Subgroups — C₂×#11720227 #13136899

Order 1728.

107 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88386	C₂	1	1	yes	maximal, characteristic
2	#11720227	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic
4	#88386	C₄	1	1	yes	characteristic
5	#88386	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₃⋊C₁₆	—	6	2	no	—
7	C₂×C₄×H(3,𝔽₃)	C₈	1	1	yes	characteristic
8	C₈×H(3,𝔽₃)	C₈	1	1	yes	characteristic
9	C₈×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
10	C₂×C₃×C₃₂	—	9	1	no	maximal
11	C₂×C₃²×C₈	—	2	2	no	—
12	C₃×C₃⋊C₁₆	—	6	2	no	—
13	C₃×C₃⋊C₁₆	—	6	2	no	—
14	C₂²×H(3,𝔽₃)	C₁₆	1	1	yes	characteristic
15	C₄×H(3,𝔽₃)	C₁₆	1	1	yes	characteristic
16	C₄×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
17	C₂×C₃×C₁₆	—	9	1	no	—
18	C₂×C₃⋊C₁₆	—	6	2	no	—
19	C₃×C₃₂	—	9	2	no	—
20	C₂×C₃²×C₄	—	2	2	no	—
21	C₃²×C₈	—	2	2	no	—
22	C₃²×C₈	—	2	2	no	—
23	C₂×C₃₂	—	9	1	no	Sylow
24	C₂×H(3,𝔽₃)	C₃₂	1	2	yes	complemented
25	C₂×H(3,𝔽₃)	C₂×C₁₆	1	1	yes	characteristic
26	C₂×C₃×C₈	—	6	2	no	—
27	C₂×C₃×C₈	#157	1	1	yes	characteristic
28	C₃×C₁₆	—	9	1	no	—
29	C₃×C₁₆	—	9	1	no	—
30	C₃⋊C₁₆	—	6	2	no	—
31	C₃⋊C₁₆	—	6	2	no	—
32	C₂²×C₃²	—	2	2	no	—
33	C₃²×C₄	—	2	2	no	—
34	C₃²×C₄	—	2	2	no	—
35	C₂×C₁₆	—	9	1	no	—
36	C₃₂	—	9	2	no	—
37	H(3,𝔽₃)	C₂×C₃₂	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₃×C₄	—	6	2	no	—
39	C₂×C₃×C₄	#622	1	1	yes	characteristic
40	C₃×C₈	—	6	2	no	—
41	C₃×C₈	—	6	2	no	—
42	C₃×C₈	#622	1	1	yes	characteristic
43	C₃×C₈	C₂×#157	1	1	yes	characteristic
44	C₂×C₃²	—	2	2	no	—
45	C₂×C₃²	—	2	4	no	—
46	C₂×C₈	#1088	1	1	yes	characteristic
47	C₁₆	—	9	1	no	—
48	C₁₆	—	9	1	no	—
49	C₂²×C₃	—	6	2	no	—
50	C₂²×C₃	#3708	1	1	yes	characteristic
51	C₃×C₄	—	6	2	no	—
52	C₃×C₄	—	6	2	no	—
53	C₃×C₄	#3708	1	1	yes	characteristic
54	C₃×C₄	C₂×#622	1	1	yes	characteristic
55	C₃²	—	2	2	no	—
56	C₂×C₄	#6188	1	1	yes	characteristic
57	C₈	#6188	1	1	yes	characteristic
58	C₈	C₂×#1088	1	1	yes	characteristic
59	C₂×C₃	—	6	4	no	—
60	C₂×C₃	—	6	2	no	—
61	C₂×C₃	#63540	1	2	yes	—
62	C₂×C₃	C₂×#3708	1	1	yes	characteristic
63	C₂²	#88410	1	1	yes	characteristic
64	C₄	#88410	1	1	yes	characteristic
65	C₄	C₂×#6188	1	1	yes	characteristic
66	C₃	—	6	2	no	—
67	C₃	C₂×#63540	1	1	yes	characteristic
68	C₂	#11720227	1	2	yes	direct product factor
69	C₂	C₂×#88410	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₈	#157
commutator, hypocenter	H(3,𝔽₃)	C₂×C₃₂
Frattini	C₃×C₈	C₂×#157
Fitting	C₂×C₈×H(3,𝔽₃)	C₄
socle	C₂²×C₃	#3708
prime radical	C₂²×H(3,𝔽₃)	C₁₆
cosocle	#88386	C₂²
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. #11720227	C₂	direct (N × H)
24. C₂×H(3,𝔽₃)	C₃₂	semidirect
37. H(3,𝔽₃)	C₂×C₃₂	semidirect
68. C₂	#11720227	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#88386	2
#11720227	2
C₂×C₃×C₃₂	9

Subgroups — C2×#11720227 #13136899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720227 #13136899