Finite Group Databasebeta

Subgroups — #33751

Order 256.

274 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 107 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	#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	#3169	C₂	1	1	yes	maximal, characteristic
3	#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	#2220	C₂	1	4	yes	maximal, complemented
8	#2175	C₂	1	4	yes	maximal, complemented
9	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#533	—	2	1	no	—
11	#532	—	2	1	no	—
12	C₂×SD₃₂	C₂²	1	4	yes	—
13	C₂×M₃₂	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	1	yes	characteristic
16	#494	C₂²	1	1	yes	characteristic
17	#493	C₂²	1	1	yes	characteristic
18	#487	—	2	1	no	—
19	#486	—	2	1	no	—
20	#467	—	2	1	no	—
21	#467	C₂²	1	2	yes	—
22	#466	—	2	1	no	—
23	#466	C₂²	1	2	yes	—
24	C₄×Q₁₆	—	2	1	no	—
25	C₄×Q₁₆	C₂²	1	2	yes	—
26	C₄×D₈	—	2	1	no	—
27	C₄×D₈	C₂²	1	2	yes	—
28	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
29	#366	C₂²	1	4	yes	—
30	#358	C₂²	1	4	yes	—
31	#357	C₂²	1	4	yes	—
32	#346	C₂²	1	2	yes	—
33	C₄×C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×#41	—	4	1	no	—
36	C₂×Q₁₆	—	2	1	no	—
37	C₂×Q₁₆	C₂³	1	2	yes	—
38	C₂×D₈	—	2	1	no	—
39	C₂×D₈	C₂³	1	2	yes	—
40	C₂²×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈	—	2	1	no	—
43	C₄⋊Q₈	—	2	1	no	—
44	C₄⋊Q₈	—	2	1	no	—
45	#127	—	2	1	no	—
46	#125	—	4	1	no	—
47	#124	—	4	1	no	—
48	C₂²⋊Q₈	—	2	2	no	—
49	C₂²⋊Q₈	—	4	1	no	—
50	#121	—	2	2	no	—
51	#121	—	4	1	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	4	1	no	—
54	C₄×D₄	—	2	2	no	—
55	C₄×D₄	—	4	1	no	—
56	#117	—	4	1	no	—
57	C₂×C₄²	D₄	1	1	yes	characteristic
58	SD₃₂	—	2	8	no	—
59	M₃₂	—	2	4	no	—
60	C₂×C₁₆	C₂³	1	2	yes	—
61	C₂×C₁₆	C₂³	1	2	yes	—
62	#107	—	2	1	no	—
63	#107	—	2	1	no	—
64	#107	C₂³	1	2	yes	—
65	#107	C₂³	1	2	yes	—
66	#103	—	2	2	no	—
67	#103	—	2	2	no	—
68	#102	—	2	2	no	—
69	#102	—	2	2	no	—
70	C₄×C₈	D₄	1	1	yes	characteristic
71	C₄×C₈	D₄	1	1	yes	characteristic
72	C₄×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₈	C₂³	1	1	yes	characteristic
74	#41	—	8	4	no	—
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	4	1	no	—
77	C₂×Q₈	—	4	1	no	—
78	C₂×D₄	—	2	2	no	—
79	C₂×D₄	—	4	2	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂×D₄	—	4	1	no	—
82	C₂²×C₄	—	4	2	no	—
83	C₂²×C₄	D₈	1	1	yes	characteristic
84	C₂²×C₄	D₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	4	2	no	—
88	D₈	—	2	4	no	—
89	D₈	—	4	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₂²⋊C₄	—	4	2	no	—
105	C₂²⋊C₄	—	4	2	no	—
106	C₂²⋊C₄	—	4	2	no	—
107	C₄²	—	4	2	no	—
108	C₄²	D₈	1	1	yes	characteristic
109	C₄²	D₈	1	1	yes	characteristic
110	C₄²	C₂×D₄	1	1	yes	characteristic
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₁₆	—	2	2	no	—
113	C₁₆	#41	1	4	yes	complemented
114	C₂³	—	4	2	no	—
115	C₂³	C₂×D₈	1	1	yes	characteristic
116	Q₈	—	4	2	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	8	2	no	—
119	Q₈	—	8	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	8	4	no	—
123	D₄	—	8	2	no	—
124	D₄	—	8	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	8	4	no	—
133	C₂×C₄	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	#135	1	2	yes	—
142	C₈	C₂×#41	1	2	yes	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	8	2	no	—
147	C₂²	—	8	2	no	—
148	C₂²	—	8	2	no	—
149	C₂²	C₂²×D₈	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	8	2	no	—
154	C₄	—	8	2	no	—
155	C₄	—	8	2	no	—
156	C₄	#509	1	2	yes	—
157	C₄	#510	1	2	yes	—
158	C₄	#532	1	1	yes	characteristic
159	C₄	C₂×#135	1	1	yes	characteristic
160	C₂	—	4	1	no	—
161	C₂	—	16	2	no	—
162	C₂	#3169	1	1	yes	characteristic
163	C₂	C₂×#509	1	1	yes	characteristic
164	C₂	C₂×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3170	C₂	semidirect
3. #2252	C₂	semidirect
4. #2248	C₂	semidirect
5. #2241	C₂	semidirect
6. #2240	C₂	semidirect
7. #2220	C₂	semidirect
8. #2175	C₂	semidirect
9. C₄×M₃₂	C₂	semidirect
113. C₁₆	#41	semidirect

Maximal subgroups

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

subgroup	index
#3170	2
#3169	2
#2252	2
#2248	2
#2241	2
#2240	2
#2220	2
#2175	2
C₄×M₃₂	2