Subgroups — #10795366

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2873	C₂	1	1	yes	maximal, characteristic
2	C₂×#5125	C₂	1	1	yes	maximal, characteristic
3	#83752	C₂	1	1	yes	maximal, characteristic
4	C₂×#4556	C₂	1	2	yes	maximal
5	C₂×#4531	C₂	1	2	yes	maximal
6	#69896	C₂	1	8	yes	maximal, complemented
7	#20757	—	3	1	no	maximal, Sylow
8	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×#117	C₂²	1	1	yes	characteristic
11	C₂×#926	C₂²	1	2	yes	—
12	C₂²×#208	C₂²	1	2	yes	—
13	C₃×#405	C₄	1	4	yes	complemented
14	C₃×#405	C₂²	1	4	yes	—
15	C₃×C₄×M₁₆	C₄	1	4	yes	complemented
16	#5125	—	2	4	no	—
17	#5121	—	2	4	no	—
18	C₂×#819	C₂²	1	4	yes	—
19	#5106	C₂²	1	4	yes	—
20	#4556	C₂²	1	8	yes	—
21	#4531	C₂²	1	8	yes	—
22	C₂×#430	—	3	1	no	—
23	#2911	—	3	1	no	—
24	#2873	D₃	1	1	yes	characteristic
25	#1833	—	3	8	no	—
26	C₂×#344	—	3	2	no	—
27	C₂×H(3,ℤ/4)	—	3	2	no	—
28	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃×M₁₆	—	2	2	no	—
30	C₂×C₃×M₁₆	—	2	2	no	—
31	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₃×M₁₆	D₄	1	4	yes	—
35	C₂×C₃×M₁₆	Q₈	1	4	yes	—
36	C₂×C₃×M₁₆	C₂³	1	2	yes	—
37	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₈	C₂³	1	2	yes	—
39	C₃×#117	—	2	1	no	—
40	C₃×#117	—	2	1	no	—
41	C₃×#117	C₂×C₄	1	2	yes	—
42	C₃×#117	C₂³	1	2	yes	—
43	C₂×C₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
45	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
46	C₂×#217	—	2	2	no	—
47	#926	—	2	2	no	—
48	#926	—	2	2	no	—
49	#926	—	2	2	no	—
50	#926	C₂³	1	4	yes	—
51	C₂×C₄⋊Dic₃	—	2	2	no	—
52	C₂×C₄×Dic₃	—	2	2	no	—
53	C₂×#208	—	2	2	no	—
54	C₂×#208	—	2	2	no	—
55	C₂×#208	—	2	2	no	—
56	C₂×#208	—	2	2	no	—
57	C₂×#208	—	2	2	no	—
58	C₂×#208	C₂³	1	4	yes	—
59	C₂²×C₃⋊C₈	—	2	2	no	—
60	C₃×#97	C₂×C₄	1	8	yes	complemented
61	C₃×C₄×C₈	C₂×C₄	1	8	yes	complemented
62	#819	—	2	8	no	—
63	#818	—	2	4	no	—
64	#817	—	2	4	no	—
65	C₂²×M₁₆	—	3	2	no	—
66	C₂²×M₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂²×M₁₆	C₂×D₃	1	1	yes	characteristic
68	C₂×#117	—	3	2	no	—
69	C₂×#117	C₂×D₃	1	1	yes	characteristic
70	#430	—	6	4	no	—
71	C₂×#108	—	3	4	no	—
72	#428	—	6	4	no	—
73	#427	—	3	4	no	—
74	#405	Dic₃	1	4	yes	complemented
75	#405	C₂×D₃	1	4	yes	—
76	C₄×M₁₆	Dic₃	1	4	yes	complemented
77	#344	—	3	8	no	—
78	H(3,ℤ/4)	—	3	8	no	—
79	C₂⁴×C₃	#41	1	1	yes	characteristic
80	C₂²×C₃×C₄	—	2	2	no	—
81	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
84	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
85	C₂²×C₃×C₄	#41	1	1	yes	characteristic
86	C₂²×C₃×C₄	#41	1	2	yes	—
87	C₂²×C₃×C₄	#41	1	2	yes	—
88	C₂²×C₃×C₄	#41	1	1	yes	characteristic
89	C₂²×C₃×C₄	#41	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₃	—	4	2	no	—
92	C₃×M₁₆	—	2	4	no	—
93	C₃×M₁₆	—	2	2	no	—
94	C₃×M₁₆	—	2	2	no	—
95	C₃×M₁₆	C₄⋊C₄	1	8	yes	—
96	C₃×M₁₆	C₄⋊C₄	1	8	yes	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
101	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
102	C₃×C₄⋊C₄	—	2	2	no	—
103	C₃×C₄⋊C₄	—	2	2	no	—
104	C₃×C₂²⋊C₄	—	2	2	no	—
105	C₃×C₂²⋊C₄	—	2	2	no	—
106	C₃×C₄²	—	2	2	no	—
107	C₃×C₄²	C₂²×C₄	1	4	yes	—
108	#217	—	2	4	no	—
109	#217	—	4	2	no	—
