Finite Group Databasebeta

Subgroups — #53373

Order 256.

869 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 395 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	#2860	C₂	1	3	yes	maximal, complemented
2	#2801	C₂	1	6	yes	maximal, complemented
3	#2727	C₂	1	6	yes	maximal, complemented
4	#2727	C₂	1	3	yes	maximal, complemented
5	#2685	C₂	1	6	yes	maximal, complemented
6	#2685	C₂	1	6	yes	maximal, complemented
7	#2679	C₂	1	1	yes	maximal, characteristic
8	C₂×#126	C₂²	1	6	yes	complemented
9	C₂×#126	C₂²	1	6	yes	complemented
10	C₂×#120	C₂²	1	6	yes	—
11	#401	C₂²	1	2	yes	complemented
12	#399	C₂²	1	6	yes	complemented
13	#399	C₂²	1	6	yes	complemented
14	#396	C₂²	1	6	yes	complemented
15	#396	C₂²	1	6	yes	complemented
16	#394	C₂²	1	6	yes	—
17	#394	C₂²	1	6	yes	—
18	#394	C₂²	1	6	yes	—
19	#394	C₂²	1	6	yes	—
20	#394	C₂²	1	6	yes	—
21	#390	C₂²	1	3	yes	—
22	#390	C₂²	1	3	yes	—
23	#390	C₂²	1	3	yes	—
24	#390	C₂²	1	6	yes	—
25	#389	C₂²	1	3	yes	complemented
26	#389	C₂²	1	3	yes	complemented
27	#389	C₂²	1	3	yes	complemented
28	#389	C₂²	1	6	yes	complemented
29	#389	C₂²	1	6	yes	complemented
30	#388	C₂²	1	6	yes	complemented
31	#388	C₂²	1	6	yes	complemented
32	#388	C₂²	1	6	yes	complemented
33	#388	C₂²	1	6	yes	complemented
34	#388	C₂²	1	6	yes	complemented
35	#388	C₂²	1	6	yes	complemented
36	#379	C₂²	1	6	yes	complemented
37	#379	C₂²	1	2	yes	complemented
38	C₄³	C₂²	1	1	yes	characteristic, complemented
39	C₂⁵	C₂³	1	2	yes	—
40	C₂²×D₄	C₂³	1	3	yes	—
41	C₂²×D₄	C₂³	1	6	yes	—
42	#126	—	4	12	no	—
43	#126	—	4	12	no	—
44	#120	—	4	12	no	—
45	C₂×C₄⋊C₄	C₂³	1	6	yes	—
46	C₂×C₄⋊C₄	C₂³	1	3	yes	—
47	C₂×C₄⋊C₄	C₂³	1	6	yes	—
48	C₂×C₄⋊C₄	C₂³	1	6	yes	—
49	C₂×C₄⋊C₄	C₂³	1	6	yes	—
50	C₂×C₄⋊C₄	C₂³	1	3	yes	—
51	C₂×C₄⋊C₄	C₂³	1	6	yes	—
52	C₂×C₄⋊C₄	C₂³	1	3	yes	—
53	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
54	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
55	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
56	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
57	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
58	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
59	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
60	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
61	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
62	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
63	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
64	C₂×C₄²	C₂³	1	3	yes	—
65	C₂×C₄²	C₂³	1	6	yes	—
66	C₂×C₄²	C₂³	1	3	yes	—
67	C₂×C₄²	C₂³	1	1	yes	characteristic
68	#95	C₂³	1	6	yes	—
69	#95	C₂³	1	6	yes	—
70	#95	C₂³	1	6	yes	—
71	#95	C₂³	1	6	yes	—
72	#95	C₂³	1	2	yes	—
73	C₂⁴	—	4	6	no	—
74	C₂⁴	—	4	6	no	—
