Subgroups — #13146092

Order 1728.

840 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	#11722206	C₂	1	2	yes	maximal
3	#10589414	—	3	13	no	maximal
4	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic
5	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
6	C₃³×C₄⋊C₄	—	2	2	no	—
7	C₃³×C₄²	C₄	1	1	yes	characteristic
8	#88855	—	2	2	no	—
9	C₃²×C₄⋊Q₈	D₃	1	13	yes	—
10	#63663	—	3	26	no	—
11	C₃³×Q₈	—	4	1	no	—
12	C₃³×Q₈	—	4	1	no	—
13	C₂×C₃³×C₄	—	2	1	no	—
14	C₂×C₃³×C₄	—	2	1	no	—
15	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃³×C₄	D₄	1	2	yes	—
17	C₃⋊(C₃⋊(C₃⋊C₈))	—	4	2	no	—
18	#4548	—	9	13	no	—
19	C₂×C₃²×Q₈	Dic₃	1	13	yes	—
20	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
21	C₃²×C₄⋊C₄	—	2	26	no	—
22	C₃²×C₄²	Dic₃	1	13	yes	—
23	#3748	—	6	26	no	—
24	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
25	C₃³×C₄	—	2	2	no	—
26	C₃³×C₄	—	2	2	no	—
27	C₃³×C₄	—	2	1	no	—
28	C₃³×C₄	—	4	1	no	—
29	C₃×C₄⋊Q₈	C₃⋊D₃	1	13	yes	—
30	#836	—	9	26	no	—
31	C₃²×Q₈	—	4	13	no	—
32	C₃²×Q₈	—	4	13	no	—
33	C₂×C₃²×C₄	—	2	13	no	—
34	C₂×C₃²×C₄	—	2	13	no	—
35	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
36	C₂×C₃²×C₄	#67	1	26	yes	—
37	C₃⋊(C₃⋊C₈)	—	12	26	no	—
38	#356	—	27	1	no	Sylow
39	C₂×C₃³	—	2	1	no	—
40	C₂×C₃³	#99	1	1	yes	characteristic
41	C₂×C₃×Q₈	C₃⋊Dic₃	1	13	yes	—
42	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
43	C₃×C₄⋊C₄	—	2	26	no	—
44	C₃×C₄²	C₃⋊Dic₃	1	13	yes	—
45	#208	—	18	26	no	—
46	C₂²×C₃²	#217	1	13	yes	—
47	C₃²×C₄	—	2	26	no	—
48	C₃²×C₄	—	2	26	no	—
49	C₃²×C₄	—	2	13	no	—
50	C₃²×C₄	—	4	13	no	—
51	C₄⋊Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
52	#101	—	27	2	no	—
53	C₃³	#356	1	1	yes	characteristic, complemented, Sylow
54	C₃×Q₈	—	4	13	no	—
55	C₃×Q₈	—	4	13	no	—
56	C₂×C₃×C₄	—	2	13	no	—
57	C₂×C₃×C₄	—	2	13	no	—
58	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
59	C₂×C₃×C₄	#638	1	26	yes	—
60	C₃⋊C₈	—	36	26	no	—
61	C₂×C₃²	—	2	13	no	—
62	C₂×C₃²	#834	1	13	yes	—
63	C₂×Q₈	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
64	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
65	M₁₆	—	54	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄²	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
68	C₂²×C₃	#3757	1	13	yes	—
69	C₃×C₄	—	2	26	no	—
70	C₃×C₄	—	2	26	no	—
71	C₃×C₄	—	2	13	no	—
72	C₃×C₄	—	4	13	no	—
73	C₃²	#4548	1	13	yes	complemented
74	Q₈	—	4	1	no	—
75	Q₈	—	4	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
79	C₂×C₄	#6312	1	2	yes	—
80	C₈	—	108	2	no	—
81	C₂×C₃	—	2	13	no	—
82	C₂×C₃	#63661	1	13	yes	—
83	C₂²	#88864	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₃	#10589414	1	13	yes	complemented
89	C₂	—	2	1	no	—
90	C₂	#11722204	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11722204
commutator	C₂×C₃³×C₄	C₂×C₄
Frattini	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₄⋊Q₈	C₂
socle	C₂×C₃³	#99
prime radical	C₂²×C₃³	C₂²⋊C₄
cosocle	C₂×C₃³×Q₈	C₂²
hypercenter	C₄⋊Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#356
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂²×C₃³ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
53. C₃³	#356	semidirect
73. C₃²	#4548	semidirect
88. C₃	#10589414	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
#11722206	2
#10589414	3