Subgroups — C₂×#3428 #62648

Order 256.

5767 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41	C₂	1	1	yes	maximal, characteristic
2	#3428	C₂	1	32	yes	maximal, direct product factor
3	C₂×#520	C₂	1	8	yes	maximal, complemented
4	C₂×#519	C₂	1	2	yes	maximal, complemented
5	C₂×#518	C₂	1	6	yes	maximal, complemented
6	C₂×C₄×#41	C₂	1	8	yes	maximal, complemented
7	C₂²×#117	C₂	1	6	yes	maximal, complemented
8	C₂²×#41	C₄	1	32	yes	complemented
9	C₂²×#41	C₂²	1	4	yes	—
10	C₂²×#41	C₂²	1	4	yes	—
11	C₂²×#41	C₂²	1	16	yes	—
12	C₂³×Q₈	C₂²	1	1	yes	characteristic
13	C₂³×D₄	C₂²	1	3	yes	—
14	C₂⁴×C₄	C₂²	1	3	yes	—
15	#520	C₂²	1	128	yes	complemented
16	#519	C₂²	1	32	yes	complemented
17	#518	C₂²	1	96	yes	complemented
18	C₄×#41	C₂²	1	128	yes	complemented
19	C₂×C₄×Q₈	C₂²	1	16	yes	complemented
20	C₂×C₄×D₄	C₂²	1	48	yes	complemented
21	C₂×#117	C₂²	1	4	yes	—
22	C₂×#117	C₂²	1	12	yes	—
23	C₂×#117	C₂²	1	12	yes	complemented
24	C₂×#117	C₂²	1	12	yes	complemented
25	C₂×#117	C₂²	1	96	yes	complemented
26	C₂²×C₄⋊C₄	C₂²	1	12	yes	complemented
27	C₂²×C₂²⋊C₄	C₂²	1	12	yes	—
28	C₂²×C₄²	C₂²	1	12	yes	complemented
29	C₂⁵	C₂³	1	3	yes	—
30	C₂×#41	C₂×C₄	1	64	yes	complemented
31	C₂×#41	C₂×C₄	1	64	yes	complemented
32	C₂×#41	C₂×C₄	1	256	yes	complemented
33	C₂×#41	C₂³	1	32	yes	—
34	C₂×#41	C₂³	1	32	yes	—
35	C₂²×Q₈	C₂×C₄	1	16	yes	complemented
36	C₂²×Q₈	C₂³	1	4	yes	—
37	C₂²×Q₈	C₂³	1	8	yes	—
38	C₂²×D₄	C₂×C₄	1	48	yes	complemented
39	C₂²×D₄	C₂³	1	12	yes	—
40	C₂²×D₄	C₂³	1	24	yes	—
41	C₂³×C₄	C₂×C₄	1	48	yes	complemented
42	C₂³×C₄	C₂³	1	24	yes	—
43	C₂³×C₄	C₂³	1	6	yes	—
44	C₂³×C₄	C₂³	1	3	yes	—
45	C₂³×C₄	C₂³	1	1	yes	characteristic
46	C₂³×C₄	C₂³	1	6	yes	—
47	C₂³×C₄	C₂³	1	24	yes	—
48	C₄×Q₈	C₂³	1	128	yes	complemented
49	C₄×D₄	C₂³	1	384	yes	complemented
50	#117	C₂³	1	32	yes	—
51	#117	C₂³	1	96	yes	—
52	#117	C₂³	1	96	yes	complemented
53	#117	C₂³	1	96	yes	complemented
54	C₂×C₄⋊C₄	C₂³	1	8	yes	—
55	C₂×C₄⋊C₄	C₂³	1	24	yes	—
56	C₂×C₄⋊C₄	C₂³	1	96	yes	complemented
57	C₂×C₂²⋊C₄	C₂³	1	8	yes	—
58	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
59	C₂×C₂²⋊C₄	C₂³	1	96	yes	—
60	C₂×C₄²	C₂³	1	24	yes	—
61	C₂×C₄²	C₂³	1	8	yes	—
62	C₂×C₄²	C₂³	1	96	yes	complemented
63	C₂⁴	—	4	12	no	—
64	C₂⁴	C₂²×C₄	1	24	yes	—
65	C₂⁴	C₂⁴	1	1	yes	characteristic
66	C₂⁴	C₂⁴	1	6	yes	—
67	C₂⁴	C₂⁴	1	12	yes	—
68	#41	C₂²×C₄	1	256	yes	complemented
69	#41	C₂²×C₄	1	256	yes	complemented
70	C₂×Q₈	C₂²×C₄	1	32	yes	—
71	C₂×Q₈	C₂²×C₄	1	64	yes	complemented
72	C₂×Q₈	C₂⁴	1	16	yes	—
73	C₂×D₄	C₂²×C₄	1	96	yes	—
74	C₂×D₄	C₂²×C₄	1	192	yes	complemented
75	C₂×D₄	C₂⁴	1	48	yes	—
76	C₂²×C₄	—	4	48	no	—
77	C₂²×C₄	C₂²×C₄	1	8	yes	—
78	C₂²×C₄	C₂²×C₄	1	24	yes	—
79	C₂²×C₄	C₂²×C₄	1	48	yes	—
80	C₂²×C₄	C₂²×C₄	1	48	yes	—
81	C₂²×C₄	C₂²×C₄	1	192	yes	complemented
82	C₂²×C₄	C₂⁴	1	16	yes	—
83	C₂²×C₄	C₂⁴	1	96	yes	—
84	C₂²×C₄	C₂⁴	1	6	yes	—
