteristic, complemented
181	#405	C₂²×C₃	1	8	yes	complemented
182	C₂×#97	C₂²×C₃	1	2	yes	complemented
183	C₂×C₄×C₈	C₂²×C₃	1	2	yes	complemented
184	C₂⁴×C₃	—	2	2	no	—
185	C₂⁴×C₃	—	2	1	no	—
186	C₂⁴×C₃	—	2	4	no	—
187	C₂⁴×C₃	—	4	4	no	—
188	C₂⁴×C₃	—	8	2	no	—
189	C₂⁴×C₃	—	8	2	no	—
190	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
191	C₃×#41	—	2	8	no	—
192	C₃×#41	—	2	8	no	—
193	C₃×#41	—	4	8	no	—
194	C₂×C₃×Q₈	—	2	1	no	—
195	C₂×C₃×Q₈	—	2	8	no	—
196	C₂×C₃×Q₈	—	2	2	no	—
197	C₂×C₃×Q₈	—	2	2	no	—
198	C₂×C₃×Q₈	—	8	4	no	—
199	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
200	C₂×C₃×D₄	—	2	4	no	—
201	C₂×C₃×D₄	—	2	4	no	—
202	C₂×C₃×D₄	—	2	2	no	—
203	C₂×C₃×D₄	—	2	1	no	—
204	C₂×C₃×D₄	—	2	8	no	—
205	C₂×C₃×D₄	—	2	8	no	—
206	C₂×C₃×D₄	—	2	4	no	—
207	C₂×C₃×D₄	—	2	4	no	—
208	C₂×C₃×D₄	—	2	8	no	—
209	C₂×C₃×D₄	—	2	8	no	—
210	C₂×C₃×D₄	—	2	2	no	—
211	C₂×C₃×D₄	—	2	2	no	—
212	C₂×C₃×D₄	—	4	4	no	—
213	C₂×C₃×D₄	—	4	4	no	—
214	C₂×C₃×D₄	—	4	8	no	—
215	C₂×C₃×D₄	—	8	4	no	—
216	C₂×C₃×D₄	—	8	4	no	—
217	C₂×C₃×D₄	C₂⁴	1	4	yes	—
218	C₂×C₃×D₄	C₂⁴	1	2	yes	—
219	C₂²×C₃×C₄	—	2	1	no	—
220	C₂²×C₃×C₄	—	2	1	no	—
221	C₂²×C₃×C₄	—	2	1	no	—
222	C₂²×C₃×C₄	—	2	2	no	—
223	C₂²×C₃×C₄	—	2	4	no	—
224	C₂²×C₃×C₄	—	2	4	no	—
225	C₂²×C₃×C₄	—	4	4	no	—
226	C₂²×C₃×C₄	—	8	2	no	—
227	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
228	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
229	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
230	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
231	C₂²×C₃×C₄	#41	1	4	yes	—
232	C₂²×C₃×C₄	#41	1	4	yes	—
233	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
234	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
235	C₃×M₁₆	—	2	8	no	—
236	C₂×C₃×C₈	—	2	2	no	—
237	C₂×C₃×C₈	—	2	2	no	—
238	C₂×C₃×C₈	—	2	8	no	—
239	C₂×C₃×C₈	—	2	4	no	—
240	C₂×C₃×C₈	C₂⁴	1	2	yes	—
241	C₂×C₃×C₈	C₂⁴	1	4	yes	—
242	C₂×C₃×C₈	C₂⁴	1	2	yes	—
243	C₃×C₄⋊C₄	—	2	4	no	—
244	C₃×C₄⋊C₄	—	2	2	no	—
245	C₃×C₄⋊C₄	—	2	8	no	—
246	C₃×C₄⋊C₄	—	2	8	no	—
247	C₃×C₄⋊C₄	—	2	8	no	—
248	C₃×C₄⋊C₄	—	2	1	no	—
249	C₃×C₄⋊C₄	—	2	1	no	—
250	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
251	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
252	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
253	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
254	C₃×C₄⋊C₄	C₂⁴	1	2	yes	—
255	C₃×C₂²⋊C₄	—	2	4	no	—
256	C₃×C₂²⋊C₄	—	2	8	no	—
257	C₃×C₂²⋊C₄	—	2	8	no	—
258	C₃×C₂²⋊C₄	C₂×D₄	1	8	yes	—
259	C₃×C₄²	—	2	4	no	—
260	C₃×C₄²	C₂×D₄	1	4	yes	—
261	C₃×C₄²	C₂⁴	1	4	yes	—
262	C₂⁵	—	2	1	no	—
263	C₂×#41	—	2	2	no	—
264	C₂×#41	—	2	4	no	—
265	C₂×#41	—	2	4	no	—
266	C₂×#41	C₂³×C₃	1	4	yes	—
267	C₂²×Q₈	—	2	2	no	—
268	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
269	C₂²×D₄	—	2	1	no	—
270	C₂²×D₄	—	2	1	no	—
