Subgroups — C₂×#2988 #60255

Order 256.

1217 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	C₂×C₄×#41	C₂	1	1	yes	maximal, characteristic
2	#2988	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₈×Q₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄	C₂	1	6	yes	maximal, complemented
5	C₂×#431	C₂	1	3	yes	maximal, complemented
6	C₂²×C₄⋊C₈	C₂	1	3	yes	maximal, complemented
7	C₂²×#41	C₄	1	1	yes	characteristic
8	C₂³×C₈	C₂²	1	6	yes	—
9	C₄×#41	C₄	1	8	yes	—
10	C₄×#41	C₂²	1	8	yes	—
11	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄	C₄	1	3	yes	—
14	C₂×C₄×D₄	C₂²	1	3	yes	—
15	C₂×#117	C₄	1	3	yes	—
16	C₂²×C₄²	C₂²	1	3	yes	—
17	C₈×Q₈	C₂²	1	16	yes	complemented
18	C₈×D₄	C₂²	1	48	yes	complemented
19	#431	C₂²	1	24	yes	complemented
20	C₂×C₄⋊C₈	C₂²	1	3	yes	—
21	C₂×C₄⋊C₈	C₂²	1	6	yes	—
22	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈	C₂²	1	24	yes	complemented
24	C₂×C₂²⋊C₈	C₂²	1	6	yes	—
25	C₂×C₄×C₈	C₂²	1	6	yes	—
26	C₂×#41	C₈	1	16	yes	complemented
27	C₂×#41	C₂×C₄	1	8	yes	—
28	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₄	C₂×C₄	1	3	yes	—
30	C₂³×C₄	C₂×C₄	1	3	yes	—
31	C₂³×C₄	C₂×C₄	1	3	yes	—
32	C₂³×C₄	C₂³	1	3	yes	—
33	C₂²×C₈	—	4	12	no	—
34	C₂²×C₈	C₂³	1	6	yes	—
35	C₂²×C₈	C₂³	1	2	yes	—
36	C₂²×C₈	C₂³	1	24	yes	—
37	C₄×Q₈	C₂×C₄	1	8	yes	—
38	C₄×Q₈	C₂×C₄	1	4	yes	—
39	C₄×Q₈	C₂³	1	4	yes	—
40	C₄×D₄	C₂×C₄	1	24	yes	—
41	C₄×D₄	C₂×C₄	1	12	yes	—
42	C₄×D₄	C₂³	1	12	yes	—
43	#117	C₂×C₄	1	24	yes	—
44	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
45	C₂×C₄⋊C₄	C₂×C₄	1	3	yes	—
46	C₂×C₂²⋊C₄	C₂×C₄	1	6	yes	—
47	C₂×C₄²	C₂×C₄	1	3	yes	—
48	C₂×C₄²	C₂×C₄	1	3	yes	—
49	C₂×C₄²	C₂×C₄	1	12	yes	—
50	C₂×C₄²	C₂³	1	12	yes	—
51	C₂×C₄²	C₂³	1	3	yes	—
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	C₄⋊C₈	C₂³	1	12	yes	—
54	C₄⋊C₈	C₂³	1	24	yes	—
55	C₄⋊C₈	C₂³	1	4	yes	—
56	C₂²⋊C₈	C₂³	1	24	yes	—
57	C₄×C₈	C₂³	1	24	yes	—
58	C₂⁴	C₂²×C₄	1	3	yes	—
59	#41	C₂×C₈	1	64	yes	complemented
60	C₂×Q₈	C₂×C₈	1	8	yes	complemented
61	C₂×Q₈	C₂²×C₄	1	4	yes	—
62	C₂×D₄	C₂×C₈	1	24	yes	complemented
63	C₂×D₄	C₂²×C₄	1	12	yes	—
64	C₂²×C₄	—	2	6	no	—
65	C₂²×C₄	—	2	6	no	—
66	C₂²×C₄	—	4	6	no	—
67	C₂²×C₄	C₂×C₈	1	24	yes	complemented
68	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄	C₂²×C₄	1	3	yes	—
70	C₂²×C₄	C₂²×C₄	1	6	yes	—
71	C₂²×C₄	C₂²×C₄	1	3	yes	—
72	C₂²×C₄	C₂²×C₄	1	12	yes	—
73	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²×C₄	1	3	yes	—
75	C₂²×C₄	C₂²×C₄	1	12	yes	—
76	C₂²×C₄	C₂⁴	1	6	yes	—
77	C₂²×C₄	C₂⁴	1	1	yes	characteristic
78	C₂×C₈	—	2	12	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	4	24	no	—
