Subgroups — #11720560

Order 864.

253 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#272	C₂	1	1	yes	maximal, characteristic
2	M₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#88538	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#823	—	3	4	no	maximal
5	C₂²×#272	C₄	1	2	yes	—
6	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₄⋊#272	—	2	1	no	—
8	C₄⋊#272	—	2	1	no	—
9	C₈×H(3,𝔽₃)	—	2	1	no	—
10	#6180	—	2	1	no	—
11	C₂×C₃×D₁₂	—	3	4	no	—
12	C₃²×M₁₆	D₃	1	4	yes	complemented
13	C₃×#208	—	3	4	no	—
14	C₂²×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₂×#272	—	2	2	no	—
16	C₂×#272	—	4	2	no	—
17	C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
18	C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
19	C₃×#100	—	9	1	no	—
20	#823	—	3	4	no	—
21	C₂²×C₃×D₃	—	3	8	no	—
22	C₂×C₃²×C₄	C₂×D₃	1	4	yes	—
23	C₃×D₁₂	—	6	4	no	—
24	C₃×D₁₂	—	6	4	no	—
25	C₃²×C₈	—	2	4	no	—
26	C₃×C₃⋊C₈	—	6	4	no	—
27	C₂×H(3,𝔽₃)	—	2	1	no	—
28	C₂×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
29	#272	—	4	2	no	—
30	C₂×C₃×D₄	—	9	1	no	—
31	C₂×D₁₂	—	3	4	no	—
32	C₃×M₁₆	—	6	4	no	—
33	C₃×M₁₆	—	9	1	no	—
34	C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
35	#208	—	3	4	no	—
36	C₂²×C₃²	C₄×D₃	1	4	yes	—
37	C₂×C₃×D₃	—	6	8	no	—
38	C₂×C₃×D₃	—	12	8	no	—
39	C₃²×C₄	D₁₂	1	4	yes	—
40	C₃²×C₄	#67	1	4	yes	—
41	#100	—	9	1	no	Sylow
42	H(3,𝔽₃)	#100	1	1	yes	characteristic, complemented, Sylow
43	C₂³×C₃	—	9	2	no	—
44	C₂²×D₃	—	3	8	no	—
45	C₃×D₄	—	18	1	no	—
46	C₃×D₄	—	18	1	no	—
47	C₂×C₃×C₄	—	6	4	no	—
48	C₂×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
49	D₁₂	—	6	4	no	—
50	D₁₂	—	6	4	no	—
51	C₃×C₈	—	2	1	no	—
52	C₃×C₈	—	6	4	no	—
53	C₃×C₈	—	18	1	no	—
54	C₃⋊C₈	—	6	4	no	—
55	C₂×C₃²	—	2	4	no	—
56	C₂×C₃²	(C₂×D₃)⋊C₄	1	4	yes	—
57	C₃×D₃	—	12	8	no	—
58	C₂×D₄	—	9	1	no	—
59	M₁₆	—	9	1	no	—
60	M₁₆	#272	1	1	yes	characteristic, complemented
61	C₂²×C₃	—	6	4	no	—
62	C₂²×C₃	—	18	2	no	—
63	C₂²×C₃	—	36	2	no	—
64	C₂²×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
65	C₂×D₃	—	6	8	no	—
66	C₂×D₃	—	12	8	no	—
67	C₃×C₄	—	6	4	no	—
68	C₃×C₄	—	6	4	no	—
69	C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
70	C₃×C₄	#638	1	1	yes	characteristic
71	C₃²	#823	1	4	yes	complemented
72	C₂³	—	9	2	no	—
73	D₄	—	18	1	no	—
74	D₄	—	18	1	no	—
75	C₂×C₄	C₂×#272	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	18	1	no	—
78	C₂×C₃	—	2	1	no	—
79	C₂×C₃	—	6	4	no	—
80	C₂×C₃	—	12	4	no	—
81	C₂×C₃	—	36	2	no	—
82	C₂×C₃	#3752	1	1	yes	characteristic
83	D₃	—	12	8	no	—
84	C₂²	—	18	2	no	—
85	C₂²	—	36	2	no	—
86	C₂²	C₄×#272	1	1	yes	characteristic
87	C₄	C₄⋊#272	1	1	yes	characteristic
88	C₄	#6235	1	1	yes	characteristic
89	C₃	—	6	4	no	—
90	C₃	#63650	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	36	2	no	—
93	C₂	#88542	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#3752
commutator	C₂²×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₄	C₂×C₃⋊D₃
Fitting	M₁₆×H(3,𝔽₃)	C₂
prime radical	C₂×C₄⋊#272	C₂
cosocle	C₂×C₄×H(3,𝔽₃)	C₂²
hypercenter	C₃×M₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#100
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. M₁₆×H(3,𝔽₃)	C₂	semidirect
3. #88538	C₂	semidirect
12. C₃²×M₁₆	D₃	semidirect
42. H(3,𝔽₃)	#100	semidirect
60. M₁₆	#272	semidirect
71. C₃²	#823	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#272	2
M₁₆×H(3,𝔽₃)	2
#88538	2
C₃×#823	3