271	C₂²×D₄	—	2	2	no	—
272	C₂²×D₄	—	2	2	no	—
273	C₂²×D₄	—	2	2	no	—
274	C₂²×D₄	—	2	4	no	—
275	C₂²×D₄	—	2	4	no	—
276	C₂²×D₄	C₂³×C₃	1	2	yes	—
277	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
278	C₂²×D₄	C₂³×C₃	1	4	yes	—
279	C₂³×C₄	—	2	1	no	—
280	C₂³×C₄	—	2	1	no	—
281	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
282	C₂×M₁₆	—	2	4	no	—
283	C₂×M₁₆	C₂³×C₃	1	4	yes	—
284	C₂²×C₈	—	2	4	no	—
285	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₃	1	4	yes	—
288	C₂²×C₈	C₂³×C₃	1	2	yes	—
289	C₄⋊Q₈	—	2	8	no	—
290	C₄⋊Q₈	—	2	4	no	—
291	C₄⋊Q₈	C₂³×C₃	1	8	yes	complemented
292	#127	—	2	4	no	—
293	#127	C₂³×C₃	1	8	yes	—
294	#125	—	2	8	no	—
295	#125	—	2	4	no	—
296	#125	C₂³×C₃	1	8	yes	complemented
297	#124	—	2	4	no	—
298	#124	C₂³×C₃	1	8	yes	—
299	C₂²⋊Q₈	—	2	16	no	—
300	#121	—	2	8	no	—
301	#121	—	2	8	no	—
302	#120	—	2	16	no	—
303	#117	—	2	2	no	—
304	#117	C₃×D₄	1	4	yes	—
305	#117	C₂³×C₃	1	4	yes	—
306	#117	C₂³×C₃	1	4	yes	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	2	no	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
316	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
317	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₂²⋊C₄	C₃×D₄	1	2	yes	—
322	C₂×C₄²	—	2	1	no	—
323	C₂×C₄²	C₂³×C₃	1	2	yes	—
324	#103	C₂³×C₃	1	4	yes	complemented
325	#103	C₂³×C₃	1	4	yes	complemented
326	#103	C₂³×C₃	1	8	yes	complemented
327	#102	C₂³×C₃	1	8	yes	—
328	#102	C₂³×C₃	1	8	yes	—
329	#102	C₂³×C₃	1	8	yes	—
330	#102	C₂³×C₃	1	8	yes	—
331	#102	C₂³×C₃	1	4	yes	complemented
332	#102	C₂³×C₃	1	4	yes	complemented
333	#102	C₂³×C₃	1	8	yes	complemented
334	#97	C₂³×C₃	1	8	yes	complemented
335	C₄×C₈	C₂³×C₃	1	8	yes	complemented
336	C₂³×C₃	—	2	2	no	—
337	C₂³×C₃	—	2	2	no	—
338	C₂³×C₃	—	2	4	no	—
339	C₂³×C₃	—	2	2	no	—
340	C₂³×C₃	—	4	2	no	—
341	C₂³×C₃	—	4	8	no	—
342	C₂³×C₃	—	4	4	no	—
343	C₂³×C₃	—	4	2	no	—
344	C₂³×C₃	—	4	4	no	—
345	C₂³×C₃	—	4	2	no	—
346	C₂³×C₃	—	4	4	no	—
347	C₂³×C₃	—	8	4	no	—
348	C₂³×C₃	—	8	4	no	—
349	C₂³×C₃	—	8	16	no	—
350	C₂³×C₃	#124	1	4	yes	—
351	C₂³×C₃	C₂²×D₄	1	2	yes	—
352	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
353	C₃×Q₈	—	2	2	no	—
354	C₃×Q₈	—	2	2	no	—
355	C₃×Q₈	—	4	2	no	—
356	C₃×Q₈	—	8	8	no	—
357	C₃×D₄	—	2	4	no	—
358	C₃×D₄	—	2	4	no	—
359	C₃×D₄	—	2	2	no	—
360	C₃×D₄	—	2	2	no	—
361	C₃×D₄	—	4	8	no	—
362	C₃×D₄	—	4	8	no	—
363	C₃×D₄	—	4	4	no	—
364	C₃×D₄	—	4	2	no	—
365	C₃×D₄	—	8	8	no	—
366	C₃×D₄	—	8	8	no	—
367	C₂×C₃×C₄	—	2	4	no	—
368	C₂×C₃×C₄	—	2	4	no	—
369	C₂×C₃×C₄	—	2	2	no	—
370	C₂×C₃×C₄	—	2	2	no	—
371	C₂×C₃×C₄	—	2	2	no	—
372	C₂×C₃×C₄	—	2	4	no	—
373	C₂×C₃×C₄	—	4	2	no	—
374	C₂×C₃×C₄	—	4	4	no	—
375	C₂×C₃×C₄	—	4	2	no	—
376	C₂×C₃×C₄	—	4	2	no	—
377	C₂×C₃×C₄	—	4	8	no	—
378	C₂×C₃×C₄	—	8	4	no	—
379	C₂×C₃×C₄	#124	1	4	yes	—
380	C₂×C₃×C₄	#124	1	8	yes	—
