Subgroups — #11720567

Order 864.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#88525	C₂	1	1	yes	maximal, characteristic
3	C₃⋊(C₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#63655	—	3	1	no	maximal
5	#63387	—	3	3	no	maximal
6	C₃×Q₈×C₉	—	2	1	no	—
7	C₄⋊(C₃⋊D₉)	—	2	1	no	—
8	C₃×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₃²×Q₁₆	D₃	1	1	yes	characteristic
10	#3745	—	3	1	no	—
11	C₃⋊(C₃⋊C₁₆)	—	3	1	no	—
12	C₉×Q₁₆	D₃	1	3	yes	complemented
13	D₇₂	—	3	3	no	—
14	C₉⋊C₁₆	—	3	3	no	—
15	C₂×C₃⋊D₉	—	4	1	no	—
16	C₃×C₄×C₉	—	4	1	no	—
17	C₃×C₄×C₉	D₄	1	1	yes	characteristic
18	#828	—	9	3	no	—
19	#828	—	9	1	no	—
20	C₃²×Q₈	—	2	1	no	—
21	C₄⋊(C₃⋊D₃)	—	6	1	no	—
22	C₃²×C₈	C₂×D₃	1	1	yes	characteristic
23	Q₈×C₉	—	2	3	no	—
24	D₃₆	—	6	3	no	—
25	C₈×C₉	C₂×D₃	1	3	yes	—
26	C₂×C₃×C₉	D₈	1	1	yes	characteristic
27	C₃⋊D₉	—	8	1	no	—
28	C₃×Q₁₆	D₉	1	3	yes	complemented
29	C₃×Q₁₆	C₃⋊D₃	1	1	yes	characteristic
30	D₂₄	—	9	3	no	—
31	D₂₄	—	9	1	no	—
32	C₃⋊C₁₆	—	9	3	no	—
33	C₃⋊C₁₆	—	9	1	no	—
34	C₂×C₃⋊D₃	—	12	1	no	—
35	C₃²×C₄	—	4	1	no	—
36	C₃²×C₄	#67	1	1	yes	characteristic
37	C₂×D₉	—	12	3	no	—
38	C₄×C₉	—	4	3	no	—
39	C₄×C₉	#67	1	3	yes	—
40	SD₃₂	—	27	1	no	Sylow
41	C₃×C₉	SD₃₂	1	1	yes	characteristic, complemented, Sylow
42	C₃×Q₈	—	2	3	no	—
43	C₃×Q₈	—	2	1	no	—
44	D₁₂	—	18	3	no	—
45	D₁₂	—	18	1	no	—
46	C₃×C₈	C₂×D₉	1	3	yes	—
47	C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
48	C₂×C₃²	#213	1	1	yes	characteristic
49	C₃⋊D₃	—	24	1	no	—
50	C₂×C₉	#213	1	3	yes	—
51	D₉	—	24	3	no	—
52	Q₁₆	C₃⋊D₉	1	1	yes	characteristic, complemented
53	D₈	—	27	1	no	—
54	C₁₆	—	27	1	no	—
55	C₂×D₃	—	36	3	no	—
56	C₂×D₃	—	36	1	no	—
57	C₃×C₄	—	4	3	no	—
58	C₃×C₄	—	4	1	no	—
59	C₃×C₄	#611	1	3	yes	—
60	C₃×C₄	#638	1	1	yes	characteristic
61	C₃²	#828	1	1	yes	characteristic
62	C₉	#828	1	3	yes	complemented
63	Q₈	—	2	1	no	—
64	D₄	—	54	1	no	—
65	C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
66	C₂×C₃	#3673	1	3	yes	—
67	C₂×C₃	#3753	1	1	yes	characteristic
68	D₃	—	72	3	no	—
69	D₃	—	72	1	no	—
70	C₂²	—	108	1	no	—
71	C₄	—	4	1	no	—
72	C₄	#6233	1	1	yes	characteristic
73	C₃	#63387	1	3	yes	complemented
74	C₃	#63655	1	1	yes	characteristic
75	C₂	—	216	1	no	—
76	C₂	#88544	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#88544
commutator, cosocle	C₃×C₈×C₉	C₂²
Frattini	C₃×C₈	C₂×C₃⋊D₃
Fitting	C₃×C₉×Q₁₆	C₂
socle	C₂×C₃²	#213
prime radical	#88525	C₂
hypercenter	Q₁₆	C₃⋊D₉
hypocenter	C₃×C₉	SD₃₂
LCS	G ▷ C₃×C₈×C₉ ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×Q₁₆	C₂	semidirect
3. C₃⋊(C₉⋊C₁₆)	C₂	semidirect
12. C₉×Q₁₆	D₃	semidirect
28. C₃×Q₁₆	D₉	semidirect
41. C₃×C₉	SD₃₂	semidirect
52. Q₁₆	C₃⋊D₉	semidirect
62. C₉	#828	semidirect
73. C₃	#63387	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×Q₁₆	2
#88525	2
C₃⋊(C₉⋊C₁₆)	2
#63655	3
#63387	3