Subgroups — #3341

Order 128.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555	C₂	1	2	yes	maximal, complemented
2	#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#482	C₂	1	4	yes	maximal, complemented
4	#476	C₂	1	2	yes	maximal, complemented
5	#466	C₂	1	2	yes	maximal, complemented
6	C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#427	C₂	1	2	yes	maximal, complemented
9	C₂×D₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	2	yes	—
11	C₄⋊Q₈	C₂²	1	2	yes	—
12	#124	—	2	4	no	—
13	#121	—	2	2	no	—
14	#121	C₂²	1	4	yes	—
15	C₄×D₄	C₂²	1	2	yes	—
16	C₄×D₄	C₂²	1	1	yes	characteristic
17	#117	—	2	2	no	—
18	#117	C₂²	1	4	yes	—
19	#107	C₂²	1	1	yes	characteristic
20	#107	C₂²	1	2	yes	—
21	#107	C₂²	1	2	yes	—
22	#106	C₂²	1	4	yes	—
23	C₄⋊C₈	C₂²	1	1	yes	characteristic
24	#102	C₂²	1	2	yes	—
25	#102	C₂²	1	4	yes	—
26	C₂²⋊C₈	C₂²	1	2	yes	—
27	C₄×C₈	C₂²	1	1	yes	characteristic
28	C₂×Q₈	—	2	2	no	—
29	C₂×D₄	—	2	4	no	—
30	C₂×D₄	D₄	1	1	yes	characteristic
31	C₂×D₄	C₂³	1	2	yes	—
32	C₂²×C₄	—	2	2	no	—
33	C₂²×C₄	C₂³	1	2	yes	—
34	D₈	—	2	2	no	—
35	C₂×C₈	—	2	4	no	—
36	C₂×C₈	C₂³	1	1	yes	characteristic
37	C₂×C₈	C₂³	1	2	yes	—
38	C₂×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₄⋊C₄	C₂³	1	2	yes	—
42	C₄⋊C₄	C₂³	1	4	yes	—
43	C₂²⋊C₄	—	2	4	no	—
44	C₂²⋊C₄	—	2	4	no	—
45	C₂²⋊C₄	—	2	2	no	—
46	C₂²⋊C₄	D₄	1	2	yes	—
47	C₄²	—	2	4	no	—
48	C₄²	C₂³	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	C₂³	C₂×D₄	1	2	yes	—
51	Q₈	—	8	2	no	—
52	D₄	—	2	2	no	—
53	D₄	—	2	2	no	—
54	D₄	—	2	2	no	—
55	D₄	—	8	4	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	4	4	no	—
61	C₂×C₄	C₂×D₄	1	2	yes	—
62	C₂×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄	C₂⁴	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₈	—	2	1	no	—
67	C₈	#41	1	2	yes	—
68	C₂²	—	2	2	no	—
69	C₂²	—	2	2	no	—
70	C₂²	—	2	2	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	4	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	C₂²×D₄	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	4	4	no	—
79	C₄	#135	1	2	yes	—
80	C₄	C₂×#41	1	1	yes	characteristic
81	C₄	2₋¹⁺⁴	1	1	yes	characteristic
82	C₂	—	4	2	no	—
83	C₂	—	8	2	no	—
84	C₂	#547	1	1	yes	characteristic
85	C₂	C₂×#135	1	1	yes	characteristic
86	C₂	#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #555	C₂	semidirect
2. #500	C₂	semidirect
3. #482	C₂	semidirect
4. #476	C₂	semidirect
5. #466	C₂	semidirect
6. C₄×D₈	C₂	semidirect
7. C₈×D₄	C₂	semidirect
8. #427	C₂	semidirect

Maximal subgroups

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

subgroup	index
#555	2
#500	2
#482	2
#476	2
#466	2
C₄×D₈	2
C₈×D₄	2
#427	2