Subgroups — 3₋¹⁺²×#100 #11720604

Order 864.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	M₁₆×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
3	C₃²×#100	C₃	1	1	yes	maximal, characteristic
4	C₉×#100	C₃	1	3	yes	maximal, complemented
5	C₂³×3₋¹⁺²	C₄	1	2	yes	—
6	D₄×3₋¹⁺²	—	2	2	no	—
7	C₂×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₈×3₋¹⁺²	—	2	2	no	—
9	C₂×C₃²×D₄	C₂×C₃	1	1	yes	characteristic
10	C₃²×M₁₆	C₂×C₃	1	2	yes	—
11	C₂×D₄×C₉	C₂×C₃	1	3	yes	—
12	C₉×M₁₆	C₂×C₃	1	6	yes	complemented
13	C₂²×3₋¹⁺²	—	2	2	no	—
14	C₂²×3₋¹⁺²	—	4	2	no	—
15	C₂²×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
16	C₄×3₋¹⁺²	D₄	1	2	yes	—
17	C₃×#100	—	3	1	no	—
18	C₃×#100	C₃²	1	1	yes	characteristic
19	C₂³×C₃²	C₃×C₄	1	2	yes	—
20	C₃²×D₄	—	2	2	no	—
21	C₂×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
22	C₂³×C₉	C₃×C₄	1	6	yes	—
23	C₃²×C₈	—	2	2	no	—
24	D₄×C₉	—	2	6	no	—
25	C₂×C₄×C₉	C₂²×C₃	1	3	yes	—
26	C₈×C₉	—	2	6	no	—
27	C₂×3₋¹⁺²	—	2	1	no	—
28	C₂×3₋¹⁺²	—	4	2	no	—
29	C₂×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₃×D₄	—	3	1	no	—
31	C₂×C₃×D₄	C₂×C₃²	1	1	yes	characteristic
32	C₃×M₁₆	—	3	2	no	—
33	C₃×M₁₆	C₂×C₃²	1	2	yes	—
34	C₂²×C₃²	—	2	2	no	—
35	C₂²×C₃²	—	4	2	no	—
36	C₂²×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
37	C₃²×C₄	C₃×D₄	1	2	yes	—
38	C₂²×C₉	—	2	6	no	—
39	C₂²×C₉	—	4	6	no	—
40	C₂²×C₉	C₂×C₃×C₄	1	3	yes	—
41	C₄×C₉	C₃×D₄	1	6	yes	—
42	#100	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
43	3₋¹⁺²	#100	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃	—	3	2	no	—
45	C₂³×C₃	C₃²×C₄	1	2	yes	—
46	C₃×D₄	—	2	2	no	—
47	C₃×D₄	—	6	2	no	—
48	C₂×C₃×C₄	—	3	1	no	—
49	C₂×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
50	C₃×C₈	—	2	2	no	—
51	C₃×C₈	—	6	2	no	—
52	C₂×C₃²	—	2	1	no	—
53	C₂×C₃²	—	4	2	no	—
54	C₂×C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₉	—	2	3	no	—
56	C₂×C₉	—	4	6	no	—
57	C₂×C₉	C₃×C₂²⋊C₄	1	3	yes	—
58	C₂×D₄	C₂×3₋¹⁺²	1	1	yes	characteristic
59	M₁₆	C₂×3₋¹⁺²	1	2	yes	complemented
60	C₂²×C₃	—	2	2	no	—
61	C₂²×C₃	—	3	1	no	—
62	C₂²×C₃	—	4	2	no	—
63	C₂²×C₃	—	6	2	no	—
64	C₂²×C₃	—	12	2	no	—
65	C₂²×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
66	C₃×C₄	—	3	2	no	—
67	C₃×C₄	C₃²×D₄	1	2	yes	—
68	C₃²	C₃×#100	1	1	yes	characteristic
69	C₉	C₃×#100	1	3	yes	complemented
70	C₂³	C₄×3₋¹⁺²	1	2	yes	—
71	D₄	—	2	2	no	—
72	C₂×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	—	3	1	no	—
76	C₂×C₃	—	4	2	no	—
77	C₂×C₃	—	6	1	no	—
78	C₂×C₃	—	12	2	no	—
79	C₂×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
80	C₂²	—	2	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
83	C₄	D₄×3₋¹⁺²	1	2	yes	—
84	C₃	—	3	1	no	—
85	C₃	C₃²×#100	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	4	2	no	—
88	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	C₂×C₃²×D₄	C₂×C₃
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
2. M₁₆×3₋¹⁺²	C₂	semidirect
4. C₉×#100	C₃	semidirect
12. C₉×M₁₆	C₂×C₃	semidirect
42. #100	3₋¹⁺²	direct (N × H)
43. 3₋¹⁺²	#100	direct (N × H)
59. M₁₆	C₂×3₋¹⁺²	semidirect
69. C₉	C₃×#100	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×3₋¹⁺²	2
M₁₆×3₋¹⁺²	2
C₃²×#100	3
C₉×#100	3

Subgroups — 3−1+2×#100 #11720604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#100 #11720604