Subgroups — #11772664

Order 1152.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63540	C₂	1	2	yes	maximal
2	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
4	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
5	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
6	#63540	C₄	1	4	yes	complemented
7	C₄×C₃⋊C₁₆	—	6	2	no	—
8	C₂×C₃²×C₈	C₈	1	1	yes	characteristic
9	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
10	C₃²×C₄²	C₈	1	1	yes	characteristic
11	C₃⋊(C₃⋊C₁₆)	C₈	1	2	yes	—
12	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	—
13	#1397	—	9	1	no	maximal, Sylow
14	C₃×C₄×C₈	—	2	2	no	—
15	C₂×C₃⋊C₁₆	—	6	2	no	—
16	C₂×C₃⋊C₁₆	—	6	2	no	—
17	C₂×C₃²×C₄	C₁₆	1	2	yes	—
18	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
19	C₃²×C₈	C₁₆	1	2	yes	—
20	C₃²×C₈	C₄²	1	1	yes	characteristic
21	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
22	C₂×C₃₂	—	9	2	no	—
23	C₄×C₁₆	—	9	1	no	—
24	C₂×C₃×C₈	—	2	2	no	—
25	C₂×C₃×C₈	—	2	2	no	—
26	C₃×C₄²	—	2	2	no	—
27	C₃⋊C₁₆	—	6	4	no	—
28	C₃⋊C₁₆	—	6	4	no	—
29	C₂²×C₃²	C₂×C₁₆	1	1	yes	characteristic
30	C₃²×C₄	—	2	2	no	—
31	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
32	C₃²×C₄	C₂×C₁₆	1	1	yes	characteristic
33	C₂×C₁₆	—	9	1	no	—
34	C₂×C₁₆	—	9	1	no	—
35	C₄×C₈	#157	1	1	yes	characteristic
36	C₃₂	—	9	4	no	—
37	C₂×C₃×C₄	—	2	4	no	—
38	C₂×C₃×C₄	—	2	2	no	—
39	C₃×C₈	—	2	4	no	—
40	C₃×C₈	—	2	2	no	—
41	C₃×C₈	—	2	2	no	—
42	C₂×C₃²	C₄×C₁₆	1	1	yes	characteristic
43	C₂×C₃²	M₆₄	1	2	yes	—
44	C₂×C₈	#622	1	1	yes	characteristic
45	C₂×C₈	C₂×#157	1	1	yes	characteristic
46	C₄²	#622	1	1	yes	characteristic
47	C₁₆	—	9	2	no	—
48	C₁₆	—	9	2	no	—
49	C₂²×C₃	—	2	2	no	—
50	C₃×C₄	—	2	8	no	—
51	C₃×C₄	—	2	2	no	—
52	C₃×C₄	—	2	2	no	—
53	C₃²	#1397	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄	#3708	1	2	yes	—
55	C₂×C₄	C₂×#622	1	1	yes	characteristic
56	C₈	#3708	1	2	yes	—
57	C₈	#3787	1	1	yes	characteristic
58	C₈	C₄×#157	1	1	yes	characteristic
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	—	2	4	no	—
61	C₂²	C₂×#3708	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	C₂×#3708	1	1	yes	characteristic
64	C₄	C₄×#622	1	1	yes	characteristic
65	C₃	—	2	2	no	—
66	C₂	C₄×#3708	1	1	yes	characteristic
67	C₂	#10590166	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#157
commutator	C₂×C₃²	C₄×C₁₆
Fitting	C₃²×C₄×C₈	C₄
socle, prime radical	C₂²×C₃²	C₂×C₁₆
cosocle	C₂×C₃⋊(C₃⋊C₁₆)	C₂²
hypercenter	C₄×C₈	#157
hypocenter	C₃²	#1397
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #63540	C₄	semidirect
53. C₃²	#1397	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63540	2
C₄×C₃⋊(C₃⋊C₁₆)	2
#1397	9