Subgroups — #11664876

Order 768.

1762 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#85829	C₂	1	2	yes	maximal
2	C₃×C₄×#131	C₂	1	1	yes	maximal, characteristic
3	#83785	C₂	1	2	yes	maximal
4	#83784	C₂	1	2	yes	maximal
5	#78546	C₂	1	6	yes	maximal, complemented
6	#78545	C₂	1	6	yes	maximal, complemented
7	#78507	C₂	1	6	yes	maximal, complemented
8	#78442	C₂	1	3	yes	maximal, complemented
9	C₂×#4683	C₂	1	3	yes	maximal, complemented
10	#60666	—	3	1	no	maximal, Sylow
11	C₂×C₃×#131	C₂²	1	1	yes	characteristic
12	C₂×C₃×#131	C₂²	1	1	yes	characteristic
13	C₃×C₄×#41	C₂²	1	1	yes	characteristic
14	#5827	C₂²	1	4	yes	—
15	#5569	—	2	4	no	—
16	#5539	—	2	12	no	—
17	#5519	C₂²	1	12	yes	—
18	#5470	C₂²	1	6	yes	—
19	C₂×#870	C₂²	1	6	yes	—
20	C₃×#405	C₂²	1	3	yes	—
21	C₃×#405	C₂²	1	3	yes	—
22	C₃×C₄×M₁₆	C₂²	1	3	yes	—
23	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
24	#5121	C₂²	1	12	yes	—
25	#5121	C₂²	1	12	yes	—
26	#5106	C₂²	1	6	yes	—
27	#5106	C₂²	1	6	yes	—
28	C₂×#818	C₂²	1	6	yes	—
29	C₂×#817	C₂²	1	6	yes	—
30	#4702	C₂²	1	24	yes	—
31	#4683	C₂²	1	3	yes	—
32	#4683	C₂²	1	6	yes	—
33	#4683	C₂²	1	6	yes	—
34	#4683	C₂²	1	6	yes	—
35	#4683	C₂²	1	1	yes	characteristic
36	#4683	C₂²	1	6	yes	—
37	#4682	C₂²	1	6	yes	—
38	#4680	C₂²	1	6	yes	—
39	#3504	—	3	2	no	—
40	#3166	—	3	6	no	—
41	#3164	—	3	6	no	—
42	#3163	—	3	6	no	—
43	#3160	—	3	3	no	—
44	C₂×#499	—	3	3	no	—
45	#2913	—	3	2	no	—
46	#2912	—	3	2	no	—
47	C₄×#131	D₃	1	1	yes	characteristic
48	C₂×C₃×#41	C₂³	1	1	yes	characteristic
49	C₃×#131	Q₈	1	8	yes	complemented
50	C₃×#131	Q₈	1	8	yes	complemented
51	C₂×C₃×M₁₆	C₂³	1	3	yes	—
52	C₂×C₃×M₁₆	C₂³	1	3	yes	—
53	C₂²×C₃×C₈	C₂³	1	3	yes	—
54	C₂²×C₃×C₈	C₂³	1	3	yes	—
55	C₃×C₄×Q₈	D₄	1	2	yes	—
56	C₃×C₄×D₄	D₄	1	6	yes	—
57	C₃×#117	C₂³	1	3	yes	—
58	C₂×C₃×C₄²	C₂³	1	3	yes	—
59	Q₈×Dic₃	—	2	4	no	—
60	D₄×Dic₃	—	2	12	no	—
61	#926	C₂³	1	12	yes	—
62	C₂×C₄⋊Dic₃	C₂³	1	12	yes	—
63	#924	—	2	12	no	—
64	C₂×Dic₃⋊C₄	—	2	12	no	—
65	#890	—	2	12	no	—
66	#888	—	2	12	no	—
67	#878	—	2	24	no	—
68	#870	—	2	12	no	—
69	#870	C₂³	1	6	yes	—
70	#870	C₂³	1	2	yes	—
71	#870	C₂³	1	12	yes	—
72	#869	C₂³	1	12	yes	—
73	C₄×Dic₆	—	2	12	no	—
74	C₃×#97	C₂³	1	6	yes	—
75	C₃×C₄×C₈	C₂³	1	3	yes	—
