Finite Group Databasebeta

Subgroups — #33844

Order 256.

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

normal subgroups only

This group has 337 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	#3287	C₂	1	1	yes	maximal, characteristic
3	D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	#2220	C₂	1	2	yes	maximal, complemented
6	#2209	C₂	1	1	yes	maximal, characteristic, complemented
7	#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	#2189	C₂	1	2	yes	maximal, complemented
9	#2185	C₂	1	2	yes	maximal, complemented
10	C₄×Q₃₂	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	#550	—	2	1	no	—
15	D₄×Q₈	—	2	1	no	—
16	#510	—	2	8	no	—
17	C₂×Q₃₂	C₂²	1	2	yes	—
18	C₂×Q₃₂	C₂²	1	1	yes	characteristic
19	C₂×SD₃₂	C₂²	1	2	yes	—
20	C₂×SD₃₂	C₂²	1	2	yes	—
21	C₂×SD₃₂	C₂²	1	2	yes	—
22	C₂×M₃₂	C₂²	1	2	yes	—
23	#494	C₂²	1	1	yes	characteristic
24	#493	C₂²	1	1	yes	characteristic
25	#467	C₂²	1	2	yes	—
26	#466	C₂²	1	2	yes	—
27	#462	—	2	1	no	—
28	#460	—	2	1	no	—
29	#451	—	2	2	no	—
30	#449	—	2	2	no	—
31	C₄×Q₁₆	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆	C₂²	1	1	yes	characteristic
33	C₈×D₄	C₂²	1	1	yes	characteristic
34	#366	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
36	#358	C₂²	1	1	yes	characteristic
37	#358	C₂²	1	1	yes	characteristic
38	#358	C₂²	1	2	yes	—
39	#357	C₂²	1	2	yes	—
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	—	4	2	no	—
50	C₂×Q₁₆	C₂³	1	2	yes	—
51	C₂×Q₁₆	C₂³	1	1	yes	characteristic
52	C₂×Q₁₆	C₂³	1	1	yes	characteristic
53	C₂×SD₁₆	—	2	2	no	—
54	C₂×D₈	—	2	1	no	—
55	C₂×D₈	C₂³	1	2	yes	—
56	C₂²×C₈	C₂³	1	2	yes	—
57	C₄⋊Q₈	—	2	1	no	—
58	C₄⋊Q₈	—	4	1	no	—
59	#127	—	2	1	no	—
60	#127	—	4	1	no	—
61	C₂²⋊Q₈	—	2	2	no	—
62	C₂²⋊Q₈	—	4	2	no	—
63	#121	—	2	2	no	—
64	#121	—	4	2	no	—
65	C₄×Q₈	—	2	1	no	—
66	C₄×Q₈	—	2	1	no	—
67	C₄×D₄	—	4	1	no	—
68	C₄×D₄	—	4	1	no	—
69	C₄×D₄	D₄	1	1	yes	characteristic
70	Q₃₂	—	2	4	no	—
71	Q₃₂	D₄	1	4	yes	complemented
72	SD₃₂	—	2	4	no	—
73	SD₃₂	—	2	4	no	—
74	SD₃₂	—	4	2	no	—
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	1	no	—
84	#103	—	2	2	no	—
85	#102	—	2	2	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	1	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	2	no	—
97	C₂×D₄	—	4	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	D₈	1	1	yes	characteristic
102	C₂²×C₄	—	4	2	no	—
103	C₂²×C₄	—	4	2	no	—
104	C₂²×C₄	C₂×D₄	1	2	yes	—
105	Q₁₆	—	2	2	no	—
106	Q₁₆	—	2	1	no	—
107	Q₁₆	—	2	1	no	—
108	Q₁₆	—	2	2	no	—
109	Q₁₆	—	4	4	no	—
110	Q₁₆	—	8	1	no	—
111	Q₁₆	C₂×D₄	1	2	yes	—
112	Q₁₆	C₂×D₄	1	2	yes	—
113	SD₁₆	—	4	4	no	—
114	D₈	—	2	2	no	—
115	D₈	—	2	2	no	—
116	D₈	—	8	1	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₈	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	4	1	no	—
125	C₄⋊C₄	—	4	1	no	—
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	4	2	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	D₈	1	2	yes	—
133	C₄²	—	4	1	no	—
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	2	no	—
140	C₂³	C₂×D₈	1	2	yes	—
141	Q₈	—	2	2	no	—
142	Q₈	—	2	2	no	—
143	Q₈	—	4	1	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	2	no	—
146	Q₈	—	4	2	no	—
147	Q₈	—	8	4	no	—
148	D₄	—	2	2	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	2	no	—
151	D₄	—	8	4	no	—
152	D₄	—	8	2	no	—
153	D₄	—	16	1	no	—
154	D₄	—	16	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄	C₂×D₈	1	2	yes	—
166	C₂×C₄	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	1	no	—
170	C₈	C₂²×D₄	1	1	yes	characteristic
171	C₈	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	8	2	no	—
176	C₂²	—	8	2	no	—
177	C₂²	—	8	2	no	—
178	C₂²	C₂²×D₈	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	8	1	no	—
185	C₄	—	8	2	no	—
186	C₄	#510	1	2	yes	complemented
187	C₄	D₄²	1	1	yes	characteristic
188	C₄	#576	1	1	yes	characteristic
189	C₂	—	4	2	no	—
190	C₂	—	16	2	no	—
191	C₂	D₄×D₈	1	1	yes	characteristic
192	C₂	C₂×#510	1	1	yes	characteristic
193	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3287	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
3. D₄×Q₁₆	C₂	semidirect
4. #2248	C₂	semidirect
5. #2220	C₂	semidirect
6. #2209	C₂	semidirect
7. #2207	C₂	semidirect
8. #2189	C₂	semidirect
9. #2185	C₂	semidirect
10. C₄×Q₃₂	C₂	semidirect
11. #2169	C₂	semidirect
71. Q₃₂	D₄	semidirect
186. 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
#3287	2
D₄×Q₁₆	2
#2248	2
#2220	2
#2209	2
#2207	2
#2189	2
#2185	2
C₄×Q₃₂	2
#2169	2