Finite Group Databasebeta

Subgroups — #60472

Order 256.

1289 conjugacy classes of subgroups; only the 517 normal subgroups are shown.

normal subgroups only

This group has 1289 conjugacy classes of subgroups — too many to draw. Only its 517 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	#3586	C₂	1	2	yes	maximal, complemented
2	C₂×#578	C₂	1	1	yes	maximal, characteristic, complemented
3	#3573	C₂	1	1	yes	maximal, characteristic, complemented
4	#3069	C₂	1	12	yes	maximal, complemented
5	#3067	C₂	1	6	yes	maximal, complemented
6	C₂×#473	C₂	1	6	yes	maximal, complemented
7	#2916	C₂	1	1	yes	maximal, characteristic, complemented
8	#2889	C₂	1	2	yes	maximal, complemented
9	#585	C₂²	1	1	yes	characteristic
10	C₂×2₋¹⁺⁴	C₂²	1	2	yes	—
11	#578	—	2	8	no	—
12	#578	—	2	4	no	—
13	#578	—	2	4	no	—
14	#577	—	2	8	no	—
15	#575	C₂²	1	12	yes	—
16	C₂×#137	C₂²	1	12	yes	—
17	C₂×#137	C₂²	1	6	yes	complemented
18	C₂×#135	C₂²	1	3	yes	complemented
19	C₂²×Q₁₆	C₂²	1	3	yes	complemented
20	#568	—	2	4	no	—
21	#568	C₂²	1	6	yes	—
22	#568	C₂²	1	2	yes	—
23	#568	C₂²	1	1	yes	characteristic
24	C₂×#131	—	2	4	no	—
25	C₂×#131	C₂²	1	6	yes	—
26	C₂×#131	C₂²	1	1	yes	characteristic
27	#473	C₂²	1	12	yes	complemented
28	#473	C₂²	1	24	yes	complemented
29	#472	C₂²	1	4	yes	complemented
30	#471	C₂²	1	24	yes	complemented
31	#430	C₂²	1	6	yes	complemented
32	C₂×#108	C₂²	1	6	yes	complemented
33	#413	C₂²	1	12	yes	complemented
34	C₂×#101	C₂²	1	12	yes	complemented
35	2₋¹⁺⁴	—	2	1	no	—
36	2₋¹⁺⁴	—	2	8	no	—
37	2₋¹⁺⁴	—	2	8	no	—
38	2₋¹⁺⁴	D₄	1	4	yes	—
39	H(5,𝔽₂)	—	2	3	no	—
40	H(5,𝔽₂)	D₄	1	4	yes	—
41	C₂×#41	—	2	4	no	—
42	C₂×#41	—	2	6	no	—
43	C₂×#41	C₂³	1	6	yes	—
44	C₂×#41	C₂³	1	6	yes	—
45	C₂×#41	C₂³	1	1	yes	characteristic
46	C₂²×Q₈	—	2	2	no	—
47	C₂²×Q₈	C₂³	1	6	yes	—
48	#137	—	2	24	no	—
49	#137	—	2	12	no	—
50	#137	—	2	24	no	—
51	#137	C₂³	1	24	yes	—
52	AGL(1,ℤ/8)	—	2	12	no	—
53	AGL(1,ℤ/8)	C₂³	1	8	yes	—
54	#135	—	2	24	no	—
55	#135	—	2	6	no	—
56	#135	—	2	6	no	—
57	C₂×Q₁₆	—	2	12	no	—
58	C₂×Q₁₆	—	2	6	no	—
59	C₂×Q₁₆	—	2	6	no	—
60	C₂×Q₁₆	C₂³	1	3	yes	—
61	C₂×Q₁₆	C₂³	1	6	yes	—
62	C₂×SD₁₆	—	2	12	no	—
63	C₂×SD₁₆	C₂³	1	6	yes	—
64	C₂×D₈	C₂³	1	1	yes	characteristic
65	#131	—	2	4	no	—
66	#131	—	2	8	no	—
67	#131	—	2	12	no	—
68	#131	—	2	6	no	—
69	#131	—	2	6	no	—
70	#131	D₄	1	4	yes	—
71	#131	D₄	1	4	yes	—
72	#131	C₂³	1	4	yes	—
73	#131	C₂³	1	12	yes	—
74	C₂×M₁₆	—	2	12	no	—
75	C₂×M₁₆	—	2	6	no	—
76	C₂×M₁₆	C₂³	1	6	yes	—
77	C₂×M₁₆	C₂³	1	12	yes	—
78	C₂×M₁₆	C₂³	1	3	yes	—
79	C₂²×C₈	—	2	6	no	—
80	C₂²×C₈	C₂³	1	3	yes	—
81	#108	C₂³	1	4	yes	—
82	#108	C₂³	1	12	yes	—
83	#101	C₂³	1	24	yes	—
84	#100	C₂³	1	8	yes	—
85	#41	—	2	4	no	—
86	#41	—	2	12	no	—
87	#41	—	2	6	no	—
88	#41	—	2	6	no	—
89	#41	—	2	12	no	—
90	#41	—	2	12	no	—
91	#41	—	4	24	no	—
92	#41	C₂×D₄	1	12	yes	—
93	#41	C₂×D₄	1	4	yes	—
94	#41	C₂×D₄	1	4	yes	—
95	#41	C₂⁴	1	4	yes	—
96	C₂×Q₈	—	2	4	no	—
