Subgroups — #13146094

Order 1728.

896 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	1	2	yes	maximal
3	#10589412	—	3	13	no	maximal
4	C₂×C₃³×Q₈	C₄	1	2	yes	—
5	C₃³×C₄⋊C₄	—	2	2	no	—
6	C₃³×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
8	C₃²×C₄⋊Q₈	D₃	1	13	yes	—
9	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	26	no	—
10	C₃³×Q₈	—	4	2	no	—
11	C₂×C₃³×C₄	—	2	2	no	—
12	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃³×C₄	D₄	1	2	yes	—
14	C₃⋊(C₃⋊(C₃⋊C₈))	—	4	2	no	—
15	#4546	—	9	13	no	—
16	C₂×C₃²×Q₈	Dic₃	1	26	yes	—
17	C₃²×C₄⋊C₄	—	2	26	no	—
18	C₃²×C₄²	C₂×D₃	1	13	yes	—
19	C₂×C₃⋊(C₃⋊C₈)	—	6	26	no	—
20	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
21	C₃³×C₄	—	2	1	no	—
22	C₃³×C₄	—	4	2	no	—
23	C₃³×C₄	SD₁₆	1	2	yes	—
24	C₃³×C₄	Q₁₆	1	2	yes	—
25	C₃×C₄⋊Q₈	C₃⋊D₃	1	13	yes	—
26	C₄⋊(C₃⋊C₈)	—	9	26	no	—
27	C₃²×Q₈	—	4	26	no	—
28	C₂×C₃²×C₄	—	2	26	no	—
29	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
30	C₂×C₃²×C₄	#67	1	26	yes	—
31	C₃⋊(C₃⋊C₈)	—	12	26	no	—
32	#333	—	27	1	no	Sylow
33	C₂×C₃³	#100	1	1	yes	characteristic
34	C₂×C₃³	#103	1	2	yes	—
35	C₂×C₃×Q₈	C₃⋊Dic₃	1	26	yes	—
36	C₃×C₄⋊C₄	—	2	26	no	—
37	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
38	C₂×C₃⋊C₈	—	18	26	no	—
39	C₂²×C₃²	#217	1	13	yes	—
40	C₃²×C₄	—	2	13	no	—
41	C₃²×C₄	—	4	26	no	—
42	C₃²×C₄	#215	1	26	yes	—
43	C₃²×C₄	#216	1	26	yes	—
44	C₄⋊Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
45	C₄⋊C₈	—	27	2	no	—
46	C₃³	#333	1	1	yes	characteristic, complemented, Sylow
47	C₃×Q₈	—	4	26	no	—
48	C₂×C₃×C₄	—	2	26	no	—
49	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
50	C₂×C₃×C₄	#638	1	26	yes	—
51	C₃⋊C₈	—	36	26	no	—
52	C₂×C₃²	#833	1	13	yes	—
53	C₂×C₃²	#835	1	26	yes	—
54	C₂×Q₈	C₃⋊(C₃⋊Dic₃)	1	2	yes	—
55	C₂×C₈	—	54	2	no	—
56	C₄⋊C₄	—	2	2	no	—
57	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
58	C₂²×C₃	#3757	1	13	yes	—
59	C₃×C₄	—	2	13	no	—
60	C₃×C₄	—	4	26	no	—
61	C₃×C₄	#3755	1	26	yes	—
62	C₃×C₄	#3756	1	26	yes	—
63	C₃²	#4546	1	13	yes	complemented
64	Q₈	—	4	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
67	C₂×C₄	#6312	1	2	yes	—
68	C₈	—	108	2	no	—
69	C₂×C₃	#63660	1	13	yes	—
70	C₂×C₃	#63662	1	26	yes	—
71	C₂²	#88864	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	4	2	no	—
74	C₄	#88862	1	2	yes	—
75	C₄	#88863	1	2	yes	—
76	C₃	#10589412	1	13	yes	complemented
77	C₂	#11722203	1	1	yes	characteristic
78	C₂	#11722205	1	2	yes	—

	subgroup	G/H
center	C₂²	#88864
commutator	C₂×C₃³×C₄	C₂×C₄
Frattini	C₄²	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂²⋊C₄
cosocle	C₃³×C₄²	C₂²
hypercenter	C₄⋊Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#333
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
46. C₃³	#333	semidirect
63. C₃²	#4546	semidirect
76. C₃	#10589412	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	2
#10589412	3