Subgroups — #11722012

Order 864.

244 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#88852	C₂	1	1	yes	maximal, characteristic
2	C₃×#3746	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#63627	—	3	1	no	maximal
5	#63549	—	3	4	no	maximal
6	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	1	no	—
7	C₃×#634	—	2	1	no	—
8	C₃³×C₈	C₂²	1	1	yes	characteristic
9	#3746	D₃	1	1	yes	characteristic, complemented
10	#3745	—	3	1	no	—
11	#3745	—	3	4	no	—
12	#3745	—	6	4	no	—
13	C₃×Dic₁₂	—	3	4	no	—
14	C₃²×C₁₆	—	3	1	no	—
15	C₃×C₃⋊C₁₆	D₃	1	4	yes	complemented
16	C₂×C₃⋊(C₃⋊D₃)	—	4	1	no	—
17	C₃³×C₄	D₄	1	1	yes	characteristic
18	C₃×C₃⋊Dic₃	—	4	1	no	—
19	#828	—	9	1	no	—
20	#800	—	9	4	no	—
21	C₄⋊(C₃⋊D₃)	—	6	8	no	—
22	C₄⋊(C₃⋊D₃)	—	6	4	no	—
23	C₄⋊(C₃⋊D₃)	—	6	1	no	—
24	#634	—	2	1	no	—
25	C₃×Dic₆	—	6	4	no	—
26	C₃²×C₈	—	2	4	no	—
27	C₃²×C₈	C₂×D₃	1	4	yes	—
28	C₃²×C₈	C₂×D₃	1	1	yes	characteristic
29	C₂×C₃³	D₈	1	1	yes	characteristic
30	C₃⋊(C₃⋊D₃)	—	8	1	no	—
31	C₃×Q₁₆	—	9	1	no	—
32	Dic₁₂	—	3	4	no	—
33	D₂₄	—	9	4	no	—
34	D₂₄	—	9	1	no	—
35	D₂₄	—	18	4	no	—
36	C₃×C₁₆	—	3	4	no	—
37	C₃⋊C₁₆	C₃⋊D₃	1	1	yes	characteristic, complemented
38	C₂×C₃⋊D₃	—	12	8	no	—
39	C₂×C₃⋊D₃	—	12	4	no	—
40	C₂×C₃⋊D₃	—	12	1	no	—
41	C₃²×C₄	—	2	4	no	—
42	C₃²×C₄	D₁₂	1	4	yes	—
43	C₃²×C₄	#67	1	1	yes	characteristic
44	C₃⋊Dic₃	—	4	1	no	—
45	C₃×Dic₃	—	12	4	no	—
46	SD₃₂	—	27	1	no	Sylow
47	C₃³	SD₃₂	1	1	yes	characteristic, complemented, Sylow
48	C₃×Q₈	—	18	1	no	—
49	D₁₂	—	18	8	no	—
50	D₁₂	—	18	4	no	—
51	D₁₂	—	18	1	no	—
52	Dic₆	—	6	4	no	—
53	C₃×C₈	—	2	4	no	—
54	C₃×C₈	D₃²	1	4	yes	—
55	C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
56	C₂×C₃²	—	2	4	no	—
57	C₂×C₃²	D₂₄	1	4	yes	—
58	C₂×C₃²	#213	1	1	yes	characteristic
59	C₃⋊D₃	—	24	8	no	—
60	C₃⋊D₃	—	24	4	no	—
61	C₃⋊D₃	—	24	1	no	—
62	Q₁₆	—	9	1	no	—
63	D₈	—	27	1	no	—
64	C₁₆	—	3	1	no	—
65	C₂×D₃	—	36	8	no	—
66	C₂×D₃	—	36	4	no	—
67	C₂×D₃	—	36	1	no	—
68	C₃×C₄	—	2	4	no	—
69	C₃×C₄	—	36	1	no	—
70	C₃×C₄	#626	1	4	yes	—
71	C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
72	Dic₃	—	12	4	no	—
73	C₃²	—	2	4	no	—
74	C₃²	#800	1	4	yes	complemented
75	C₃²	#828	1	1	yes	characteristic, complemented
76	Q₈	—	18	1	no	—
77	D₄	—	54	1	no	—
78	C₈	D₃×C₃⋊D₃	1	1	yes	characteristic
79	C₂×C₃	—	2	4	no	—
80	C₂×C₃	#3714	1	4	yes	—
81	C₂×C₃	#3745	1	1	yes	characteristic
82	D₃	—	72	8	no	—
83	D₃	—	72	4	no	—
84	D₃	—	72	1	no	—
85	C₂²	—	108	1	no	—
86	C₄	—	36	1	no	—
87	C₄	#6292	1	1	yes	characteristic
88	C₃	—	2	4	no	—
89	C₃	#63549	1	4	yes	complemented
90	C₃	#63627	1	1	yes	characteristic, complemented
91	C₂	—	216	1	no	—
92	C₂	#88791	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#88791
commutator, Fitting, cosocle	C₃³×C₈	C₂²
Frattini, hypercenter	C₈	D₃×C₃⋊D₃
socle	C₂×C₃³	D₈
prime radical	#88852	C₂
hypocenter	C₃³	SD₃₂
LCS	G ▷ C₃³×C₈ ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₃×#3746	C₂	semidirect
3. C₃²×C₃⋊C₁₆	C₂	semidirect
9. #3746	D₃	semidirect
15. C₃×C₃⋊C₁₆	D₃	semidirect
37. C₃⋊C₁₆	C₃⋊D₃	semidirect
47. C₃³	SD₃₂	semidirect
74. C₃²	#800	semidirect
75. C₃²	#828	semidirect
89. C₃	#63549	semidirect
90. C₃	#63627	semidirect

Maximal subgroups

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

subgroup	index
#88852	2
C₃×#3746	2
C₃²×C₃⋊C₁₆	2
#63627	3
#63549	3