Subgroups — #10710558

Order 768.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1770	C₂	1	1	yes	maximal, characteristic
2	C₂×#4466	C₂	1	2	yes	maximal
3	#69743	C₂	1	1	yes	maximal, characteristic
4	C₄×#812	C₂	1	1	yes	maximal, characteristic
5	#67663	C₂	1	2	yes	maximal
6	#12666	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₃×#97	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₂×#812	C₂²	1	1	yes	characteristic
12	C₂×#812	C₂²	1	2	yes	—
13	C₂×#812	C₂²	1	1	yes	characteristic
14	C₃×#336	C₄	1	2	yes	—
15	#4466	—	2	4	no	—
16	C₄⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
17	C₄⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
18	#4461	C₄	1	2	yes	complemented
19	C₄×C₃⋊C₁₆	C₄	1	2	yes	complemented
20	#2150	—	3	1	no	—
21	C₂×#347	—	3	2	no	—
22	C₄×M₃₂	—	3	1	no	—
23	#1770	D₃	1	1	yes	characteristic
24	#1375	—	3	2	no	—
25	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄⋊C₈	—	2	1	no	—
32	C₃×C₄⋊C₈	—	2	1	no	—
33	C₃×#97	—	2	2	no	—
34	C₃×#97	—	2	2	no	—
35	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×#95	—	2	1	no	—
40	#812	—	2	2	no	—
41	#812	—	2	1	no	—
42	#812	—	2	1	no	—
43	#812	—	2	2	no	—
44	#812	D₄	1	2	yes	—
45	#812	Q₈	1	2	yes	—
46	C₂×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
48	C₂×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
49	C₂×M₃₂	—	3	1	no	—
50	C₂×M₃₂	—	3	1	no	—
51	C₂×M₃₂	—	3	2	no	—
52	C₂×C₄⋊C₈	Dic₃	1	1	yes	characteristic
53	C₂×#97	C₂×D₃	1	2	yes	—
54	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
55	C₄×C₄⋊C₄	Dic₃	1	1	yes	characteristic
56	C₄⋊C₁₆	—	3	2	no	—
57	C₄⋊C₁₆	—	3	2	no	—
58	#347	—	6	4	no	—
59	#346	—	3	2	no	—
60	C₄×C₁₆	—	3	2	no	—
61	#336	Dic₃	1	2	yes	—
62	C₂²×C₃×C₄	—	2	2	no	—
63	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
64	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₈	—	2	1	no	—
67	C₂×C₃×C₈	—	2	1	no	—
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
72	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃×C₈	#41	1	1	yes	characteristic
78	C₂×C₃×C₈	#41	1	1	yes	characteristic
79	C₃×C₄⋊C₄	—	4	2	no	—
80	C₃×C₄²	—	2	2	no	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	C₄²	1	1	yes	characteristic
83	C₃×C₄²	C₄²	1	1	yes	characteristic
84	C₃×C₄²	M₁₆	1	1	yes	characteristic
85	C₃×C₄²	M₁₆	1	1	yes	characteristic
86	C₃⋊C₁₆	—	2	2	no	—
87	C₃⋊C₁₆	—	2	4	no	—
88	C₃⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
89	C₂²×C₈	C₂×Dic₃	1	2	yes	—
90	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄²	C₂×Dic₃	1	2	yes	—
94	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
95	M₃₂	—	3	2	no	—
96	M₃₂	—	3	2	no	—
97	M₃₂	—	6	2	no	—
98	M₃₂	—	6	1	no	—
99	M₃₂	—	6	1	no	—
100	M₃₂	—	6	2	no	—
101	C₂×C₁₆	—	3	2	no	—
102	C₂×C₁₆	—	3	2	no	—
103	C₂×C₁₆	—	3	4	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	2	1	no	—
106	#97	—	2	2	no	—
107	#97	—	2	2	no	—
108	C₄×C₈	C₄×D₃	1	1	yes	characteristic
109	C₄×C₈	C₄×D₃	1	1	yes	characteristic
110	C₄×C₈	C₄×D₃	1	1	yes	characteristic
111	C₄×C₈	C₄×D₃	1	1	yes	characteristic
112	#95	—	2	1	no	—
113	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	4	1	no	—
116	C₂×C₃×C₄	—	4	2	no	—
117	C₂×C₃×C₄	—	4	2	no	—
118	C₂×C₃×C₄	—	4	2	no	—
119	C₂×C₃×C₄	#97	1	1	yes	characteristic
120	C₂×C₃×C₄	#97	1	1	yes	characteristic
121	C₂×C₃×C₄	#97	1	1	yes	characteristic
122	C₂×C₃×C₄	#97	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₃×C₄	#131	1	1	yes	characteristic
128	C₂×C₃×C₄	#131	1	1	yes	characteristic
