Subgroups — #10693421

Order 768.

378 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#70757	C₂	1	1	yes	maximal, characteristic
2	C₂×#4483	C₂	1	1	yes	maximal, characteristic
3	#70420	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#823	C₂	1	1	yes	maximal, characteristic
5	M₁₆×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1283	C₂	1	1	yes	maximal, characteristic, complemented
7	#67739	C₂	1	1	yes	maximal, characteristic, complemented
8	#9550	—	3	1	no	maximal, Sylow
9	C₂×C₄×D₁₂	C₂²	1	1	yes	characteristic
10	C₃×#431	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×#823	C₄	1	1	yes	characteristic
14	C₂×#823	C₄	1	1	yes	characteristic
15	#4925	C₂²	1	1	yes	characteristic
16	C₄×#208	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	#4554	C₄	1	1	yes	characteristic
19	#4554	C₄	1	1	yes	characteristic
20	#4483	—	2	1	no	—
21	#4483	—	2	1	no	—
22	#4480	—	2	1	no	—
23	#4479	—	2	1	no	—
24	#4459	—	2	1	no	—
25	#4459	—	2	1	no	—
26	#4455	—	2	1	no	—
27	C₈×C₃⋊C₈	—	2	1	no	—
28	C₄×#100	—	3	1	no	—
29	#1565	—	3	1	no	—
30	#1478	—	3	1	no	—
31	C₂×D₄⋊C₈	—	3	1	no	—
32	C₈×M₁₆	—	3	1	no	—
33	#1283	D₃	1	1	yes	characteristic, complemented
34	#1278	—	3	1	no	—
35	C₂²×D₁₂	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×D₃	—	2	1	no	—
37	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
41	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
42	C₂×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#208	C₂×C₄	1	1	yes	characteristic
44	C₂×#208	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₃⋊C₈	—	2	1	no	—
46	C₄×D₁₂	—	2	1	no	—
47	C₄×D₁₂	—	2	1	no	—
48	C₄×D₁₂	—	2	1	no	—
49	C₄×D₁₂	—	2	1	no	—
50	C₃×C₄⋊C₈	—	2	1	no	—
51	C₃×C₄⋊C₈	—	2	1	no	—
52	C₃×C₄⋊C₈	—	2	1	no	—
53	C₃×C₂²⋊C₈	—	2	1	no	—
54	C₃×#97	—	2	1	no	—
55	C₃×C₄×C₈	—	2	1	no	—
56	C₃×C₄×C₈	—	2	1	no	—
57	#830	—	2	1	no	—
58	#823	—	2	2	no	—
59	#823	C₈	1	4	yes	complemented
60	C₄⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
61	C₄⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
62	#803	—	2	1	no	—
63	C₄×C₃⋊C₈	—	2	1	no	—
64	C₄×C₃⋊C₈	—	2	1	no	—
65	C₄×C₃⋊C₈	—	2	1	no	—
66	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
67	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
68	C₂×C₄×D₄	—	3	1	no	—
69	#431	—	3	1	no	—
70	#431	C₂×D₃	1	1	yes	characteristic
71	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
72	C₂×#100	—	3	1	no	—
73	C₂×#100	—	3	1	no	—
74	C₄×M₁₆	—	3	1	no	—
75	C₄×M₁₆	C₂×D₃	1	1	yes	characteristic
76	C₂×C₄×C₈	—	3	1	no	—
77	#343	—	3	1	no	—
78	#343	—	3	1	no	—
79	#335	—	6	1	no	—
80	#334	—	6	1	no	—
81	D₄⋊C₈	—	6	1	no	—
82	D₄⋊C₈	—	6	1	no	—
83	D₄⋊C₈	—	6	1	no	—
84	D₄⋊C₈	—	6	1	no	—
85	#322	—	6	1	no	—
86	C₈²	—	6	1	no	—
87	C₂³×D₃	—	2	1	no	—
88	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	#41	1	1	yes	characteristic
90	C₂²×C₃×C₄	#41	1	1	yes	characteristic
91	C₂²×Dic₃	—	2	1	no	—
92	C₂×D₁₂	—	2	1	no	—
93	C₂×D₁₂	—	2	1	no	—
94	C₂×D₁₂	—	2	1	no	—
95	C₂×D₁₂	—	2	1	no	—
96	C₂×D₁₂	—	2	1	no	—
97	C₂×D₁₂	C₂×C₈	1	2	yes	—
98	C₂×C₄×D₃	—	4	1	no	—
99	C₂×C₄×D₃	—	4	1	no	—
100	C₂×C₄×D₃	—	8	1	no	—
101	C₂×C₄×D₃	—	8	1	no	—
102	C₃×M₁₆	—	2	1	no	—
103	C₃×M₁₆	—	2	1	no	—
104	C₃×M₁₆	C₂×C₈	1	2	yes	—
105	C₃×M₁₆	M₁₆	1	2	yes	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	4	1	no	—
