Finite Group Databasebeta

← C₂×#3414 #62631

Subgroups — C₂×#3414 #62631

Order 256.

1049 conjugacy classes of subgroups; only the 479 normal subgroups are shown.

normal subgroups only

This group has 1049 conjugacy classes of subgroups — too many to draw. Only its 479 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	C₂²×#135	C₂	1	2	yes	maximal, complemented
2	#3414	C₂	1	16	yes	maximal, direct product factor
3	C₂×#510	C₂	1	4	yes	maximal, complemented
4	C₂×#509	C₂	1	4	yes	maximal, complemented
5	C₂×#508	C₂	1	4	yes	maximal, complemented
6	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41	—	2	2	no	—
8	C₂×#135	—	2	2	no	—
9	C₂×#135	—	2	2	no	—
10	C₂×#135	—	2	8	no	—
11	C₂×#135	C₂²	1	4	yes	—
12	C₂×#135	C₂²	1	4	yes	complemented
13	C₂×#135	C₂²	1	16	yes	complemented
14	C₂²×Q₁₆	C₂²	1	2	yes	complemented
15	C₂²×SD₁₆	—	2	2	no	—
16	C₂²×D₈	C₂²	1	2	yes	complemented
17	C₂³×C₈	C₂²	1	1	yes	characteristic
18	#510	C₂²	1	32	yes	complemented
19	#509	C₂²	1	32	yes	complemented
20	#508	C₂²	1	32	yes	complemented
21	C₂×Q₃₂	C₂²	1	4	yes	complemented
22	C₂×SD₃₂	C₂²	1	8	yes	complemented
23	C₂×D₁₆	C₂²	1	4	yes	complemented
24	C₂×M₃₂	C₂²	1	4	yes	complemented
25	C₂×M₃₂	C₂²	1	8	yes	complemented
26	C₂²×C₁₆	C₂²	1	2	yes	complemented
27	C₂×#41	—	2	4	no	—
28	C₂×#41	—	2	16	no	—
29	C₂×#41	—	2	4	no	—
30	C₂×#41	—	4	2	no	—
31	C₂×#41	—	4	8	no	—
32	C₂×#41	—	4	2	no	—
33	C₂²×Q₈	—	2	2	no	—
34	C₂²×D₄	—	2	2	no	—
35	C₂²×D₄	—	4	2	no	—
36	C₂³×C₄	—	4	2	no	—
37	C₂³×C₄	D₄	1	1	yes	characteristic
38	#135	—	2	8	no	—
39	#135	—	2	16	no	—
40	#135	—	2	8	no	—
41	#135	—	2	16	no	—
42	#135	C₂³	1	16	yes	—
43	#135	C₂³	1	16	yes	complemented
44	C₂×Q₁₆	—	2	2	no	—
45	C₂×Q₁₆	—	2	4	no	—
46	C₂×Q₁₆	C₂³	1	4	yes	—
47	C₂×Q₁₆	C₂³	1	8	yes	complemented
48	C₂×SD₁₆	—	2	4	no	—
49	C₂×SD₁₆	—	2	4	no	—
50	C₂×SD₁₆	—	2	16	no	—
51	C₂×D₈	—	2	2	no	—
52	C₂×D₈	—	2	4	no	—
53	C₂×D₈	C₂³	1	4	yes	—
54	C₂×D₈	C₂³	1	8	yes	complemented
55	C₂²×C₈	D₄	1	2	yes	—
56	C₂²×C₈	D₄	1	1	yes	characteristic
57	C₂²×C₈	D₄	1	4	yes	—
58	C₂²×C₈	C₂³	1	2	yes	—
59	C₂²×C₈	C₂³	1	1	yes	characteristic
60	C₂²×C₈	C₂³	1	4	yes	—
61	Q₃₂	C₂³	1	16	yes	complemented
62	SD₃₂	C₂³	1	32	yes	complemented
63	D₁₆	C₂³	1	16	yes	complemented
64	M₃₂	C₂³	1	16	yes	complemented
65	C₂×C₁₆	C₂³	1	4	yes	—
66	C₂×C₁₆	C₂³	1	8	yes	complemented
67	C₂⁴	—	4	2	no	—
68	C₂⁴	D₈	1	1	yes	characteristic
69	#41	—	2	16	no	—
70	#41	—	2	16	no	—
71	#41	—	4	16	no	—
72	#41	—	4	8	no	—
73	#41	—	4	8	no	—
74	#41	—	4	16	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	8	no	—
77	C₂×Q₈	—	4	2	no	—
78	C₂×Q₈	—	4	4	no	—
79	C₂×D₄	—	2	4	no	—
80	C₂×D₄	—	2	8	no	—
81	C₂×D₄	—	4	4	no	—
82	C₂×D₄	—	4	4	no	—
83	C₂×D₄	—	4	16	no	—
84	C₂×D₄	—	4	2	no	—
85	C₂×D₄	—	4	4	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	4	no	—
88	C₂²×C₄	—	4	16	no	—
89	C₂²×C₄	—	4	4	no	—
90	C₂²×C₄	D₈	1	1	yes	characteristic