129	C₃×C₈	—	2	1	no	—
130	C₃×C₈	—	2	1	no	—
131	C₃×C₈	—	4	2	no	—
132	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₃×C₈	#117	1	1	yes	characteristic
134	C₃×C₈	C₄×D₄	1	1	yes	characteristic
135	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	#208	1	1	yes	characteristic
138	C₂²×C₄	#208	1	1	yes	characteristic
139	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
146	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
147	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×C₈	D₃×D₄	1	1	yes	characteristic
150	C₂×C₈	#237	1	1	yes	characteristic
151	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
152	C₂×C₈	#239	1	1	yes	characteristic
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	#208	1	1	yes	characteristic
157	C₄²	#208	1	1	yes	characteristic
158	C₄²	C₄×Dic₃	1	1	yes	characteristic
159	C₄²	C₄×Dic₃	1	1	yes	characteristic
160	C₁₆	—	3	4	no	—
161	C₁₆	—	6	2	no	—
162	C₁₆	—	6	4	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	C₂×#97	1	1	yes	characteristic
166	C₂²×C₃	#436	1	1	yes	characteristic
167	C₂²×C₃	#446	1	1	yes	characteristic
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	8	2	no	—
172	C₃×C₄	#347	1	2	yes	complemented
173	C₃×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
174	C₃×C₄	#405	1	1	yes	characteristic
175	C₃×C₄	#436	1	1	yes	characteristic
176	C₃×C₄	#446	1	1	yes	characteristic
177	C₂³	C₂×#208	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	#803	1	1	yes	characteristic
184	C₂×C₄	#803	1	1	yes	characteristic
185	C₂×C₄	#803	1	1	yes	characteristic
186	C₂×C₄	#803	1	1	yes	characteristic
187	C₂×C₄	C₂×#208	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
189	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
190	C₂×C₄	Q₈×Dic₃	1	1	yes	characteristic
191	C₂×C₄	#948	1	1	yes	characteristic
192	C₂×C₄	#948	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	2	no	—
196	C₈	#887	1	1	yes	characteristic
197	C₈	D₃×C₄⋊C₄	1	1	yes	characteristic
198	C₈	#892	1	1	yes	characteristic
199	C₈	#893	1	1	yes	characteristic
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	#1770	1	1	yes	characteristic
203	C₂×C₃	C₂×#347	1	1	yes	characteristic
204	C₂×C₃	#2110	1	1	yes	characteristic
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	C₂×#803	1	1	yes	characteristic
208	C₂²	#5012	1	1	yes	characteristic
209	C₂²	#5024	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	8	2	no	—
214	C₄	#4466	1	2	yes	complemented
215	C₄	Dic₃×C₄⋊C₄	1	1	yes	characteristic
216	C₄	#4997	1	1	yes	characteristic
217	C₄	#5012	1	1	yes	characteristic
218	C₄	#5024	1	1	yes	characteristic
219	C₃	#12666	1	1	yes	characteristic, complemented, Sylow
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#69477	1	1	yes	characteristic
223	C₂	C₂×#4466	1	1	yes	characteristic
224	C₂	#69884	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₃
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1770	C₂
socle	C₂²×C₃	C₂×#97
prime radical	C₂³×C₃	C₂×M₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1770	D₃
hypocenter	C₃	#12666
LCS	G ▷ C₂×C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
16. C₄⋊(C₃⋊C₁₆)	C₄	semidirect
17. C₄⋊(C₃⋊C₁₆)	C₄	semidirect
18. #4461	C₄	semidirect
19. C₄×C₃⋊C₁₆	C₄	semidirect
88. C₃⋊C₁₆	C₄⋊C₄	semidirect
172. C₃×C₄	#347	semidirect
214. C₄	#4466	semidirect
219. C₃	#12666	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1770	2
C₂×#4466	2
#69743	2
C₄×#812	2
#67663	2
#12666	3