Subgroups — #11772228

Order 1152.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#3708	C₂	1	1	yes	maximal, characteristic
2	#10589100	C₂	1	2	yes	maximal
3	C₂×#3708	C₄	1	1	yes	characteristic
4	C₂×#3708	C₄	1	1	yes	characteristic
5	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
6	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
7	#4479	—	6	2	no	—
8	C₃²×C₄²	D₄	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
11	C₂×C₃×C₃⋊C₈	—	4	2	no	—
12	#3708	—	2	1	no	—
13	#3708	C₈	1	2	yes	complemented
14	#1371	—	9	1	no	maximal, Sylow
15	C₃×C₄⋊C₈	—	6	2	no	—
16	C₄⋊(C₃⋊C₈)	—	2	2	no	—
17	C₄×C₃⋊C₈	—	6	2	no	—
18	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
20	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
21	C₃⋊(C₃⋊C₈)	—	2	1	no	—
22	C₃⋊(C₃⋊C₈)	C₂×C₈	1	1	yes	characteristic
23	C₃⋊(C₃⋊C₈)	M₁₆	1	1	yes	characteristic
24	C₃×C₃⋊C₈	—	8	2	no	—
25	C₄×C₁₆	—	9	1	no	—
26	#335	—	9	2	no	—
27	C₂×C₃×C₈	—	12	2	no	—
28	C₃×C₄²	—	2	2	no	—
29	C₂×C₃⋊C₈	—	4	2	no	—
30	C₂×C₃⋊C₈	—	6	2	no	—
31	C₂×C₃⋊C₈	—	6	2	no	—
32	C₂²×C₃²	#107	1	1	yes	characteristic
33	C₃²×C₄	—	2	1	no	—
34	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
35	C₃²×C₄	#108	1	1	yes	characteristic
36	C₃²×C₄	D₁₆	1	1	yes	characteristic
37	C₃²×C₄	Q₃₂	1	1	yes	characteristic
38	C₂×C₁₆	—	9	1	no	—
39	C₂×C₁₆	—	9	1	no	—
40	C₄⋊C₈	—	6	2	no	—
41	C₄×C₈	—	9	1	no	—
42	C₂×C₃×C₄	—	2	2	no	—
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	—	2	2	no	—
45	C₃×C₈	—	24	2	no	—
46	C₃⋊C₈	—	6	2	no	—
47	C₃⋊C₈	—	6	2	no	—
48	C₃⋊C₈	—	8	2	no	—
49	C₃⋊C₈	—	12	2	no	—
50	C₂×C₃²	#335	1	1	yes	characteristic
51	C₂×C₃²	#366	1	1	yes	characteristic
52	C₂×C₃²	#368	1	1	yes	characteristic
53	C₂×C₈	—	9	1	no	—
54	C₂×C₈	—	9	1	no	—
55	C₂×C₈	—	12	2	no	—
56	C₄²	GO⁺(4,𝔽₂)	1	1	yes	characteristic
57	C₁₆	—	9	2	no	—
58	C₁₆	—	18	1	no	—
59	C₂²×C₃	—	2	2	no	—
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	—	2	2	no	—
63	C₃×C₄	—	2	2	no	—
64	C₃×C₄	—	4	2	no	—
65	C₃²	#1371	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄	#3773	1	1	yes	characteristic
67	C₂×C₄	#3774	1	1	yes	characteristic
68	C₂×C₄	#3776	1	1	yes	characteristic
69	C₈	—	9	1	no	—
70	C₈	—	9	1	no	—
71	C₈	—	18	1	no	—
72	C₈	—	24	2	no	—
73	C₂×C₃	—	2	2	no	—
74	C₂×C₃	—	2	2	no	—
75	C₂×C₃	—	2	2	no	—
76	C₂²	#63743	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#63732	1	1	yes	characteristic
79	C₄	#63733	1	1	yes	characteristic
80	C₄	#63734	1	1	yes	characteristic
81	C₄	#63736	1	1	yes	characteristic
82	C₃	—	2	2	no	—
83	C₂	#10590042	1	1	yes	characteristic
84	C₂	#10590068	1	1	yes	characteristic
85	C₂	#10590070	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3774
commutator	C₃⋊(C₃⋊C₈)	C₂×C₈
Frattini, hypercenter	C₄²	GO⁺(4,𝔽₂)
Fitting	C₃²×C₄²	D₄
socle, prime radical	C₂²×C₃²	#107
cosocle	C₄×C₃⋊(C₃⋊C₈)	C₂²
hypocenter	C₃²	#1371
LCS	G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
13. #3708	C₈	semidirect
65. C₃²	#1371	semidirect

Maximal subgroups

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

subgroup	index
C₄×#3708	2
#10589100	2
#1371	9