Subgroups — #2237

Order 128.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#467	C₂	1	1	yes	maximal, characteristic
2	#427	C₂	1	1	yes	maximal, characteristic
3	#367	C₂	1	1	yes	maximal, characteristic, complemented
4	#366	C₂	1	1	yes	maximal, characteristic, complemented
5	#358	C₂	1	1	yes	maximal, characteristic, complemented
6	#358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₂²⋊Q₈	—	2	1	no	—
11	#117	—	2	1	no	—
12	C₂×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₆	C₂²	1	1	yes	characteristic
14	#107	C₂²	1	1	yes	characteristic
15	#107	C₂²	1	1	yes	characteristic
16	#107	C₂²	1	1	yes	characteristic
17	#106	—	2	1	no	—
18	#103	—	2	1	no	—
19	C₂×Q₈	—	2	1	no	—
20	C₂²×C₄	D₄	1	1	yes	characteristic
21	Q₁₆	—	2	1	no	—
22	Q₁₆	—	2	1	no	—
23	C₂×C₈	—	2	1	no	—
24	C₂×C₈	—	2	1	no	—
25	C₂×C₈	D₄	1	1	yes	characteristic
26	C₂×C₈	C₂³	1	1	yes	characteristic
27	C₄⋊C₄	—	2	1	no	—
28	C₄⋊C₄	—	2	1	no	—
29	C₄⋊C₄	—	2	1	no	—
30	C₄⋊C₄	—	4	1	no	—
31	C₂²⋊C₄	—	4	1	no	—
32	C₂²⋊C₄	—	4	1	no	—
33	C₄²	—	4	1	no	—
34	C₁₆	—	2	1	no	—
35	C₁₆	—	2	1	no	—
36	C₂³	D₈	1	1	yes	characteristic
37	Q₈	—	4	1	no	—
38	Q₈	—	4	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	4	1	no	—
45	C₂×C₄	D₈	1	1	yes	characteristic
46	C₂×C₄	C₂×D₄	1	1	yes	characteristic
47	C₈	—	2	1	no	—
48	C₈	#41	1	1	yes	characteristic
49	C₈	#41	1	1	yes	characteristic
50	C₂²	—	2	1	no	—
51	C₂²	—	2	1	no	—
52	C₂²	—	2	1	no	—
53	C₂²	C₂×D₈	1	1	yes	characteristic
54	C₄	—	2	1	no	—
55	C₄	—	8	1	no	—
56	C₄	—	8	1	no	—
57	C₄	—	8	1	no	—
58	C₄	—	8	1	no	—
59	C₄	#123	1	1	yes	characteristic
60	C₄	#137	1	1	yes	characteristic
61	C₂	—	4	1	no	—
62	C₂	#480	1	1	yes	characteristic
63	C₂	#508	1	1	yes	characteristic
64	C₂	#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂³	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #367	C₂	semidirect
4. #366	C₂	semidirect
5. #358	C₂	semidirect
6. #358	C₂	semidirect

Maximal subgroups

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

subgroup	index
#467	2
#427	2
#367	2
#366	2
#358	2
C₂²⋊C₁₆	2