Subgroups — #10950782

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78568	C₂	1	2	yes	maximal, complemented
2	C₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
3	#71444	C₂	1	2	yes	maximal, complemented
4	#71416	C₂	1	1	yes	maximal, characteristic, complemented
5	#71413	C₂	1	1	yes	maximal, characteristic
6	#69079	C₂	1	2	yes	maximal, complemented
7	#69077	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#69062	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#69044	C₂	1	1	yes	maximal, characteristic, complemented
12	#33739	—	3	1	no	maximal, Sylow
13	#5534	—	2	2	no	—
14	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₃×#405	C₂²	1	1	yes	characteristic
17	#5134	—	2	2	no	—
18	#5116	C₂²	1	2	yes	—
19	#5115	C₂²	1	2	yes	—
20	C₂×Dic₂₄	C₂²	1	1	yes	characteristic
21	C₂×#800	C₂²	1	2	yes	—
22	C₂×D₄₈	C₂²	1	1	yes	characteristic
23	#4714	—	2	2	no	—
24	#4710	—	2	2	no	—
25	C₄×Dic₁₂	C₂²	1	2	yes	—
26	#4696	C₂²	1	2	yes	—
27	C₄×D₂₄	C₂²	1	2	yes	—
28	#4682	C₂²	1	2	yes	—
29	C₃×#346	C₂²	1	1	yes	characteristic
30	C₃×#346	C₂²	1	1	yes	characteristic
31	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
33	#4509	C₂²	1	2	yes	—
34	#4509	C₂²	1	2	yes	—
35	#4504	C₂²	1	2	yes	—
36	#4504	C₂²	1	1	yes	characteristic
37	#4504	C₂²	1	1	yes	characteristic
38	#4503	C₂²	1	2	yes	—
39	#4503	C₂²	1	2	yes	—
40	#3175	—	3	2	no	—
41	#2252	—	3	1	no	—
42	#2249	—	3	1	no	—
43	#2243	—	3	2	no	—
44	#2220	—	3	2	no	—
45	#2216	—	3	1	no	—
46	#2215	—	3	1	no	—
47	#2175	—	3	2	no	—
48	C₄×Q₃₂	—	3	1	no	—
49	C₄×D₁₆	—	3	1	no	—
50	#2108	D₃	1	1	yes	characteristic, complemented
51	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
53	C₃×#117	D₄	1	1	yes	characteristic
54	#930	—	2	2	no	—
55	#924	—	2	2	no	—
56	C₂×Dic₁₂	C₂³	1	2	yes	—
57	C₂×D₂₄	C₂³	1	2	yes	—
58	C₂×#204	—	2	2	no	—
59	#898	—	4	2	no	—
60	#896	—	4	2	no	—
61	#885	—	4	2	no	—
62	#883	—	4	2	no	—
63	#875	—	2	2	no	—
64	C₄×D₁₂	—	2	2	no	—
65	#869	—	2	2	no	—
66	C₄×Dic₆	—	2	2	no	—
67	C₃×M₃₂	—	2	2	no	—
68	C₂×C₃×C₁₆	—	2	2	no	—
69	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
73	C₃×#97	D₄	1	1	yes	characteristic
74	C₃×#97	D₄	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
77	#821	—	2	2	no	—
78	#821	—	2	2	no	—
79	#818	—	2	2	no	—
80	#818	C₂³	1	2	yes	—
81	#818	C₂³	1	2	yes	—
82	#816	—	2	2	no	—
83	#816	—	2	2	no	—
84	Dic₂₄	—	2	2	no	—
85	#800	—	2	4	no	—
86	D₄₈	—	2	2	no	—
87	#542	—	6	2	no	—
88	C₂×Q₃₂	—	3	1	no	—
89	C₂×SD₃₂	—	3	2	no	—
90	C₂×D₁₆	—	3	1	no	—
91	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
92	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
93	#500	—	3	2	no	—
94	#495	—	3	2	no	—
95	#489	—	6	2	no	—
96	#468	—	6	2	no	—
97	#467	—	3	2	no	—
98	#466	—	3	2	no	—
99	#440	—	6	2	no	—
100	C₄×Q₁₆	—	3	2	no	—
101	C₄×D₈	—	3	2	no	—
102	#405	C₂×D₃	1	1	yes	characteristic
103	#366	—	3	1	no	—
104	#366	—	3	1	no	—
105	#366	—	3	2	no	—
106	#358	—	3	2	no	—
107	#358	—	3	2	no	—
108	#357	—	3	2	no	—
109	#357	—	3	2	no	—
