Subgroups — C₃²×#1204 #11762034

Order 1080.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃³×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₃²×Dic₁₀	—	3	1	no	maximal
5	C₃×#1204	C₃	1	4	yes	maximal, direct product factor
6	C₂×C₃³×C₅	C₂²	1	1	yes	characteristic
7	C₃²×Dic₆	—	5	1	no	maximal, Hall
8	C₃²×C₄×C₅	—	3	1	no	—
9	C₃×Dic₁₅	C₂×C₃	1	4	yes	—
10	C₃×C₅×Dic₃	C₂×C₃	1	4	yes	—
11	C₃²×Dic₅	—	2	4	no	—
12	C₃²×Dic₅	—	3	1	no	—
13	C₃²×Dic₅	D₃	1	1	yes	characteristic
14	C₃²×Dic₅	C₂×C₃	1	4	yes	—
15	C₃³×C₅	Q₈	1	1	yes	characteristic, complemented, Hall
16	C₃×Dic₁₀	—	3	4	no	—
17	#1204	C₃²	1	1	yes	characteristic, direct product factor
18	C₃³×C₄	—	5	1	no	—
19	C₃²×Dic₃	—	5	1	no	—
20	C₃²×Dic₃	D₅	1	1	yes	characteristic
21	C₂×C₃²×C₅	—	2	4	no	—
22	C₂×C₃²×C₅	C₂×D₃	1	1	yes	characteristic
23	C₂×C₃²×C₅	C₂²×C₃	1	4	yes	—
24	C₃²×Q₈	—	15	1	no	—
25	C₃×Dic₆	—	5	4	no	—
26	C₃×C₄×C₅	—	3	4	no	—
27	Dic₁₅	C₂×C₃²	1	1	yes	characteristic
28	C₃×Dic₅	—	2	4	no	—
29	C₃×Dic₅	—	3	4	no	—
30	C₃×Dic₅	C₃×D₃	1	4	yes	—
31	C₃×Dic₅	C₂×C₃²	1	1	yes	characteristic
32	C₅×Dic₃	C₂×C₃²	1	1	yes	characteristic
33	C₂×C₃³	C₂×D₅	1	1	yes	characteristic
34	C₃²×C₅	—	2	4	no	—
35	C₃²×C₅	Dic₆	1	1	yes	characteristic, complemented
36	C₃²×C₅	C₃×Q₈	1	4	yes	complemented
37	Dic₁₀	—	3	1	no	Hall
38	C₃²×C₄	—	3	1	no	—
39	C₃²×C₄	—	5	1	no	—
40	C₃²×C₄	—	5	4	no	—
41	C₃²×C₄	—	10	4	no	—
42	C₃²×C₄	—	15	1	no	—
43	C₃×Dic₃	—	5	4	no	—
44	C₃×Dic₃	C₃×D₅	1	4	yes	—
45	C₂×C₃×C₅	—	2	4	no	—
46	C₂×C₃×C₅	C₂×C₃×D₃	1	4	yes	—
47	C₂×C₃×C₅	C₂²×C₃²	1	1	yes	characteristic
48	C₃³	Dic₁₀	1	1	yes	characteristic, complemented, Sylow
49	C₃×Q₈	—	15	4	no	—
50	Dic₆	—	5	1	no	—
51	C₄×C₅	—	3	1	no	—
52	Dic₅	—	3	1	no	—
53	Dic₅	C₃²×D₃	1	1	yes	characteristic
54	C₂×C₃²	—	2	4	no	—
55	C₂×C₃²	D₃×D₅	1	1	yes	characteristic
56	C₂×C₃²	C₂×C₃×D₅	1	4	yes	—
57	C₃×C₅	—	2	4	no	—
58	C₃×C₅	C₃×Dic₆	1	4	yes	complemented
59	C₃×C₅	C₃²×Q₈	1	1	yes	characteristic, complemented
60	C₃×C₄	—	3	4	no	—
61	C₃×C₄	—	5	4	no	—
62	C₃×C₄	—	5	1	no	—
63	C₃×C₄	—	10	4	no	—
64	C₃×C₄	—	15	4	no	—
65	Dic₃	—	5	1	no	—
66	Dic₃	C₃²×D₅	1	1	yes	characteristic
67	C₂×C₅	C₂×C₃²×D₃	1	1	yes	characteristic
68	C₃²	—	2	4	no	—
69	C₃²	#1204	1	1	yes	characteristic, direct product factor
70	C₃²	C₃×Dic₁₀	1	4	yes	complemented
71	Q₈	—	15	1	no	Sylow
72	C₂×C₃	—	2	4	no	—
73	C₂×C₃	C₃×D₃×D₅	1	4	yes	—
74	C₂×C₃	C₂×C₃²×D₅	1	1	yes	characteristic
75	C₅	C₃²×Dic₆	1	1	yes	characteristic, complemented, Sylow
76	C₄	—	3	1	no	—
77	C₄	—	5	1	no	—
78	C₄	—	15	1	no	—
79	C₃	—	2	4	no	—
80	C₃	C₃×#1204	1	4	yes	direct product factor
81	C₃	C₃²×Dic₁₀	1	1	yes	characteristic, complemented
82	C₂	C₃²×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²	D₃×D₅
commutator, cosocle	C₂×C₃×C₅	C₂²×C₃²
Frattini	C₂	C₃²×D₃×D₅
Fitting, socle, prime radical	C₂×C₃³×C₅	C₂²
hypocenter	C₃×C₅	C₃²×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
5. C₃×#1204	C₃	direct (N × H)
15. C₃³×C₅	Q₈	semidirect
17. #1204	C₃²	direct (N × H)
35. C₃²×C₅	Dic₆	semidirect
36. C₃²×C₅	C₃×Q₈	semidirect
48. C₃³	Dic₁₀	semidirect
58. C₃×C₅	C₃×Dic₆	semidirect
59. C₃×C₅	C₃²×Q₈	semidirect
69. C₃²	#1204	direct (N × H)
70. C₃²	C₃×Dic₁₀	semidirect
75. C₅	C₃²×Dic₆	semidirect
80. C₃	C₃×#1204	direct (N × H)
81. C₃	C₃²×Dic₁₀	semidirect

Maximal subgroups

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

subgroup	index
C₃²×Dic₁₅	2
C₃²×C₅×Dic₃	2
C₃³×Dic₅	2
C₃²×Dic₁₀	3
C₃×#1204	3
C₃²×Dic₆	5

Subgroups — C32×#1204 #11762034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1204 #11762034