Subgroups — #8132

Order 256.

239 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1744	C₂	1	1	yes	maximal, characteristic
2	C₄×#97	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	#1273	C₂	1	4	yes	maximal
5	C₂×#97	C₄	1	2	yes	—
6	C₂×#97	C₄	1	2	yes	—
7	C₂×C₄×C₈	C₄	1	4	yes	—
8	C₂×C₄×C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	C₄³	C₂²	1	1	yes	characteristic
12	#336	C₄	1	4	yes	—
13	C₂²×C₈	C₂×C₄	1	4	yes	—
14	C₂²×C₈	C₂×C₄	1	4	yes	—
15	C₂²×C₈	C₂×C₄	1	4	yes	—
16	C₂×C₄²	C₂×C₄	1	2	yes	—
17	C₂×C₄²	C₂×C₄	1	2	yes	—
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²	C₂³	1	1	yes	characteristic
21	#97	—	2	4	no	—
22	#97	C₈	1	8	yes	complemented
23	C₄×C₈	—	2	8	no	—
24	C₄×C₈	—	2	4	no	—
25	C₄×C₈	—	2	4	no	—
26	C₄×C₈	—	2	4	no	—
27	C₂²×C₄	C₄²	1	2	yes	—
28	C₂²×C₄	C₄²	1	2	yes	—
29	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈	—	2	4	no	—
33	C₂×C₈	—	2	4	no	—
34	C₂×C₈	—	2	4	no	—
35	C₂×C₈	—	2	4	no	—
36	C₂×C₈	—	2	4	no	—
37	C₂×C₈	—	2	8	no	—
38	C₂×C₈	C₂×C₈	1	8	yes	—
39	C₂×C₈	M₁₆	1	8	yes	—
40	C₄²	—	2	4	no	—
41	C₄²	—	2	4	no	—
42	C₄²	C₂×C₈	1	4	yes	—
43	C₄²	M₁₆	1	4	yes	—
44	C₄²	#41	1	4	yes	—
45	C₂³	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	C₄×C₈	1	4	yes	—
51	C₂×C₄	#97	1	4	yes	—
52	C₂×C₄	#117	1	2	yes	—
53	C₂×C₄	#117	1	2	yes	—
54	C₂×C₄	#117	1	2	yes	—
55	C₂×C₄	C₂²×C₈	1	2	yes	—
56	C₂×C₄	C₂×M₁₆	1	2	yes	—
57	C₂×C₄	C₂×M₁₆	1	4	yes	—
58	C₂×C₄	#131	1	4	yes	—
59	C₈	—	2	8	no	—
60	C₈	—	2	8	no	—
61	C₈	—	4	4	no	—
62	C₂²	#376	1	1	yes	characteristic
63	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂²	C₂×#97	1	1	yes	characteristic
65	C₂²	C₄×M₁₆	1	2	yes	—
66	C₂²	#405	1	2	yes	—
67	C₄	—	2	4	no	—
68	C₄	—	2	4	no	—
69	C₄	#431	1	4	yes	—
70	C₄	#432	1	4	yes	—
71	C₄	#433	1	4	yes	—
72	C₂	C₈×M₁₆	1	2	yes	—
73	C₂	#1451	1	2	yes	—
74	C₂	#1744	1	1	yes	characteristic
75	C₂	#1745	1	1	yes	characteristic
76	C₂	#1746	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
22. #97	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1744	2
C₄×#97	2
C₄²×C₈	2
#1273	2