Subgroups — 3₋¹⁺²×#101 #11720607

Order 864.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	M₁₆×3₋¹⁺²	C₂	1	2	yes	maximal
3	C₃²×#101	C₃	1	1	yes	maximal, characteristic
4	C₉×#101	C₃	1	3	yes	maximal, complemented
5	Q₈×3₋¹⁺²	—	2	2	no	—
6	C₂×C₄×3₋¹⁺²	C₄	1	2	yes	—
7	C₂×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₈×3₋¹⁺²	—	2	2	no	—
9	C₂×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₃²×M₁₆	C₂×C₃	1	2	yes	—
11	C₂×Q₈×C₉	C₂×C₃	1	3	yes	—
12	C₉×M₁₆	C₂×C₃	1	6	yes	—
13	C₂²×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
14	C₄×3₋¹⁺²	—	2	2	no	—
15	C₄×3₋¹⁺²	D₄	1	2	yes	—
16	C₃×#101	—	3	1	no	—
17	C₃×#101	C₃²	1	1	yes	characteristic
18	C₃²×Q₈	—	2	2	no	—
19	C₂×C₃²×C₄	C₃×C₄	1	2	yes	—
20	C₂×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
21	C₃²×C₈	—	2	2	no	—
22	Q₈×C₉	—	2	6	no	—
23	C₂×C₄×C₉	C₃×C₄	1	6	yes	—
24	C₂×C₄×C₉	C₂²×C₃	1	3	yes	—
25	C₈×C₉	—	2	6	no	—
26	C₂×3₋¹⁺²	—	2	1	no	—
27	C₂×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₃×Q₈	—	3	1	no	—
29	C₂×C₃×Q₈	C₂×C₃²	1	1	yes	characteristic
30	C₃×M₁₆	—	3	2	no	—
31	C₃×M₁₆	C₂×C₃²	1	2	yes	—
32	C₂²×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
33	C₃²×C₄	—	2	2	no	—
34	C₃²×C₄	C₃×D₄	1	2	yes	—
35	C₂²×C₉	C₂×C₃×C₄	1	3	yes	—
36	C₄×C₉	—	2	6	no	—
37	C₄×C₉	C₃×D₄	1	6	yes	—
38	#101	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
39	3₋¹⁺²	#101	1	1	yes	characteristic, direct product factor, Sylow
40	C₃×Q₈	—	2	2	no	—
41	C₃×Q₈	—	6	2	no	—
42	C₂×C₃×C₄	—	3	2	no	—
43	C₂×C₃×C₄	—	3	1	no	—
44	C₂×C₃×C₄	C₃²×C₄	1	2	yes	—
45	C₂×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
46	C₃×C₈	—	2	2	no	—
47	C₃×C₈	—	6	2	no	—
48	C₂×C₃²	—	2	1	no	—
49	C₂×C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₉	—	2	3	no	—
51	C₂×C₉	C₃×C₂²⋊C₄	1	3	yes	—
52	C₂×Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic
53	M₁₆	C₂×3₋¹⁺²	1	2	yes	—
54	C₂²×C₃	—	3	1	no	—
55	C₂²×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
56	C₃×C₄	—	2	2	no	—
57	C₃×C₄	—	3	2	no	—
58	C₃×C₄	—	6	2	no	—
59	C₃×C₄	C₃²×D₄	1	2	yes	—
60	C₃²	C₃×#101	1	1	yes	characteristic
61	C₉	C₃×#101	1	3	yes	complemented
62	Q₈	—	2	2	no	—
63	C₂×C₄	C₄×3₋¹⁺²	1	2	yes	—
64	C₂×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₂×C₃	—	2	1	no	—
67	C₂×C₃	—	3	1	no	—
68	C₂×C₃	—	6	1	no	—
69	C₂×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	D₄×3₋¹⁺²	1	2	yes	—
73	C₃	—	3	1	no	—
74	C₃	C₃²×#101	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	C₂²⋊C₄×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×C₂²⋊C₄
commutator	C₂²×C₃	C₂×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₄	C₂²×C₃²
prime radical	C₂²×C₃²	C₂×C₃×C₄
LCS	G ▷ C₂²×C₃ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×#101	C₃	semidirect
38. #101	3₋¹⁺²	direct (N × H)
39. 3₋¹⁺²	#101	direct (N × H)
61. C₉	C₃×#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×3₋¹⁺²	2
M₁₆×3₋¹⁺²	2
C₃²×#101	3
C₉×#101	3

Subgroups — 3−1+2×#101 #11720607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#101 #11720607