Subgroups — #10948286

Order 768.

582 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78549	C₂	1	1	yes	maximal, characteristic
2	C₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	#71432	C₂	1	1	yes	maximal, characteristic
4	Dic₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	#71409	C₂	1	1	yes	maximal, characteristic
6	C₂×#4502	C₂	1	1	yes	maximal, characteristic, complemented
7	#69882	C₂	1	1	yes	maximal, characteristic, complemented
8	#69276	C₂	1	4	yes	maximal, complemented
9	#69275	C₂	1	4	yes	maximal, complemented
10	#33421	—	3	1	no	maximal, Sylow
11	C₂×#907	C₄	1	1	yes	characteristic
12	C₂×#901	C₂²	1	1	yes	characteristic
13	#5522	C₄	1	1	yes	characteristic
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	Dic₃×M₁₆	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×#812	C₂²	1	1	yes	characteristic
20	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	#4997	C₄	1	1	yes	characteristic
22	#4708	C₄	1	2	yes	—
23	#4707	C₄	1	2	yes	—
24	#4689	C₂²	1	2	yes	—
25	C₄×D₃×C₈	C₂²	1	2	yes	—
26	C₃×C₄⋊C₁₆	C₂²	1	4	yes	—
27	(C₂×D₃)⋊C₁₆	C₂²	1	4	yes	—
28	(C₂×D₃)⋊C₁₆	C₂²	1	4	yes	—
29	#4502	C₂²	1	2	yes	—
30	#4502	C₂²	1	2	yes	—
31	#4502	C₂²	1	2	yes	—
32	Dic₃×C₁₆	C₂²	1	2	yes	—
33	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	—
34	C₄×#131	—	3	1	no	—
35	#2164	—	3	4	no	—
36	#2163	—	3	4	no	—
37	#2151	—	3	1	no	—
38	#2151	D₃	1	1	yes	characteristic, complemented
39	#2113	—	3	1	no	—
40	#2113	—	3	1	no	—
41	C₄×M₃₂	—	3	1	no	—
42	C₂×#346	—	3	1	no	—
43	C₂×#235	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
46	C₃×#117	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₂×#208	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
50	#907	—	2	4	no	—
51	#901	—	2	2	no	—
52	#901	—	2	2	no	—
53	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
54	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
55	C₂×D₃×C₈	C₂×C₄	1	2	yes	—
56	C₂×D₃×C₈	C₂³	1	2	yes	—
57	#893	C₈	1	2	yes	—
58	#887	C₈	1	2	yes	—
59	#67⋊C₄	C₈	1	4	yes	—
60	#872	C₂×C₄	1	2	yes	—
61	C₄²×D₃	C₂×C₄	1	2	yes	—
62	C₃×M₃₂	—	2	2	no	—
63	C₂×C₃×C₁₆	—	2	2	no	—
64	C₂×C₃×C₁₆	C₂³	1	2	yes	—
65	C₂×C₃×C₁₆	C₂³	1	2	yes	—
66	C₃×#97	C₂×C₄	1	2	yes	—
67	C₃×C₄×C₈	C₂³	1	2	yes	—
68	#815	C₂×C₄	1	2	yes	—
69	C₈×Dic₃	C₂×C₄	1	2	yes	—
70	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₃	C₂³	1	1	yes	characteristic
73	C₈×Dic₃	C₂³	1	1	yes	characteristic
74	#812	—	2	2	no	—
75	C₂×C₃⋊C₁₆	—	2	2	no	—
76	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
77	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
78	#803	C₂×C₄	1	2	yes	—
79	C₄×C₃⋊C₈	C₂×C₄	1	2	yes	—
80	C₂×#131	—	3	1	no	—
81	C₂×#131	—	3	1	no	—
82	C₄×#41	—	3	1	no	—
83	C₂×M₃₂	—	3	1	no	—
84	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
85	C₂²×C₁₆	—	3	1	no	—
86	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
87	#405	—	3	1	no	—
88	#405	—	3	2	no	—
89	#405	—	3	2	no	—
90	#405	C₂×D₃	1	1	yes	characteristic
91	C₄×M₁₆	—	3	1	no	—
92	C₄×M₁₆	—	3	2	no	—
93	C₂×C₄×C₈	—	3	1	no	—
94	C₂×C₄×C₈	—	3	2	no	—
95	C₄⋊C₁₆	—	3	4	no	—
96	C₄⋊C₁₆	C₂×D₃	1	4	yes	—
97	C₂²⋊C₁₆	—	3	4	no	—
98	C₂²⋊C₁₆	—	3	4	no	—
99	#346	—	3	2	no	—
