Subgroups — C₂×#3498 #62795

Order 256.

6491 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	#3498	C₂	1	32	yes	maximal, direct product factor
3	C₂×#550	C₂	1	4	yes	maximal, complemented
4	C₂×D₄²	C₂	1	12	yes	maximal, complemented
5	C₂×#535	C₂	1	6	yes	maximal, complemented
6	C₂×#532	C₂	1	3	yes	maximal, complemented
7	C₂²×#127	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)	—	2	32	no	—
10	C₂×H(5,𝔽₂)	C₂²	1	32	yes	complemented
11	C₂²×#41	C₂²	1	1	yes	characteristic
12	C₂²×#41	C₂²	1	6	yes	complemented
13	C₂²×#41	C₂²	1	4	yes	complemented
14	C₂³×D₄	C₂²	1	6	yes	complemented
15	C₂³×D₄	C₂²	1	12	yes	complemented
16	C₂³×D₄	C₂²	1	12	yes	complemented
17	#550	C₂²	1	64	yes	complemented
18	D₄²	C₂²	1	192	yes	complemented
19	#535	C₂²	1	96	yes	complemented
20	#532	C₂²	1	48	yes	complemented
21	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×#127	C₂²	1	3	yes	complemented
23	C₂×#127	C₂²	1	6	yes	complemented
24	C₂×#127	C₂²	1	12	yes	complemented
25	C₂×#127	C₂²	1	48	yes	complemented
26	C₂×#124	C₂²	1	6	yes	complemented
27	C₂×#121	C₂²	1	12	yes	complemented
28	C₂×#121	C₂²	1	24	yes	complemented
29	C₂×#120	C₂²	1	24	yes	complemented
30	C₄×#41	C₂²	1	16	yes	complemented
31	C₂×C₄×Q₈	C₂²	1	2	yes	complemented
32	C₂×C₄×D₄	C₂²	1	6	yes	complemented
33	C₂×C₄×D₄	C₂²	1	12	yes	complemented
34	C₂×#117	C₂²	1	3	yes	complemented
35	C₂²×C₄²	C₂²	1	3	yes	complemented
36	C₂⁵	C₂³	1	12	yes	—
37	H(5,𝔽₂)	—	2	256	no	—
38	C₂×#41	—	2	32	no	—
39	C₂×#41	—	2	48	no	—
40	C₂×#41	D₄	1	16	yes	complemented
41	C₂×#41	C₂³	1	8	yes	—
42	C₂×#41	C₂³	1	48	yes	complemented
43	C₂×#41	C₂³	1	32	yes	complemented
44	C₂²×Q₈	C₂³	1	2	yes	—
45	C₂²×Q₈	C₂³	1	1	yes	characteristic
46	C₂²×D₄	—	2	96	no	—
47	C₂²×D₄	—	2	48	no	—
48	C₂²×D₄	—	4	48	no	—
49	C₂²×D₄	C₂³	1	12	yes	—
50	C₂²×D₄	C₂³	1	12	yes	—
51	C₂²×D₄	C₂³	1	12	yes	—
52	C₂²×D₄	C₂³	1	6	yes	—
53	C₂²×D₄	C₂³	1	24	yes	—
54	C₂²×D₄	C₂³	1	3	yes	—
55	C₂²×D₄	C₂³	1	48	yes	complemented
56	C₂²×D₄	C₂³	1	96	yes	complemented
57	C₂²×D₄	C₂³	1	96	yes	complemented
58	C₂³×C₄	C₂³	1	6	yes	—
59	C₂³×C₄	C₂³	1	12	yes	—
60	C₂³×C₄	C₂³	1	3	yes	—
61	C₄⋊Q₈	C₂³	1	8	yes	complemented
62	#127	C₂³	1	24	yes	complemented
63	#127	C₂³	1	48	yes	complemented
64	#127	C₂³	1	96	yes	complemented
65	#124	C₂³	1	48	yes	complemented
66	#121	C₂³	1	96	yes	complemented
67	#121	C₂³	1	192	yes	complemented
68	#120	C₂³	1	192	yes	complemented
