Finite Group Databasebeta

Subgroups — #33851

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×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#3285	C₂	1	2	yes	maximal, complemented
4	#2249	C₂	1	1	yes	maximal, characteristic, complemented
5	#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	#2211	C₂	1	2	yes	maximal, 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	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	C₂²	1	2	yes	—
13	C₂²×Q₁₆	C₂²	1	2	yes	—
14	#548	—	2	2	no	—
15	#508	—	2	4	no	—
16	C₂×Q₃₂	—	2	4	no	—
17	C₂×Q₃₂	C₂²	1	2	yes	—
18	C₂×Q₃₂	C₂²	1	1	yes	characteristic
19	C₂×Q₃₂	C₂²	1	1	yes	characteristic
20	C₂×SD₃₂	C₂²	1	2	yes	—
21	C₂×SD₃₂	C₂²	1	2	yes	—
22	C₂×D₁₆	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆	C₂²	1	1	yes	characteristic
25	#495	C₂²	1	2	yes	—
26	#467	C₂²	1	2	yes	—
27	#466	C₂²	1	2	yes	—
28	#464	—	2	2	no	—
29	#452	—	2	2	no	—
30	#448	—	2	2	no	—
31	C₄×Q₁₆	C₂²	1	2	yes	—
32	C₈×D₄	C₂²	1	1	yes	characteristic
33	#366	C₂²	1	1	yes	characteristic
34	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	#358	C₂²	1	2	yes	—
36	#358	C₂²	1	2	yes	—
37	#357	C₂²	1	2	yes	—
38	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×#41	—	2	2	no	—
42	C₂²×Q₈	—	2	2	no	—
43	#135	—	2	4	no	—
44	#135	—	4	2	no	—
45	C₂×Q₁₆	—	2	4	no	—
46	C₂×Q₁₆	—	2	2	no	—
47	C₂×Q₁₆	—	4	2	no	—
48	C₂×Q₁₆	C₂³	1	2	yes	—
49	C₂×Q₁₆	C₂³	1	2	yes	—
50	C₂×SD₁₆	—	2	2	no	—
51	C₂×SD₁₆	—	2	2	no	—
52	C₂×D₈	C₂³	1	2	yes	—
53	C₂²×C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₈	C₂³	1	1	yes	characteristic
55	#124	—	2	2	no	—
56	#124	—	4	2	no	—
57	C₂²⋊Q₈	—	2	2	no	—
58	C₂²⋊Q₈	—	4	2	no	—
59	#121	—	2	2	no	—
60	#121	—	4	2	no	—
61	C₄×Q₈	—	2	2	no	—
62	C₄×D₄	—	4	2	no	—
63	C₄×D₄	D₄	1	1	yes	characteristic
64	Q₃₂	—	2	4	no	—
65	Q₃₂	—	2	2	no	—
66	Q₃₂	D₄	1	4	yes	complemented
67	SD₃₂	—	2	4	no	—
68	SD₃₂	—	4	2	no	—
69	D₁₆	—	2	2	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₁₆	C₂³	1	1	yes	characteristic
75	C₂×C₁₆	C₂³	1	1	yes	characteristic
76	#107	C₂³	1	2	yes	—
77	C₄⋊C₈	D₄	1	1	yes	characteristic
78	#103	—	2	2	no	—
79	#103	—	2	2	no	—
80	#102	—	2	2	no	—
81	C₂²⋊C₈	D₄	1	1	yes	characteristic
82	C₂²⋊C₈	D₄	1	1	yes	characteristic
83	C₄×C₈	C₂³	1	1	yes	characteristic
84	#41	—	4	4	no	—
85	#41	—	8	2	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	4	4	no	—
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	8	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	4	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	D₈	1	1	yes	characteristic
95	C₂²×C₄	—	4	2	no	—
96	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
99	Q₁₆	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	2	2	no	—
102	Q₁₆	—	4	4	no	—
103	Q₁₆	C₂×D₄	1	4	yes	—
104	SD₁₆	—	4	4	no	—
105	SD₁₆	—	8	2	no	—
106	D₈	—	2	2	no	—
107	D₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	4	2	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	D₈	1	1	yes	characteristic
120	C₂²⋊C₄	—	4	2	no	—
121	C₂²⋊C₄	—	4	2	no	—
122	C₂²⋊C₄	—	4	2	no	—
123	C₂²⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	D₈	1	1	yes	characteristic
125	C₂²⋊C₄	D₈	1	1	yes	characteristic
126	C₄²	—	4	2	no	—
127	C₄²	C₂×D₄	1	1	yes	characteristic
128	C₁₆	—	2	1	no	—
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	1	no	—
131	C₁₆	C₂×D₄	1	2	yes	—
132	C₂³	—	4	2	no	—
133	C₂³	C₂×D₈	1	1	yes	characteristic
134	C₂³	C₂×D₈	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	4	2	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	2	no	—
139	Q₈	—	8	4	no	—
140	D₄	—	2	2	no	—
141	D₄	—	4	2	no	—
142	D₄	—	4	2	no	—
143	D₄	—	8	4	no	—
144	D₄	—	8	4	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	—	8	4	no	—
154	C₂×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	C₂²×D₄	1	1	yes	characteristic
163	C₈	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	8	2	no	—
170	C₂²	—	8	2	no	—
171	C₂²	—	8	2	no	—
172	C₂²	#508	1	2	yes	complemented
173	C₂²	C₂²×D₈	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	4	4	no	—
179	C₄	—	8	2	no	—
180	C₄	—	8	2	no	—
181	C₄	D₄²	1	1	yes	characteristic
182	C₄	#576	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	4	1	no	—
185	C₂	—	16	2	no	—
186	C₂	D₄×D₈	1	1	yes	characteristic
187	C₂	C₂×#508	1	1	yes	characteristic
188	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
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×#508	C₂	semidirect
2. C₂²×Q₃₂	C₂	semidirect
3. #3285	C₂	semidirect
4. #2249	C₂	semidirect
5. #2216	C₂	semidirect
6. #2215	C₂	semidirect
7. #2211	C₂	semidirect
8. #2189	C₂	semidirect
9. #2185	C₂	semidirect
10. C₄×Q₃₂	C₂	semidirect
11. D₄×C₁₆	C₂	semidirect
66. Q₃₂	D₄	semidirect
172. C₂²	#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508	2
C₂²×Q₃₂	2
#3285	2
#2249	2
#2216	2
#2215	2
#2211	2
#2189	2
#2185	2
C₄×Q₃₂	2
D₄×C₁₆	2