381	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
382	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
383	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
384	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
385	C₂×C₃×C₄	C₂×#41	1	2	yes	—
386	C₂×C₃×C₄	C₂×#41	1	2	yes	—
387	C₂×C₃×C₄	C₂×#41	1	4	yes	—
388	C₂×C₃×C₄	C₂×#41	1	4	yes	—
389	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
390	C₃×C₈	—	2	2	no	—
391	C₃×C₈	—	2	2	no	—
392	C₃×C₈	—	2	4	no	—
393	C₂⁴	—	2	2	no	—
394	C₂⁴	—	2	1	no	—
395	C₂⁴	—	2	4	no	—
396	C₂⁴	—	4	4	no	—
397	C₂⁴	—	8	2	no	—
398	C₂⁴	—	8	2	no	—
399	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
400	#41	—	2	8	no	—
401	#41	—	2	8	no	—
402	#41	—	4	8	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	8	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	8	4	no	—
408	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	4	4	no	—
422	C₂×D₄	—	4	4	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	8	4	no	—
425	C₂×D₄	—	8	4	no	—
426	C₂×D₄	C₂⁴×C₃	1	4	yes	—
427	C₂×D₄	C₂⁴×C₃	1	2	yes	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	—	8	2	no	—
436	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
437	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
438	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
439	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
440	C₂²×C₄	C₃×#41	1	4	yes	—
441	C₂²×C₄	C₃×#41	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
443	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
444	M₁₆	—	2	8	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	2	8	no	—
448	C₂×C₈	—	2	4	no	—
449	C₂×C₈	C₂⁴×C₃	1	2	yes	—
450	C₂×C₈	C₂⁴×C₃	1	4	yes	—
451	C₂×C₈	C₂⁴×C₃	1	2	yes	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
460	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
461	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
462	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
463	C₄⋊C₄	C₂⁴×C₃	1	2	yes	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	C₂×C₃×D₄	1	8	yes	—
468	C₄²	—	2	4	no	—
469	C₄²	C₂×C₃×D₄	1	4	yes	—
470	C₄²	C₂⁴×C₃	1	4	yes	—
471	C₂²×C₃	—	2	8	no	—
472	C₂²×C₃	—	2	2	no	—
473	C₂²×C₃	—	2	2	no	—
474	C₂²×C₃	—	4	2	no	—
475	C₂²×C₃	—	4	4	no	—
476	C₂²×C₃	—	4	2	no	—
477	C₂²×C₃	—	4	4	no	—
478	C₂²×C₃	—	4	2	no	—
479	C₂²×C₃	—	4	4	no	—
480	C₂²×C₃	—	8	16	no	—
481	C₂²×C₃	—	8	4	no	—
482	C₂²×C₃	—	8	8	no	—
483	C₂²×C₃	C₂×#124	1	4	yes	—
484	C₂²×C₃	C₂×#124	1	2	yes	—
485	C₂²×C₃	#576	1	2	yes	—
486	C₂²×C₃	#577	1	2	yes	—
487	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
488	C₃×C₄	—	2	4	no	—
489	C₃×C₄	—	2	4	no	—
490	C₃×C₄	—	4	2	no	—
491	C₃×C₄	—	4	4	no	—
492	C₃×C₄	—	4	2	no	—
493	C₃×C₄	—	4	2	no	—
494	C₃×C₄	C₂×#124	1	4	yes	—
495	C₃×C₄	C₂×#124	1	2	yes	—