69	C₄×Q₈	C₂³	1	16	yes	complemented
70	C₄×D₄	C₂³	1	48	yes	complemented
71	C₄×D₄	C₂³	1	96	yes	complemented
72	#117	C₂³	1	24	yes	complemented
73	C₂×C₄⋊C₄	C₂³	1	6	yes	—
74	C₂×C₄⋊C₄	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
76	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
77	C₂×C₄²	C₂³	1	3	yes	—
78	C₂×C₄²	C₂³	1	6	yes	—
79	C₂×C₄²	C₂³	1	1	yes	characteristic
80	C₂×C₄²	C₂³	1	24	yes	complemented
81	C₂⁴	—	2	48	no	—
82	C₂⁴	—	4	48	no	—
83	C₂⁴	C₂⁴	1	12	yes	—
84	C₂⁴	C₂⁴	1	3	yes	—
85	C₂⁴	C₂⁴	1	48	yes	—
86	#41	—	2	128	no	—
87	#41	—	2	192	no	—
88	#41	C₂×D₄	1	64	yes	complemented
89	C₂×Q₈	—	2	8	no	—
90	C₂×Q₈	C₂×D₄	1	8	yes	complemented
91	C₂×Q₈	C₂⁴	1	8	yes	—
92	C₂×Q₈	C₂⁴	1	4	yes	—
93	C₂×D₄	—	2	48	no	—
94	C₂×D₄	—	2	384	no	—
95	C₂×D₄	—	2	48	no	—
96	C₂×D₄	—	2	24	no	—
97	C₂×D₄	—	2	192	no	—
98	C₂×D₄	—	2	48	no	—
99	C₂×D₄	—	4	192	no	—
100	C₂×D₄	—	4	48	no	—
101	C₂×D₄	C₂×D₄	1	24	yes	complemented
102	C₂×D₄	C₂⁴	1	48	yes	—
103	C₂×D₄	C₂⁴	1	48	yes	—
104	C₂×D₄	C₂⁴	1	48	yes	—
105	C₂×D₄	C₂⁴	1	24	yes	—
106	C₂×D₄	C₂⁴	1	96	yes	—
107	C₂×D₄	C₂⁴	1	12	yes	—
108	C₂²×C₄	—	2	48	no	—
109	C₂²×C₄	—	2	24	no	—
110	C₂²×C₄	C₂×D₄	1	24	yes	complemented
111	C₂²×C₄	C₂⁴	1	24	yes	—
112	C₂²×C₄	C₂⁴	1	48	yes	—
113	C₂²×C₄	C₂⁴	1	6	yes	—
114	C₂²×C₄	C₂⁴	1	4	yes	—
115	C₂²×C₄	C₂⁴	1	1	yes	characteristic
116	C₂²×C₄	C₂⁴	1	3	yes	—
117	C₂²×C₄	C₂⁴	1	12	yes	—
118	C₂²×C₄	C₂⁴	1	2	yes	—
119	C₄⋊C₄	C₂⁴	1	24	yes	—
120	C₄⋊C₄	C₂⁴	1	16	yes	—
121	C₂²⋊C₄	C₂⁴	1	24	yes	—
122	C₂²⋊C₄	C₂⁴	1	96	yes	—
123	C₄²	C₂⁴	1	4	yes	—
124	C₄²	C₂⁴	1	12	yes	—
125	C₄²	C₂⁴	1	24	yes	—
126	C₂³	—	2	12	no	—
127	C₂³	—	2	24	no	—
128	C₂³	—	2	48	no	—
129	C₂³	—	2	96	no	—
130	C₂³	—	4	96	no	—
131	C₂³	—	8	96	no	—
132	C₂³	C₂²×D₄	1	12	yes	—
133	C₂³	C₂⁵	1	24	yes	—
134	C₂³	C₂⁵	1	6	yes	—
135	C₂³	C₂⁵	1	1	yes	characteristic
136	Q₈	—	2	16	no	—
137	Q₈	C₂²×D₄	1	16	yes	complemented
138	D₄	—	2	96	no	—
139	D₄	—	2	96	no	—
140	D₄	—	2	48	no	—
141	D₄	—	2	96	no	—
142	D₄	—	4	96	no	—
143	D₄	C₂²×D₄	1	48	yes	complemented
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	12	no	—
146	C₂×C₄	—	2	96	no	—
147	C₂×C₄	—	2	48	no	—
148	C₂×C₄	C₂²×D₄	1	12	yes	—
149	C₂×C₄	C₂²×D₄	1	4	yes	—
150	C₂×C₄	C₂²×D₄	1	48	yes	complemented
151	C₂×C₄	H(5,𝔽₂)	1	8	yes	complemented
152	C₂×C₄	C₂⁵	1	12	yes	—
153	C₂×C₄	C₂⁵	1	8	yes	—
154	C₂×C₄	C₂⁵	1	2	yes	—
155	C₂×C₄	C₂⁵	1	6	yes	—
156	C₂×C₄	C₂⁵	1	4	yes	—
157	C₂²	—	2	12	no	—
158	C₂²	—	2	24	no	—
