Subgroups — #13046829

Order 1296.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#732	C₂	1	1	yes	maximal, characteristic
2	#10619666	C₂	1	1	yes	maximal, characteristic
3	#10619511	C₂	1	1	yes	maximal, characteristic
4	#3756⋊C₃	—	3	3	no	maximal
5	C₃×#3674	—	3	1	no	maximal
6	C₄×#732	—	2	1	no	—
7	C₄×#732	C₂²	1	1	yes	characteristic
8	#66332	—	2	1	no	—
9	Q₈×H(3,𝔽₃)	D₃	1	3	yes	—
10	C₃×Q₈×C₉	D₃	1	1	yes	characteristic
11	#634⋊C₃	—	3	3	no	—
12	C₃×Dic₁₈	—	3	1	no	—
13	(C₃⋊(C₃⋊C₈))⋊C₃	—	3	3	no	—
14	C₃×C₉⋊C₈	—	3	1	no	—
15	C₂×#732	D₄	1	1	yes	characteristic
16	#3756	—	3	3	no	—
17	C₃×#216	—	9	1	no	—
18	#3674	—	3	1	no	—
19	C₄×H(3,𝔽₃)	—	2	3	no	—
20	C₄×H(3,𝔽₃)	C₂×D₃	1	3	yes	—
21	C₃×C₄×C₉	—	2	1	no	—
22	C₃×C₄×C₉	C₂×D₃	1	1	yes	characteristic
23	(C₃⋊Dic₃)⋊C₃	—	6	3	no	—
24	C₃×Dic₉	—	6	1	no	—
25	#732	Q₁₆	1	1	yes	characteristic, complemented, Sylow
26	C₃²×Q₈	—	3	3	no	—
27	C₃²×Q₈	C₃⋊D₃	1	1	yes	characteristic
28	#634	—	3	3	no	—
29	C₃×Dic₆	—	9	1	no	—
30	C₃⋊(C₃⋊C₈)	—	3	3	no	—
31	C₃×C₃⋊C₈	—	9	1	no	—
32	Q₈×C₉	—	3	1	no	—
33	Dic₁₈	—	3	1	no	—
34	C₉⋊C₈	—	3	1	no	—
35	C₂×H(3,𝔽₃)	#67	1	3	yes	—
36	C₂×C₃×C₉	#67	1	1	yes	characteristic
37	C₃×Q₁₆	—	27	1	no	—
38	#216	—	9	1	no	—
39	#216	—	27	3	no	—
40	C₃²×C₄	—	2	1	no	—
41	C₃²×C₄	—	3	3	no	—
42	C₃²×C₄	—	6	3	no	—
43	C₃²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
44	C₃⋊Dic₃	—	6	3	no	—
45	C₃×Dic₃	—	18	1	no	—
46	C₄×C₉	—	3	1	no	—
47	C₄×C₉	—	6	1	no	—
48	Dic₉	—	6	1	no	—
49	H(3,𝔽₃)	#216	1	3	yes	complemented
50	C₃×C₉	#216	1	1	yes	characteristic, complemented
51	C₃×Q₈	—	3	1	no	—
52	C₃×Q₈	—	9	3	no	—
53	C₃×Q₈	—	27	1	no	—
54	C₃×Q₈	#272	1	1	yes	characteristic
55	Dic₆	—	9	1	no	—
56	Dic₆	—	27	3	no	—
57	C₃×C₈	—	27	1	no	—
58	C₃⋊C₈	—	9	1	no	—
59	C₃⋊C₈	—	27	3	no	—
60	C₂×C₃²	—	3	3	no	—
61	C₂×C₃²	#638	1	1	yes	characteristic
62	C₂×C₉	—	3	1	no	—
63	Q₁₆	—	27	1	no	Sylow
64	C₃×C₄	—	2	1	no	—
65	C₃×C₄	—	3	1	no	—
66	C₃×C₄	—	6	1	no	—
67	C₃×C₄	—	9	3	no	—
68	C₃×C₄	—	18	3	no	—
69	C₃×C₄	—	54	1	no	—
70	C₃×C₄	C₂×#272	1	1	yes	characteristic
71	Dic₃	—	18	1	no	—
72	Dic₃	—	54	3	no	—
73	C₃²	—	3	3	no	—
74	C₃²	#3756	1	1	yes	characteristic
75	C₉	—	3	1	no	—
76	Q₈	—	27	1	no	—
77	Q₈	#4185	1	1	yes	characteristic
78	C₈	—	27	1	no	—
79	C₂×C₃	—	3	1	no	—
80	C₂×C₃	—	9	3	no	—
81	C₂×C₃	#6235	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	54	1	no	—
84	C₄	C₂×#4185	1	1	yes	characteristic
85	C₃	—	3	1	no	—
86	C₃	—	9	3	no	—
87	C₃	#88550	1	1	yes	characteristic
88	C₂	#10619683	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10619683
commutator, cosocle	C₄×#732	C₂²
Frattini	C₃²×C₄	C₂×C₃⋊D₃
Fitting	Q₈×#732	C₂
socle	C₂×C₃	#6235
prime radical	C₂×#732	D₄
hypercenter	Q₈	#4185
hypocenter	#732	Q₁₆
LCS	G ▷ C₄×#732 ▷ C₂×#732 ▷ #732
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
25. #732	Q₁₆	semidirect
49. H(3,𝔽₃)	#216	semidirect
50. C₃×C₉	#216	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#732	2
#10619666	2
#10619511	2
#3756⋊C₃	3
C₃×#3674	3