Finite Group Databasebeta

← C₂×#3292 #61103

Subgroups — C₂×#3292 #61103

Order 256.

1383 conjugacy classes of subgroups; only the 487 normal subgroups are shown.

normal subgroups only

This group has 1383 conjugacy classes of subgroups — too many to draw. Only its 487 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576	C₂	1	4	yes	maximal, complemented
2	#3292	C₂	1	16	yes	maximal, direct product factor
3	C₂×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#472	C₂	1	4	yes	maximal, complemented
5	C₂×#453	C₂	1	4	yes	maximal, complemented
6	C₂×#443	C₂	1	2	yes	maximal, complemented
7	C₂×H(5,𝔽₂)	C₂²	1	4	yes	complemented
8	#576	—	2	16	no	—
9	#576	C₂²	1	32	yes	complemented
10	C₂×AGL(1,ℤ/8)	—	2	8	no	—
11	C₂×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
12	C₂×#135	—	2	4	no	—
13	C₂×#135	C₂²	1	2	yes	—
14	C₂²×D₈	—	2	4	no	—
15	C₂²×D₈	C₂²	1	4	yes	complemented
16	C₂×#131	C₂²	1	4	yes	complemented
17	C₂×#127	C₂²	1	2	yes	complemented
18	#493	C₂²	1	8	yes	complemented
19	#472	C₂²	1	32	yes	complemented
20	#453	C₂²	1	32	yes	complemented
21	#443	C₂²	1	16	yes	complemented
22	C₂×#108	C₂²	1	2	yes	complemented
23	C₂×#104	C₂²	1	4	yes	complemented
24	C₂×#100	C₂²	1	4	yes	complemented
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
26	H(5,𝔽₂)	—	2	16	no	—
27	H(5,𝔽₂)	—	4	4	no	—
28	H(5,𝔽₂)	C₂³	1	16	yes	complemented
29	C₂×#41	—	2	8	no	—
30	C₂×#41	C₂³	1	4	yes	—
31	C₂²×D₄	—	2	8	no	—
32	C₂²×D₄	—	4	2	no	—
33	C₂²×D₄	—	4	4	no	—
34	C₂²×D₄	C₂³	1	4	yes	—
35	AGL(1,ℤ/8)	—	2	16	no	—
36	AGL(1,ℤ/8)	—	2	64	no	—
37	AGL(1,ℤ/8)	C₂³	1	32	yes	complemented
38	#135	—	2	32	no	—
39	#135	D₄	1	8	yes	complemented
40	#135	C₂³	1	8	yes	—
41	C₂×Q₁₆	D₄	1	2	yes	complemented
42	C₂×SD₁₆	—	2	8	no	—
43	C₂×SD₁₆	D₄	1	4	yes	complemented
44	C₂×D₈	—	2	8	no	—
45	C₂×D₈	—	2	32	no	—
46	C₂×D₈	—	2	4	no	—
47	C₂×D₈	—	2	4	no	—
48	C₂×D₈	—	2	8	no	—
49	C₂×D₈	D₄	1	2	yes	complemented
50	C₂×D₈	C₂³	1	16	yes	complemented
51	#131	—	2	8	no	—
52	#131	C₂³	1	16	yes	complemented
53	C₂×M₁₆	—	2	4	no	—
54	C₂×M₁₆	C₂³	1	4	yes	—
55	C₂²×C₈	—	2	4	no	—
56	C₂²×C₈	C₂³	1	2	yes	—
57	#127	—	4	2	no	—
58	#127	C₂³	1	8	yes	complemented
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	#108	C₂³	1	8	yes	complemented
61	#104	C₂³	1	16	yes	complemented
62	#100	C₂³	1	16	yes	complemented
63	C₄×C₈	C₂³	1	4	yes	complemented
64	C₂⁴	—	2	4	no	—
65	C₂⁴	—	4	4	no	—
66	#41	—	2	4	no	—
67	#41	—	2	4	no	—
68	#41	—	2	32	no	—
69	#41	—	4	16	no	—
70	#41	C₂×D₄	1	8	yes	—
71	#41	C₂⁴	1	8	yes	—
72	C₂×Q₈	C₂×D₄	1	4	yes	—
73	C₂×D₄	—	2	4	no	—
74	C₂×D₄	—	2	32	no	—
75	C₂×D₄	—	2	8	no	—
76	C₂×D₄	—	2	8	no	—
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	—	4	16	no	—
79	C₂×D₄	—	4	8	no	—
80	C₂×D₄	—	4	8	no	—
81	C₂×D₄	—	4	16	no	—
82	C₂×D₄	—	4	16	no	—
83	C₂×D₄	—	8	8	no	—
84	C₂×D₄	C₂×D₄	1	4	yes	—
85	C₂×D₄	C₂⁴	1	8	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	4	4	no	—
