Subgroups — #13047048

Order 1296.

123 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10619579	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#272	C₃	1	1	yes	maximal, characteristic, complemented
3	D₃×Q₈⋊C₉	—	3	1	no	maximal
4	C₂×#4178	—	4	1	no	maximal
5	Q₈×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
6	C₄×#272	—	3	1	no	—
7	#6229	—	3	1	no	—
8	Q₈⋊H(3,𝔽₃)	—	2	1	no	—
9	C₃×Q₈⋊C₉	D₃	1	1	yes	characteristic, complemented
10	C₂×#732	—	4	1	no	—
11	#4178	—	4	2	no	—
12	C₃×D₃×Q₈	—	3	1	no	—
13	C₃×D₃×Q₈	—	9	1	no	—
14	C₂×Q₈⋊C₉	—	9	1	no	—
15	C₂×#272	A₄	1	1	yes	characteristic
16	C₂×D₃×C₉	—	12	1	no	—
17	C₄×H(3,𝔽₃)	—	3	1	no	—
18	#1084	—	3	1	no	—
19	#732	—	4	1	no	Sylow
20	C₃²×Q₈	—	3	1	no	—
21	C₃²×Q₈	C₃×D₃	1	1	yes	characteristic
22	C₃×C₄×D₃	—	9	1	no	—
23	C₃×C₄×D₃	—	9	3	no	—
24	C₃×Dic₆	—	9	3	no	—
25	C₃×Dic₆	—	9	1	no	—
26	C₃×SL(2,𝔽₃)	—	6	1	no	—
27	Q₈⋊C₉	—	3	1	no	—
28	C₂×H(3,𝔽₃)	—	8	1	no	—
29	C₂×H(3,𝔽₃)	C₂×A₄	1	1	yes	characteristic
30	C₂×C₃×C₉	—	4	1	no	—
31	#272	SL(2,𝔽₃)	1	2	yes	complemented
32	D₃×C₉	—	12	2	no	—
33	C₂×C₃×Q₈	—	9	1	no	—
34	D₃×Q₈	—	3	1	no	—
35	D₃×Q₈	—	9	1	no	—
36	C₂×C₃×D₃	—	3	1	no	—
37	C₂×C₃×D₃	—	9	1	no	—
38	C₃²×C₄	—	3	3	no	—
39	C₃²×C₄	—	3	1	no	—
40	C₃×Dic₃	—	9	3	no	—
41	C₃×Dic₃	—	9	1	no	—
42	C₂²×C₉	—	36	1	no	—
43	H(3,𝔽₃)	—	8	1	no	—
44	H(3,𝔽₃)	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
45	C₃×C₉	—	4	1	no	—
46	C₃×Q₈	—	3	1	no	—
47	C₃×Q₈	—	9	1	no	—
48	C₃×Q₈	—	27	1	no	—
49	C₃×Q₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
50	C₂×C₃×C₄	—	27	1	no	—
51	C₄×D₃	—	9	1	no	—
52	C₄×D₃	—	9	3	no	—
53	Dic₆	—	9	3	no	—
54	Dic₆	—	9	1	no	—
55	SL(2,𝔽₃)	—	18	1	no	—
56	C₂×C₃²	—	3	1	no	—
57	C₂×C₃²	—	24	1	no	—
58	C₂×C₃²	D₃×A₄	1	1	yes	characteristic
59	C₃×D₃	—	3	2	no	—
60	C₃×D₃	—	9	2	no	—
61	C₂×C₉	—	12	1	no	—
62	C₂×C₉	—	36	2	no	—
63	C₂×Q₈	—	9	1	no	Sylow
64	C₂²×C₃	—	9	1	no	—
65	C₂×D₃	—	3	1	no	—
66	C₂×D₃	—	9	1	no	—
67	C₃×C₄	—	3	1	no	—
68	C₃×C₄	—	9	1	no	—
69	C₃×C₄	—	9	3	no	—
70	C₃×C₄	—	27	1	no	—
71	Dic₃	—	9	1	no	—
72	Dic₃	—	9	3	no	—
73	C₃²	—	3	1	no	—
74	C₃²	—	24	1	no	—
75	C₃²	D₃×SL(2,𝔽₃)	1	1	yes	characteristic
76	C₉	—	12	1	no	—
77	Q₈	—	27	1	no	—
78	Q₈	#4178	1	1	yes	characteristic, complemented
79	C₂×C₄	—	27	1	no	—
80	C₂×C₃	—	3	1	no	—
81	C₂×C₃	—	9	2	no	—
82	C₂×C₃	—	9	1	no	—
83	C₂×C₃	—	72	1	no	—
84	C₂×C₃	(C₃⋊D₃)⋊A₄	1	1	yes	characteristic
85	D₃	—	3	2	no	—
86	D₃	—	9	2	no	—
87	C₂²	—	9	1	no	—
88	C₄	—	3	1	no	—
89	C₄	—	27	1	no	—
90	C₃	—	3	1	no	—
91	C₃	—	9	1	no	—
92	C₃	—	72	1	no	—
93	C₃	#88623	1	1	yes	characteristic
94	C₂	—	9	2	no	—
95	C₂	#10619771	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂×C₃	(C₃⋊D₃)⋊A₄
commutator, Fitting, cosocle, hypocenter	Q₈×H(3,𝔽₃)	C₂×C₃
Frattini	C₂×C₃²	D₃×A₄
LCS	G ▷ Q₈×H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. #10619579	C₂	semidirect
2. Q₈×#272	C₃	semidirect
9. C₃×Q₈⋊C₉	D₃	semidirect
31. #272	SL(2,𝔽₃)	semidirect
78. Q₈	#4178	semidirect

Maximal subgroups

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

subgroup	index
#10619579	2
Q₈×#272	3
D₃×Q₈⋊C₉	3
C₂×#4178	4