110	#346	C₂×D₃	1	1	yes	characteristic
111	#346	C₂×D₃	1	1	yes	characteristic
112	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
113	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
115	C₂×#67	—	4	2	no	—
116	C₂×D₁₂	—	2	2	no	—
117	C₂×C₄×D₃	—	4	2	no	—
118	C₂×Dic₆	—	2	2	no	—
119	C₂×Dic₆	—	4	2	no	—
120	C₃×M₁₆	—	2	2	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
127	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
128	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
129	C₃×C₂²⋊C₄	D₈	1	1	yes	characteristic
130	C₃×C₂²⋊C₄	D₈	1	1	yes	characteristic
131	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
132	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
133	#217	—	4	2	no	—
134	(C₂×D₃)⋊C₄	—	4	2	no	—
135	(C₂×D₃)⋊C₄	—	4	2	no	—
136	(C₂×D₃)⋊C₄	—	4	2	no	—
137	(C₂×D₃)⋊C₄	—	4	2	no	—
138	C₄⋊Dic₃	—	2	2	no	—
139	C₄⋊Dic₃	—	2	2	no	—
140	C₄⋊Dic₃	—	4	2	no	—
141	Dic₃⋊C₄	—	4	2	no	—
142	Dic₃⋊C₄	—	4	2	no	—
143	C₄×Dic₃	—	4	2	no	—
144	Dic₁₂	—	2	2	no	—
145	Dic₁₂	—	2	2	no	—
146	D₂₄	—	2	2	no	—
147	D₂₄	—	2	2	no	—
148	#204	—	4	4	no	—
149	C₃×C₁₆	—	2	1	no	—
150	C₃×C₁₆	—	2	1	no	—
151	C₃×C₁₆	#41	1	2	yes	—
152	C₃×C₁₆	#41	1	2	yes	—
153	C₂×Q₁₆	—	3	2	no	—
154	C₂×SD₁₆	—	6	2	no	—
155	C₂×D₈	—	3	2	no	—
156	C₂×M₁₆	D₁₂	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
158	C₄⋊Q₈	—	6	2	no	—
159	#126	—	12	2	no	—
160	#124	—	6	2	no	—
161	#124	—	12	2	no	—
162	#123	—	12	2	no	—
163	C₂²⋊Q₈	—	6	2	no	—
164	C₂²⋊Q₈	—	12	2	no	—
165	#121	—	6	2	no	—
166	C₄×Q₈	—	6	2	no	—
167	C₄×D₄	—	6	2	no	—
168	#117	D₁₂	1	1	yes	characteristic
169	Q₃₂	—	6	2	no	—
170	SD₃₂	—	6	4	no	—
171	D₁₆	—	6	2	no	—
172	M₃₂	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
176	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
178	#107	—	3	2	no	—
179	#107	—	3	2	no	—
180	#107	—	6	2	no	—
181	#103	—	6	2	no	—
182	#103	—	6	2	no	—
183	#102	—	6	2	no	—
184	#102	—	6	2	no	—
185	#97	D₁₂	1	1	yes	characteristic
186	#97	D₁₂	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
189	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
190	C₂²×D₃	—	4	2	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
199	#67	—	16	2	no	—
200	C₂×Dic₃	—	4	2	no	—
201	C₂×Dic₃	—	4	2	no	—
202	C₂×Dic₃	—	4	2	no	—
203	D₁₂	—	4	2	no	—
204	D₁₂	—	4	2	no	—
205	C₄×D₃	—	16	2	no	—
206	Dic₆	—	4	2	no	—
207	Dic₆	—	4	2	no	—
208	Dic₆	—	16	2	no	—
209	C₃×C₈	—	2	1	no	—
210	C₃×C₈	—	2	1	no	—
211	C₃×C₈	#135	1	2	yes	—
212	C₃×C₈	C₂×#41	1	1	yes	characteristic
213	C₃×C₈	C₂×#41	1	1	yes	characteristic
214	C₂×Q₈	—	6	2	no	—
215	C₂×Q₈	—	12	2	no	—
216	C₂×D₄	—	6	2	no	—
217	C₂×D₄	—	12	2	no	—
218	C₂²×C₄	—	12	2	no	—
219	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
220	Q₁₆	—	6	2	no	—
221	Q₁₆	—	6	2	no	—
222	SD₁₆	—	12	4	no	—
223	D₈	—	6	2	no	—
224	D₈	—	6	2	no	—
225	M₁₆	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
229	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
230	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
232	C₄⋊C₄	—	6	2	no	—
