Subgroups — #13145698

Order 1728.

458 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	1	3	yes	maximal
2	C₃⋊C₈²	—	3	3	no	maximal
3	C₃³×C₄²	C₂²	1	1	yes	characteristic
4	C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	1	6	yes	—
5	C₄×C₃⋊(C₃⋊C₈)	D₃	1	3	yes	—
6	C₃×C₄×C₃⋊C₈	—	3	6	no	—
7	C₃×C₄×C₃⋊C₈	—	6	3	no	—
8	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
9	C₃×C₃⋊(C₃⋊C₈)	C₈	1	12	yes	complemented
10	C₈×C₃⋊C₈	—	9	3	no	—
11	C₃²×C₄²	—	2	3	no	—
12	C₃²×C₄²	—	4	1	no	—
13	C₃²×C₄²	C₂×D₃	1	3	yes	—
14	C₂×C₃⋊(C₃⋊C₈)	Dic₃	1	6	yes	—
15	C₂×C₃×C₃⋊C₈	—	3	12	no	—
16	C₂×C₃×C₃⋊C₈	—	6	6	no	—
17	C₂²×C₃³	C₄²	1	1	yes	characteristic
18	C₃³×C₄	C₂×C₈	1	6	yes	—
19	C₃×C₄×C₈	—	9	3	no	—
20	C₄×C₃⋊C₈	—	3	6	no	—
21	C₄×C₃⋊C₈	—	6	3	no	—
22	C₂×C₃²×C₄	—	2	6	no	—
23	C₂×C₃²×C₄	—	2	3	no	—
24	C₂×C₃²×C₄	—	4	3	no	—
25	C₂×C₃²×C₄	C₄×D₃	1	6	yes	—
26	C₂×C₃²×C₄	C₂×Dic₃	1	3	yes	—
27	C₃⋊(C₃⋊C₈)	C₃⋊C₈	1	12	yes	complemented
28	C₃×C₃⋊C₈	—	3	24	no	—
29	C₃×C₃⋊C₈	—	6	12	no	—
30	C₈²	—	27	1	no	Sylow
31	C₂×C₃³	C₄×C₈	1	3	yes	—
32	C₂×C₃×C₈	—	9	6	no	—
33	C₃×C₄²	—	2	3	no	—
34	C₃×C₄²	—	4	1	no	—
35	C₃×C₄²	D₃²	1	3	yes	—
36	C₂×C₃⋊C₈	—	3	12	no	—
37	C₂×C₃⋊C₈	—	6	6	no	—
38	C₂²×C₃²	—	2	3	no	—
39	C₂²×C₃²	—	4	1	no	—
40	C₂²×C₃²	C₄×Dic₃	1	3	yes	—
41	C₃²×C₄	—	2	12	no	—
42	C₃²×C₄	—	2	6	no	—
43	C₃²×C₄	—	4	6	no	—
44	C₃²×C₄	D₃×C₈	1	12	yes	—
45	C₃²×C₄	C₂×C₃⋊C₈	1	6	yes	—
46	C₄×C₈	—	9	3	no	—
47	C₃³	C₈²	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃×C₄	—	2	6	no	—
49	C₂×C₃×C₄	—	2	3	no	—
50	C₂×C₃×C₄	—	4	3	no	—
51	C₂×C₃×C₄	D₃×Dic₃	1	6	yes	—
52	C₂×C₃×C₄	#624	1	3	yes	—
53	C₃×C₈	—	9	12	no	—
54	C₃⋊C₈	—	3	24	no	—
55	C₃⋊C₈	—	6	12	no	—
56	C₂×C₃²	—	2	6	no	—
57	C₂×C₃²	—	2	3	no	—
58	C₂×C₃²	—	4	3	no	—
59	C₂×C₃²	C₄×C₃⋊C₈	1	3	yes	—
60	C₂×C₃²	C₈×Dic₃	1	6	yes	—
61	C₂×C₈	—	9	6	no	—
62	C₄²	#1113	1	1	yes	characteristic
63	C₂²×C₃	—	2	3	no	—
64	C₂²×C₃	—	4	1	no	—
65	C₂²×C₃	Dic₃²	1	3	yes	—
66	C₃×C₄	—	2	6	no	—
67	C₃×C₄	—	2	12	no	—
68	C₃×C₄	—	4	6	no	—
69	C₃×C₄	D₃×C₃⋊C₈	1	12	yes	—
70	C₃×C₄	#3710	1	6	yes	—
71	C₃²	—	2	3	no	—
72	C₃²	—	4	1	no	—
73	C₃²	C₈×C₃⋊C₈	1	3	yes	complemented
74	C₂×C₄	#6294	1	3	yes	—
75	C₈	—	9	12	no	—
76	C₂×C₃	—	2	3	no	—
77	C₂×C₃	—	2	6	no	—
78	C₂×C₃	—	4	3	no	—
79	C₂×C₃	Dic₃×C₃⋊C₈	1	6	yes	—
80	C₂×C₃	#63553	1	3	yes	—
81	C₂²	#88813	1	1	yes	characteristic
82	C₄	#88808	1	6	yes	—
83	C₃	—	2	3	no	—
84	C₃	—	4	1	no	—
85	C₃	C₃⋊C₈²	1	3	yes	complemented
86	C₂	#11722070	1	3	yes	—

	subgroup	G/H
center, Frattini, hypercenter	C₄²	#1113
commutator, hypocenter	C₃³	C₈²
Fitting, cosocle	C₃³×C₄²	C₂²
socle, prime radical	C₂²×C₃³	C₄²
LCS	G ▷ C₃³
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
9. C₃×C₃⋊(C₃⋊C₈)	C₈	semidirect
27. C₃⋊(C₃⋊C₈)	C₃⋊C₈	semidirect
47. C₃³	C₈²	semidirect
73. C₃²	C₈×C₃⋊C₈	semidirect
85. C₃	C₃⋊C₈²	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₃⋊(C₃⋊C₈)	2
C₃⋊C₈²	3