Subgroups — #13285

Order 256.

175 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	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1848	C₂	1	1	yes	maximal, characteristic
4	#1382	C₂	1	4	yes	maximal
5	#533	—	2	1	no	—
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	#500	C₄	1	2	yes	—
8	#499	—	2	1	no	—
9	#499	C₄	1	2	yes	—
10	#498	—	2	1	no	—
11	C₂×#108	—	2	1	no	—
12	C₂×#108	C₂²	1	2	yes	—
13	#427	—	2	1	no	—
14	#427	C₂²	1	2	yes	—
15	#423	—	2	1	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₄×C₁₆	—	2	1	no	—
18	C₄×C₁₆	—	2	1	no	—
19	C₂×M₁₆	—	2	2	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	C₂³	1	1	yes	characteristic
22	C₄⋊Q₈	—	2	2	no	—
23	#125	—	2	2	no	—
24	C₂²⋊Q₈	—	4	2	no	—
25	C₄×Q₈	—	4	2	no	—
26	#117	—	2	2	no	—
27	C₂×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	—	2	2	no	—
30	C₂×C₁₆	Q₈	1	4	yes	complemented
31	#108	—	2	4	no	—
32	#108	—	4	2	no	—
33	#107	—	2	2	no	—
34	#107	C₂×C₄	1	4	yes	—
35	#106	—	2	2	no	—
36	#106	—	2	2	no	—
37	C₄⋊C₈	—	4	2	no	—
38	C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈	D₄	1	1	yes	characteristic
41	C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×Q₈	—	4	2	no	—
43	C₂²×C₄	SD₁₆	1	2	yes	—
44	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
45	M₁₆	—	4	2	no	—
46	M₁₆	—	4	2	no	—
47	C₂×C₈	—	4	2	no	—
48	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈	SD₁₆	1	2	yes	—
51	C₂×C₈	Q₁₆	1	2	yes	—
52	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈	C₂×Q₈	1	2	yes	—
55	C₂×C₈	#41	1	2	yes	—
56	C₄⋊C₄	—	2	4	no	—
57	C₄⋊C₄	—	4	4	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₂²⋊C₄	—	4	2	no	—
60	C₄²	—	4	2	no	—
61	C₄²	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²	D₈	1	2	yes	—
64	C₁₆	—	2	4	no	—
65	C₂³	C₂×SD₁₆	1	1	yes	characteristic
66	Q₈	—	8	4	no	—
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	—	4	4	no	—
69	C₂×C₄	#102	1	2	yes	—
70	C₂×C₄	#102	1	2	yes	—
71	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄	#124	1	1	yes	characteristic
73	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
77	C₈	—	8	2	no	—
78	C₈	#103	1	2	yes	—
79	C₈	#103	1	2	yes	—
80	C₈	C₄×Q₈	1	2	yes	—
81	C₈	C₂²⋊Q₈	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	C₂×#102	1	1	yes	characteristic
84	C₂²	#487	1	1	yes	characteristic
85	C₂²	#498	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	8	4	no	—
88	C₄	#359	1	4	yes	—
89	C₄	#391	1	1	yes	characteristic
90	C₄	C₂×#103	1	1	yes	characteristic
91	C₄	#486	1	1	yes	characteristic
92	C₄	#500	1	1	yes	characteristic
93	C₂	—	2	1	no	—
94	C₂	—	2	1	no	—
95	C₂	#1990	1	1	yes	characteristic
96	C₂	C₂×#359	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#102
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
30. C₂×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3160	2
C₂×C₄×C₁₆	2
#1848	2
#1382	2