Subgroups — #3164

Order 128.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#544	C₂	1	2	yes	maximal
2	#501	C₂	1	4	yes	maximal, complemented
3	#500	C₂	1	2	yes	maximal, complemented
4	#499	C₂	1	2	yes	maximal, complemented
5	#428	C₂	1	2	yes	maximal
6	#427	C₂	1	1	yes	maximal, characteristic
7	C₂×#107	C₂	1	1	yes	maximal, characteristic
8	#405	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈	—	2	2	no	—
12	C₄⋊Q₈	C₂²	1	4	yes	—
13	#125	—	2	2	no	—
14	#125	C₂²	1	4	yes	—
15	C₂²⋊Q₈	—	2	4	no	—
16	#117	C₂²	1	2	yes	—
17	#117	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄	C₂²	1	2	yes	—
19	#107	C₂²	1	2	yes	—
20	#107	C₂²	1	2	yes	—
21	#107	C₂²	1	1	yes	characteristic
22	#107	C₂²	1	1	yes	characteristic
23	#107	C₂²	1	4	yes	—
24	#106	C₂²	1	2	yes	—
25	#106	C₂²	1	4	yes	—
26	#97	C₂²	1	2	yes	—
27	C₄×C₈	C₂²	1	2	yes	—
28	C₂×Q₈	—	2	4	no	—
29	C₂²×C₄	—	2	2	no	—
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	M₁₆	Q₈	1	4	yes	complemented
32	C₂×C₈	Q₈	1	4	yes	complemented
33	C₂×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₈	C₂³	1	2	yes	—
35	C₂×C₈	C₂³	1	1	yes	characteristic
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	2	4	no	—
38	C₄⋊C₄	—	2	4	no	—
39	C₄⋊C₄	—	2	4	no	—
40	C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₄⋊C₄	C₂³	1	2	yes	—
43	C₄⋊C₄	C₂³	1	4	yes	—
44	C₄⋊C₄	C₂³	1	2	yes	—
45	C₂²⋊C₄	—	2	2	no	—
46	C₂²⋊C₄	D₄	1	2	yes	—
47	C₄²	—	2	2	no	—
48	C₄²	C₂³	1	2	yes	—
49	C₂³	C₂×D₄	1	1	yes	characteristic
50	Q₈	—	8	4	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	4	no	—
54	C₂×C₄	—	8	2	no	—
55	C₂×C₄	C₂×D₄	1	2	yes	—
56	C₂×C₄	C₂×D₄	1	2	yes	—
57	C₂×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄	C₂×Q₈	1	2	yes	—
59	C₂×C₄	C₂×Q₈	1	2	yes	—
60	C₂×C₄	C₂⁴	1	1	yes	characteristic
61	C₈	C₂×Q₈	1	2	yes	—
62	C₈	C₂×Q₈	1	2	yes	—
63	C₈	C₂×Q₈	1	4	yes	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	C₄⋊Q₈	1	2	yes	—
67	C₂²	C₂²×D₄	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	2	2	no	—
70	C₄	—	4	4	no	—
71	C₄	—	4	2	no	—
72	C₄	—	4	2	no	—
73	C₄	C₄⋊Q₈	1	2	yes	—
74	C₄	C₂²×Q₈	1	1	yes	characteristic
75	C₄	C₂²×Q₈	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂	#576	1	1	yes	characteristic
79	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #501	C₂	semidirect
3. #500	C₂	semidirect
4. #499	C₂	semidirect
31. M₁₆	Q₈	semidirect
32. C₂×C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#544	2
#501	2
#500	2
#499	2
#428	2
#427	2
C₂×#107	2
#405	2