85	C₂²×C₄	C₂⁴	1	24	yes	—
86	C₂²×C₄	C₂⁴	1	1	yes	characteristic
87	C₂²×C₄	C₂⁴	1	24	yes	—
88	C₂²×C₄	C₂⁴	1	4	yes	—
89	C₂²×C₄	C₂⁴	1	12	yes	—
90	C₂²×C₄	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄	C₂⁴	1	32	yes	—
92	C₄⋊C₄	C₂⁴	1	96	yes	—
93	C₂²⋊C₄	C₂⁴	1	32	yes	—
94	C₂²⋊C₄	C₂⁴	1	96	yes	—
95	C₄²	C₂⁴	1	32	yes	—
96	C₄²	C₂⁴	1	96	yes	—
97	C₂³	—	2	24	no	—
98	C₂³	—	2	4	no	—
99	C₂³	—	4	48	no	—
100	C₂³	—	4	24	no	—
101	C₂³	C₂³×C₄	1	4	yes	—
102	C₂³	C₂³×C₄	1	24	yes	—
103	C₂³	C₂³×C₄	1	48	yes	—
104	C₂³	C₂⁵	1	2	yes	—
105	C₂³	C₂⁵	1	12	yes	—
106	C₂³	C₂⁵	1	1	yes	characteristic
107	Q₈	C₂³×C₄	1	64	yes	—
108	D₄	C₂³×C₄	1	192	yes	—
109	C₂×C₄	—	2	32	no	—
110	C₂×C₄	—	4	96	no	—
111	C₂×C₄	C₂³×C₄	1	4	yes	—
112	C₂×C₄	C₂³×C₄	1	4	yes	—
113	C₂×C₄	C₂³×C₄	1	24	yes	—
114	C₂×C₄	C₂³×C₄	1	16	yes	—
115	C₂×C₄	C₂³×C₄	1	48	yes	—
116	C₂×C₄	C₂³×C₄	1	96	yes	—
117	C₂×C₄	C₂³×C₄	1	96	yes	—
118	C₂×C₄	C₂⁵	1	32	yes	—
119	C₂×C₄	C₂⁵	1	2	yes	—
120	C₂×C₄	C₂⁵	1	12	yes	—
121	C₂×C₄	C₂⁵	1	2	yes	—
122	C₂²	—	2	24	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	2	8	no	—
125	C₂²	—	2	48	no	—
126	C₂²	—	4	48	no	—
127	C₂²	C₂⁴×C₄	1	4	yes	—
128	C₂²	C₂⁴×C₄	1	24	yes	—
129	C₂²	C₂⁴×C₄	1	2	yes	—
130	C₂²	#585	1	4	yes	—
131	C₂²	C₂⁶	1	1	yes	characteristic
132	C₄	—	2	32	no	—
133	C₄	C₂⁴×C₄	1	4	yes	—
134	C₄	C₂⁴×C₄	1	4	yes	—
135	C₄	C₂⁴×C₄	1	24	yes	—
136	C₂	—	2	4	no	—
137	C₂	—	2	24	no	—
138	C₂	#3428	1	4	yes	direct product factor
139	C₂	C₂⁵×C₄	1	1	yes	characteristic
140	C₂	C₂×#585	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₂²	C₂⁶
socle	C₂³	C₂⁵
prime radical	C₂³×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #3428	C₂	direct (N × H)
3. C₂×#520	C₂	semidirect
4. C₂×#519	C₂	semidirect
5. C₂×#518	C₂	semidirect
6. C₂×C₄×#41	C₂	semidirect
7. C₂²×#117	C₂	semidirect
8. C₂²×#41	C₄	semidirect
15. #520	C₂²	semidirect
16. #519	C₂²	semidirect
17. #518	C₂²	semidirect
18. C₄×#41	C₂²	semidirect
19. C₂×C₄×Q₈	C₂²	semidirect
20. C₂×C₄×D₄	C₂²	semidirect
23. C₂×#117	C₂²	semidirect
24. C₂×#117	C₂²	semidirect
25. C₂×#117	C₂²	semidirect
26. C₂²×C₄⋊C₄	C₂²	semidirect
28. C₂²×C₄²	C₂²	semidirect
30. C₂×#41	C₂×C₄	semidirect
31. C₂×#41	C₂×C₄	semidirect
32. C₂×#41	C₂×C₄	semidirect
35. C₂²×Q₈	C₂×C₄	semidirect
38. C₂²×D₄	C₂×C₄	semidirect
41. C₂³×C₄	C₂×C₄	semidirect
48. C₄×Q₈	C₂³	semidirect
49. C₄×D₄	C₂³	semidirect
52. #117	C₂³	semidirect
53. #117	C₂³	semidirect
56. C₂×C₄⋊C₄	C₂³	semidirect
62. C₂×C₄²	C₂³	semidirect
68. #41	C₂²×C₄	semidirect
69. #41	C₂²×C₄	semidirect
71. C₂×Q₈	C₂²×C₄	semidirect
74. C₂×D₄	C₂²×C₄	semidirect
81. C₂²×C₄	C₂²×C₄	semidirect
138. C₂	#3428	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41	2
#3428	2
C₂×#520	2
C₂×#519	2
C₂×#518	2
C₂×C₄×#41	2
C₂²×#117	2

Subgroups — C2×#3428 #62648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3428 #62648