Finite Group Databasebeta

Subgroups — #62415

Order 256.

1299 conjugacy classes of subgroups; only the 487 normal subgroups are shown.

normal subgroups only

This group has 1299 conjugacy classes of subgroups — too many to draw. Only its 487 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	#3498	C₂	1	2	yes	maximal, complemented
2	#3391	C₂	1	4	yes	maximal, complemented
3	D₄×D₈	C₂	1	12	yes	maximal, complemented
4	#3205	C₂	1	6	yes	maximal, complemented
5	#3169	C₂	1	3	yes	maximal, complemented
6	C₂×#493	C₂	1	3	yes	maximal, complemented
7	C₈×#41	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)	—	2	2	no	—
9	C₂²×D₈	C₂²	1	12	yes	complemented
10	#550	—	2	2	no	—
11	#550	C₂²	1	4	yes	complemented
12	D₄²	—	2	6	no	—
13	D₄²	C₂²	1	12	yes	complemented
14	#535	—	2	6	no	—
15	#532	C₂²	1	6	yes	complemented
16	C₂×#127	C₂²	1	6	yes	complemented
17	C₄×#41	C₂²	1	1	yes	characteristic
18	#500	C₂²	1	1	yes	characteristic, complemented
19	#493	C₂²	1	12	yes	complemented
20	#493	C₂²	1	3	yes	complemented
21	#493	C₂²	1	6	yes	complemented
22	#486	C₂²	1	6	yes	complemented
23	#466	C₂²	1	12	yes	complemented
24	#459	C₂²	1	24	yes	complemented
25	#447	C₂²	1	24	yes	complemented
26	C₈×Q₈	C₂²	1	2	yes	complemented
27	C₄×D₈	C₂²	1	12	yes	complemented
28	C₈×D₄	C₂²	1	6	yes	complemented
29	#431	C₂²	1	3	yes	complemented
30	C₂×C₄×C₈	C₂²	1	3	yes	complemented
31	H(5,𝔽₂)	—	4	16	no	—
32	C₂×#41	—	2	4	no	—
33	C₂×#41	—	2	6	no	—
34	C₂×#41	D₄	1	1	yes	characteristic
35	C₂²×D₄	—	2	12	no	—
36	C₂²×D₄	—	2	6	no	—
37	C₂²×D₄	—	2	6	no	—
38	C₂²×D₄	C₂³	1	12	yes	—
39	C₂×D₈	—	4	24	no	—
40	C₂×D₈	C₂³	1	12	yes	—
41	C₂×D₈	C₂³	1	24	yes	—
42	C₂²×C₈	C₂³	1	6	yes	—
43	C₄⋊Q₈	C₂³	1	2	yes	—
44	#127	—	2	6	no	—
45	#127	—	2	6	no	—
46	#127	C₂³	1	6	yes	—
47	#127	C₂³	1	12	yes	—
48	#124	—	2	6	no	—
49	#121	—	2	6	no	—
50	#121	—	2	12	no	—
51	#121	—	2	12	no	—
52	#121	C₂³	1	12	yes	—
53	#120	—	2	24	no	—
54	C₄×Q₈	D₄	1	1	yes	characteristic
55	C₄×Q₈	C₂³	1	1	yes	characteristic
56	C₄×D₄	—	2	6	no	—
57	C₄×D₄	D₄	1	3	yes	—
58	C₄×D₄	C₂³	1	12	yes	—
59	C₄×D₄	C₂³	1	3	yes	—
60	#117	D₄	1	3	yes	—
61	C₂×C₄²	C₂³	1	3	yes	—
62	#107	C₂³	1	4	yes	—
63	C₄⋊C₈	C₂³	1	6	yes	—
64	#102	C₂³	1	24	yes	—
65	C₂²⋊C₈	C₂³	1	6	yes	—
66	C₄×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₈	C₂³	1	3	yes	—
68	C₄×C₈	C₂³	1	6	yes	—
69	C₂⁴	—	2	12	no	—
70	#41	—	4	16	no	—
71	#41	—	4	24	no	—
72	#41	D₈	1	8	yes	complemented
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
75	C₂×D₄	—	2	12	no	—
76	C₂×D₄	—	2	6	no	—
77	C₂×D₄	—	2	12	no	—
78	C₂×D₄	—	2	6	no	—
79	C₂×D₄	—	2	12	no	—
80	C₂×D₄	—	2	12	no	—
81	C₂×D₄	—	4	24	no	—
82	C₂×D₄	—	4	48	no	—
83	C₂×D₄	—	4	24	no	—
84	C₂×D₄	—	8	12	no	—
85	C₂×D₄	C₂×D₄	1	3	yes	—