76	C₃×C₄×C₈	C₂³	1	3	yes	—
77	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
78	C₃×C₄×C₈	C₂³	1	3	yes	—
79	#818	C₂³	1	12	yes	—
80	#818	C₂³	1	12	yes	—
81	#818	C₂³	1	2	yes	—
82	#818	C₂³	1	6	yes	—
83	#817	C₂³	1	12	yes	—
84	#817	C₂³	1	12	yes	—
85	#817	C₂³	1	2	yes	—
86	#817	C₂³	1	6	yes	—
87	C₂×#131	C₂×D₃	1	1	yes	characteristic
88	C₂×#131	C₂×D₃	1	1	yes	characteristic
89	#557	—	6	4	no	—
90	#554	—	6	12	no	—
91	#544	—	3	12	no	—
92	#533	—	3	6	no	—
93	C₂×#125	—	3	6	no	—
94	#520	—	3	4	no	—
95	C₄×#41	C₂×D₃	1	1	yes	characteristic
96	#501	—	3	24	no	—
97	#500	—	3	6	no	—
98	#499	—	3	3	no	—
99	#499	—	3	6	no	—
100	#499	—	3	6	no	—
101	#499	—	3	6	no	—
102	#499	—	3	1	no	—
103	#499	—	3	6	no	—
104	#498	—	3	6	no	—
105	#428	—	3	12	no	—
106	#428	—	3	12	no	—
107	#427	—	3	6	no	—
108	#427	—	3	6	no	—
109	C₂×#107	—	3	6	no	—
110	C₂×#106	—	3	6	no	—
111	#405	C₂×D₃	1	3	yes	—
112	#405	C₂×D₃	1	3	yes	—
113	C₄×M₁₆	C₂×D₃	1	3	yes	—
114	C₂×C₄×C₈	C₂×D₃	1	3	yes	—
115	C₃×#41	C₂×Q₈	1	4	yes	—
116	C₃×#41	C₂×Q₈	1	4	yes	—
117	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₃×D₄	C₂×D₄	1	3	yes	—
119	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
120	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
121	C₂²×Dic₃	—	2	12	no	—
122	C₂×Dic₆	—	2	12	no	—
123	C₃×M₁₆	C₂×Q₈	1	12	yes	—
124	C₃×M₁₆	C₂×Q₈	1	12	yes	—
125	C₂×C₃×C₈	C₂×Q₈	1	12	yes	—
126	C₂×C₃×C₈	C₂×Q₈	1	12	yes	—
127	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
128	C₂×C₃×C₈	C₂⁴	1	3	yes	—
129	C₂×C₃×C₈	C₂⁴	1	3	yes	—
130	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
131	C₃×C₄⋊C₄	C₂×D₄	1	6	yes	—
132	C₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
133	C₃×C₄²	C₂×D₄	1	6	yes	—
134	C₃×C₄²	C₂⁴	1	1	yes	characteristic
135	C₃×C₄²	C₂⁴	1	3	yes	—
136	#217	—	2	12	no	—
137	C₄⋊Dic₃	—	2	12	no	—
138	C₄⋊Dic₃	—	2	12	no	—
139	C₄⋊Dic₃	C₂⁴	1	4	yes	—
140	C₄⋊Dic₃	C₂⁴	1	12	yes	—
141	Dic₃⋊C₄	—	2	4	no	—
142	Dic₃⋊C₄	—	2	12	no	—
143	Dic₃⋊C₄	—	2	24	no	—
144	C₄×Dic₃	—	2	12	no	—
145	C₂×#41	C₂²×D₃	1	1	yes	characteristic
146	#131	Dic₆	1	8	yes	complemented
147	#131	Dic₆	1	8	yes	complemented
148	C₂×M₁₆	C₂²×D₃	1	3	yes	—
149	C₂×M₁₆	C₂²×D₃	1	3	yes	—
150	C₂²×C₈	C₂²×D₃	1	3	yes	—
