Subgroups — #1483

Order 128.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#529	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×M₁₆	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×M₁₆	—	2	1	no	—
7	C₂²×C₈	—	2	1	no	—
8	#126	—	2	1	no	—
9	#125	C₄	1	1	yes	characteristic
10	#124	C₄	1	1	yes	characteristic
11	#123	—	2	1	no	—
12	C₂²⋊Q₈	C₄	1	1	yes	characteristic
13	#121	C₄	1	1	yes	characteristic
14	C₄×Q₈	C₂²	1	1	yes	characteristic
15	C₄×D₄	—	2	1	no	—
16	C₄×D₄	C₂²	1	1	yes	characteristic
17	#117	—	2	1	no	—
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	C₄⋊C₈	—	2	1	no	—
20	C₄⋊C₈	C₂²	1	2	yes	—
21	#97	—	2	1	no	—
22	C₄×C₈	C₂²	1	2	yes	—
23	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄	—	2	1	no	—
25	C₂×D₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄	—	2	1	no	—
27	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₄	D₄	1	1	yes	characteristic
30	M₁₆	—	2	4	no	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	4	2	no	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	—	2	1	no	—
38	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₂²⋊C₄	—	2	1	no	—
41	C₂²⋊C₄	—	2	1	no	—
42	C₂²⋊C₄	—	2	1	no	—
43	C₂²⋊C₄	—	2	1	no	—
44	C₂²⋊C₄	—	2	1	no	—
45	C₄²	—	2	1	no	—
46	C₄²	C₂×C₄	1	1	yes	characteristic
47	C₄²	D₄	1	1	yes	characteristic
48	C₄²	D₄	1	1	yes	characteristic
49	C₄²	C₂³	1	1	yes	characteristic
50	C₂³	—	2	1	no	—
51	C₂³	C₂²⋊C₄	1	1	yes	characteristic
52	Q₈	—	2	2	no	—
53	D₄	—	2	2	no	—
54	D₄	—	4	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄	C₂×D₄	1	1	yes	characteristic
70	C₈	—	2	4	no	—
71	C₈	—	4	2	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	4	1	no	—
75	C₂²	—	4	1	no	—
76	C₂²	—	4	1	no	—
77	C₂²	#104	1	2	yes	—
78	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	#131	1	2	yes	—
87	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₄	#137	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	—	8	1	no	—
91	C₂	#408	1	1	yes	characteristic
92	C₂	#417	1	1	yes	characteristic
93	C₂	C₂×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₄²	C₂³
socle	C₂²	C₂×C₂²⋊C₄
prime radical	#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. C₂×C₄⋊C₈	C₂	semidirect
3. C₄×M₁₆	C₂	semidirect
4. Q₈⋊C₈	C₂	semidirect
5. D₄⋊C₈	C₂	semidirect

Maximal subgroups

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

subgroup	index
#529	2
C₂×C₄⋊C₈	2
C₄×M₁₆	2
Q₈⋊C₈	2
D₄⋊C₈	2