100	#346	—	3	2	no	—
101	#346	—	3	2	no	—
102	C₄×C₁₆	—	3	2	no	—
103	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
104	C₂×#67	C₂×C₈	1	2	yes	—
105	C₂²×Dic₃	C₂×C₈	1	2	yes	—
106	#235	—	2	2	no	—
107	#235	—	2	2	no	—
108	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
109	C₂×C₄×D₃	C₂×C₈	1	4	yes	—
110	C₂×C₄×D₃	C₂²×C₄	1	2	yes	—
111	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
112	C₃×M₁₆	—	2	2	no	—
113	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₃×C₈	#41	1	2	yes	—
116	C₂×C₃×C₈	#41	1	2	yes	—
117	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
118	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
119	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
120	C₃×C₄²	C₂²×C₄	1	2	yes	—
121	(C₂×D₃)⋊C₄	C₂×C₈	1	4	yes	—
122	Dic₃⋊C₄	C₂×C₈	1	4	yes	—
123	C₄×Dic₃	C₂×C₈	1	4	yes	—
124	C₄×Dic₃	C₂×C₈	1	2	yes	—
125	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
126	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
127	#208	—	2	2	no	—
128	C₂×C₃⋊C₈	M₁₆	1	4	yes	—
129	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
130	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
132	ΓL(1,𝔽₂₅)	—	2	4	no	—
133	ΓL(1,𝔽₂₅)	—	2	4	no	—
134	D₃×C₈	—	2	2	no	—
135	D₃×C₈	—	2	2	no	—
136	D₃×C₈	—	2	4	no	—
137	C₃×C₁₆	—	2	2	no	—
138	C₃×C₁₆	—	2	2	no	—
139	C₃⋊C₁₆	—	2	2	no	—
140	C₃⋊C₁₆	—	2	2	no	—
141	C₂×#41	—	3	1	no	—
142	#131	—	6	4	no	—
143	#131	—	6	2	no	—
144	#131	—	6	2	no	—
145	C₂×M₁₆	—	3	1	no	—
146	C₂×M₁₆	—	3	2	no	—
147	C₂×M₁₆	—	3	2	no	—
148	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
149	C₂²×C₈	—	3	1	no	—
150	C₂²×C₈	—	3	2	no	—
151	C₂²×C₈	—	3	2	no	—
152	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
153	C₄×Q₈	—	3	2	no	—
154	C₄×D₄	—	3	2	no	—
155	C₄×D₄	—	3	4	no	—
156	#117	—	3	2	no	—
157	#117	C₄×D₃	1	1	yes	characteristic
158	C₂×C₄²	—	3	1	no	—
159	C₂×C₄²	—	3	2	no	—
160	M₃₂	—	2	2	no	—
161	M₃₂	—	6	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	—	3	2	no	—
164	C₂×C₁₆	—	3	2	no	—
165	C₂×C₁₆	—	6	2	no	—
166	C₂×C₁₆	C₂²×D₃	1	2	yes	—
167	C₂×C₁₆	C₂²×D₃	1	2	yes	—
168	#97	—	3	2	no	—
169	#97	—	3	2	no	—
170	#97	C₄×D₃	1	2	yes	—
171	C₄×C₈	—	3	1	no	—
172	C₄×C₈	—	3	2	no	—
173	C₄×C₈	—	3	1	no	—
174	C₄×C₈	—	3	1	no	—
175	C₄×C₈	—	3	1	no	—
176	C₄×C₈	—	3	2	no	—
177	C₄×C₈	C₂²×D₃	1	2	yes	—
178	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
179	C₂²×D₃	C₂²×C₈	1	2	yes	—
180	C₂×C₃×C₄	#117	1	2	yes	—
181	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
182	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
184	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
185	#67	—	2	4	no	—
186	C₂×Dic₃	—	4	2	no	—
187	C₂×Dic₃	C₂²×C₈	1	2	yes	—
188	C₂×Dic₃	C₂²×C₈	1	2	yes	—
189	C₂×Dic₃	C₂²×C₈	1	2	yes	—
190	D₁₂	—	2	2	no	—
191	C₄×D₃	—	2	2	no	—
192	C₄×D₃	—	2	2	no	—
193	C₄×D₃	—	4	4	no	—
194	Dic₆	—	2	2	no	—
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	#117	1	2	yes	—
197	C₃×C₈	C₂×#41	1	1	yes	characteristic
198	C₃×C₈	C₂×#41	1	1	yes	characteristic
199	C₃⋊C₈	—	2	2	no	—
200	C₃⋊C₈	C₂×M₁₆	1	2	yes	—
201	C₃⋊C₈	C₂×M₁₆	1	2	yes	—
202	#41	—	6	2	no	—
203	#41	—	6	2	no	—
204	C₂×Q₈	—	3	1	no	—
205	C₂×D₄	—	3	2	no	—