110	C₄⋊Dic₃	—	2	4	no	—
111	C₄⋊Dic₃	—	4	2	no	—
112	C₄×Dic₃	—	2	4	no	—
113	C₄×Dic₃	—	4	2	no	—
114	#208	—	2	4	no	—
115	#208	—	2	4	no	—
116	#208	—	4	2	no	—
117	#208	—	4	2	no	—
118	C₂×C₃⋊C₈	—	2	4	no	—
119	C₂×C₃⋊C₈	—	4	2	no	—
120	C₂×C₃⋊C₈	—	4	2	no	—
121	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂×M₁₆	—	3	4	no	—
125	C₂×M₁₆	—	6	2	no	—
126	C₂×M₁₆	—	6	2	no	—
127	C₂×M₁₆	—	6	2	no	—
128	C₂×M₁₆	—	6	2	no	—
129	C₂×M₁₆	—	6	2	no	—
130	C₂×M₁₆	Dic₆	1	4	yes	—
131	C₂×M₁₆	D₁₂	1	4	yes	—
132	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
133	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
134	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
135	C₂×M₁₆	C₂²×D₃	1	2	yes	—
136	C₂²×C₈	—	6	2	no	—
137	C₂²×C₈	C₂×Dic₃	1	2	yes	—
138	C₂²×C₈	C₂²×D₃	1	2	yes	—
139	#117	—	2	1	no	—
140	#117	—	2	1	no	—
141	#117	—	3	4	no	—
142	#117	—	6	2	no	—
143	#117	—	6	2	no	—
144	#117	—	6	2	no	—
145	#117	C₂×Dic₃	1	2	yes	—
146	#117	C₂²×D₃	1	2	yes	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	6	2	no	—
149	C₂×C₂²⋊C₄	—	2	1	no	—
150	C₂×C₂²⋊C₄	—	6	2	no	—
151	C₂×C₄²	—	6	2	no	—
152	C₂×C₄²	C₂×Dic₃	1	2	yes	—
153	#108	—	6	8	no	—
154	#107	—	6	4	no	—
155	#106	—	6	4	no	—
156	#97	C₂×Dic₃	1	8	yes	complemented
157	C₄×C₈	C₂×Dic₃	1	8	yes	complemented
158	C₂³×C₃	—	8	1	no	—
159	C₂³×C₃	#117	1	1	yes	characteristic
160	C₂³×C₃	#117	1	1	yes	characteristic
161	C₂³×C₃	#124	1	2	yes	—
162	C₂³×C₃	#125	1	2	yes	—
163	C₂³×C₃	C₂×#41	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	8	2	no	—
166	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
168	C₂×C₃×C₄	#117	1	2	yes	—
169	C₂×C₃×C₄	#117	1	2	yes	—
170	C₂×C₃×C₄	#117	1	1	yes	characteristic
171	C₂×C₃×C₄	#117	1	1	yes	characteristic
172	C₂×C₃×C₄	#124	1	2	yes	—
173	C₂×C₃×C₄	#124	1	2	yes	—
174	C₂×C₃×C₄	#124	1	2	yes	—
175	C₂×C₃×C₄	#125	1	2	yes	—
176	C₂×C₃×C₄	#125	1	2	yes	—
177	C₂×C₃×C₄	#125	1	2	yes	—
178	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
184	C₂×Dic₃	—	4	4	no	—
185	C₂×Dic₃	—	8	4	no	—
186	C₃×C₈	—	2	4	no	—
187	C₃×C₈	C₂×C₄⋊C₄	1	8	yes	—
188	C₃⋊C₈	—	4	4	no	—
189	C₂⁴	#235	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	12	2	no	—
192	C₂²×C₄	C₂×Dic₆	1	2	yes	—
193	C₂²×C₄	C₂×D₁₂	1	2	yes	—
194	C₂²×C₄	#235	1	1	yes	characteristic
195	C₂²×C₄	#235	1	1	yes	characteristic
196	C₂²×C₄	#235	1	1	yes	characteristic
197	C₂²×C₄	#235	1	2	yes	—
198	C₂²×C₄	#235	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
201	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	2	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	6	4	no	—
206	M₁₆	—	6	4	no	—
207	M₁₆	—	12	2	no	—
208	M₁₆	—	12	2	no	—
209	M₁₆	C₄⋊Dic₃	1	8	yes	—
210	M₁₆	C₄⋊Dic₃	1	8	yes	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	6	4	no	—
213	C₂×C₈	—	12	2	no	—
214	C₂×C₈	—	12	2	no	—
215	C₂×C₈	C₂×Dic₆	1	4	yes	—
216	C₂×C₈	C₂×D₁₂	1	4	yes	—
217	C₂×C₈	C₂²×Dic₃	1	4	yes	—
218	C₂×C₈	C₂²×Dic₃	1	4	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	6	4	no	—
222	C₄⋊C₄	—	12	2	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	6	4	no	—
226	C₂²⋊C₄	—	12	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	6	4	no	—
