Subgroups — #11721876

Order 864.

715 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#6235	C₂	1	3	yes	maximal, complemented
3	#88552	C₂	1	3	yes	maximal, complemented
4	C₃×#953	—	3	4	no	maximal
5	C₂³×H(3,𝔽₃)	—	2	6	no	—
6	C₂³×H(3,𝔽₃)	C₂²	1	3	yes	—
7	C₂²×#272	C₂²	1	1	yes	characteristic, complemented
8	#6235	—	2	6	no	—
9	C₂×#1084	C₂²	1	3	yes	complemented
10	C₂⁴×C₃²	D₃	1	4	yes	complemented
11	C₂×C₃×#67	—	3	12	no	—
12	C₃×#217	—	3	12	no	—
13	C₂²×H(3,𝔽₃)	—	2	8	no	—
14	C₂²×H(3,𝔽₃)	—	2	6	no	—
15	C₂²×H(3,𝔽₃)	D₄	1	6	yes	complemented
16	C₂²×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
17	C₂×#272	—	2	3	no	—
18	#1084	—	2	3	no	—
19	C₃×#120	—	9	1	no	—
20	#953	—	3	4	no	—
21	C₂³×C₃²	—	2	24	no	—
22	C₂³×C₃²	C₂×D₃	1	12	yes	—
23	C₂²×C₃×D₃	—	3	4	no	—
24	C₃×#67	—	6	24	no	—
25	C₂×C₃×Dic₃	—	3	12	no	—
26	C₂×H(3,𝔽₃)	—	2	6	no	—
27	C₂×H(3,𝔽₃)	C₂×D₄	1	3	yes	—
28	#272	—	4	1	no	—
29	C₂⁴×C₃	—	3	4	no	—
30	C₂⁴×C₃	C₃⋊D₃	1	1	yes	characteristic
31	C₂×C₃×D₄	—	9	3	no	—
32	C₂×#67	—	3	12	no	—
33	C₃×C₂²⋊C₄	—	9	3	no	—
34	#217	—	3	12	no	—
35	C₂²×C₃²	—	2	32	no	—
36	C₂²×C₃²	—	2	24	no	—
37	C₂²×C₃²	#67	1	24	yes	complemented
38	C₂²×C₃²	C₂²×D₃	1	4	yes	—
39	C₂×C₃×D₃	—	6	12	no	—
40	C₃×Dic₃	—	6	12	no	—
41	#120	—	9	1	no	Sylow
42	H(3,𝔽₃)	#120	1	1	yes	characteristic, complemented, Sylow
43	C₂³×C₃	—	2	6	no	—
44	C₂³×C₃	—	3	12	no	—
45	C₂³×C₃	—	6	24	no	—
46	C₂³×C₃	—	9	1	no	—
47	C₂³×C₃	C₂×C₃⋊D₃	1	3	yes	—
48	C₂²×D₃	—	3	4	no	—
49	C₃×D₄	—	18	6	no	—
50	C₂×C₃×C₄	—	9	3	no	—
51	#67	—	6	24	no	—
52	C₂×Dic₃	—	3	12	no	—
53	C₂×C₃²	—	2	24	no	—
54	C₂×C₃²	C₂×#67	1	12	yes	—
55	C₃×D₃	—	12	4	no	—
56	C₂⁴	#272	1	1	yes	characteristic, complemented
57	C₂×D₄	—	9	3	no	—
58	C₂²⋊C₄	—	9	3	no	—
59	C₂²×C₃	—	2	8	no	—
60	C₂²×C₃	—	2	6	no	—
61	C₂²×C₃	—	3	24	no	—
62	C₂²×C₃	—	3	4	no	—
63	C₂²×C₃	—	6	24	no	—
64	C₂²×C₃	—	6	32	no	—
65	C₂²×C₃	—	18	3	no	—
66	C₂²×C₃	#638	1	6	yes	—
67	C₂²×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
68	C₂×D₃	—	6	12	no	—
69	C₃×C₄	—	18	3	no	—
70	Dic₃	—	6	12	no	—
71	C₃²	#953	1	4	yes	complemented
72	C₂³	—	2	6	no	—
73	C₂³	—	9	1	no	—
74	C₂³	C₂×#272	1	3	yes	—
75	D₄	—	18	6	no	—
76	C₂×C₄	—	9	3	no	—
77	C₂×C₃	—	2	6	no	—
78	C₂×C₃	—	3	12	no	—
79	C₂×C₃	—	6	24	no	—
80	C₂×C₃	—	36	1	no	—
81	C₂×C₃	C₂×#638	1	3	yes	—
82	D₃	—	12	4	no	—
83	C₂²	—	2	8	no	—
84	C₂²	—	2	6	no	—
85	C₂²	—	18	3	no	—
86	C₂²	#6235	1	6	yes	complemented
87	C₂²	C₂²×#272	1	1	yes	characteristic
88	C₄	—	18	3	no	—
89	C₃	—	3	4	no	—
90	C₃	#64162	1	1	yes	characteristic
91	C₂	—	2	6	no	—
92	C₂	—	36	1	no	—
93	C₂	C₂×#6235	1	3	yes	—

	subgroup	G/H
center, Frattini, socle	C₂²×C₃	C₂²×C₃⋊D₃
commutator, cosocle	C₂²×H(3,𝔽₃)	C₂³
Fitting	C₂⁴×H(3,𝔽₃)	C₂
hypercenter	C₂⁴×C₃	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#120
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃

Complements

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

N	H ≅ G/N	type
1. C₂⁴×H(3,𝔽₃)	C₂	semidirect
2. C₂×#6235	C₂	semidirect
3. #88552	C₂	semidirect
7. C₂²×#272	C₂²	semidirect
9. C₂×#1084	C₂²	semidirect
10. C₂⁴×C₃²	D₃	semidirect
15. C₂²×H(3,𝔽₃)	D₄	semidirect
37. C₂²×C₃²	#67	semidirect
42. H(3,𝔽₃)	#120	semidirect
56. C₂⁴	#272	semidirect
71. C₃²	#953	semidirect
86. C₂²	#6235	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×H(3,𝔽₃)	2
C₂×#6235	2
#88552	2
C₃×#953	3