206	C₂×D₄	—	3	1	no	—
207	C₂²×C₄	—	3	2	no	—
208	C₂²×C₄	—	3	2	no	—
209	C₂²×C₄	—	3	4	no	—
210	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
211	M₁₆	—	2	2	no	—
212	M₁₆	—	6	4	no	—
213	M₁₆	—	6	2	no	—
214	M₁₆	—	6	4	no	—
215	C₂×C₈	—	3	4	no	—
216	C₂×C₈	—	3	1	no	—
217	C₂×C₈	—	3	1	no	—
218	C₂×C₈	—	3	2	no	—
219	C₂×C₈	—	6	4	no	—
220	C₂×C₈	—	6	2	no	—
221	C₂×C₈	—	6	2	no	—
222	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
224	C₂×C₈	#237	1	2	yes	—
225	C₂×C₈	#239	1	2	yes	—
226	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
227	C₄⋊C₄	—	3	4	no	—
228	C₄⋊C₄	D₃×C₈	1	2	yes	—
229	C₂²⋊C₄	—	3	4	no	—
230	C₂²⋊C₄	D₃×C₈	1	2	yes	—
231	C₄²	—	3	2	no	—
232	C₄²	—	3	4	no	—
233	C₄²	—	3	1	no	—
234	C₄²	—	3	1	no	—
235	C₄²	C₂×C₄×D₃	1	2	yes	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	2	2	no	—
238	C₁₆	—	6	2	no	—
239	C₁₆	—	6	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	#431	1	2	yes	—
242	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
243	C₂×D₃	—	2	4	no	—
244	C₂×D₃	—	2	2	no	—
245	C₃×C₄	—	2	4	no	—
246	C₃×C₄	#431	1	2	yes	—
247	C₃×C₄	C₂×#117	1	1	yes	characteristic
248	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
249	Dic₃	—	2	2	no	—
250	Dic₃	—	2	2	no	—
251	Dic₃	—	2	2	no	—
252	C₂³	—	3	2	no	—
253	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
254	Q₈	—	6	2	no	—
255	D₄	—	6	4	no	—
256	D₄	—	6	2	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	3	2	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	—	6	2	no	—
261	C₂×C₄	—	6	2	no	—
262	C₂×C₄	—	12	4	no	—
263	C₂×C₄	—	12	2	no	—
264	C₂×C₄	#892	1	2	yes	—
265	C₂×C₄	C₂×D₃×C₈	1	4	yes	—
266	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
267	C₂×C₄	D₃×M₁₆	1	2	yes	—
268	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	3	2	no	—
271	C₈	—	3	2	no	—
272	C₈	—	6	2	no	—
273	C₈	#892	1	2	yes	—
274	C₈	C₂×#237	1	1	yes	characteristic
275	C₈	C₂×#239	1	1	yes	characteristic
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	C₂×#431	1	1	yes	characteristic
278	C₂×C₃	#3407	1	2	yes	—
279	D₃	—	4	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	6	4	no	—
282	C₂²	—	6	2	no	—
283	C₂²	#4833	1	2	yes	—
284	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	6	2	no	—
287	C₄	—	6	2	no	—
288	C₄	—	6	2	no	—
289	C₄	#4833	1	2	yes	—
290	C₄	C₂×#892	1	1	yes	characteristic
291	C₄	C₂×D₃×M₁₆	1	1	yes	characteristic
292	C₃	#33421	1	1	yes	characteristic, complemented, Sylow
293	C₂	—	2	2	no	—
294	C₂	—	12	2	no	—
295	C₂	C₂×#4833	1	1	yes	characteristic
296	C₂	#82012	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2151	C₂
prime radical	C₂×#235	C₂×C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2151	D₃
hypocenter	C₃	#33421
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
2. C₃×#2151	C₂	semidirect
4. Dic₃×M₃₂	C₂	semidirect
6. C₂×#4502	C₂	semidirect
7. #69882	C₂	semidirect
8. #69276	C₂	semidirect
9. #69275	C₂	semidirect
38. #2151	D₃	semidirect
292. C₃	#33421	semidirect

Maximal subgroups

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

subgroup	index
#78549	2
C₃×#2151	2
#71432	2
Dic₃×M₃₂	2
#71409	2
C₂×#4502	2
#69882	2
#69276	2
#69275	2
#33421	3