233	C₄⋊C₄	—	6	2	no	—
234	C₄⋊C₄	—	12	2	no	—
235	C₄⋊C₄	—	12	2	no	—
236	C₄⋊C₄	—	12	2	no	—
237	C₄⋊C₄	D₂₄	1	1	yes	characteristic
238	C₄⋊C₄	D₂₄	1	1	yes	characteristic
239	C₂²⋊C₄	—	12	2	no	—
240	C₂²⋊C₄	—	12	2	no	—
241	C₂²⋊C₄	—	12	2	no	—
242	C₂²⋊C₄	—	12	2	no	—
243	C₂²⋊C₄	—	12	2	no	—
244	C₂²⋊C₄	D₂₄	1	1	yes	characteristic
245	C₂²⋊C₄	D₂₄	1	1	yes	characteristic
246	C₄²	—	12	2	no	—
247	C₄²	C₂×D₁₂	1	1	yes	characteristic
248	C₄²	C₂×D₁₂	1	1	yes	characteristic
249	C₁₆	—	2	1	no	—
250	C₁₆	—	2	1	no	—
251	C₁₆	#235	1	2	yes	—
252	C₁₆	#235	1	2	yes	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
257	C₂×D₃	—	8	2	no	—
258	C₂×D₃	—	8	2	no	—
259	C₂×D₃	—	8	2	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	#532	1	1	yes	characteristic
266	C₃×C₄	C₂×#135	1	1	yes	characteristic
267	Dic₃	—	8	2	no	—
268	Dic₃	—	8	2	no	—
269	Dic₃	—	8	2	no	—
270	C₂³	—	12	2	no	—
271	C₂³	C₂×D₂₄	1	1	yes	characteristic
272	Q₈	—	12	2	no	—
273	Q₈	—	12	2	no	—
274	Q₈	—	48	2	no	—
275	D₄	—	12	2	no	—
276	D₄	—	12	2	no	—
277	D₄	—	48	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	12	2	no	—
281	C₂×C₄	—	12	2	no	—
282	C₂×C₄	—	12	2	no	—
283	C₂×C₄	—	48	2	no	—
284	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
285	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
286	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
287	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
288	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
289	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	#904	1	2	yes	—
293	C₈	C₂×#235	1	1	yes	characteristic
294	C₈	C₂×#235	1	1	yes	characteristic
295	C₂×C₃	—	4	1	no	—
296	C₂×C₃	#3169	1	1	yes	characteristic
297	C₂×C₃	#3415	1	1	yes	characteristic
298	C₂×C₃	#3417	1	1	yes	characteristic
299	D₃	—	16	2	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	24	2	no	—
304	C₂²	—	24	2	no	—
305	C₂²	—	24	2	no	—
306	C₂²	C₂²×D₂₄	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	24	2	no	—
313	C₄	—	24	2	no	—
314	C₄	—	24	2	no	—
315	C₄	#5477	1	1	yes	characteristic
316	C₄	C₂×#904	1	1	yes	characteristic
317	C₃	#33739	1	1	yes	characteristic, complemented, Sylow
318	C₂	—	4	1	no	—
319	C₂	—	48	2	no	—
320	C₂	#78470	1	1	yes	characteristic
321	C₂	#82013	1	1	yes	characteristic
322	C₂	#82015	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×#2108	C₂
socle	C₂²×C₃	C₂²×D₈
prime radical	#71413	C₂
hypercenter	#2108	D₃
hypocenter	C₃	#33739
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
1. #78568	C₂	semidirect
2. C₃×#2108	C₂	semidirect
3. #71444	C₂	semidirect
4. #71416	C₂	semidirect
6. #69079	C₂	semidirect
7. #69077	C₂	semidirect
8. C₄×Dic₂₄	C₂	semidirect
9. #69062	C₂	semidirect
10. C₄×D₄₈	C₂	semidirect
11. #69044	C₂	semidirect
50. #2108	D₃	semidirect
317. C₃	#33739	semidirect

Maximal subgroups

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

subgroup	index
#78568	2
C₃×#2108	2
#71444	2
#71416	2
#71413	2
#69079	2
#69077	2
C₄×Dic₂₄	2
#69062	2
C₄×D₄₈	2
#69044	2
#33739	3