Subgroups — #13136729

Order 1728.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11720785	C₂	1	2	yes	maximal
2	C₄×#6188	C₂	1	1	yes	maximal, characteristic
3	C₂×#6188	—	2	2	no	—
4	C₂×#6188	C₄	1	1	yes	characteristic
5	C₂×#6188	C₄	1	1	yes	characteristic
6	C₄×#1084	C₂²	1	1	yes	characteristic
7	C₂×C₄×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
8	C₂×#1084	C₂×C₄	1	1	yes	characteristic
9	C₂×#1084	D₄	1	1	yes	characteristic
10	#6188	—	2	1	no	—
11	#6188	—	4	2	no	—
12	#6188	C₈	1	2	yes	complemented
13	C₃×#335	—	9	1	no	maximal
14	C₃×C₄×Dic₃	—	12	1	no	—
15	C₂²×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
16	C₄×H(3,𝔽₃)	—	2	1	no	—
17	#1084	C₂×C₈	1	1	yes	characteristic
18	#1084	M₁₆	1	1	yes	characteristic
19	#1084	D₈	1	1	yes	characteristic
20	#1084	Q₁₆	1	1	yes	characteristic
21	C₃×C₄⋊C₈	—	9	2	no	—
22	C₃×C₄×C₈	—	9	1	no	—
23	C₂×C₃²×C₄	—	4	1	no	—
24	C₂×C₃×Dic₃	—	12	1	no	—
25	C₂×C₃×Dic₃	—	12	1	no	—
26	#335	—	9	1	no	Sylow
27	C₂×H(3,𝔽₃)	C₄⋊C₈	1	1	yes	characteristic
28	C₂×H(3,𝔽₃)	#107	1	1	yes	characteristic
29	C₂×H(3,𝔽₃)	#108	1	1	yes	characteristic
30	C₂×C₃×C₈	—	9	1	no	—
31	C₂×C₃×C₈	—	9	1	no	—
32	C₂×C₃×C₈	—	18	2	no	—
33	C₃×C₄²	—	9	1	no	—
34	C₄×Dic₃	—	12	1	no	—
35	C₂²×C₃²	—	4	1	no	—
36	C₃²×C₄	—	4	2	no	—
37	C₃×Dic₃	—	12	1	no	—
38	C₃×Dic₃	—	12	1	no	—
39	C₃×Dic₃	—	12	1	no	—
40	C₃×Dic₃	—	12	1	no	—
41	C₄⋊C₈	—	9	2	no	—
42	C₄×C₈	—	9	1	no	—
43	H(3,𝔽₃)	#335	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×C₄	—	9	1	no	—
45	C₂×C₃×C₄	—	9	1	no	—
46	C₂×C₃×C₄	—	12	1	no	—
47	C₂×C₃×C₄	PSU(3,𝔽₂)	1	1	yes	characteristic
48	C₂×Dic₃	—	12	1	no	—
49	C₂×Dic₃	—	12	1	no	—
50	C₃×C₈	—	9	2	no	—
51	C₃×C₈	—	18	1	no	—
52	C₃×C₈	—	36	2	no	—
53	C₂×C₃²	—	4	1	no	—
54	C₂×C₃²	—	4	1	no	—
55	C₂×C₃²	—	4	1	no	—
56	C₂×C₈	—	9	1	no	—
57	C₂×C₈	—	9	1	no	—
58	C₂×C₈	—	18	2	no	—
59	C₄²	—	9	1	no	—
60	C₂²×C₃	—	12	1	no	—
61	C₂²×C₃	#3777	1	1	yes	characteristic
62	C₃×C₄	—	2	1	no	—
63	C₃×C₄	—	9	1	no	—
64	C₃×C₄	—	9	1	no	—
65	C₃×C₄	—	9	1	no	—
66	C₃×C₄	—	9	1	no	—
67	C₃×C₄	—	12	2	no	—
68	Dic₃	—	12	1	no	—
69	Dic₃	—	12	1	no	—
70	Dic₃	—	12	1	no	—
71	Dic₃	—	12	1	no	—
72	C₃²	—	4	1	no	—
73	C₂×C₄	—	9	1	no	—
74	C₂×C₄	—	9	1	no	—
75	C₂×C₄	SU(3,𝔽₂)	1	1	yes	characteristic
76	C₈	—	9	2	no	—
77	C₈	—	18	1	no	—
78	C₈	—	36	2	no	—
79	C₂×C₃	—	12	1	no	—
80	C₂×C₃	—	12	1	no	—
81	C₂×C₃	—	12	1	no	—
82	C₂×C₃	#63744	1	1	yes	characteristic
83	C₂×C₃	#63746	1	1	yes	characteristic
84	C₂×C₃	#63748	1	1	yes	characteristic
85	C₂²	#88592	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	9	1	no	—
88	C₄	—	9	1	no	—
89	C₄	—	9	1	no	—
90	C₄	—	9	1	no	—
91	C₃	—	12	1	no	—
92	C₃	#10590089	1	1	yes	characteristic
93	C₂	#11720708	1	1	yes	characteristic
94	C₂	#11720710	1	1	yes	characteristic
95	C₂	#11720712	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#3777
commutator	#1084	C₂×C₈
Frattini, hypercenter	C₂×C₃×C₄	PSU(3,𝔽₂)
Fitting	C₂×C₄×H(3,𝔽₃)	Q₈
prime radical	C₂²×H(3,𝔽₃)	C₄⋊C₄
cosocle	C₄×#1084	C₂²
hypocenter	H(3,𝔽₃)	#335
LCS	G ▷ #1084 ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
12. #6188	C₈	semidirect
43. H(3,𝔽₃)	#335	semidirect

Maximal subgroups

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

subgroup	index
#11720785	2
C₄×#6188	2
C₃×#335	9