151	C₂²×C₈	C₂²×D₃	1	3	yes	—
152	C₄⋊Q₈	—	3	12	no	—
153	C₄⋊Q₈	—	6	12	no	—
154	#125	—	3	6	no	—
155	#125	—	3	12	no	—
156	#125	—	3	2	no	—
157	#125	—	6	12	no	—
158	#125	—	6	12	no	—
159	C₂²⋊Q₈	—	6	12	no	—
160	C₂²⋊Q₈	—	6	24	no	—
161	C₄×Q₈	—	6	4	no	—
162	C₄×Q₈	—	6	12	no	—
163	C₄×Q₈	D₁₂	1	2	yes	—
164	C₄×D₄	—	6	12	no	—
165	C₄×D₄	D₁₂	1	6	yes	—
166	#117	—	3	12	no	—
167	#117	C₂²×D₃	1	3	yes	—
168	C₂×C₄⋊C₄	—	3	12	no	—
169	C₂×C₄⋊C₄	—	6	12	no	—
170	C₂×C₄²	C₂²×D₃	1	3	yes	—
171	#107	—	3	6	no	—
172	#107	—	3	12	no	—
173	#107	—	3	2	no	—
174	#107	—	3	12	no	—
175	#106	—	3	6	no	—
176	#106	—	3	12	no	—
177	#106	—	3	2	no	—
178	#106	—	3	12	no	—
179	#97	C₂²×D₃	1	6	yes	—
180	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
181	C₄×C₈	C₂²×D₃	1	3	yes	—
182	C₄×C₈	C₂²×D₃	1	3	yes	—
183	C₄×C₈	C₂²×D₃	1	3	yes	—
184	C₂³×C₃	C₂²×D₄	1	3	yes	—
185	C₃×Q₈	C₄⋊Q₈	1	4	yes	—
186	C₃×D₄	C₄⋊Q₈	1	12	yes	—
187	C₂×C₃×C₄	—	4	6	no	—
188	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
189	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
190	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
191	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
192	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
193	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
194	C₂×Dic₃	—	2	4	no	—
195	C₂×Dic₃	—	2	12	no	—
196	C₂×Dic₃	—	8	12	no	—
197	Dic₆	—	8	12	no	—
198	C₃×C₈	C₂²×Q₈	1	2	yes	—
199	C₃×C₈	C₂²×Q₈	1	2	yes	—
200	C₃×C₈	C₂²×Q₈	1	6	yes	—
201	C₃×C₈	C₂²×Q₈	1	6	yes	—
202	#41	C₂×Dic₆	1	4	yes	—
203	#41	C₂×Dic₆	1	4	yes	—
204	C₂×Q₈	—	6	12	no	—
205	C₂×Q₈	C₂×D₁₂	1	1	yes	characteristic
206	C₂×D₄	C₂×D₁₂	1	3	yes	—
207	C₂²×C₄	—	6	12	no	—
208	C₂²×C₄	C₂×D₁₂	1	6	yes	—
209	C₂²×C₄	C₂³×D₃	1	3	yes	—
210	M₁₆	C₂×Dic₆	1	12	yes	—
211	M₁₆	C₂×Dic₆	1	12	yes	—
212	C₂×C₈	C₂×Dic₆	1	12	yes	—
213	C₂×C₈	C₂×Dic₆	1	12	yes	—
214	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
215	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
216	C₂×C₈	C₂³×D₃	1	3	yes	—
217	C₂×C₈	C₂³×D₃	1	3	yes	—
218	C₄⋊C₄	—	3	12	no	—
219	C₄⋊C₄	—	3	4	no	—
220	C₄⋊C₄	—	6	12	no	—
221	C₄⋊C₄	—	6	12	no	—
222	C₄⋊C₄	—	6	24	no	—
223	C₄⋊C₄	—	6	4	no	—
224	C₄⋊C₄	—	6	12	no	—
225	C₄⋊C₄	C₂×D₁₂	1	6	yes	—
226	C₂²⋊C₄	—	6	12	no	—