229	C₄²	—	12	2	no	—
230	C₄²	C₂²×Dic₃	1	4	yes	—
231	C₂²×C₃	—	4	2	no	—
232	C₂²×C₃	—	4	1	no	—
233	C₂²×C₃	—	4	1	no	—
234	C₂²×C₃	—	4	2	no	—
235	C₂²×C₃	—	4	1	no	—
236	C₂²×C₃	#382	1	2	yes	—
237	C₂²×C₃	#382	1	2	yes	—
238	C₂²×C₃	C₂×#117	1	1	yes	characteristic
239	C₂²×C₃	C₂×#124	1	1	yes	characteristic
240	C₂²×C₃	C₂×#125	1	1	yes	characteristic
241	C₃×C₄	—	4	4	no	—
242	C₃×C₄	#382	1	2	yes	—
243	C₃×C₄	#382	1	2	yes	—
244	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
245	C₃×C₄	C₂×#117	1	1	yes	characteristic
246	C₃×C₄	C₂×#124	1	1	yes	characteristic
247	C₃×C₄	C₂×#125	1	1	yes	characteristic
248	Dic₃	—	8	4	no	—
249	C₂³	—	8	1	no	—
250	C₂³	#870	1	2	yes	—
251	C₂³	#875	1	2	yes	—
252	C₂³	#926	1	1	yes	characteristic
253	C₂³	#926	1	1	yes	characteristic
254	C₂³	C₂×#235	1	1	yes	characteristic
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	12	4	no	—
258	C₂×C₄	—	24	4	no	—
259	C₂×C₄	#870	1	2	yes	—
260	C₂×C₄	#870	1	2	yes	—
261	C₂×C₄	#870	1	2	yes	—
262	C₂×C₄	#875	1	2	yes	—
263	C₂×C₄	#875	1	2	yes	—
264	C₂×C₄	#875	1	2	yes	—
265	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
266	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
267	C₂×C₄	#926	1	1	yes	characteristic
268	C₂×C₄	#926	1	1	yes	characteristic
269	C₂×C₄	#926	1	2	yes	—
270	C₂×C₄	#926	1	2	yes	—
271	C₂×C₄	C₂²×Dic₆	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
273	C₂×C₄	C₂×#235	1	1	yes	characteristic
274	C₂×C₄	C₂×#235	1	1	yes	characteristic
275	C₂×C₄	C₂×#235	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₈	—	12	4	no	—
279	C₈	C₂×C₄⋊Dic₃	1	8	yes	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	2	2	no	—
285	C₂×C₃	C₂×#382	1	1	yes	characteristic
286	C₂²	—	4	1	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	#4936	1	2	yes	—
292	C₂²	#4936	1	2	yes	—
293	C₂²	C₂×#870	1	1	yes	characteristic
294	C₂²	C₂×#875	1	1	yes	characteristic
295	C₂²	C₂×#926	1	1	yes	characteristic
296	C₄	—	4	4	no	—
297	C₄	—	24	4	no	—
298	C₄	#4936	1	2	yes	—
299	C₄	#4936	1	2	yes	—
300	C₄	C₂×#870	1	1	yes	characteristic
301	C₄	C₂×#875	1	1	yes	characteristic
302	C₄	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
303	C₄	C₂×#926	1	1	yes	characteristic
304	C₃	#20757	1	1	yes	characteristic, complemented, Sylow
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	C₂×#4936	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#926
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₃×#2873	C₂
socle	C₂×C₃	C₂×#382
prime radical	C₂⁴×C₃	#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#2873	D₃
hypocenter	C₃	#20757
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ #2873

Complements

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

N	H ≅ G/N	type
6. #69896	C₂	semidirect
13. C₃×#405	C₄	semidirect
15. C₃×C₄×M₁₆	C₄	semidirect
60. C₃×#97	C₂×C₄	semidirect
61. C₃×C₄×C₈	C₂×C₄	semidirect
74. #405	Dic₃	semidirect
76. C₄×M₁₆	Dic₃	semidirect
156. #97	C₂×Dic₃	semidirect
157. C₄×C₈	C₂×Dic₃	semidirect
304. C₃	#20757	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2873	2
C₂×#5125	2
#83752	2
C₂×#4556	2
C₂×#4531	2
#69896	2
#20757	3