Subgroups — #1986

Order 128.

130 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	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#339	C₂	1	4	yes	maximal
5	C₂×M₁₆	C₂²	1	2	yes	—
6	C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	#125	C₄	1	2	yes	complemented
9	C₂²⋊Q₈	—	2	1	no	—
10	C₂²⋊Q₈	—	2	1	no	—
11	C₄×Q₈	—	2	1	no	—
12	C₄×Q₈	—	2	1	no	—
13	#117	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄	—	2	1	no	—
15	C₂×C₄²	C₂²	1	2	yes	—
16	C₂×C₄²	C₂²	1	1	yes	characteristic
17	#97	—	2	1	no	—
18	C₄×C₈	—	2	1	no	—
19	#95	—	2	1	no	—
20	C₂×Q₈	—	2	1	no	—
21	C₂×Q₈	—	2	1	no	—
22	C₂²×C₄	—	2	2	no	—
23	C₂²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₄	C₂³	1	1	yes	characteristic
26	M₁₆	—	2	2	no	—
27	M₁₆	Q₈	1	4	yes	complemented
28	C₂×C₈	—	2	2	no	—
29	C₄⋊C₄	—	2	4	no	—
30	C₄⋊C₄	—	2	2	no	—
31	C₄⋊C₄	—	2	2	no	—
32	C₄⋊C₄	—	2	1	no	—
33	C₄⋊C₄	—	2	1	no	—
34	C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂²⋊C₄	—	2	2	no	—
37	C₄²	—	2	2	no	—
38	C₄²	—	2	4	no	—
39	C₄²	C₂×C₄	1	1	yes	characteristic
40	C₄²	C₂×C₄	1	1	yes	characteristic
41	C₄²	D₄	1	1	yes	characteristic
42	C₄²	D₄	1	1	yes	characteristic
43	C₂³	C₂×D₄	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	4	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	—	4	2	no	—
52	C₂×C₄	—	4	2	no	—
53	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂×Q₈	1	2	yes	—
60	C₂×C₄	#41	1	2	yes	—
61	C₈	—	2	4	no	—
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²	#124	1	1	yes	characteristic
66	C₂²	C₄⋊Q₈	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	4	no	—
71	C₄	—	4	2	no	—
72	C₄	—	4	2	no	—
73	C₄	#104	1	2	yes	—
74	C₄	C₄×Q₈	1	2	yes	—
75	C₄	C₂²⋊Q₈	1	2	yes	—
76	C₂	—	2	1	no	—
77	C₂	—	2	1	no	—
78	C₂	#391	1	1	yes	characteristic
79	C₂	C₂×#104	1	1	yes	characteristic
80	C₂	#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₂×C₂²⋊C₄
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
6. C₄⋊Q₈	C₄	semidirect
7. C₄⋊Q₈	C₄	semidirect
8. #125	C₄	semidirect
27. M₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#533	2
C₄×M₁₆	2
C₄×C₄⋊C₄	2
#339	2