227	C₂²⋊C₄	C₂×D₁₂	1	6	yes	—
228	C₄²	—	6	12	no	—
229	C₄²	C₂×D₁₂	1	6	yes	—
230	C₄²	C₂³×D₃	1	1	yes	characteristic
231	C₄²	C₂³×D₃	1	3	yes	—
232	C₂²×C₃	—	2	6	no	—
233	C₂²×C₃	C₂×C₄⋊Q₈	1	6	yes	—
234	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
235	C₃×C₄	—	2	6	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	C₂×C₄⋊Q₈	1	6	yes	—
238	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
239	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
240	Dic₃	—	4	4	no	—
241	Dic₃	—	4	12	no	—
242	C₂³	C₂²×D₁₂	1	3	yes	—
243	Q₈	—	24	12	no	—
244	Q₈	#869	1	4	yes	—
245	D₄	#869	1	12	yes	—
246	C₂×C₄	—	4	6	no	—
247	C₂×C₄	—	6	4	no	—
248	C₂×C₄	—	6	12	no	—
249	C₂×C₄	—	24	12	no	—
250	C₂×C₄	C₂²×Dic₆	1	6	yes	—
251	C₂×C₄	C₂²×Dic₆	1	6	yes	—
252	C₂×C₄	C₂²×D₁₂	1	2	yes	—
253	C₂×C₄	C₂²×D₁₂	1	6	yes	—
254	C₂×C₄	C₂²×D₁₂	1	3	yes	—
255	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
256	C₈	C₂²×Dic₆	1	2	yes	—
257	C₈	C₂²×Dic₆	1	2	yes	—
258	C₈	C₂²×Dic₆	1	6	yes	—
259	C₈	C₂²×Dic₆	1	6	yes	—
260	C₂×C₃	—	2	6	no	—
261	C₂×C₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
262	C₂×C₃	#3584	1	2	yes	—
263	C₂²	—	2	6	no	—
264	C₂²	C₂×#869	1	6	yes	—
265	C₂²	C₂³×D₁₂	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	6	no	—
268	C₄	—	12	12	no	—
269	C₄	—	12	4	no	—
270	C₄	C₂×#869	1	6	yes	—
271	C₄	C₂³×Dic₆	1	1	yes	characteristic
272	C₄	C₂³×Dic₆	1	1	yes	characteristic
273	C₃	#60666	1	1	yes	characteristic, complemented, Sylow
274	C₂	—	2	6	no	—
275	C₂	C₂²×#869	1	1	yes	characteristic
276	C₂	#87163	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₃×C₄×#131	C₂
socle	C₂²×C₃	C₂³×D₄
prime radical	C₂×C₃×#41	C₂³
hypercenter	C₄×#131	D₃
hypocenter	C₃	#60666
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
5. #78546	C₂	semidirect
6. #78545	C₂	semidirect
7. #78507	C₂	semidirect
8. #78442	C₂	semidirect
9. C₂×#4683	C₂	semidirect
49. C₃×#131	Q₈	semidirect
50. C₃×#131	Q₈	semidirect
146. #131	Dic₆	semidirect
147. #131	Dic₆	semidirect
273. C₃	#60666	semidirect

Maximal subgroups

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

subgroup	index
#85829	2
C₃×C₄×#131	2
#83785	2
#83784	2
#78546	2
#78545	2
#78507	2
#78442	2
C₂×#4683	2
#60666	3