89	C₂²×C₄	C₂⁴	1	1	yes	characteristic
90	Q₁₆	C₂×D₄	1	8	yes	complemented
91	SD₁₆	—	2	32	no	—
92	SD₁₆	C₂×D₄	1	16	yes	complemented
93	D₈	—	2	32	no	—
94	D₈	—	2	16	no	—
95	D₈	—	2	16	no	—
96	D₈	C₂×D₄	1	8	yes	complemented
97	M₁₆	—	2	4	no	—
98	M₁₆	—	2	16	no	—
99	M₁₆	C₂⁴	1	8	yes	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	2	16	no	—
102	C₂×C₈	C₂×D₄	1	2	yes	—
103	C₂×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₈	C₂×D₄	1	4	yes	—
105	C₂×C₈	C₂⁴	1	4	yes	—
106	C₄²	—	2	1	no	—
107	C₄²	C₂⁴	1	2	yes	—
108	C₂³	—	2	4	no	—
109	C₂³	—	2	8	no	—
110	C₂³	—	4	16	no	—
111	C₂³	—	4	4	no	—
112	C₂³	—	4	4	no	—
113	C₂³	—	8	8	no	—
114	C₂³	—	16	8	no	—
115	C₂³	H(5,𝔽₂)	1	1	yes	characteristic
116	Q₈	—	2	4	no	—
117	Q₈	C₂²×D₄	1	8	yes	—
118	D₄	—	2	8	no	—
119	D₄	—	2	8	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	16	no	—
122	D₄	—	4	32	no	—
123	D₄	—	4	8	no	—
124	D₄	—	8	16	no	—
125	D₄	C₂²×D₄	1	8	yes	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	4	16	no	—
132	C₂×C₄	C₂²×D₄	1	2	yes	—
133	C₂×C₄	C₂²×D₄	1	2	yes	—
134	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₄	C₂⁵	1	1	yes	characteristic
136	C₈	—	2	8	no	—
137	C₈	C₂²×D₄	1	4	yes	—
138	C₈	C₂²×D₄	1	4	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	8	no	—
141	C₂²	—	4	8	no	—
142	C₂²	—	4	8	no	—
143	C₂²	—	8	8	no	—
144	C₂²	—	8	8	no	—
145	C₂²	—	16	4	no	—
146	C₂²	—	16	16	no	—
147	C₂²	D₄²	1	1	yes	characteristic
148	C₂²	D₄²	1	1	yes	characteristic
149	C₂²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
150	C₄	—	2	4	no	—
151	C₄	—	2	8	no	—
152	C₄	D₄²	1	2	yes	—
153	C₄	C₂³×D₄	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	4	8	no	—
157	C₂	—	8	8	no	—
158	C₂	#3292	1	2	yes	direct product factor
159	C₂	C₂×D₄²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#576	C₂	semidirect
2. #3292	C₂	direct (N × H)
3. C₂×#493	C₂	semidirect
4. C₂×#472	C₂	semidirect
5. C₂×#453	C₂	semidirect
6. C₂×#443	C₂	semidirect
7. C₂×H(5,𝔽₂)	C₂²	semidirect
9. #576	C₂²	semidirect
11. C₂×AGL(1,ℤ/8)	C₂²	semidirect
15. C₂²×D₈	C₂²	semidirect
16. C₂×#131	C₂²	semidirect
17. C₂×#127	C₂²	semidirect
18. #493	C₂²	semidirect
19. #472	C₂²	semidirect
20. #453	C₂²	semidirect
21. #443	C₂²	semidirect
22. C₂×#108	C₂²	semidirect
23. C₂×#104	C₂²	semidirect
24. C₂×#100	C₂²	semidirect
25. C₂×C₄×C₈	C₂²	semidirect
28. H(5,𝔽₂)	C₂³	semidirect
37. AGL(1,ℤ/8)	C₂³	semidirect
39. #135	D₄	semidirect
41. C₂×Q₁₆	D₄	semidirect
43. C₂×SD₁₆	D₄	semidirect
49. C₂×D₈	D₄	semidirect
50. C₂×D₈	C₂³	semidirect
52. #131	C₂³	semidirect
58. #127	C₂³	semidirect
60. #108	C₂³	semidirect
61. #104	C₂³	semidirect
62. #100	C₂³	semidirect
63. C₄×C₈	C₂³	semidirect
90. Q₁₆	C₂×D₄	semidirect
92. SD₁₆	C₂×D₄	semidirect
96. D₈	C₂×D₄	semidirect
158. C₂	#3292	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#576	2
#3292	2
C₂×#493	2
C₂×#472	2
C₂×#453	2
C₂×#443	2