Finite Group Databasebeta

← #24307

Subgroups — #24307

Order 256.

539 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C22×#125C211yesmaximal, characteristic
2C2×#479C212yesmaximal, complemented
3C4×C22⋊Q8C211yesmaximal, characteristic
4#2059C212yesmaximal, complemented
5#2026C212yesmaximal, complemented
6#1989C212yesmaximal, complemented
7#1872C211yesmaximal, characteristic
8#1872C211yesmaximal, characteristic
9#1853C211yesmaximal, characteristic
10#1852C211yesmaximal, characteristic
11#1797C211yesmaximal, characteristic
12C2×#125—28no—
13C2×#125—21no—
14C2×#125—21no—
15C2×#125C2212yes—
16C2×#125C2212yes—
17C2×C22⋊Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C22×C4⋊C4—21no—
20C22×C4⋊C4—21no—
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#479—28no—
25C2×#107C2212yes—
26C2×#106C2212yes—
27C2×C4⋊C8C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32#391C2211yescharacteristic
33#389C2211yescharacteristic
34#387—21no—
35#385—21no—
36#380C2211yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4C2211yescharacteristic
39C4×C22⋊C4—21no—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2312yes—
53#125—28no—
54#125—28no—
55#125—24no—
56#125—24no—
57#125—24no—
58#125D418yes—
59C22⋊Q8—22no—
60C22⋊Q8Q814yes—
61C4×Q8—24no—
62C2×C4⋊C4—28no—
63C2×C4⋊C4—28no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C42—24no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42D411yescharacteristic
86C2×C42D411yescharacteristic
87C2×C42C2311yescharacteristic
88C2×C42C2311yescharacteristic
89#107—24no—
90#106—24no—
91C4⋊C8—24no—
92#103—24no—
93#103—24no—
94C22⋊C8—22no—
95C22⋊C8—22no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—21no—
101C24C2×D411yescharacteristic
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Q812yes—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×Q812yes—
132C22×C4#4112yes—
133C22×C4C2411yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C4⋊C4—24no—
141C4⋊C4—24no—
142C4⋊C4—28no—
143C4⋊C4—28no—
144C4⋊C4—28no—
145C4⋊C4—28no—
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4C2×D414yes—
155C4⋊C4C2×D414yes—
156C4⋊C4C2×Q812yes—
157C22⋊C4—24no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42—42no—
163C42—42no—
164C42C2×D414yes—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23C22⋊Q812yes—
170C23#13512yes—
171C23#13712yes—
172C23C22×D411yescharacteristic
173Q8—24no—
174Q8—42no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—48no—
196C2×C4—48no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4#12012yes—
212C2×C4#12012yes—
213C2×C4C22⋊Q812yes—
214C2×C4C22×D412yes—
215C2×C4C22×Q811yescharacteristic
216C2×C4C2×#4111yescharacteristic
217C2×C42−1+412yes—
218C8—42no—
219C8—42no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22#47914yescomplemented
227C22C2×#12011yescharacteristic
228C22C2×C22⋊Q811yescharacteristic
229C22#53611yescharacteristic
230C22C2×#13511yescharacteristic
231C22C2×#13711yescharacteristic
232C22#57511yescharacteristic
233C22#57511yescharacteristic
234C4—22no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—42no—
239C4—41no—
240C4—44no—
241C4—41no—
242C4—44no—
243C4—81no—
244C4—81no—
245C4#54812yes—
246C4D4×Q812yes—
247C2—24no—
248C2#240911yescharacteristic
249C2#300211yescharacteristic
250C2#300711yescharacteristic
251C2C2×#47911yescharacteristic
252C2#308011yescharacteristic
253C2#323211yescharacteristic
254C2#323611yescharacteristic
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×#479C2semidirect
4. #2059C2semidirect
5. #2026C2semidirect
6. #1989C2semidirect
226. C22#479semidirect

Maximal subgroups

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

subgroupindex
C22×#1252
C2×#4792
C4×C22⋊Q82
#20592
#20262
#19892
#18722
#18532
#18522
#17972