Subgroups — #11771297

Order 1152.

288 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	#67437	—	3	4	no	maximal
4	C₃²×C₄×C₈	C₄	1	2	yes	—
5	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	4	yes	—
7	C₃×C₈²	D₃	1	4	yes	—
8	C₄×C₃⋊C₁₆	—	3	8	no	—
9	C₂×C₃²×C₈	C₈	1	4	yes	—
10	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
11	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
12	C₃⋊(C₃⋊C₁₆)	C₈	1	8	yes	complemented
13	#1311	—	9	1	no	Sylow
14	C₃×C₄×C₈	Dic₃	1	8	yes	—
15	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
16	C₂×C₃⋊C₁₆	—	3	16	no	—
17	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
18	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
19	C₃²×C₈	—	2	4	no	—
20	C₃²×C₈	C₂×C₈	1	4	yes	—
21	C₄×C₁₆	—	9	2	no	—
22	C₈²	C₃⋊D₃	1	1	yes	characteristic
23	C₂×C₃×C₈	C₃⋊C₈	1	16	yes	—
24	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
25	C₃×C₄²	C₂×Dic₃	1	4	yes	—
26	C₃⋊C₁₆	—	3	32	no	—
27	C₂²×C₃²	C₄×C₈	1	1	yes	characteristic
28	C₃²×C₄	C₄×C₈	1	2	yes	—
29	C₃²×C₄	M₃₂	1	4	yes	—
30	C₂×C₁₆	—	9	4	no	—
31	C₄×C₈	C₃⋊Dic₃	1	2	yes	—
32	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
34	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
35	C₃×C₈	—	2	16	no	—
36	C₃×C₈	D₃×C₈	1	16	yes	—
37	C₂×C₃²	C₈²	1	1	yes	characteristic
38	C₂×C₃²	#346	1	2	yes	—
39	C₂×C₈	C₃⋊(C₃⋊C₈)	1	4	yes	—
40	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
41	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
42	C₁₆	—	9	8	no	—
43	C₂²×C₃	C₄×C₃⋊C₈	1	4	yes	—
44	C₃×C₄	#812	1	16	yes	—
45	C₃×C₄	C₈×Dic₃	1	8	yes	—
46	C₃²	#1311	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
48	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
49	C₈	—	2	4	no	—
50	C₈	C₈×C₃⋊D₃	1	4	yes	—
51	C₂×C₃	C₈×C₃⋊C₈	1	4	yes	—
52	C₂×C₃	#4461	1	8	yes	—
53	C₂²	C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
54	C₄	#63639	1	4	yes	—
55	C₄	C₈×C₃⋊Dic₃	1	2	yes	—
56	C₃	#67437	1	4	yes	complemented
57	C₂	C₈×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
58	C₂	#10589327	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₈²
Fitting	C₃²×C₈²	C₂
socle, prime radical	C₂²×C₃²	C₄×C₈
cosocle	C₃²×C₄×C₈	C₂²
hypercenter	C₈²	C₃⋊D₃
hypocenter	C₃²	#1311
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
12. C₃⋊(C₃⋊C₁₆)	C₈	semidirect
46. C₃²	#1311	semidirect
56. C₃	#67437	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
#67437	3