Subgroups — #7844

Order 256.

259 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1744	C₂	1	2	yes	maximal
2	C₄×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×#322	C₂	1	2	yes	maximal
5	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
6	C₂×#97	C₄	1	4	yes	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	C₄³	C₂²	1	1	yes	characteristic
12	#336	C₄	1	8	yes	—
13	#322	C₄	1	8	yes	complemented
14	#322	C₄	1	4	yes	complemented
15	C₈²	C₄	1	4	yes	complemented
16	C₂²×C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄²	C₂×C₄	1	4	yes	—
21	C₂×C₄²	C₂×C₄	1	2	yes	—
22	C₂×C₄²	C₂³	1	1	yes	characteristic
23	#97	—	2	8	no	—
24	C₄×C₈	C₂×C₄	1	4	yes	—
25	C₄×C₈	C₂×C₄	1	2	yes	—
26	C₄×C₈	C₂×C₄	1	4	yes	—
27	C₄×C₈	C₂×C₄	1	4	yes	—
28	C₄×C₈	C₂×C₄	1	4	yes	—
29	C₄×C₈	C₂×C₄	1	4	yes	—
30	C₄×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₄	C₄²	1	4	yes	—
32	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄	C₂²×C₄	1	2	yes	—
34	C₂×C₈	—	2	4	no	—
35	C₂×C₈	—	2	8	no	—
36	C₂×C₈	—	2	8	no	—
37	C₂×C₈	C₄²	1	4	yes	—
38	C₂×C₈	C₄²	1	8	yes	—
39	C₂×C₈	C₄²	1	8	yes	—
40	C₂×C₈	C₄²	1	4	yes	—
41	C₂×C₈	M₁₆	1	8	yes	—
42	C₄²	—	2	4	no	—
43	C₄²	—	2	8	no	—
44	C₄²	C₂²×C₄	1	1	yes	characteristic
45	C₄²	C₂²×C₄	1	1	yes	characteristic
46	C₄²	C₂²×C₄	1	2	yes	—
47	C₂³	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄	—	2	4	no	—
49	C₂×C₄	—	2	8	no	—
50	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄	C₂×C₄²	1	2	yes	—
53	C₂×C₄	C₂×C₄²	1	2	yes	—
54	C₂×C₄	C₂×M₁₆	1	4	yes	—
55	C₂×C₄	#131	1	8	yes	—
56	C₈	—	2	4	no	—
57	C₈	—	2	8	no	—
58	C₈	—	2	8	no	—
59	C₈	#97	1	8	yes	—
60	C₂²	C₄³	1	1	yes	characteristic
61	C₂²	C₄×M₁₆	1	2	yes	—
62	C₂²	#405	1	4	yes	—
63	C₄	—	4	4	no	—
64	C₄	C₂×#97	1	2	yes	—
65	C₄	C₄×M₁₆	1	2	yes	—
66	C₄	#405	1	4	yes	—
67	C₄	#405	1	4	yes	—
68	C₂	#1453	1	4	yes	—
69	C₂	C₄×#97	1	1	yes	characteristic
70	C₂	#1726	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂³
commutator	C₂²	C₄³
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
13. #322	C₄	semidirect
14. #322	C₄	semidirect
15. C₈²	C₄	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
C₂×#322	2
C₂×C₈²	2