81	C₂×C₈	C₂⁴	1	4	yes	—
82	C₂×C₈	C₂⁴	1	12	yes	—
83	C₄⋊C₄	C₂²×C₄	1	12	yes	—
84	C₄⋊C₄	C₂²×C₄	1	12	yes	—
85	C₂²⋊C₄	C₂²×C₄	1	24	yes	—
86	C₄²	C₂²×C₄	1	2	yes	—
87	C₄²	C₂²×C₄	1	12	yes	—
88	C₄²	C₂²×C₄	1	12	yes	—
89	C₄²	C₂²×C₄	1	6	yes	—
90	C₄²	C₂⁴	1	6	yes	—
91	C₄²	C₂⁴	1	2	yes	—
92	C₂³	—	2	6	no	—
93	C₂³	—	2	6	no	—
94	C₂³	C₂²×C₈	1	12	yes	—
95	C₂³	C₂³×C₄	1	6	yes	—
96	C₂³	C₂³×C₄	1	1	yes	characteristic
97	Q₈	C₂²×C₈	1	16	yes	complemented
98	D₄	C₂²×C₈	1	48	yes	complemented
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	12	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	4	12	no	—
106	C₂×C₄	C₂²×C₈	1	4	yes	—
107	C₂×C₄	C₂²×C₈	1	12	yes	—
108	C₂×C₄	C₂²×C₈	1	48	yes	complemented
109	C₂×C₄	C₂³×C₄	1	2	yes	—
110	C₂×C₄	C₂³×C₄	1	6	yes	—
111	C₂×C₄	C₂³×C₄	1	6	yes	—
112	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄	C₂³×C₄	1	2	yes	—
114	C₂×C₄	C₂³×C₄	1	6	yes	—
115	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
116	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
117	C₂×C₄	C₂⁵	1	1	yes	characteristic
118	C₈	—	2	12	no	—
119	C₈	—	2	4	no	—
120	C₂²	—	2	6	no	—
121	C₂²	—	2	24	no	—
122	C₂²	—	2	6	no	—
123	C₂²	#520	1	2	yes	—
124	C₂²	C₂³×C₈	1	12	yes	—
125	C₂²	C₂³×C₈	1	2	yes	—
126	C₂²	#568	1	2	yes	—
127	C₂²	C₂⁴×C₄	1	1	yes	characteristic
128	C₄	—	2	6	no	—
129	C₄	—	2	2	no	—
130	C₄	—	2	6	no	—
131	C₄	#520	1	2	yes	—
132	C₄	C₂³×C₈	1	4	yes	—
133	C₄	C₂³×C₈	1	12	yes	—
134	C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₄	C₂×2₋¹⁺⁴	1	1	yes	characteristic
136	C₂	—	2	12	no	—
137	C₂	#2988	1	4	yes	direct product factor
138	C₂	C₂×#520	1	1	yes	characteristic
139	C₂	C₂⁴×C₈	1	1	yes	characteristic
140	C₂	C₂×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂³	C₂³×C₄
prime radical	C₂²×#41	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ 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
2. #2988	C₂	direct (N × H)
3. C₂×C₈×Q₈	C₂	semidirect
4. C₂×C₈×D₄	C₂	semidirect
5. C₂×#431	C₂	semidirect
6. C₂²×C₄⋊C₈	C₂	semidirect
17. C₈×Q₈	C₂²	semidirect
18. C₈×D₄	C₂²	semidirect
19. #431	C₂²	semidirect
23. C₂×C₄⋊C₈	C₂²	semidirect
26. C₂×#41	C₈	semidirect
59. #41	C₂×C₈	semidirect
60. C₂×Q₈	C₂×C₈	semidirect
62. C₂×D₄	C₂×C₈	semidirect
67. C₂²×C₄	C₂×C₈	semidirect
97. Q₈	C₂²×C₈	semidirect
98. D₄	C₂²×C₈	semidirect
108. C₂×C₄	C₂²×C₈	semidirect
137. C₂	#2988	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×C₄×#41	2
#2988	2
C₂×C₈×Q₈	2
C₂×C₈×D₄	2
C₂×#431	2
C₂²×C₄⋊C₈	2

Subgroups — C2×#2988 #60255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2988 #60255