496	C₃×C₄	C₂²×#41	1	1	yes	characteristic
497	C₃×C₄	C₂²×#41	1	1	yes	characteristic
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	4	4	no	—
504	C₂³	—	4	2	no	—
505	C₂³	—	4	4	no	—
506	C₂³	—	4	2	no	—
507	C₂³	—	4	8	no	—
508	C₂³	—	4	4	no	—
509	C₂³	—	8	4	no	—
510	C₂³	—	8	4	no	—
511	C₂³	—	8	16	no	—
512	C₂³	C₃×#124	1	4	yes	—
513	C₂³	C₂²×C₃×D₄	1	2	yes	—
514	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
515	Q₈	—	2	2	no	—
516	Q₈	—	2	2	no	—
517	Q₈	—	4	2	no	—
518	Q₈	—	8	8	no	—
519	D₄	—	2	4	no	—
520	D₄	—	2	4	no	—
521	D₄	—	2	2	no	—
522	D₄	—	2	2	no	—
523	D₄	—	4	8	no	—
524	D₄	—	4	8	no	—
525	D₄	—	4	4	no	—
526	D₄	—	4	2	no	—
527	D₄	—	8	8	no	—
528	D₄	—	8	8	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	4	4	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	4	8	no	—
540	C₂×C₄	—	8	4	no	—
541	C₂×C₄	C₃×#124	1	4	yes	—
542	C₂×C₄	C₃×#124	1	8	yes	—
543	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
544	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
545	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
546	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
547	C₂×C₄	C₂×C₃×#41	1	2	yes	—
548	C₂×C₄	C₂×C₃×#41	1	2	yes	—
549	C₂×C₄	C₂×C₃×#41	1	4	yes	—
550	C₂×C₄	C₂×C₃×#41	1	4	yes	—
551	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
552	C₈	—	2	2	no	—
553	C₈	—	2	2	no	—
554	C₈	—	2	4	no	—
555	C₂×C₃	—	2	4	no	—
556	C₂×C₃	—	8	2	no	—
557	C₂×C₃	—	8	4	no	—
558	C₂×C₃	#3136	1	4	yes	direct product factor
559	C₂×C₃	C₂²×#124	1	1	yes	characteristic
560	C₂×C₃	C₂×#576	1	1	yes	characteristic
561	C₂×C₃	C₂×#577	1	1	yes	characteristic
562	C₂²	—	2	8	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	4	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	4	2	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	8	16	no	—
572	C₂²	—	8	8	no	—
573	C₂²	—	8	4	no	—
574	C₂²	C₂×C₃×#124	1	4	yes	—
575	C₂²	C₂×C₃×#124	1	2	yes	—
576	C₂²	C₃×#576	1	2	yes	—
577	C₂²	C₃×#577	1	2	yes	—
578	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
579	C₄	—	2	4	no	—
580	C₄	—	2	4	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	2	no	—
585	C₄	C₂×C₃×#124	1	4	yes	—
586	C₄	C₂×C₃×#124	1	2	yes	—
587	C₄	C₂²×C₃×#41	1	1	yes	characteristic
588	C₄	C₂²×C₃×#41	1	1	yes	characteristic
589	C₃	C₂×#3136	1	1	yes	characteristic, direct product factor, Sylow
590	C₂	—	2	4	no	—
591	C₂	—	8	4	no	—
592	C₂	—	8	2	no	—
593	C₂	C₃×#3136	1	4	yes	direct product factor
594	C₂	C₂²×C₃×#124	1	1	yes	characteristic
595	C₂	C₂×C₃×#576	1	1	yes	characteristic
596	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 ◁ C₂×C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#544	C₂	semidirect
3. C₃×#3136	C₂	direct (N × H)
4. C₂×C₃×#490	C₂	semidirect
5. C₂×C₃×#489	C₂	semidirect
6. C₂×C₃×#488	C₂	semidirect
7. C₂×C₃×#486	C₂	semidirect