86	C₂×D₄	C₂⁴	1	12	yes	—
87	C₂²×C₄	—	2	12	no	—
88	C₂²×C₄	C₂×D₄	1	3	yes	—
89	C₂²×C₄	C₂×D₄	1	3	yes	—
90	C₂²×C₄	C₂⁴	1	3	yes	—
91	D₈	—	2	24	no	—
92	D₈	—	4	24	no	—
93	C₂×C₈	—	2	12	no	—
94	C₂×C₈	C₂⁴	1	6	yes	—
95	C₂×C₈	C₂⁴	1	2	yes	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	C₂×D₄	1	3	yes	—
98	C₄⋊C₄	C₂×D₄	1	3	yes	—
99	C₄⋊C₄	C₂⁴	1	4	yes	—
100	C₂²⋊C₄	—	2	12	no	—
101	C₂²⋊C₄	—	2	12	no	—
102	C₂²⋊C₄	C₂×D₄	1	6	yes	—
103	C₄²	C₂×D₄	1	3	yes	—
104	C₄²	C₂×D₄	1	3	yes	—
105	C₄²	C₂⁴	1	1	yes	characteristic
106	C₄²	C₂⁴	1	3	yes	—
107	C₂³	—	2	12	no	—
108	C₂³	—	4	24	no	—
109	C₂³	—	8	12	no	—
110	C₂³	—	8	12	no	—
111	C₂³	C₂²×D₄	1	3	yes	—
112	Q₈	—	4	4	no	—
113	Q₈	C₂×D₈	1	4	yes	complemented
114	D₄	—	2	24	no	—
115	D₄	—	4	24	no	—
116	D₄	—	4	24	no	—
117	D₄	—	4	12	no	—
118	D₄	—	4	12	no	—
119	D₄	—	8	12	no	—
120	D₄	—	8	12	no	—
121	D₄	C₂×D₈	1	12	yes	complemented
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	2	6	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	24	no	—
126	C₂×C₄	C₂×D₈	1	12	yes	complemented
127	C₂×C₄	C₂²×D₄	1	3	yes	—
128	C₂×C₄	C₂²×D₄	1	3	yes	—
129	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄	C₂²×D₄	1	3	yes	—
131	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂⁵	1	1	yes	characteristic
133	C₈	—	2	6	no	—
134	C₈	H(5,𝔽₂)	1	4	yes	complemented
135	C₂²	—	2	3	no	—
136	C₂²	—	2	3	no	—
137	C₂²	—	4	12	no	—
138	C₂²	—	4	12	no	—
139	C₂²	—	4	12	no	—
140	C₂²	—	16	24	no	—
141	C₂²	C₂²×D₈	1	6	yes	—
142	C₂²	C₂³×D₄	1	1	yes	characteristic
143	C₄	—	2	3	no	—
144	C₄	—	2	3	no	—
145	C₄	—	4	4	no	—
146	C₄	C₂²×D₈	1	6	yes	—
147	C₄	C₂²×D₈	1	2	yes	—
148	C₄	#576	1	2	yes	—
149	C₄	C₂×H(5,𝔽₂)	1	2	yes	—
150	C₂	—	2	6	no	—
151	C₂	—	8	12	no	—
152	C₂	#3498	1	1	yes	characteristic
153	C₂	C₂³×D₈	1	1	yes	characteristic
154	C₂	C₂×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3498	C₂	semidirect
2. #3391	C₂	semidirect
3. D₄×D₈	C₂	semidirect
4. #3205	C₂	semidirect
5. #3169	C₂	semidirect
6. C₂×#493	C₂	semidirect
7. C₈×#41	C₂	semidirect
9. C₂²×D₈	C₂²	semidirect
11. #550	C₂²	semidirect
13. D₄²	C₂²	semidirect
15. #532	C₂²	semidirect
16. C₂×#127	C₂²	semidirect
18. #500	C₂²	semidirect
19. #493	C₂²	semidirect
20. #493	C₂²	semidirect
21. #493	C₂²	semidirect
22. #486	C₂²	semidirect
23. #466	C₂²	semidirect
24. #459	C₂²	semidirect
25. #447	C₂²	semidirect
26. C₈×Q₈	C₂²	semidirect
27. C₄×D₈	C₂²	semidirect
28. C₈×D₄	C₂²	semidirect
29. #431	C₂²	semidirect
30. C₂×C₄×C₈	C₂²	semidirect
72. #41	D₈	semidirect
113. Q₈	C₂×D₈	semidirect
121. D₄	C₂×D₈	semidirect
126. C₂×C₄	C₂×D₈	semidirect
134. C₈	H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#3498	2
#3391	2
D₄×D₈	2
#3205	2
#3169	2
C₂×#493	2
C₈×#41	2