Subgroups — #11772164

Order 1152.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	2	yes	maximal
3	#67439	—	3	4	no	maximal
4	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
5	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
6	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
8	C₃×C₈²	D₃	1	4	yes	—
9	C₄⋊(C₃⋊C₁₆)	—	3	8	no	—
10	C₂×C₃²×C₈	—	2	1	no	—
11	C₂×C₃²×C₈	C₈	1	1	yes	characteristic
12	C₂×C₃²×C₈	C₈	1	1	yes	characteristic
13	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
14	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
15	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₃⋊(C₃⋊C₁₆)	—	4	2	no	—
17	#1367	—	9	1	no	Sylow
18	C₃×C₄×C₈	Dic₃	1	4	yes	—
19	C₃×C₄×C₈	Dic₃	1	4	yes	—
20	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
21	C₂×C₃⋊C₁₆	—	6	8	no	—
22	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
23	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
24	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
25	C₃²×C₈	—	2	2	no	—
26	C₃²×C₈	—	2	1	no	—
27	C₃²×C₈	—	2	1	no	—
28	C₃²×C₈	C₁₆	1	2	yes	complemented
29	C₃²×C₈	SD₁₆	1	2	yes	—
30	C₄⋊C₁₆	—	9	2	no	—
31	C₈²	C₃⋊D₃	1	1	yes	characteristic
32	C₂×C₃×C₈	—	2	4	no	—
33	C₂×C₃×C₈	C₃⋊C₈	1	4	yes	—
34	C₂×C₃×C₈	C₃⋊C₈	1	4	yes	—
35	C₂×C₃×C₈	Dic₆	1	4	yes	—
36	C₂×C₃×C₈	D₁₂	1	4	yes	—
37	C₃×C₄²	C₂×Dic₃	1	4	yes	—
38	C₃⋊C₁₆	—	12	8	no	—
39	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
40	C₃²×C₄	—	2	1	no	—
41	C₃²×C₄	#106	1	1	yes	characteristic
42	C₃²×C₄	#108	1	1	yes	characteristic
43	C₃²×C₄	C₂×C₁₆	1	1	yes	characteristic
44	C₃²×C₄	M₃₂	1	1	yes	characteristic
45	C₂×C₁₆	—	18	2	no	—
46	C₄×C₈	C₃⋊Dic₃	1	1	yes	characteristic
47	C₄×C₈	C₃⋊Dic₃	1	1	yes	characteristic
48	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
49	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
50	C₂×C₃×C₄	#208	1	4	yes	—
51	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
52	C₃×C₈	—	2	8	no	—
53	C₃×C₈	—	2	4	no	—
54	C₃×C₈	—	2	4	no	—
55	C₃×C₈	C₃⋊C₁₆	1	8	yes	complemented
56	C₃×C₈	#204	1	8	yes	—
57	C₂×C₃²	#334	1	1	yes	characteristic
58	C₂×C₃²	C₄⋊C₁₆	1	1	yes	characteristic
59	C₂×C₃²	#364	1	1	yes	characteristic
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
62	C₂×C₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
63	C₂×C₈	#634	1	1	yes	characteristic
64	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
65	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
66	C₁₆	—	36	2	no	—
67	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	4	yes	—
68	C₃×C₄	—	2	4	no	—
69	C₃×C₄	C₂×C₃⋊C₁₆	1	4	yes	—
70	C₃×C₄	#812	1	4	yes	—
71	C₃×C₄	#817	1	4	yes	—
72	C₃×C₄	#819	1	4	yes	—
73	C₃²	#1367	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
75	C₂×C₄	#3748	1	1	yes	characteristic
76	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	2	1	no	—
79	C₈	—	2	1	no	—
80	C₈	C₃⋊(C₃⋊C₁₆)	1	2	yes	complemented
81	C₈	#3744	1	2	yes	—
82	C₂×C₃	#4457	1	4	yes	—
83	C₂×C₃	C₄⋊(C₃⋊C₁₆)	1	4	yes	—
84	C₂×C₃	#4463	1	4	yes	—
85	C₂²	C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	C₂×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
88	C₄	#63639	1	1	yes	characteristic
89	C₄	#63644	1	1	yes	characteristic
90	C₄	#63646	1	1	yes	characteristic
91	C₃	#67439	1	4	yes	complemented
92	C₂	#10589323	1	1	yes	characteristic
93	C₂	C₄⋊(C₃⋊(C₃⋊C₁₆))	1	1	yes	characteristic
94	C₂	#10589329	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂×C₁₆
Frattini	C₄×C₈	C₂×C₃⋊D₃
Fitting	C₃²×C₈²	C₂
socle, prime radical	C₂²×C₃²	C₄⋊C₈
cosocle	C₃²×C₄×C₈	C₂²
hypercenter	C₈²	C₃⋊D₃
hypocenter	C₃²	#1367
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
28. C₃²×C₈	C₁₆	semidirect
55. C₃×C₈	C₃⋊C₁₆	semidirect
73. C₃²	#1367	semidirect
80. C₈	C₃⋊(C₃⋊C₁₆)	semidirect
91. C₃	#67439	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₈²	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	2
#67439	3