91	C₂²×C₄	D₈	1	4	yes	—
92	C₂²×C₄	D₈	1	2	yes	—
93	C₂²×C₄	C₂×D₄	1	2	yes	—
94	C₂²×C₄	C₂×D₄	1	4	yes	—
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	Q₁₆	—	2	4	no	—
97	Q₁₆	—	2	8	no	—
98	Q₁₆	C₂⁴	1	8	yes	—
99	SD₁₆	—	2	16	no	—
100	SD₁₆	—	2	16	no	—
101	D₈	—	2	4	no	—
102	D₈	—	2	8	no	—
103	D₈	C₂⁴	1	8	yes	—
104	C₂×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₈	C₂×D₄	1	4	yes	—
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	8	yes	—
108	C₂×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₈	C₂×D₄	1	4	yes	—
110	C₂×C₈	C₂⁴	1	1	yes	characteristic
111	C₂×C₈	C₂⁴	1	2	yes	—
112	C₂×C₈	C₂⁴	1	4	yes	—
113	C₁₆	C₂⁴	1	8	yes	—
114	C₂³	—	4	2	no	—
115	C₂³	—	4	8	no	—
116	C₂³	—	4	4	no	—
117	C₂³	—	8	8	no	—
118	C₂³	C₂×D₈	1	2	yes	—
119	C₂³	C₂×D₈	1	1	yes	characteristic
120	C₂³	C₂×D₈	1	4	yes	—
121	Q₈	—	2	8	no	—
122	Q₈	—	4	8	no	—
123	Q₈	—	4	4	no	—
124	D₄	—	2	8	no	—
125	D₄	—	4	16	no	—
126	D₄	—	4	16	no	—
127	D₄	—	4	8	no	—
128	D₄	—	4	4	no	—
129	C₂×C₄	—	4	8	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	4	16	no	—
132	C₂×C₄	—	4	16	no	—
133	C₂×C₄	C₂×D₈	1	2	yes	—
134	C₂×C₄	C₂×D₈	1	8	yes	—
135	C₂×C₄	C₂×D₈	1	4	yes	—
136	C₂×C₄	C₂×D₈	1	2	yes	—
137	C₂×C₄	C₂×D₈	1	4	yes	—
138	C₂×C₄	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄	C₂²×D₄	1	4	yes	—
141	C₂×C₄	C₂²×D₄	1	2	yes	—
142	C₈	C₂²×D₄	1	1	yes	characteristic
143	C₈	C₂²×D₄	1	4	yes	—
144	C₈	C₂²×D₄	1	2	yes	—
145	C₈	C₂⁵	1	1	yes	characteristic
146	C₂²	—	2	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	8	no	—
150	C₂²	—	8	16	no	—
151	C₂²	—	8	8	no	—
152	C₂²	C₂²×D₈	1	4	yes	—
153	C₂²	C₂²×D₈	1	2	yes	—
154	C₂²	C₂²×D₈	1	1	yes	characteristic
155	C₄	—	4	8	no	—
156	C₄	C₂²×D₈	1	2	yes	—
157	C₄	C₂²×D₈	1	4	yes	—
158	C₄	C₂²×D₈	1	1	yes	characteristic
159	C₄	C₂³×D₄	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	—	2	2	no	—
162	C₂	—	8	8	no	—
163	C₂	#3414	1	2	yes	direct product factor
164	C₂	C₂³×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁵
socle	C₂²	C₂²×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#135	C₂	semidirect
2. #3414	C₂	direct (N × H)
3. C₂×#510	C₂	semidirect
4. C₂×#509	C₂	semidirect
5. C₂×#508	C₂	semidirect
6. C₂²×M₃₂	C₂	semidirect
12. C₂×#135	C₂²	semidirect
13. C₂×#135	C₂²	semidirect
14. C₂²×Q₁₆	C₂²	semidirect
16. C₂²×D₈	C₂²	semidirect
18. #510	C₂²	semidirect
19. #509	C₂²	semidirect
20. #508	C₂²	semidirect
21. C₂×Q₃₂	C₂²	semidirect
22. C₂×SD₃₂	C₂²	semidirect
23. C₂×D₁₆	C₂²	semidirect
24. C₂×M₃₂	C₂²	semidirect
25. C₂×M₃₂	C₂²	semidirect
26. C₂²×C₁₆	C₂²	semidirect
43. #135	C₂³	semidirect
47. C₂×Q₁₆	C₂³	semidirect
54. C₂×D₈	C₂³	semidirect
61. Q₃₂	C₂³	semidirect
62. SD₃₂	C₂³	semidirect
63. D₁₆	C₂³	semidirect
64. M₃₂	C₂³	semidirect
66. C₂×C₁₆	C₂³	semidirect
163. C₂	#3414	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#135	2
#3414	2
C₂×#510	2
C₂×#509	2
C₂×#508	2
C₂²×M₃₂	2