109	C₂×C₃×C₈	—	4	1	no	—
110	C₂×C₃×C₈	—	8	1	no	—
111	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
113	C₃×C₄²	#41	1	1	yes	characteristic
114	C₃×C₄²	#41	1	1	yes	characteristic
115	(C₂×D₃)⋊C₄	—	4	1	no	—
116	(C₂×D₃)⋊C₄	—	4	1	no	—
117	C₄⋊Dic₃	—	2	1	no	—
118	C₄⋊Dic₃	—	2	1	no	—
119	#208	—	2	1	no	—
120	#208	—	2	1	no	—
121	#208	—	2	1	no	—
122	#208	C₂×C₈	1	2	yes	—
123	C₂×C₃⋊C₈	—	2	1	no	—
124	C₂×C₃⋊C₈	—	2	1	no	—
125	C₂×C₃⋊C₈	—	2	1	no	—
126	C₂×C₃⋊C₈	—	2	1	no	—
127	C₂×C₃⋊C₈	—	2	1	no	—
128	C₂×C₃⋊C₈	—	2	1	no	—
129	C₂×C₃⋊C₈	—	4	2	no	—
130	C₂²×D₄	—	3	1	no	—
131	C₂³×C₄	—	6	1	no	—
132	C₂×M₁₆	—	3	1	no	—
133	C₂×M₁₆	—	3	1	no	—
134	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
135	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	6	1	no	—
138	C₄×D₄	—	6	1	no	—
139	C₄×D₄	—	6	1	no	—
140	C₄×D₄	—	6	1	no	—
141	C₄×D₄	—	6	1	no	—
142	C₂×C₄⋊C₄	—	3	1	no	—
143	C₂×C₂²⋊C₄	—	6	1	no	—
144	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
145	C₄⋊C₈	—	2	1	no	—
146	C₄⋊C₈	—	2	1	no	—
147	C₄⋊C₈	—	2	1	no	—
148	C₄⋊C₈	—	3	1	no	—
149	C₄⋊C₈	—	3	1	no	—
150	#100	—	3	4	no	—
151	#100	—	6	2	no	—
152	C₂²⋊C₈	—	2	1	no	—
153	C₂²⋊C₈	—	6	1	no	—
154	#97	—	2	1	no	—
155	#97	—	6	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	—	3	1	no	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	—	6	1	no	—
161	C₄×C₈	—	6	1	no	—
162	C₄×C₈	—	6	1	no	—
163	C₂³×C₃	#117	1	1	yes	characteristic
164	C₂²×D₃	—	2	2	no	—
165	C₂²×D₃	—	4	1	no	—
166	C₂²×D₃	—	4	1	no	—
167	C₂²×D₃	—	8	1	no	—
168	C₂²×D₃	—	8	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	4	1	no	—
172	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	#121	1	1	yes	characteristic
175	C₂×C₃×C₄	#123	1	1	yes	characteristic
176	C₂×C₃×C₄	#124	1	1	yes	characteristic
177	C₂×C₃×C₄	#126	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
180	C₂×C₃×C₄	#131	1	1	yes	characteristic
181	C₂×C₃×C₄	#131	1	1	yes	characteristic
182	C₂×Dic₃	—	4	1	no	—
183	C₂×Dic₃	—	8	1	no	—
184	D₁₂	—	2	2	no	—
185	D₁₂	—	2	2	no	—
186	D₁₂	—	4	1	no	—
187	D₁₂	—	4	1	no	—
188	C₄×D₃	—	8	2	no	—
189	C₄×D₃	—	8	1	no	—
190	C₄×D₃	—	8	1	no	—
191	C₃×C₈	—	2	2	no	—
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	—	8	1	no	—
194	C₃×C₈	—	8	1	no	—
195	C₃⋊C₈	—	2	2	no	—
196	C₃⋊C₈	—	2	2	no	—
197	C₃⋊C₈	—	4	1	no	—
198	C₃⋊C₈	—	4	1	no	—
199	C₃⋊C₈	—	4	1	no	—
200	C₃⋊C₈	—	4	1	no	—
201	C₂⁴	—	6	1	no	—
202	C₂×D₄	—	3	2	no	—
203	C₂×D₄	—	6	1	no	—
204	C₂×D₄	—	6	1	no	—
205	C₂×D₄	—	6	1	no	—
206	C₂×D₄	—	6	1	no	—
207	C₂×D₄	—	6	1	no	—
208	C₂²×C₄	—	6	1	no	—
209	C₂²×C₄	—	12	1	no	—
210	C₂²×C₄	—	12	1	no	—
211	C₂²×C₄	—	24	1	no	—
212	C₂²×C₄	—	24	1	no	—
213	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
214	C₂²×C₄	#235	1	1	yes	characteristic
215	C₂²×C₄	#235	1	1	yes	characteristic
216	M₁₆	—	2	1	no	—
217	M₁₆	—	2	1	no	—
218	M₁₆	—	3	2	no	—
219	M₁₆	—	6	1	no	—
220	M₁₆	—	6	1	no	—
221	M₁₆	—	6	1	no	—
222	M₁₆	D₃×C₈	1	2	yes	—
223	M₁₆	ΓL(1,𝔽₂₅)	1	2	yes	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	6	1	no	—
229	C₂×C₈	—	6	1	no	—
230	C₂×C₈	—	6	1	no	—
231	C₂×C₈	—	6	1	no	—
232	C₂×C₈	—	6	1	no	—
233	C₂×C₈	—	6	1	no	—
234	C₂×C₈	—	8	1	no	—
235	C₂×C₈	—	12	2	no	—
236	C₄⋊C₄	—	6	1	no	—
237	C₄⋊C₄	—	6	1	no	—
238	C₂²⋊C₄	—	12	1	no	—
239	C₂²⋊C₄	—	12	1	no	—
