Subgroups — #13048936

Order 1296.

388 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#737	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#4208	C₂	1	3	yes	maximal, complemented
3	C₄×#4208	C₂	1	3	yes	maximal, complemented
4	#88748	C₃	1	1	yes	maximal, characteristic
5	C₉×#239	—	3	4	no	maximal
6	C₂×#4208	C₂²	1	3	yes	—
7	C₄×#737	C₂²	1	3	yes	—
8	#66410	C₂²	1	1	yes	characteristic
9	Q₈×3₋¹⁺²	—	2	4	no	—
10	Q₈×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
11	C₃×Q₈×C₉	D₃	1	4	yes	complemented
12	C₄⋊#272	C₂×C₃	1	3	yes	—
13	C₄×#272	C₂×C₃	1	3	yes	—
14	C₉×D₁₂	—	3	12	no	—
15	C₄×D₃×C₉	—	3	12	no	—
16	C₂×#737	C₂³	1	1	yes	characteristic
17	#4208	—	2	3	no	—
18	C₃×#239	—	3	4	no	—
19	C₉×#41	—	9	1	no	—
20	C₂×#272	C₂²×C₃	1	3	yes	—
21	C₂×D₃×C₉	—	3	12	no	—
22	C₄×3₋¹⁺²	—	2	12	no	—
23	C₄×H(3,𝔽₃)	C₂²×C₃	1	3	yes	—
24	C₃×C₄×C₉	C₂×D₃	1	12	yes	—
25	#1084	C₂²×C₃	1	1	yes	characteristic
26	C₉×Dic₃	—	3	4	no	—
27	#737	#41	1	1	yes	characteristic, complemented, Sylow
28	C₃²×Q₈	C₃×D₃	1	4	yes	—
29	C₃×D₁₂	—	3	12	no	—
30	C₃×C₄×D₃	—	3	12	no	—
31	Q₈×C₉	—	2	4	no	—
32	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic
33	D₄×C₉	—	9	3	no	—
34	C₂×C₄×C₉	—	9	3	no	—
35	C₂×3₋¹⁺²	—	2	4	no	—
36	C₂×H(3,𝔽₃)	C₂³×C₃	1	1	yes	characteristic
37	C₂×C₃×C₉	C₂²×D₃	1	4	yes	—
38	#272	—	2	3	no	—
39	D₃×C₉	—	6	12	no	—
40	C₃×#41	—	9	1	no	—
41	#239	—	3	4	no	—
42	C₂×C₃×D₃	—	3	12	no	—
43	C₃²×C₄	C₂×C₃×D₃	1	12	yes	—
44	C₃×Dic₃	—	3	4	no	—
45	C₂²×C₉	—	9	3	no	—
46	C₄×C₉	—	2	12	no	—
47	C₄×C₉	—	9	1	no	—
48	C₄×C₉	C₂×C₃⋊D₃	1	3	yes	—
49	3₋¹⁺²	—	2	4	no	—
50	H(3,𝔽₃)	C₃×#41	1	1	yes	characteristic
51	C₃×C₉	#239	1	4	yes	complemented
52	C₃×Q₈	—	3	4	no	—
53	C₃×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
54	C₃×D₄	—	9	3	no	—
55	C₂×C₃×C₄	—	9	3	no	—
56	D₁₂	—	3	12	no	—
57	C₄×D₃	—	3	12	no	—
58	C₂×C₃²	C₂²×C₃×D₃	1	4	yes	—
59	C₃×D₃	—	6	12	no	—
60	C₂×C₉	—	2	4	no	—
61	C₂×C₉	—	18	3	no	—
62	C₂×C₉	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	#41	—	9	1	no	Sylow
64	C₂²×C₃	—	9	3	no	—
65	C₂×D₃	—	3	12	no	—
66	C₃×C₄	—	3	12	no	—
67	C₃×C₄	—	9	1	no	—
68	C₃×C₄	C₂×C₃×C₃⋊D₃	1	3	yes	—
69	Dic₃	—	3	4	no	—
70	C₃²	C₃×#239	1	4	yes	—
71	C₉	—	2	4	no	—
72	C₉	#3832	1	1	yes	characteristic
73	Q₈	#4208	1	1	yes	characteristic, complemented
74	D₄	—	9	3	no	—
75	C₂×C₄	—	9	3	no	—
76	C₂×C₃	—	3	4	no	—
77	C₂×C₃	—	18	3	no	—
78	C₂×C₃	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
79	D₃	—	6	12	no	—
80	C₂²	—	9	3	no	—
81	C₄	—	9	1	no	—
82	C₄	C₂×#4208	1	3	yes	—
83	C₃	—	3	4	no	—
84	C₃	C₃×#3832	1	1	yes	characteristic
85	C₂	—	18	3	no	—
86	C₂	C₂²×#4208	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂²×C₃⋊D₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³×C₃
Frattini, socle	C₂×C₃	C₂²×C₃×C₃⋊D₃
Fitting	Q₈×#737	C₂
prime radical	#88748	C₃
hypercenter	Q₈×C₉	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₃×#41
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₉ ◁ Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. Q₈×#737	C₂	semidirect
2. C₄⋊#4208	C₂	semidirect
3. C₄×#4208	C₂	semidirect
11. C₃×Q₈×C₉	D₃	semidirect
27. #737	#41	semidirect
51. C₃×C₉	#239	semidirect
73. Q₈	#4208	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#737	2
C₄⋊#4208	2
C₄×#4208	2
#88748	3
C₉×#239	3