Subgroups — 3₋¹⁺²×#126 #11721915

Order 864.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×3₋¹⁺²	C₂	1	3	yes	maximal, complemented
2	C₂²⋊C₄×3₋¹⁺²	C₂	1	3	yes	maximal
3	C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#126	C₃	1	1	yes	maximal, characteristic
5	C₉×#126	C₃	1	3	yes	maximal, complemented
6	C₂³×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₂×C₄×3₋¹⁺²	C₂²	1	3	yes	—
8	C₂×C₄×3₋¹⁺²	C₂²	1	3	yes	—
9	C₃²×C₄⋊C₄	C₂×C₃	1	3	yes	—
10	C₃²×C₂²⋊C₄	C₂×C₃	1	3	yes	—
11	C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
12	C₉×C₄⋊C₄	C₂×C₃	1	9	yes	complemented
13	C₉×C₂²⋊C₄	C₂×C₃	1	9	yes	—
14	C₄²×C₉	C₂×C₃	1	3	yes	complemented
15	C₂²×3₋¹⁺²	—	2	3	no	—
16	C₂²×3₋¹⁺²	C₂³	1	1	yes	characteristic
17	C₄×3₋¹⁺²	—	2	3	no	—
18	C₄×3₋¹⁺²	—	2	3	no	—
19	C₃×#126	—	3	1	no	—
20	C₃×#126	C₃²	1	1	yes	characteristic
21	C₂³×C₃²	C₂²×C₃	1	1	yes	characteristic
22	C₂×C₃²×C₄	C₂²×C₃	1	3	yes	—
23	C₂×C₃²×C₄	C₂²×C₃	1	3	yes	—
24	C₂³×C₉	C₂²×C₃	1	3	yes	—
25	C₂×C₄×C₉	C₂²×C₃	1	9	yes	—
26	C₂×C₄×C₉	C₂²×C₃	1	9	yes	—
27	C₂×3₋¹⁺²	—	4	1	no	—
28	C₂×3₋¹⁺²	#41	1	3	yes	—
29	C₃×C₄⋊C₄	—	3	3	no	—
30	C₃×C₄⋊C₄	C₂×C₃²	1	3	yes	—
31	C₃×C₂²⋊C₄	—	3	3	no	—
32	C₃×C₂²⋊C₄	C₂×C₃²	1	3	yes	—
33	C₃×C₄²	—	3	1	no	—
34	C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
35	C₂²×C₃²	—	2	3	no	—
36	C₂²×C₃²	C₂³×C₃	1	1	yes	characteristic
37	C₃²×C₄	—	2	3	no	—
38	C₃²×C₄	—	2	3	no	—
39	C₂²×C₉	—	2	9	no	—
40	C₂²×C₉	C₂³×C₃	1	3	yes	—
41	C₄×C₉	—	2	9	no	—
42	C₄×C₉	—	2	9	no	—
43	#126	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
44	3₋¹⁺²	#126	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₃	—	3	1	no	—
46	C₂³×C₃	C₂²×C₃²	1	1	yes	characteristic
47	C₂×C₃×C₄	—	3	3	no	—
48	C₂×C₃×C₄	—	3	3	no	—
49	C₂×C₃×C₄	C₂²×C₃²	1	3	yes	—
50	C₂×C₃×C₄	C₂²×C₃²	1	3	yes	—
51	C₂×C₃²	—	4	1	no	—
52	C₂×C₃²	C₃×#41	1	3	yes	—
53	C₂×C₉	—	4	3	no	—
54	C₂×C₉	C₃×#41	1	9	yes	—
55	C₄⋊C₄	C₂×3₋¹⁺²	1	3	yes	complemented
56	C₂²⋊C₄	C₂×3₋¹⁺²	1	3	yes	—
57	C₄²	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
58	C₂²×C₃	—	2	3	no	—
59	C₂²×C₃	—	3	1	no	—
60	C₂²×C₃	—	6	3	no	—
61	C₂²×C₃	C₂³×C₃²	1	1	yes	characteristic
62	C₃×C₄	—	2	3	no	—
63	C₃×C₄	—	2	3	no	—
64	C₃×C₄	—	6	3	no	—
65	C₃×C₄	—	6	3	no	—
66	C₃²	C₃×#126	1	1	yes	characteristic
67	C₉	C₃×#126	1	3	yes	complemented
68	C₂³	C₂²×3₋¹⁺²	1	1	yes	characteristic
69	C₂×C₄	C₂²×3₋¹⁺²	1	3	yes	—
70	C₂×C₄	C₂²×3₋¹⁺²	1	3	yes	—
71	C₂×C₃	—	3	3	no	—
72	C₂×C₃	—	4	1	no	—
73	C₂×C₃	—	12	1	no	—
74	C₂×C₃	C₃²×#41	1	3	yes	—
75	C₂²	—	2	3	no	—
76	C₂²	C₂³×3₋¹⁺²	1	1	yes	characteristic
77	C₄	—	2	3	no	—
78	C₄	—	2	3	no	—
79	C₃	—	3	1	no	—
80	C₃	C₃²×#126	1	1	yes	characteristic
81	C₂	—	4	1	no	—
82	C₂	#41×3₋¹⁺²	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²×C₃	C₂³×C₃²
prime radical	C₂³×C₃²	C₂²×C₃
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄⋊C₄×3₋¹⁺²	C₂	semidirect
3. C₄²×3₋¹⁺²	C₂	semidirect
5. C₉×#126	C₃	semidirect
12. C₉×C₄⋊C₄	C₂×C₃	semidirect
14. C₄²×C₉	C₂×C₃	semidirect
43. #126	3₋¹⁺²	direct (N × H)
44. 3₋¹⁺²	#126	direct (N × H)
55. C₄⋊C₄	C₂×3₋¹⁺²	semidirect
57. C₄²	C₂×3₋¹⁺²	semidirect
67. C₉	C₃×#126	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×3₋¹⁺²	2
C₂²⋊C₄×3₋¹⁺²	2
C₄²×3₋¹⁺²	2
C₃²×#126	3
C₉×#126	3

Subgroups — 3−1+2×#126 #11721915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#126 #11721915