Subgroups — #13047988

Order 1296.

218 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	1	yes	maximal, characteristic
3	#10619725	C₂	1	1	yes	maximal, characteristic, complemented
4	#88549	C₃	1	1	yes	maximal, characteristic
5	C₉×#215	—	3	4	no	maximal
6	C₂×#4208	—	2	1	no	—
7	C₄×#737	—	2	1	no	—
8	C₄×#737	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	1	yes	characteristic
13	C₉×D₁₂	—	3	4	no	—
14	#6180	C₂×C₃	1	1	yes	characteristic
15	C₉×C₃⋊C₈	—	3	4	no	—
16	C₂×#737	D₄	1	1	yes	characteristic
17	#4208	—	4	1	no	—
18	C₃×#215	—	3	4	no	—
19	C₉×SD₁₆	—	9	1	no	—
20	C₂×#272	—	2	1	no	—
21	C₂×D₃×C₉	—	6	4	no	—
22	C₄×3₋¹⁺²	—	2	8	no	—
23	C₄×3₋¹⁺²	—	2	4	no	—
24	C₄×H(3,𝔽₃)	—	2	1	no	—
25	C₄×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
26	C₃×C₄×C₉	—	2	4	no	—
27	C₃×C₄×C₉	C₂×D₃	1	4	yes	—
28	#737	SD₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃²×Q₈	C₃×D₃	1	4	yes	—
30	C₃×D₁₂	—	3	4	no	—
31	C₃×C₃⋊C₈	—	3	4	no	—
32	Q₈×C₉	—	2	4	no	—
33	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic
34	D₄×C₉	—	9	1	no	—
35	C₈×C₉	—	9	1	no	—
36	C₂×3₋¹⁺²	—	2	4	no	—
37	C₂×H(3,𝔽₃)	C₃×D₄	1	1	yes	characteristic
38	C₂×C₃×C₉	#67	1	4	yes	—
39	#272	—	4	1	no	—
40	D₃×C₉	—	12	4	no	—
41	C₃×SD₁₆	—	9	1	no	—
42	#215	—	3	4	no	—
43	C₂×C₃×D₃	—	6	4	no	—
44	C₃²×C₄	—	2	4	no	—
45	C₃²×C₄	C₂×C₃×D₃	1	4	yes	—
46	C₂²×C₉	—	18	1	no	—
47	C₄×C₉	—	2	8	no	—
48	C₄×C₉	—	2	1	no	—
49	C₄×C₉	—	2	4	no	—
50	C₄×C₉	C₂×C₃⋊D₃	1	1	yes	characteristic
51	3₋¹⁺²	—	2	4	no	—
52	H(3,𝔽₃)	C₃×SD₁₆	1	1	yes	characteristic
53	C₃×C₉	#215	1	4	yes	complemented
54	C₃×Q₈	—	3	4	no	—
55	C₃×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
56	C₃×D₄	—	9	1	no	—
57	D₁₂	—	3	4	no	—
58	C₃×C₈	—	9	1	no	—
59	C₃⋊C₈	—	3	4	no	—
60	C₂×C₃²	C₃×#67	1	4	yes	—
61	C₃×D₃	—	12	4	no	—
62	C₂×C₉	—	2	4	no	—
63	C₂×C₉	—	36	1	no	—
64	C₂×C₉	#638	1	1	yes	characteristic
65	SD₁₆	—	9	1	no	Sylow
66	C₂²×C₃	—	18	1	no	—
67	C₂×D₃	—	6	4	no	—
68	C₃×C₄	—	2	1	no	—
69	C₃×C₄	—	3	4	no	—
70	C₃×C₄	—	6	4	no	—
71	C₃×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
72	C₃²	C₃×#215	1	4	yes	—
73	C₉	—	2	4	no	—
74	C₉	#3755	1	1	yes	characteristic
75	Q₈	#4208	1	1	yes	characteristic, complemented
76	D₄	—	9	1	no	—
77	C₈	—	9	1	no	—
78	C₂×C₃	—	3	4	no	—
79	C₂×C₃	—	36	1	no	—
80	C₂×C₃	C₃×#638	1	1	yes	characteristic
81	D₃	—	12	4	no	—
82	C₂²	—	18	1	no	—
83	C₄	—	2	1	no	—
84	C₄	C₂×#4208	1	1	yes	characteristic
85	C₃	—	3	4	no	—
86	C₃	C₃×#3755	1	1	yes	characteristic
87	C₂	—	36	1	no	—
88	C₂	#10619984	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#638
commutator, cosocle	C₄×H(3,𝔽₃)	C₂²×C₃
Frattini	C₃×C₄	C₂×C₃×C₃⋊D₃
Fitting	Q₈×#737	C₂
socle	C₂×C₃	C₃×#638
prime radical	C₄⋊#272	C₂×C₃
hypercenter	Q₈×C₉	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₃×SD₁₆
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. Q₈×#737	C₂	semidirect
3. #10619725	C₂	semidirect
11. C₃×Q₈×C₉	D₃	semidirect
28. #737	SD₁₆	semidirect
53. C₃×C₉	#215	semidirect
75. 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
#10619725	2
#88549	3
C₉×#215	3