Subgroups — #22258

Order 256.

289 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160	C₂	1	1	yes	maximal, characteristic
2	#2961	C₂	1	2	yes	maximal
3	#1995	C₂	1	2	yes	maximal
4	#1922	C₂	1	8	yes	maximal
5	#1837	C₂	1	1	yes	maximal, characteristic
6	#1746	C₂	1	1	yes	maximal, characteristic
7	#533	C₂²	1	2	yes	—
8	#500	C₄	1	2	yes	complemented
9	#499	C₄	1	4	yes	complemented
10	#498	C₄	1	2	yes	complemented
11	#446	—	2	8	no	—
12	#427	C₂²	1	4	yes	—
13	#424	C₂²	1	4	yes	—
14	#423	C₂²	1	2	yes	—
15	#405	C₂²	1	4	yes	—
16	C₂×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	#376	C₂²	1	1	yes	characteristic
19	#341	C₂²	1	4	yes	—
20	#339	C₂²	1	8	yes	—
21	#336	C₂²	1	4	yes	—
22	C₂×M₁₆	C₂³	1	4	yes	—
23	C₂²×C₈	C₂³	1	2	yes	—
24	C₂²×C₈	C₂³	1	2	yes	—
25	C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	#125	C₂×C₄	1	4	yes	—
27	C₂²⋊Q₈	—	2	4	no	—
28	C₄×Q₈	—	2	4	no	—
29	#117	C₂³	1	4	yes	—
30	C₂×C₄²	C₂³	1	2	yes	—
31	C₂×C₄²	C₂³	1	1	yes	characteristic
32	#107	C₂×C₄	1	8	yes	—
33	#106	C₂×C₄	1	8	yes	—
34	C₄⋊C₈	—	2	8	no	—
35	C₄⋊C₈	—	2	4	no	—
36	#97	—	2	4	no	—
37	#97	D₄	1	4	yes	—
38	C₄×C₈	—	2	4	no	—
39	C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	#95	—	2	2	no	—
43	C₂×Q₈	—	2	4	no	—
44	C₂²×C₄	—	2	2	no	—
45	C₂²×C₄	#41	1	2	yes	—
46	C₂²×C₄	C₂⁴	1	1	yes	characteristic
47	M₁₆	—	2	8	no	—
48	C₂×C₈	—	2	4	no	—
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	C₂²×C₄	1	2	yes	—
52	C₂×C₈	C₂²×C₄	1	2	yes	—
53	C₂×C₈	C₂×D₄	1	4	yes	—
54	C₄⋊C₄	—	2	8	no	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	C₂²×C₄	1	8	yes	—
57	C₂²⋊C₄	—	2	4	no	—
58	C₄²	—	2	4	no	—
59	C₄²	—	2	4	no	—
60	C₄²	C₂²×C₄	1	1	yes	characteristic
61	C₄²	C₂²×C₄	1	1	yes	characteristic
62	C₄²	C₂×D₄	1	2	yes	—
63	C₂³	C₂×#41	1	1	yes	characteristic
64	Q₈	—	8	4	no	—
65	C₂×C₄	—	2	8	no	—
66	C₂×C₄	—	4	2	no	—
67	C₂×C₄	—	4	4	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	C₄×D₄	1	2	yes	—
70	C₂×C₄	C₄×D₄	1	2	yes	—
71	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄	2₋¹⁺⁴	1	4	yes	—
74	C₈	—	2	2	no	—
75	C₈	—	2	2	no	—
76	C₈	—	2	4	no	—
77	C₈	—	4	4	no	—
78	C₂²	—	2	2	no	—
79	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
80	C₂²	#541	1	2	yes	—
81	C₄	—	2	2	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	8	no	—
84	C₄	—	8	2	no	—
85	C₄	#519	1	2	yes	—
86	C₄	#536	1	2	yes	—
87	C₂	—	2	2	no	—
88	C₂	#2327	1	1	yes	characteristic
89	C₂	#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂³	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. #500	C₄	semidirect
9. #499	C₄	semidirect
10. #498	C₄	semidirect

Maximal subgroups

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

subgroup	index
#3160	2
#2961	2
#1995	2
#1922	2
#1837	2
#1746	2