159	C₂²	—	2	24	no	—
160	C₂²	—	2	48	no	—
161	C₂²	—	4	48	no	—
162	C₂²	—	8	96	no	—
163	C₂²	C₂³×D₄	1	12	yes	—
164	C₂²	C₂³×D₄	1	2	yes	—
165	C₂²	C₂×H(5,𝔽₂)	1	4	yes	—
166	C₂²	C₂⁶	1	1	yes	characteristic
167	C₄	—	2	8	no	—
168	C₄	—	2	12	no	—
169	C₄	C₂³×D₄	1	12	yes	—
170	C₄	C₂³×D₄	1	4	yes	—
171	C₄	C₂×H(5,𝔽₂)	1	8	yes	complemented
172	C₂	—	2	12	no	—
173	C₂	—	4	24	no	—
174	C₂	#3498	1	4	yes	direct product factor
175	C₂	C₂⁴×D₄	1	1	yes	characteristic
176	C₂	C₂²×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×H(5,𝔽₂)	C₂	semidirect
2. #3498	C₂	direct (N × H)
3. C₂×#550	C₂	semidirect
4. C₂×D₄²	C₂	semidirect
5. C₂×#535	C₂	semidirect
6. C₂×#532	C₂	semidirect
7. C₂²×#127	C₂	semidirect
8. C₂×C₄×#41	C₂	semidirect
10. C₂×H(5,𝔽₂)	C₂²	semidirect
12. C₂²×#41	C₂²	semidirect
13. C₂²×#41	C₂²	semidirect
14. C₂³×D₄	C₂²	semidirect
15. C₂³×D₄	C₂²	semidirect
16. C₂³×D₄	C₂²	semidirect
17. #550	C₂²	semidirect
18. D₄²	C₂²	semidirect
19. #535	C₂²	semidirect
20. #532	C₂²	semidirect
21. C₂×C₄⋊Q₈	C₂²	semidirect
22. C₂×#127	C₂²	semidirect
23. C₂×#127	C₂²	semidirect
24. C₂×#127	C₂²	semidirect
25. C₂×#127	C₂²	semidirect
26. C₂×#124	C₂²	semidirect
27. C₂×#121	C₂²	semidirect
28. C₂×#121	C₂²	semidirect
29. C₂×#120	C₂²	semidirect
30. C₄×#41	C₂²	semidirect
31. C₂×C₄×Q₈	C₂²	semidirect
32. C₂×C₄×D₄	C₂²	semidirect
33. C₂×C₄×D₄	C₂²	semidirect
34. C₂×#117	C₂²	semidirect
35. C₂²×C₄²	C₂²	semidirect
40. C₂×#41	D₄	semidirect
42. C₂×#41	C₂³	semidirect
43. C₂×#41	C₂³	semidirect
55. C₂²×D₄	C₂³	semidirect
56. C₂²×D₄	C₂³	semidirect
57. C₂²×D₄	C₂³	semidirect
61. C₄⋊Q₈	C₂³	semidirect
62. #127	C₂³	semidirect
63. #127	C₂³	semidirect
64. #127	C₂³	semidirect
65. #124	C₂³	semidirect
66. #121	C₂³	semidirect
67. #121	C₂³	semidirect
68. #120	C₂³	semidirect
69. C₄×Q₈	C₂³	semidirect
70. C₄×D₄	C₂³	semidirect
71. C₄×D₄	C₂³	semidirect
72. #117	C₂³	semidirect
80. C₂×C₄²	C₂³	semidirect
88. #41	C₂×D₄	semidirect
90. C₂×Q₈	C₂×D₄	semidirect
101. C₂×D₄	C₂×D₄	semidirect
110. C₂²×C₄	C₂×D₄	semidirect
137. Q₈	C₂²×D₄	semidirect
143. D₄	C₂²×D₄	semidirect
150. C₂×C₄	C₂²×D₄	semidirect
151. C₂×C₄	H(5,𝔽₂)	semidirect
171. C₄	C₂×H(5,𝔽₂)	semidirect
174. C₂	#3498	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×H(5,𝔽₂)	2
#3498	2
C₂×#550	2
C₂×D₄²	2
C₂×#535	2
C₂×#532	2
C₂²×#127	2
C₂×C₄×#41	2

Subgroups — C2×#3498 #62795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3498 #62795