75	C₂⁴	—	4	2	no	—
76	C₂⁴	C₂⁴	1	6	yes	—
77	C₂×D₄	—	2	6	no	—
78	C₂×D₄	—	4	6	no	—
79	C₂×D₄	—	4	6	no	—
80	C₂×D₄	—	4	3	no	—
81	C₂×D₄	—	4	3	no	—
82	C₂×D₄	—	4	6	no	—
83	C₂²×C₄	C₂⁴	1	1	yes	characteristic
84	C₂²×C₄	C₂⁴	1	6	yes	—
85	C₂²×C₄	C₂⁴	1	6	yes	—
86	C₂²×C₄	C₂⁴	1	3	yes	—
87	C₂²×C₄	C₂⁴	1	6	yes	—
88	C₂²×C₄	C₂⁴	1	3	yes	—
89	C₄⋊C₄	—	2	6	no	—
90	C₄⋊C₄	—	2	12	no	—
91	C₄⋊C₄	—	4	3	no	—
92	C₄⋊C₄	—	4	3	no	—
93	C₄⋊C₄	—	4	6	no	—
94	C₄⋊C₄	—	4	6	no	—
95	C₄⋊C₄	—	4	6	no	—
96	C₄⋊C₄	—	4	6	no	—
97	C₂²⋊C₄	—	2	12	no	—
98	C₂²⋊C₄	—	2	12	no	—
99	C₂²⋊C₄	—	4	6	no	—
100	C₂²⋊C₄	—	4	6	no	—
101	C₂²⋊C₄	—	4	6	no	—
102	C₂²⋊C₄	—	4	6	no	—
103	C₂²⋊C₄	—	4	6	no	—
104	C₂²⋊C₄	—	4	6	no	—
105	C₂²⋊C₄	—	4	6	no	—
106	C₂²⋊C₄	—	4	6	no	—
107	C₂²⋊C₄	—	4	6	no	—
108	C₄²	—	2	6	no	—
109	C₄²	—	4	1	no	—
110	C₄²	—	4	3	no	—
111	C₄²	—	4	6	no	—
112	C₂³	—	2	6	no	—
113	C₂³	—	2	6	no	—
114	C₂³	—	2	6	no	—
115	C₂³	—	2	6	no	—
116	C₂³	—	2	6	no	—
117	C₂³	—	2	6	no	—
118	C₂³	—	2	6	no	—
119	C₂³	—	8	12	no	—
120	C₂³	—	8	12	no	—
121	C₂³	—	8	4	no	—
122	C₂³	C₂⁵	1	1	yes	characteristic
123	D₄	—	8	6	no	—
124	D₄	—	8	12	no	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	6	no	—
130	C₂×C₄	—	2	6	no	—
131	C₂×C₄	—	2	6	no	—
132	C₂×C₄	—	2	6	no	—
133	C₂×C₄	—	2	6	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	2	6	no	—
137	C₂×C₄	—	2	6	no	—
138	C₂×C₄	—	2	6	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	H(5,𝔽₂)	1	6	yes	—
141	C₂²	—	4	6	no	—
142	C₂²	—	4	6	no	—
143	C₂²	—	4	6	no	—
144	C₂²	—	4	6	no	—
145	C₂²	—	4	6	no	—
146	C₂²	—	4	6	no	—
147	C₂²	—	4	6	no	—
148	C₂²	—	8	16	no	—
149	C₂²	C₂×H(5,𝔽₂)	1	3	yes	—
150	C₂²	#585	1	3	yes	—
151	C₂²	#585	1	1	yes	characteristic
152	C₄	—	2	6	no	—
153	C₄	—	4	3	no	—
154	C₄	—	4	6	no	—
155	C₄	—	4	6	no	—
156	C₄	—	4	6	no	—
157	C₄	—	4	1	no	—
158	C₂	—	8	6	no	—
159	C₂	#3546	1	3	yes	—
160	C₂	#3560	1	3	yes	—
161	C₂	#3568	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	#2679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2860	C₂	semidirect
2. #2801	C₂	semidirect
3. #2727	C₂	semidirect
4. #2727	C₂	semidirect
5. #2685	C₂	semidirect
6. #2685	C₂	semidirect
8. C₂×#126	C₂²	semidirect
9. C₂×#126	C₂²	semidirect
11. #401	C₂²	semidirect
12. #399	C₂²	semidirect
13. #399	C₂²	semidirect
14. #396	C₂²	semidirect
15. #396	C₂²	semidirect
25. #389	C₂²	semidirect
26. #389	C₂²	semidirect
27. #389	C₂²	semidirect
28. #389	C₂²	semidirect
29. #389	C₂²	semidirect
30. #388	C₂²	semidirect
31. #388	C₂²	semidirect
32. #388	C₂²	semidirect
33. #388	C₂²	semidirect
34. #388	C₂²	semidirect
35. #388	C₂²	semidirect
36. #379	C₂²	semidirect
37. #379	C₂²	semidirect
38. C₄³	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#2860	2
#2801	2
#2727	2
#2685	2
#2679	2