Subgroups — #3139

Order 128.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	#491	C₂	1	4	yes	maximal, complemented
4	#487	C₂	1	4	yes	maximal, complemented
5	#419	C₂	1	4	yes	maximal
6	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈	C₂²	1	2	yes	—
8	C₂×M₁₆	C₂²	1	2	yes	—
9	C₄⋊Q₈	—	2	2	no	—
10	C₄⋊Q₈	C₂²	1	2	yes	—
11	C₄⋊Q₈	C₂²	1	2	yes	—
12	C₄⋊Q₈	C₂²	1	2	yes	—
13	#125	C₂²	1	2	yes	—
14	C₂²⋊Q₈	—	2	2	no	—
15	C₄×Q₈	—	2	2	no	—
16	#117	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	#103	C₂²	1	16	yes	—
20	#97	C₂²	1	2	yes	—
21	C₄×C₈	C₂²	1	2	yes	—
22	C₂×Q₈	—	2	2	no	—
23	C₂×Q₈	—	2	2	no	—
24	C₂×Q₈	—	2	8	no	—
25	C₂×Q₈	C₂³	1	4	yes	—
26	C₂²×C₄	—	2	2	no	—
27	C₂²×C₄	D₄	1	2	yes	—
28	C₂²×C₄	C₂³	1	1	yes	characteristic
29	M₁₆	—	2	4	no	—
30	C₂×C₈	C₂³	1	4	yes	—
31	C₄⋊C₄	—	2	4	no	—
32	C₄⋊C₄	—	2	4	no	—
33	C₄⋊C₄	—	2	4	no	—
34	C₄⋊C₄	—	2	2	no	—
35	C₄⋊C₄	C₂³	1	4	yes	—
36	C₂²⋊C₄	—	2	2	no	—
37	C₄²	—	2	2	no	—
38	C₄²	D₄	1	2	yes	—
39	C₄²	C₂³	1	1	yes	characteristic
40	C₄²	C₂³	1	1	yes	characteristic
41	C₂³	C₂×D₄	1	1	yes	characteristic
42	Q₈	—	2	8	no	—
43	Q₈	—	4	4	no	—
44	Q₈	—	4	4	no	—
45	C₂×C₄	—	2	4	no	—
46	C₂×C₄	—	2	4	no	—
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	—	4	4	no	—
49	C₂×C₄	C₂×D₄	1	2	yes	—
50	C₂×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄	C₂×D₄	1	2	yes	—
52	C₂×C₄	#41	1	4	yes	—
53	C₂×C₄	C₂⁴	1	1	yes	characteristic
54	C₈	—	2	4	no	—
55	C₂²	—	2	2	no	—
56	C₂²	#124	1	2	yes	—
57	C₂²	C₂²×D₄	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	4	4	no	—
60	C₄	—	4	4	no	—
61	C₄	#124	1	2	yes	—
62	C₄	#137	1	4	yes	—
63	C₄	C₂×#41	1	2	yes	—
64	C₂	—	2	2	no	—
65	C₂	C₂×#124	1	1	yes	characteristic
66	C₂	C₂×#137	1	2	yes	—

	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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #491	C₂	semidirect
4. #487	C₂	semidirect

Maximal subgroups

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

subgroup	index
#533	2
C₂×C₄⋊Q₈	2
#491	2
#487	2
#419	2
C₄×M₁₆	2