97	C₂×Q₈	—	2	6	no	—
98	C₂×Q₈	—	2	12	no	—
99	C₂×Q₈	—	2	12	no	—
100	C₂×Q₈	—	4	8	no	—
101	C₂×Q₈	C₂×D₄	1	6	yes	—
102	C₂×Q₈	#41	1	1	yes	characteristic
103	C₂×Q₈	C₂⁴	1	6	yes	—
104	C₂×D₄	—	2	6	no	—
105	C₂×D₄	—	2	12	no	—
106	C₂×D₄	—	2	6	no	—
107	C₂×D₄	C₂×D₄	1	6	yes	—
108	C₂×D₄	#41	1	3	yes	—
109	C₂×D₄	C₂⁴	1	2	yes	—
110	C₂²×C₄	—	2	6	no	—
111	C₂²×C₄	—	2	6	no	—
112	C₂²×C₄	—	2	6	no	—
113	C₂²×C₄	C₂⁴	1	3	yes	—
114	Q₁₆	—	2	24	no	—
115	Q₁₆	—	2	3	no	—
116	Q₁₆	—	2	3	no	—
117	Q₁₆	—	2	12	no	—
118	SD₁₆	—	2	8	no	—
119	SD₁₆	—	2	24	no	—
120	SD₁₆	—	2	12	no	—
121	D₈	—	2	8	no	—
122	D₈	—	2	1	no	—
123	D₈	—	2	1	no	—
124	M₁₆	—	2	6	no	—
125	M₁₆	—	2	12	no	—
126	M₁₆	—	2	12	no	—
127	M₁₆	—	2	4	no	—
128	M₁₆	C₂×D₄	1	12	yes	—
129	M₁₆	C₂⁴	1	4	yes	—
130	M₁₆	C₂⁴	1	8	yes	—
131	C₂×C₈	—	2	6	no	—
132	C₂×C₈	—	2	12	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	12	no	—
135	C₂×C₈	C₂×D₄	1	6	yes	—
136	C₂×C₈	C₂×D₄	1	6	yes	—
137	C₂×C₈	C₂⁴	1	1	yes	characteristic
138	C₂×C₈	C₂⁴	1	3	yes	—
139	C₂³	—	2	6	no	—
140	C₂³	—	2	2	no	—
141	C₂³	C₂×#41	1	3	yes	—
142	Q₈	—	2	6	no	—
143	Q₈	—	2	6	no	—
144	Q₈	—	2	6	no	—
145	Q₈	—	4	12	no	—
146	Q₈	#121	1	4	yes	—
147	Q₈	C₂²×D₄	1	4	yes	—
148	D₄	—	2	12	no	—
149	D₄	—	2	4	no	—
150	D₄	—	2	6	no	—
151	D₄	—	2	2	no	—
152	D₄	—	2	2	no	—
153	D₄	—	4	12	no	—
154	D₄	#121	1	12	yes	—
155	D₄	C₂²×D₄	1	4	yes	—
156	C₂×C₄	—	2	6	no	—
157	C₂×C₄	—	2	12	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	12	no	—
161	C₂×C₄	—	4	12	no	—
162	C₂×C₄	C₂²×D₄	1	6	yes	—
163	C₂×C₄	C₂²×D₄	1	6	yes	—
164	C₂×C₄	C₂×#41	1	3	yes	—
165	C₂×C₄	C₂⁵	1	1	yes	characteristic
166	C₈	—	2	4	no	—
167	C₈	—	2	8	no	—
168	C₈	C₂²×D₄	1	1	yes	characteristic
169	C₈	C₂²×D₄	1	6	yes	—
170	C₈	C₂²×D₄	1	1	yes	characteristic
171	C₂²	—	2	4	no	—
172	C₂²	—	4	3	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	8	6	no	—
175	C₂²	—	8	2	no	—
176	C₂²	C₂×#121	1	6	yes	—
177	C₂²	C₂²×#41	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	4	6	no	—
180	C₄	C₂×#121	1	6	yes	—
181	C₄	C₂³×D₄	1	1	yes	characteristic
182	C₄	C₂³×D₄	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	6	no	—
185	C₂	—	4	4	no	—
186	C₂	—	8	2	no	—
187	C₂	C₂²×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂²×#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#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. #3586	C₂	semidirect
2. C₂×#578	C₂	semidirect
3. #3573	C₂	semidirect
4. #3069	C₂	semidirect
5. #3067	C₂	semidirect
6. C₂×#473	C₂	semidirect
7. #2916	C₂	semidirect
8. #2889	C₂	semidirect
17. C₂×#137	C₂²	semidirect
18. C₂×#135	C₂²	semidirect
19. C₂²×Q₁₆	C₂²	semidirect
27. #473	C₂²	semidirect
28. #473	C₂²	semidirect
29. #472	C₂²	semidirect
30. #471	C₂²	semidirect
31. #430	C₂²	semidirect
32. C₂×#108	C₂²	semidirect
33. #413	C₂²	semidirect
34. C₂×#101	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3586	2
C₂×#578	2
#3573	2
#3069	2
#3067	2
C₂×#473	2
#2916	2
#2889	2