8. C₂×C₃×#417	C₂	semidirect
9. C₂×C₃×#418	C₂	semidirect
10. C₂×C₃×#416	C₂	semidirect
11. C₂²×C₃×#102	C₂	semidirect
12. C₂×C₃×#405	C₂	semidirect
13. C₂×#3136	C₃	direct (N × H)
19. C₃×#544	C₂²	semidirect
23. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
26. C₂×C₃×#125	C₂²	semidirect
35. C₃×#490	C₂²	semidirect
36. C₃×#489	C₂²	semidirect
37. C₃×#488	C₂²	semidirect
38. C₃×#486	C₂²	semidirect
39. C₃×#418	C₂²	semidirect
40. C₃×#417	C₂²	semidirect
41. C₃×#416	C₂²	semidirect
42. C₂×C₃×#103	C₂²	semidirect
43. C₂×C₃×#103	C₂²	semidirect
44. C₂×C₃×#103	C₂²	semidirect
49. C₂×C₃×#102	C₂²	semidirect
50. C₂×C₃×#102	C₂²	semidirect
51. C₂×C₃×#102	C₂²	semidirect
52. C₂×C₃×#102	C₂²	semidirect
53. C₃×#405	C₂²	semidirect
54. C₂×C₃×#97	C₂²	semidirect
55. C₂×C₃×C₄×C₈	C₂²	semidirect
56. C₂×#544	C₂×C₃	semidirect
58. #3136	C₂×C₃	direct (N × H)
59. C₂×#490	C₂×C₃	semidirect
60. C₂×#489	C₂×C₃	semidirect
61. C₂×#488	C₂×C₃	semidirect
62. C₂×#486	C₂×C₃	semidirect
63. C₂×#417	C₂×C₃	semidirect
64. C₂×#418	C₂×C₃	semidirect
65. C₂×#416	C₂×C₃	semidirect
66. C₂²×#102	C₂×C₃	semidirect
67. C₂×#405	C₂×C₃	semidirect
97. C₃×C₄⋊Q₈	C₂³	semidirect
102. C₃×#125	C₂³	semidirect
130. C₃×#103	C₂³	semidirect
131. C₃×#103	C₂³	semidirect
132. C₃×#103	C₂³	semidirect
137. C₃×#102	C₂³	semidirect
138. C₃×#102	C₂³	semidirect
139. C₃×#102	C₂³	semidirect
140. C₃×#97	C₂³	semidirect
141. C₃×C₄×C₈	C₂³	semidirect
147. #544	C₂²×C₃	semidirect
151. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
154. C₂×#125	C₂²×C₃	semidirect
163. #490	C₂²×C₃	semidirect
164. #489	C₂²×C₃	semidirect
165. #488	C₂²×C₃	semidirect
166. #486	C₂²×C₃	semidirect
167. #418	C₂²×C₃	semidirect
168. #417	C₂²×C₃	semidirect
169. #416	C₂²×C₃	semidirect
170. C₂×#103	C₂²×C₃	semidirect
171. C₂×#103	C₂²×C₃	semidirect
172. C₂×#103	C₂²×C₃	semidirect
177. C₂×#102	C₂²×C₃	semidirect
178. C₂×#102	C₂²×C₃	semidirect
179. C₂×#102	C₂²×C₃	semidirect
180. C₂×#102	C₂²×C₃	semidirect
181. #405	C₂²×C₃	semidirect
182. C₂×#97	C₂²×C₃	semidirect
183. C₂×C₄×C₈	C₂²×C₃	semidirect
291. C₄⋊Q₈	C₂³×C₃	semidirect
296. #125	C₂³×C₃	semidirect
324. #103	C₂³×C₃	semidirect
325. #103	C₂³×C₃	semidirect
326. #103	C₂³×C₃	semidirect
331. #102	C₂³×C₃	semidirect
332. #102	C₂³×C₃	semidirect
333. #102	C₂³×C₃	semidirect
334. #97	C₂³×C₃	semidirect
335. C₄×C₈	C₂³×C₃	semidirect
558. C₂×C₃	#3136	direct (N × H)
589. C₃	C₂×#3136	direct (N × H)
593. C₂	C₃×#3136	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×#544	2
C₂×C₃×#535	2
C₃×#3136	2
C₂×C₃×#490	2
C₂×C₃×#489	2
C₂×C₃×#488	2
C₂×C₃×#486	2
C₂×C₃×#417	2
C₂×C₃×#418	2
C₂×C₃×#416	2
C₂²×C₃×#102	2
C₂×C₃×#405	2
C₂×#3136	3

Subgroups — C2×C3×#3136 #10679217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#3136 #10679217