Subgroups — #11772924

Order 1152.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63628	C₂	1	1	yes	maximal, characteristic
2	C₃²×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊(C₃⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#67578	—	3	4	no	maximal
5	C₂×#3746	—	2	1	no	—
6	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
7	C₃²×#107	—	2	1	no	—
8	#63628	—	2	1	no	—
9	#63628	C₄	1	2	yes	complemented
10	C₃⋊(C₃⋊C₃₂)	—	2	1	no	—
11	C₂×Dic₂₄	—	3	4	no	—
12	C₃×#366	D₃	1	4	yes	—
13	C₂×C₃⋊C₃₂	—	3	4	no	—
14	C₂×#634	—	4	1	no	—
15	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
16	C₃²×C₄⋊C₄	—	4	1	no	—
17	#3746	—	2	2	no	—
18	#3746	—	4	1	no	—
19	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₃²×C₁₆	D₄	1	1	yes	characteristic
21	#1416	—	9	1	no	Sylow
22	C₂×Dic₁₂	—	6	4	no	—
23	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
24	C₃×#107	—	2	4	no	—
25	Dic₂₄	—	3	8	no	—
26	Dic₂₄	—	6	4	no	—
27	C₃⋊C₃₂	—	6	4	no	—
28	C₂×C₃²×C₄	—	8	1	no	—
29	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
30	C₂×C₃⋊Dic₃	—	8	1	no	—
31	#634	—	4	2	no	—
32	#634	—	8	1	no	—
33	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
34	C₃²×C₈	SD₁₆	1	1	yes	characteristic
35	C₂×Q₃₂	—	9	1	no	—
36	C₂×C₃₂	—	9	1	no	—
37	#366	C₃⋊D₃	1	1	yes	characteristic
38	C₂×Dic₆	—	12	4	no	—
39	C₂×C₃×C₈	#67	1	4	yes	—
40	C₃×C₄⋊C₄	—	4	4	no	—
41	Dic₁₂	—	6	8	no	—
42	Dic₁₂	—	12	4	no	—
43	C₃×C₁₆	C₄×D₃	1	4	yes	—
44	C₃×C₁₆	D₁₂	1	4	yes	—
45	C₂²×C₃²	D₁₆	1	1	yes	characteristic
46	C₃²×C₄	—	16	1	no	—
47	C₃²×C₄	#102	1	1	yes	characteristic
48	C₃²×C₄	SD₃₂	1	1	yes	characteristic
49	C₃⋊Dic₃	—	8	2	no	—
50	C₂×Q₁₆	—	18	1	no	—
51	Q₃₂	—	9	2	no	—
52	Q₃₂	—	18	1	no	—
53	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
54	#107	—	2	1	no	—
55	C₃₂	—	18	1	no	—
56	C₂×C₃×C₄	—	8	4	no	—
57	C₂×C₃×C₄	#213	1	4	yes	—
58	C₂×Dic₃	—	24	4	no	—
59	Dic₆	—	12	8	no	—
60	Dic₆	—	24	4	no	—
61	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
62	C₃×C₈	#215	1	4	yes	—
63	C₂×C₃²	#357	1	1	yes	characteristic
64	C₂×C₃²	SD₆₄	1	1	yes	characteristic
65	C₂×C₃²	Q₆₄	1	1	yes	characteristic
66	C₂×Q₈	—	36	1	no	—
67	Q₁₆	—	18	2	no	—
68	Q₁₆	—	36	1	no	—
69	C₂×C₈	#638	1	1	yes	characteristic
70	C₄⋊C₄	—	4	1	no	—
71	C₁₆	C₄×C₃⋊D₃	1	1	yes	characteristic
72	C₁₆	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
73	C₂²×C₃	#826	1	4	yes	—
74	C₃×C₄	—	16	4	no	—
75	C₃×C₄	#809	1	4	yes	—
76	C₃×C₄	#828	1	4	yes	—
77	Dic₃	—	24	8	no	—
78	C₃²	#1416	1	1	yes	characteristic, complemented, Sylow
79	Q₈	—	36	2	no	—
80	Q₈	—	72	1	no	—
81	C₂×C₄	—	8	1	no	—
82	C₂×C₄	—	72	1	no	—
83	C₂×C₄	#3753	1	1	yes	characteristic
84	C₈	#3752	1	1	yes	characteristic
85	C₈	#3755	1	1	yes	characteristic
86	C₂×C₃	#4491	1	4	yes	—
87	C₂×C₃	#4520	1	4	yes	—
88	C₂×C₃	#4522	1	4	yes	—
89	C₂²	#63653	1	1	yes	characteristic
90	C₄	—	16	1	no	—
91	C₄	—	72	2	no	—
92	C₄	#63636	1	1	yes	characteristic
93	C₄	#63655	1	1	yes	characteristic
94	C₃	#67578	1	4	yes	complemented
95	C₂	#10589357	1	1	yes	characteristic
96	C₂	#10589386	1	1	yes	characteristic
97	C₂	#10589388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63653
commutator	C₃²×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×C₃⋊D₃
Fitting	C₃²×#366	C₂
socle, prime radical	C₂²×C₃²	D₁₆
cosocle	C₂×C₃²×C₁₆	C₂²
hypercenter	#366	C₃⋊D₃
hypocenter	C₃²	#1416
LCS	G ▷ C₃²×C₁₆ ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
9. #63628	C₄	semidirect
78. C₃²	#1416	semidirect
94. C₃	#67578	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63628	2
C₃²×#366	2
C₂×C₃⋊(C₃⋊C₃₂)	2
#67578	3