Finite Group Databasebeta

← C₂×#3416 #62633

Subgroups — C₂×#3416 #62633

Order 256.

1037 conjugacy classes of subgroups; only the 479 normal subgroups are shown.

normal subgroups only

This group has 1037 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	C₂×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#576	C₂	1	1	yes	maximal, characteristic, complemented
3	#3416	C₂	1	16	yes	maximal, direct product factor
4	C₂×#510	C₂	1	3	yes	maximal, complemented
5	C₂×#509	C₂	1	3	yes	maximal, complemented
6	C₂×#508	C₂	1	3	yes	maximal, complemented
7	C₂²×SD₃₂	C₂	1	3	yes	maximal, complemented
8	C₂×#504	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×2₋¹⁺⁴	—	2	1	no	—
10	C₂×H(5,𝔽₂)	—	2	1	no	—
11	#578	—	2	4	no	—
12	#578	C₂²	1	8	yes	complemented
13	#576	—	2	4	no	—
14	#576	C₂²	1	8	yes	complemented
15	C₂×#137	—	2	3	no	—
16	C₂×AGL(1,ℤ/8)	—	2	3	no	—
17	C₂×#135	C₂²	1	3	yes	—
18	C₂×#135	C₂²	1	3	yes	complemented
19	C₂²×Q₁₆	C₂²	1	3	yes	complemented
20	C₂²×D₈	C₂²	1	3	yes	—
21	C₂×#131	C₂²	1	1	yes	characteristic
22	#510	C₂²	1	24	yes	complemented
23	#509	C₂²	1	24	yes	complemented
24	#508	C₂²	1	24	yes	complemented
25	C₂×Q₃₂	C₂²	1	3	yes	complemented
26	C₂×SD₃₂	C₂²	1	3	yes	complemented
27	C₂×SD₃₂	C₂²	1	6	yes	complemented
28	C₂×SD₃₂	C₂²	1	1	yes	characteristic, complemented
29	C₂×SD₃₂	C₂²	1	24	yes	complemented
30	C₂×D₁₆	C₂²	1	3	yes	complemented
31	#504	C₂²	1	8	yes	complemented
32	C₂×M₃₂	C₂²	1	3	yes	complemented
33	C₂²×C₁₆	C₂²	1	3	yes	complemented
34	2₋¹⁺⁴	—	2	8	no	—
35	2₋¹⁺⁴	—	4	4	no	—
36	H(5,𝔽₂)	—	2	8	no	—
37	H(5,𝔽₂)	—	4	4	no	—
38	C₂×#41	—	2	3	no	—
39	C₂×#41	—	2	3	no	—
40	C₂×#41	—	4	1	no	—
41	C₂×#41	—	4	3	no	—
42	C₂×#41	D₄	1	1	yes	characteristic
43	C₂²×Q₈	—	2	3	no	—
44	C₂²×Q₈	—	4	1	no	—
45	C₂²×D₄	—	2	3	no	—
46	C₂²×D₄	—	4	3	no	—
47	#137	—	2	24	no	—
48	AGL(1,ℤ/8)	—	2	24	no	—
49	#135	—	2	6	no	—
50	#135	—	2	6	no	—
51	#135	C₂³	1	12	yes	—
52	#135	C₂³	1	12	yes	complemented
53	C₂×Q₁₆	—	2	3	no	—
54	C₂×Q₁₆	—	2	6	no	—
55	C₂×Q₁₆	C₂³	1	3	yes	—
56	C₂×Q₁₆	C₂³	1	1	yes	characteristic
57	C₂×Q₁₆	C₂³	1	12	yes	complemented
58	C₂×SD₁₆	—	2	3	no	—
59	C₂×SD₁₆	—	2	3	no	—
60	C₂×D₈	—	2	3	no	—
61	C₂×D₈	—	2	6	no	—
62	C₂×D₈	C₂³	1	3	yes	—
63	C₂×D₈	C₂³	1	12	yes	—
64	C₂×D₈	C₂³	1	1	yes	characteristic
65	#131	D₄	1	4	yes	—
66	#131	C₂³	1	4	yes	—
67	C₂×M₁₆	D₄	1	3	yes	—
68	C₂²×C₈	C₂³	1	3	yes	—
69	Q₃₂	C₂³	1	12	yes	complemented
70	SD₃₂	C₂³	1	12	yes	complemented
71	SD₃₂	C₂³	1	24	yes	complemented
72	SD₃₂	C₂³	1	4	yes	complemented
73	D₁₆	C₂³	1	12	yes	complemented
74	M₃₂	C₂³	1	12	yes	complemented
75	C₂×C₁₆	C₂³	1	1	yes	characteristic
76	C₂×C₁₆	C₂³	1	3	yes	—
77	C₂×C₁₆	C₂³	1	12	yes	complemented
78	C₂⁴	—	4	3	no	—
79	#41	—	2	12	no	—
80	#41	—	2	12	no	—
81	#41	—	4	8	no	—
82	#41	—	4	24	no	—
83	#41	—	4	6	no	—
84	#41	—	4	6	no	—
85	#41	D₈	1	4	yes	—
86	#41	C₂×D₄	1	4	yes	—
87	C₂×Q₈	—	2	1	no	—
88	C₂×Q₈	—	2	3	no	—
89	C₂×Q₈	—	2	12	no	—
90	C₂×Q₈	—	4	8	no	—
91	C₂×Q₈	—	4	3	no	—
92	C₂×Q₈	—	4	6	no	—
93	C₂×Q₈	D₈	1	1	yes	characteristic
94	C₂×D₄	—	2	3	no	—
