Subgroups — #1476

Order 128.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41	C₂	1	1	yes	maximal, characteristic
2	#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
5	D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×#41	C₄	1	1	yes	characteristic
7	C₂²×C₈	—	2	1	no	—
8	C₄×Q₈	C₄	1	1	yes	characteristic
9	C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₄×D₄	—	2	1	no	—
11	C₄×D₄	—	2	1	no	—
12	C₄×D₄	C₄	1	1	yes	characteristic
13	C₄×D₄	C₂²	1	1	yes	characteristic
14	#117	—	2	1	no	—
15	#117	C₄	1	1	yes	characteristic
16	C₂×C₄²	—	2	1	no	—
17	C₂×C₄²	C₂²	1	1	yes	characteristic
18	C₄⋊C₈	C₂²	1	2	yes	—
19	C₂²⋊C₈	—	2	1	no	—
20	C₄×C₈	—	2	1	no	—
21	C₄×C₈	—	2	1	no	—
22	C₄×C₈	C₂²	1	2	yes	—
23	#41	—	2	2	no	—
24	#41	C₈	1	4	yes	complemented
25	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄	—	2	1	no	—
27	C₂×D₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄	—	2	1	no	—
29	C₂²×C₄	—	2	1	no	—
30	C₂²×C₄	—	2	1	no	—
31	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₄	D₄	1	1	yes	characteristic
34	C₂×C₈	—	2	4	no	—
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	—	2	2	no	—
37	C₄⋊C₄	—	2	1	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂²⋊C₄	—	2	1	no	—
42	C₂²⋊C₄	—	2	1	no	—
43	C₂²⋊C₄	—	2	1	no	—
44	C₄²	—	2	1	no	—
45	C₄²	—	2	1	no	—
46	C₄²	—	2	1	no	—
47	C₄²	C₂×C₄	1	1	yes	characteristic
48	C₄²	D₄	1	1	yes	characteristic
49	C₄²	D₄	1	1	yes	characteristic
50	C₄²	C₂³	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	C₂³	C₂²⋊C₄	1	1	yes	characteristic
53	Q₈	—	2	1	no	—
54	Q₈	C₂×C₈	1	2	yes	complemented
55	D₄	—	2	4	no	—
56	D₄	—	2	1	no	—
57	D₄	C₂×C₈	1	2	yes	complemented
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄	C₂×C₈	1	2	yes	—
70	C₂×C₄	M₁₆	1	2	yes	—
71	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄	C₂×D₄	1	1	yes	characteristic
74	C₈	—	2	4	no	—
75	C₈	—	4	2	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	C₂²⋊C₈	1	2	yes	—
82	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	C₂²⋊C₈	1	2	yes	—
89	C₄	#104	1	2	yes	—
90	C₄	C₂²×C₈	1	1	yes	characteristic
91	C₄	C₂×M₁₆	1	1	yes	characteristic
92	C₄	#135	1	2	yes	—
93	C₂	—	2	2	no	—
94	C₂	—	4	2	no	—
95	C₂	C₂×C₂²⋊C₈	1	1	yes	characteristic
96	C₂	#416	1	1	yes	characteristic
97	C₂	C₂×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄
commutator	C₄	C₂²×C₈
Frattini, cosocle	C₄²	C₂³
socle	C₂²	C₂×C₂²⋊C₄
prime radical	C₂×#41	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #431	C₂	semidirect
3. C₂×C₄×C₈	C₂	semidirect
4. Q₈⋊C₈	C₂	semidirect
5. D₄⋊C₈	C₂	semidirect
24. #41	C₈	semidirect
54. Q₈	C₂×C₈	semidirect
57. D₄	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#41	2
#431	2
C₂×C₄×C₈	2
Q₈⋊C₈	2
D₄⋊C₈	2