Subgroups — #3503

Order 128.

530 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²	C₂	1	2	yes	maximal, complemented
2	#557	C₂	1	2	yes	maximal, complemented
3	#554	C₂	1	6	yes	maximal, complemented
4	D₄×Q₈	C₂	1	6	yes	maximal, complemented
5	#544	C₂	1	6	yes	maximal, complemented
6	#533	C₂	1	3	yes	maximal, complemented
7	C₂×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
8	#520	C₂	1	2	yes	maximal
9	C₄×#41	C₂	1	1	yes	maximal, characteristic
10	C₂×#41	C₂²	1	1	yes	characteristic
11	C₂²×Q₈	C₂²	1	6	yes	—
12	C₄⋊Q₈	C₂²	1	3	yes	—
13	C₄⋊Q₈	C₂²	1	6	yes	—
14	C₄⋊Q₈	C₂²	1	6	yes	—
15	C₄⋊Q₈	C₂²	1	6	yes	—
16	C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈	C₂²	1	12	yes	—
18	#125	C₂²	1	6	yes	—
19	#125	C₂²	1	12	yes	—
20	C₂²⋊Q₈	C₂²	1	6	yes	—
21	C₂²⋊Q₈	C₂²	1	6	yes	—
22	C₂²⋊Q₈	C₂²	1	24	yes	—
23	C₄×Q₈	C₂²	1	1	yes	characteristic
24	C₄×Q₈	C₂²	1	1	yes	characteristic
25	C₄×Q₈	C₂²	1	4	yes	—
26	C₄×Q₈	C₂²	1	6	yes	—
27	C₄×Q₈	C₂²	1	6	yes	—
28	C₄×D₄	C₂²	1	3	yes	—
29	C₄×D₄	C₂²	1	3	yes	—
30	C₄×D₄	C₂²	1	12	yes	—
31	#117	C₂²	1	3	yes	—
32	#117	C₂²	1	6	yes	—
33	C₂×C₄⋊C₄	C₂²	1	6	yes	—
34	C₂×C₄⋊C₄	C₂²	1	6	yes	—
35	C₂×C₄²	C₂²	1	3	yes	—
36	#41	Q₈	1	8	yes	complemented
37	C₂×Q₈	—	4	12	no	—
38	C₂×Q₈	C₂³	1	2	yes	—
39	C₂×Q₈	C₂³	1	6	yes	—
40	C₂×Q₈	C₂³	1	12	yes	—
41	C₂×Q₈	C₂³	1	1	yes	characteristic
42	C₂×D₄	C₂³	1	3	yes	—
43	C₂²×C₄	C₂³	1	3	yes	—
44	C₂²×C₄	C₂³	1	3	yes	—
45	C₂²×C₄	C₂³	1	12	yes	—
46	C₂²×C₄	C₂³	1	3	yes	—
47	C₄⋊C₄	C₂³	1	6	yes	—
48	C₄⋊C₄	C₂³	1	6	yes	—
49	C₄⋊C₄	C₂³	1	12	yes	—
50	C₄⋊C₄	C₂³	1	12	yes	—
51	C₄⋊C₄	C₂³	1	2	yes	—
52	C₄⋊C₄	C₂³	1	3	yes	—
53	C₄⋊C₄	C₂³	1	3	yes	—
54	C₄⋊C₄	C₂³	1	2	yes	—
55	C₄⋊C₄	C₂³	1	12	yes	—
56	C₄⋊C₄	C₂³	1	12	yes	—
57	C₂²⋊C₄	C₂³	1	12	yes	—
58	C₂²⋊C₄	C₂³	1	6	yes	—
59	C₄²	C₂³	1	1	yes	characteristic
60	C₄²	C₂³	1	3	yes	—
61	C₄²	C₂³	1	3	yes	—
62	C₄²	C₂³	1	3	yes	—
63	C₄²	C₂³	1	12	yes	—
64	C₂³	C₂⁴	1	3	yes	—
65	Q₈	—	2	12	no	—
66	Q₈	—	2	4	no	—
67	Q₈	—	4	12	no	—
68	Q₈	C₂×Q₈	1	4	yes	complemented
69	D₄	C₂×Q₈	1	12	yes	complemented
70	C₂×C₄	—	2	6	no	—
71	C₂×C₄	—	2	6	no	—
72	C₂×C₄	—	4	12	no	—
73	C₂×C₄	C₂×Q₈	1	12	yes	complemented
74	C₂×C₄	C₂⁴	1	4	yes	—
75	C₂×C₄	C₂⁴	1	3	yes	—
76	C₂×C₄	C₂⁴	1	3	yes	—
77	C₂×C₄	C₂⁴	1	12	yes	—
78	C₂×C₄	C₂⁴	1	3	yes	—
79	C₂×C₄	C₂⁴	1	1	yes	characteristic
80	C₂×C₄	C₂⁴	1	1	yes	characteristic
81	C₂×C₄	C₂⁴	1	1	yes	characteristic
82	C₂²	—	2	3	no	—
83	C₂²	—	2	3	no	—
84	C₂²	C₂²×Q₈	1	6	yes	—
85	C₂²	C₂⁵	1	1	yes	characteristic
86	C₄	—	2	3	no	—
87	C₄	—	2	12	no	—
88	C₄	—	2	3	no	—
89	C₄	—	2	4	no	—
90	C₄	C₂²×Q₈	1	2	yes	—
91	C₄	C₂²×Q₈	1	6	yes	—
92	C₄	H(5,𝔽₂)	1	2	yes	—
93	C₄	2₋¹⁺⁴	1	2	yes	—
94	C₂	—	2	6	no	—
95	C₂	C₂³×Q₈	1	1	yes	characteristic
96	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
97	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. Q₈²	C₂	semidirect
2. #557	C₂	semidirect
3. #554	C₂	semidirect
4. D₄×Q₈	C₂	semidirect
5. #544	C₂	semidirect
6. #533	C₂	semidirect
7. C₂×C₄⋊Q₈	C₂	semidirect
36. #41	Q₈	semidirect
68. Q₈	C₂×Q₈	semidirect
69. D₄	C₂×Q₈	semidirect
73. C₂×C₄	C₂×Q₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈²	2
#557	2
#554	2
D₄×Q₈	2
#544	2
#533	2
C₂×C₄⋊Q₈	2
#520	2
C₄×#41	2