Subgroups — #1990

Order 128.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	#384	C₂	1	1	yes	maximal, characteristic
4	#340	C₂	1	4	yes	maximal
5	C₂²×Q₈	—	2	1	no	—
6	C₂²×C₈	C₂²	1	2	yes	—
7	C₄⋊Q₈	—	2	2	no	—
8	C₄⋊Q₈	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄	—	2	1	no	—
10	C₂×C₄⋊C₄	—	2	1	no	—
11	C₂×C₄⋊C₄	—	2	1	no	—
12	C₂×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	C₄×C₈	—	2	2	no	—
16	C₂×Q₈	—	2	4	no	—
17	C₂×Q₈	—	4	2	no	—
18	C₂×Q₈	—	4	2	no	—
19	C₂²×C₄	—	2	2	no	—
20	C₂²×C₄	—	2	2	no	—
21	C₂²×C₄	D₄	1	1	yes	characteristic
22	C₂²×C₄	D₄	1	1	yes	characteristic
23	C₂²×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₈	—	2	2	no	—
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	Q₈	1	4	yes	complemented
27	C₄⋊C₄	—	2	4	no	—
28	C₄⋊C₄	—	2	2	no	—
29	C₄⋊C₄	—	2	4	no	—
30	C₄⋊C₄	—	4	2	no	—
31	C₄⋊C₄	—	4	2	no	—
32	C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₄²	C₂×C₄	1	2	yes	—
34	C₄²	D₄	1	2	yes	—
35	C₂³	C₂×D₄	1	1	yes	characteristic
36	Q₈	—	4	8	no	—
37	C₂×C₄	—	2	4	no	—
38	C₂×C₄	—	4	2	no	—
39	C₂×C₄	—	4	2	no	—
40	C₂×C₄	—	4	2	no	—
41	C₂×C₄	—	4	2	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄	D₈	1	2	yes	—
46	C₂×C₄	SD₁₆	1	2	yes	—
47	C₂×C₄	SD₁₆	1	2	yes	—
48	C₂×C₄	Q₁₆	1	2	yes	—
49	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄	C₂×Q₈	1	2	yes	—
52	C₂×C₄	#41	1	2	yes	—
53	C₈	—	2	4	no	—
54	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²	#124	1	1	yes	characteristic
56	C₂²	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²	C₂×D₈	1	1	yes	characteristic
58	C₂²	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²	C₂×Q₁₆	1	1	yes	characteristic
61	C₄	—	4	4	no	—
62	C₄	—	8	2	no	—
63	C₄	#102	1	4	yes	—
64	C₄	#103	1	4	yes	—
65	C₄	C₄×Q₈	1	2	yes	—
66	C₄	C₂²⋊Q₈	1	2	yes	—
67	C₂	#391	1	1	yes	characteristic
68	C₂	C₂×#102	1	1	yes	characteristic
69	C₂	C₂×#103	1	1	yes	characteristic
70	C₂	#486	1	1	yes	characteristic
71	C₂	#487	1	1	yes	characteristic
72	C₂	#498	1	1	yes	characteristic
73	C₂	#500	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. C₄⋊Q₈	C₄	semidirect
26. 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
C₂×C₄⋊Q₈	2
C₂×C₄×C₈	2
#384	2
#340	2