Finite Group Databasebeta

← #25945

Subgroups — #25945

Order 256.

536 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 137 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×#554C211yesmaximal, characteristic
2C2×#491C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic
4#2568C211yesmaximal, characteristic, complemented
5#2026C212yesmaximal, complemented
6#2024C212yesmaximal, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1877C212yesmaximal
10#1877C212yesmaximal
11#1762C211yesmaximal, characteristic
12C22×SD16C2212yes—
13#554—24no—
14#554—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8—21no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4—21no—
27#491—24no—
28#461—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×#97C2211yescharacteristic
35#400—21no—
36#400—21no—
37#391—21no—
38#391C2212yes—
39#391C2212yes—
40#382—21no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8D411yescharacteristic
49C22×Q8C2312yes—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C2×SD16—24no—
55C2×SD16—28no—
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62#125—22no—
63#125—22no—
64#125—22no—
65#125D414yes—
66C22⋊Q8—28no—
67C22⋊Q8—28no—
68C22⋊Q8—28no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4D414yes—
77C2×C4⋊C4—28no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2312yes—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C4⋊C8—22no—
100#103—24no—
101#103—24no—
102#103—24no—
103#102—24no—
104#97—22no—
105#95—21no—
106#95—22no—
107#95—22no—
108#95—22no—
109#95—22no—
110C24—21no—
111C2×Q8—24no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—24no—
117C2×Q8—24no—
118C2×Q8—41no—
119C2×Q8—82no—
120C2×D4—21no—
121C2×D4—22no—
122C2×D4C2×D412yes—
123C2×D4#4114yes—
124C22×C4—24no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
143SD16—48no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—24no—
148C2×C8—22no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×D412yes—
166C4⋊C4C2×D412yes—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—42no—
175C42C2×D412yes—
176C42C2×D412yes—
177C23—22no—
178C23—24no—
179C23—41no—
180C23—41no—
181C23—42no—
182C23C22×D411yescharacteristic
183Q8—44no—
184Q8—42no—
185Q8—42no—
186Q8—44no—
187Q8—84no—
188D4—22no—
189D4—24no—
190D4#12414yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—48no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4#12012yes—
219C2×C4#12012yes—
220C2×C4#13712yes—
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C2×#4112yes—
224C2×C42−1+412yes—
225C8—42no—
226C8—42no—
227C22—24no—
228C22—44no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22C2×#12011yescharacteristic
233C22#54812yes—
234C22C2×#13711yescharacteristic
235C22#57511yescharacteristic
236C22#57712yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—41no—
241C4—42no—
242C4—41no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—82no—
248C4C2×#12411yescharacteristic
249C4#53611yescharacteristic
250C4#54812yes—
251C2—44no—
252C2#243611yescharacteristic
253C2#300711yescharacteristic
254C2#300911yescharacteristic
255C2#337212yes—
256C2#337612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#491C2semidirect
4. #2568C2semidirect
5. #2026C2semidirect
6. #2024C2semidirect
7. #1989C2semidirect
8. #1985C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4912
C2×#4612
#25682
#20262
#20242
#19892
#19852
#18772
#17622