95	C₂×D₄	—	2	12	no	—
96	C₂×D₄	—	2	1	no	—
97	C₂×D₄	—	4	3	no	—
98	C₂×D₄	—	4	3	no	—
99	C₂×D₄	—	4	24	no	—
100	C₂×D₄	—	4	3	no	—
101	C₂×D₄	—	4	6	no	—
102	C₂×D₄	D₈	1	3	yes	—
103	C₂²×C₄	—	4	3	no	—
104	C₂²×C₄	—	4	3	no	—
105	C₂²×C₄	—	4	3	no	—
106	C₂²×C₄	C₂×D₄	1	3	yes	—
107	Q₁₆	—	2	3	no	—
108	Q₁₆	—	2	12	no	—
109	Q₁₆	—	2	1	no	—
110	Q₁₆	C₂⁴	1	6	yes	—
111	Q₁₆	C₂⁴	1	2	yes	—
112	SD₁₆	—	2	12	no	—
113	SD₁₆	—	2	12	no	—
114	D₈	—	2	3	no	—
115	D₈	—	2	12	no	—
116	D₈	—	2	1	no	—
117	D₈	C₂⁴	1	2	yes	—
118	D₈	C₂⁴	1	6	yes	—
119	M₁₆	C₂×D₄	1	12	yes	—
120	C₂×C₈	C₂×D₄	1	3	yes	—
121	C₂×C₈	C₂×D₄	1	6	yes	—
122	C₂×C₈	C₂⁴	1	1	yes	characteristic
123	C₂×C₈	C₂⁴	1	6	yes	—
124	C₁₆	C₂⁴	1	2	yes	—
125	C₁₆	C₂⁴	1	6	yes	—
126	C₂³	—	4	1	no	—
127	C₂³	—	4	3	no	—
128	C₂³	—	4	12	no	—
129	C₂³	—	16	6	no	—
130	C₂³	C₂×D₈	1	3	yes	—
131	Q₈	—	2	2	no	—
132	Q₈	—	2	6	no	—
133	Q₈	—	4	12	no	—
134	Q₈	—	4	3	no	—
135	Q₈	—	4	1	no	—
136	Q₈	C₂×D₈	1	4	yes	—
137	D₄	—	2	6	no	—
138	D₄	—	2	2	no	—
139	D₄	—	4	12	no	—
140	D₄	—	4	12	no	—
141	D₄	—	4	12	no	—
142	D₄	—	4	3	no	—
143	D₄	—	4	1	no	—
144	D₄	C₂×D₈	1	12	yes	—
145	C₂×C₄	—	4	12	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	3	no	—
148	C₂×C₄	—	4	12	no	—
149	C₂×C₄	—	4	12	no	—
150	C₂×C₄	C₂×D₈	1	6	yes	—
151	C₂×C₄	C₂×D₈	1	3	yes	—
152	C₂×C₄	C₂²×D₄	1	6	yes	—
153	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₈	C₂²×D₄	1	1	yes	characteristic
155	C₈	C₂²×D₄	1	6	yes	—
156	C₈	C₂⁵	1	1	yes	characteristic
157	C₂²	—	2	6	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	6	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	6	no	—
162	C₂²	—	16	12	no	—
163	C₂²	C₂²×D₈	1	6	yes	—
164	C₂²	C₂²×D₈	1	1	yes	characteristic
165	C₄	—	4	2	no	—
166	C₄	—	4	6	no	—
167	C₄	C₂²×D₈	1	6	yes	—
168	C₄	C₂²×D₈	1	1	yes	characteristic
169	C₄	C₂³×D₄	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	—	8	2	no	—
172	C₂	—	8	6	no	—
173	C₂	#3416	1	2	yes	direct product factor
174	C₂	C₂³×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁵
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#578	C₂	semidirect
2. C₂×#576	C₂	semidirect
3. #3416	C₂	direct (N × H)
4. C₂×#510	C₂	semidirect
5. C₂×#509	C₂	semidirect
6. C₂×#508	C₂	semidirect
7. C₂²×SD₃₂	C₂	semidirect
8. C₂×#504	C₂	semidirect
12. #578	C₂²	semidirect
14. #576	C₂²	semidirect
18. C₂×#135	C₂²	semidirect
19. C₂²×Q₁₆	C₂²	semidirect
22. #510	C₂²	semidirect
23. #509	C₂²	semidirect
24. #508	C₂²	semidirect
25. C₂×Q₃₂	C₂²	semidirect
26. C₂×SD₃₂	C₂²	semidirect
27. C₂×SD₃₂	C₂²	semidirect
28. C₂×SD₃₂	C₂²	semidirect
29. C₂×SD₃₂	C₂²	semidirect
30. C₂×D₁₆	C₂²	semidirect
31. #504	C₂²	semidirect
32. C₂×M₃₂	C₂²	semidirect
33. C₂²×C₁₆	C₂²	semidirect
52. #135	C₂³	semidirect
57. C₂×Q₁₆	C₂³	semidirect
69. Q₃₂	C₂³	semidirect
70. SD₃₂	C₂³	semidirect
71. SD₃₂	C₂³	semidirect
72. SD₃₂	C₂³	semidirect
73. D₁₆	C₂³	semidirect
74. M₃₂	C₂³	semidirect
77. C₂×C₁₆	C₂³	semidirect
173. C₂	#3416	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×#578	2
C₂×#576	2
#3416	2
C₂×#510	2
C₂×#509	2
C₂×#508	2
C₂²×SD₃₂	2
C₂×#504	2