Finite Group Databasebeta

← #23577

Subgroups — #23577

Order 256.

572 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#533C211yesmaximal, characteristic
2C2×#441C212yesmaximal, complemented
3C4×C22⋊Q8C211yesmaximal, characteristic
4#2058C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2035C212yesmaximal
7#2024C211yesmaximal, characteristic, complemented
8#2016C211yesmaximal, characteristic, complemented
9#2001C212yesmaximal
10#1763C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic
12C22×Q16C2212yes—
13#533—28no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2212yes—
23C2×#117—21no—
24C22×C42C2211yescharacteristic
25#441—28no—
26C2×#106C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×C22⋊C8C2212yes—
32C2×#97C2212yes—
33#391C2211yescharacteristic
34#389C2211yescharacteristic
35#387—21no—
36#385—21no—
37#380C2211yescharacteristic
38C4×C4⋊C4—21no—
39C4×C4⋊C4C2211yescharacteristic
40C4×C22⋊C4—21no—
41#340C2212yes—
42#340C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C22×Q8—21no—
46C22×Q8C2312yes—
47C22×Q8C2311yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C2×Q16—24no—
52C2×Q16—24no—
53C2×Q16—24no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—24no—
57C4⋊Q8—24no—
58#125—28no—
59C22⋊Q8—28no—
60C22⋊Q8—24no—
61C22⋊Q8—22no—
62C22⋊Q8—24no—
63C22⋊Q8—22no—
64C4×Q8—28no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×Q8D418yes—
68#117—28no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—22no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C42—22no—
85C2×C42—24no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42D411yescharacteristic
89C2×C42D411yescharacteristic
90C2×C42C2311yescharacteristic
91C2×C42C2311yescharacteristic
92#106—24no—
93#103—24no—
94#103—24no—
95#103—24no—
96#103—24no—
97C22⋊C8—24no—
98#97—24no—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104C24C2×D411yescharacteristic
105C2×Q8—24no—
106C2×Q8—22no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8C2×D414yes—
117C2×Q8#4112yes—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2411yescharacteristic
140Q16—216no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8#4114yes—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—28no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—28no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4C2×D414yes—
160C4⋊C4#4112yes—
161C22⋊C4—28no—
162C22⋊C4—24no—
163C22⋊C4—42no—
164C42—22no—
165C42—22no—
166C42—28no—
167C42—22no—
168C42—42no—
169C42—42no—
170C42C2×D414yes—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23#13712yes—
178C23C22×D411yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—24no—
182Q8—48no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186C2×C4—28no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—44no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4#12012yes—
220C2×C4#12012yes—
221C2×C4#13712yes—
222C2×C4#13712yes—
223C2×C4C22×D412yes—
224C2×C4C2×#4111yescharacteristic
225C2×C4C2×#4112yes—
226C2×C4C2×#4111yescharacteristic
227C8—24no—
228C8—42no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22C2×#12011yescharacteristic
237C22#52511yescharacteristic
238C22#53211yescharacteristic
239C22C2×#13711yescharacteristic
240C22C2×#13711yescharacteristic
241C22C2×#13711yescharacteristic
242C22#57511yescharacteristic
243C4—22no—
244C4—22no—
245C4—22no—
246C4—22no—
247C4—42no—
248C4—41no—
249C4—44no—
250C4—42no—
251C4—44no—
252C4—81no—
253C4—81no—
254C4#52512yes—
255C4#53212yes—
256C2—42no—
257C2#238811yescharacteristic
258C2#300411yescharacteristic
259C2#300711yescharacteristic
260C2#311311yescharacteristic
261C2#311611yescharacteristic
262C2#318111yescharacteristic
263C2#318511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#441C2semidirect
4. #2058C2semidirect
5. #2043C2semidirect
7. #2024C2semidirect
8. #2016C2semidirect
10. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4412
C4×C22⋊Q82
#20582
#20432
#20352
#20242
#20162
#20012
#17632
#17492