240	C₄²	D₃×D₄	1	1	yes	characteristic
241	C₄²	D₃×D₄	1	1	yes	characteristic
242	C₄²	#237	1	1	yes	characteristic
243	C₄²	#239	1	1	yes	characteristic
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	#388	1	1	yes	characteristic
247	C₂²×C₃	#431	1	1	yes	characteristic
248	C₂²×C₃	#433	1	1	yes	characteristic
249	C₂×D₃	—	4	2	no	—
250	C₂×D₃	—	8	2	no	—
251	C₂×D₃	—	8	1	no	—
252	C₂×D₃	—	8	1	no	—
253	C₂×D₃	—	8	2	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	C₈×D₄	1	1	yes	characteristic
258	C₃×C₄	C₈×D₄	1	1	yes	characteristic
259	C₃×C₄	#435	1	1	yes	characteristic
260	C₃×C₄	#436	1	1	yes	characteristic
261	C₃×C₄	#471	1	1	yes	characteristic
262	C₃×C₄	#472	1	1	yes	characteristic
263	Dic₃	—	8	1	no	—
264	C₂³	—	6	2	no	—
265	C₂³	—	12	1	no	—
266	C₂³	—	12	1	no	—
267	C₂³	—	24	1	no	—
268	C₂³	—	24	1	no	—
269	C₂³	#872	1	1	yes	characteristic
270	D₄	—	6	2	no	—
271	D₄	—	6	2	no	—
272	D₄	—	12	1	no	—
273	D₄	—	12	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	12	1	no	—
278	C₂×C₄	—	24	2	no	—
279	C₂×C₄	—	24	1	no	—
280	C₂×C₄	—	24	1	no	—
281	C₂×C₄	—	24	1	no	—
282	C₂×C₄	#67⋊C₄	1	1	yes	characteristic
283	C₂×C₄	#884	1	1	yes	characteristic
284	C₂×C₄	#885	1	1	yes	characteristic
285	C₂×C₄	#893	1	1	yes	characteristic
286	C₂×C₄	#894	1	1	yes	characteristic
287	C₂×C₄	#898	1	1	yes	characteristic
288	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
289	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
290	C₂×C₄	#907	1	1	yes	characteristic
291	C₂×C₄	#907	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	2	2	no	—
294	C₈	—	6	2	no	—
295	C₈	—	6	2	no	—
296	C₈	—	8	1	no	—
297	C₈	—	8	1	no	—
298	C₈	—	12	1	no	—
299	C₈	—	12	1	no	—
300	C₈	—	12	1	no	—
301	C₈	—	12	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	#1923	1	1	yes	characteristic
305	C₂×C₃	#1931	1	1	yes	characteristic
306	C₂×C₃	#1932	1	1	yes	characteristic
307	D₃	—	8	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	12	2	no	—
311	C₂²	—	24	2	no	—
312	C₂²	—	24	2	no	—
313	C₂²	—	24	1	no	—
314	C₂²	—	24	1	no	—
315	C₂²	#4669	1	1	yes	characteristic
316	C₂²	#4685	1	1	yes	characteristic
317	C₂²	#4709	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	24	1	no	—
322	C₄	#67⋊C₈	1	1	yes	characteristic
323	C₄	#4730	1	1	yes	characteristic
324	C₄	#4749	1	1	yes	characteristic
325	C₄	#4754	1	1	yes	characteristic
326	C₄	#4834	1	1	yes	characteristic
327	C₄	#4838	1	1	yes	characteristic
328	C₃	#9550	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	24	2	no	—
332	C₂	#68657	1	1	yes	characteristic
333	C₂	#68983	1	1	yes	characteristic
334	C₂	#69012	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#884
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1283	C₂
socle	C₂²×C₃	#388
prime radical	C₂²×D₁₂	C₂×C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1283	D₃
hypocenter	C₃	#9550
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
3. #70420	C₂	semidirect
5. M₁₆×C₃⋊C₈	C₂	semidirect
6. C₃×#1283	C₂	semidirect
7. #67739	C₂	semidirect
33. #1283	D₃	semidirect
59. #823	C₈	semidirect
328. C₃	#9550	semidirect

Maximal subgroups

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

subgroup	index
#70757	2
C₂×#4483	2
#70420	2
C₄×#823	2
M₁₆×C₃⋊C₈	2
C₃×#1283	2
#67739	2
#9550	3