Finite Group Databasebeta

Subgroups — #33842

Order 256.

332 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#510	C₂	1	2	yes	maximal, complemented
2	D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#3283	C₂	1	1	yes	maximal, characteristic
4	#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	#2219	C₂	1	2	yes	maximal, complemented
6	#2209	C₂	1	1	yes	maximal, characteristic, complemented
7	#2208	C₂	1	1	yes	maximal, characteristic, complemented
8	#2189	C₂	1	2	yes	maximal, complemented
9	#2188	C₂	1	2	yes	maximal, complemented
10	C₄×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	C₂²	1	2	yes	—
13	C₂²×Q₁₆	C₂²	1	2	yes	—
14	D₄×Q₈	—	2	1	no	—
15	#547	—	2	1	no	—
16	#510	—	2	8	no	—
17	C₂×Q₃₂	C₂²	1	2	yes	—
18	C₂×Q₃₂	C₂²	1	2	yes	—
19	C₂×Q₃₂	C₂²	1	2	yes	—
20	C₂×SD₃₂	C₂²	1	2	yes	—
21	C₂×SD₃₂	C₂²	1	1	yes	characteristic
22	C₂×M₃₂	C₂²	1	2	yes	—
23	#494	C₂²	1	1	yes	characteristic
24	#494	C₂²	1	1	yes	characteristic
25	#467	C₂²	1	2	yes	—
26	#467	C₂²	1	2	yes	—
27	#462	—	2	1	no	—
28	#461	—	2	1	no	—
29	#452	—	2	2	no	—
30	#451	—	2	2	no	—
31	C₄×Q₁₆	C₂²	1	1	yes	characteristic
32	C₄×D₈	C₂²	1	1	yes	characteristic
33	C₈×D₄	C₂²	1	1	yes	characteristic
34	#367	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
36	#358	C₂²	1	1	yes	characteristic
37	#358	C₂²	1	2	yes	—
38	#358	C₂²	1	2	yes	—
39	#357	C₂²	1	1	yes	characteristic
40	C₂²⋊C₁₆	C₂²	1	2	yes	—
41	C₄×C₁₆	C₂²	1	1	yes	characteristic
42	C₂×#41	—	2	2	no	—
43	C₂²×Q₈	—	2	2	no	—
44	#135	—	2	4	no	—
45	#135	—	4	2	no	—
46	C₂×Q₁₆	—	2	4	no	—
47	C₂×Q₁₆	—	2	2	no	—
48	C₂×Q₁₆	—	2	1	no	—
49	C₂×Q₁₆	—	2	1	no	—
50	C₂×Q₁₆	—	4	2	no	—
51	C₂×Q₁₆	C₂³	1	2	yes	—
52	C₂×Q₁₆	C₂³	1	2	yes	—
53	C₂×Q₁₆	C₂³	1	1	yes	characteristic
54	C₂×SD₁₆	—	2	2	no	—
55	C₂×D₈	C₂³	1	1	yes	characteristic
56	C₂²×C₈	C₂³	1	2	yes	—
57	C₄⋊Q₈	—	2	1	no	—
58	C₄⋊Q₈	—	2	1	no	—
59	C₄⋊Q₈	—	4	1	no	—
60	#123	—	4	2	no	—
61	C₂²⋊Q₈	—	2	2	no	—
62	C₂²⋊Q₈	—	2	2	no	—
63	C₂²⋊Q₈	—	4	2	no	—
64	#121	—	4	1	no	—
65	C₄×Q₈	—	2	1	no	—
66	C₄×D₄	—	2	1	no	—
67	C₄×D₄	—	4	1	no	—
68	C₄×D₄	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄	—	4	1	no	—
70	Q₃₂	—	2	4	no	—
71	Q₃₂	—	2	4	no	—
72	Q₃₂	—	4	2	no	—
73	SD₃₂	—	2	4	no	—
74	SD₃₂	D₄	1	4	yes	complemented
75	M₃₂	—	2	4	no	—
76	C₂×C₁₆	C₂³	1	1	yes	characteristic
77	C₂×C₁₆	C₂³	1	2	yes	—
78	C₂×C₁₆	C₂³	1	1	yes	characteristic
79	#107	C₂³	1	1	yes	characteristic
80	#107	C₂³	1	1	yes	characteristic
81	C₄⋊C₈	D₄	1	1	yes	characteristic
82	#103	—	2	1	no	—
83	#103	—	2	2	no	—
84	#103	—	2	2	no	—
85	#102	—	2	1	no	—
86	C₂²⋊C₈	D₄	1	2	yes	—
87	C₄×C₈	C₂³	1	1	yes	characteristic
88	#41	—	4	4	no	—
89	#41	—	8	2	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	4	4	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	8	2	no	—
96	C₂×D₄	—	2	1	no	—
97	C₂×D₄	—	4	2	no	—
98	C₂×D₄	D₈	1	1	yes	characteristic
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	4	1	no	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	C₂×D₄	1	2	yes	—
103	Q₁₆	—	2	2	no	—
104	Q₁₆	—	2	2	no	—
105	Q₁₆	—	2	1	no	—
106	Q₁₆	—	2	2	no	—
107	Q₁₆	—	2	2	no	—
108	Q₁₆	—	4	4	no	—
109	Q₁₆	—	8	1	no	—
110	Q₁₆	—	8	1	no	—
111	Q₁₆	C₂×D₄	1	2	yes	—
112	SD₁₆	—	4	4	no	—
113	D₈	—	2	1	no	—
114	D₈	C₂×D₄	1	2	yes	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈	C₂×D₄	1	2	yes	—
119	C₂×C₈	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	1	no	—
125	C₄⋊C₄	—	4	1	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	4	1	no	—
129	C₄⋊C₄	D₈	1	1	yes	characteristic
130	C₂²⋊C₄	—	4	2	no	—
131	C₂²⋊C₄	—	4	2	no	—
132	C₂²⋊C₄	—	4	1	no	—
133	C₂²⋊C₄	D₈	1	2	yes	—
134	C₄²	—	4	1	no	—
135	C₄²	C₂×D₄	1	1	yes	characteristic
136	C₁₆	—	2	2	no	—
137	C₁₆	—	2	1	no	—
138	C₁₆	C₂×D₄	1	2	yes	—
139	C₂³	—	4	1	no	—
140	C₂³	C₂×D₈	1	2	yes	—
141	Q₈	—	2	2	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	4	2	no	—
145	Q₈	—	4	2	no	—
146	Q₈	—	4	2	no	—
147	Q₈	—	8	4	no	—
148	D₄	—	2	2	no	—
149	D₄	—	2	2	no	—
150	D₄	—	4	1	no	—
151	D₄	—	8	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄	C₂×D₈	1	2	yes	—
164	C₂×C₄	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	2	1	no	—
168	C₈	C₂²×D₄	1	1	yes	characteristic
169	C₈	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	8	1	no	—
175	C₂²	—	8	1	no	—
176	C₂²	C₂²×D₈	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	8	1	no	—
181	C₄	—	8	1	no	—
182	C₄	—	8	2	no	—
183	C₄	—	8	2	no	—
184	C₄	#510	1	2	yes	complemented
185	C₄	D₄²	1	1	yes	characteristic
186	C₄	#576	1	1	yes	characteristic
187	C₂	—	4	2	no	—
188	C₂	—	8	2	no	—
189	C₂	D₄×D₈	1	1	yes	characteristic
190	C₂	C₂×#510	1	1	yes	characteristic
191	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3283	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ 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. C₂×#510	C₂	semidirect
2. D₄×Q₁₆	C₂	semidirect
4. #2247	C₂	semidirect
5. #2219	C₂	semidirect
6. #2209	C₂	semidirect
7. #2208	C₂	semidirect
8. #2189	C₂	semidirect
9. #2188	C₂	semidirect
10. C₄×SD₃₂	C₂	semidirect
11. #2169	C₂	semidirect
74. SD₃₂	D₄	semidirect
184. C₄	#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510	2
D₄×Q₁₆	2
#3283	2
#2247	2
#2219	2
#2209	2
#2208	2
#2189	2
#2188	2
C₄×SD₃₂	2
#2169	2