Finite Group Databasebeta

← #25613

Subgroups — #25613

Order 256.

558 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 131 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×#542C211yesmaximal, characteristic
2C2×#497C211yesmaximal, characteristic, complemented
3C2×#487C211yesmaximal, characteristic, complemented
4#2621C211yesmaximal, characteristic, complemented
5#2001C212yesmaximal, complemented
6#2000C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1958C211yesmaximal, characteristic, complemented
9#1878C212yesmaximal, complemented
10#1877C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14#542—24no—
15#542—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#126—21no—
19C2×#124—21no—
20C2×#124C2211yescharacteristic
21C2×#123—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2212yes—
25C2×#121C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#497—24no—
30#487—24no—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×#97C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#398—21no—
39#398—21no—
40#397—21no—
41#397—21no—
42#391C2212yes—
43#382C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2312yes—
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C2×Q16—24no—
55C2×Q16—24no—
56C2×Q16—42no—
57C2×SD16—24no—
58C2×SD16—24no—
59C2×SD16—42no—
60C22×C8C2312yes—
61C22×C8C2312yes—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65C4⋊Q8D414yes—
66#126—28no—
67#124—28no—
68#124—22no—
69#124D414yes—
70#123—28no—
71C22⋊Q8—28no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—42no—
75C22⋊Q8—42no—
76#121—22no—
77#121—22no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×D4—22no—
81C4×D4—22no—
82#117—22no—
83#117—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#103—24no—
107#103—24no—
108#103—42no—
109#102—42no—
110C22⋊C8—24no—
111#97—22no—
112C4×C8—22no—
113#95—21no—
114#95—21no—
115#95—22no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—82no—
130C2×Q8C2×D412yes—
131C2×D4—24no—
132C2×D4—21no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—82no—
136C2×D4C2×D412yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—24no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—82no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2411yescharacteristic
154Q16—48no—
155SD16—48no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8#4114yes—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4C2×D412yes—
175C4⋊C4C2×D412yes—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—44no—
184C42—24no—
185C42—21no—
186C42—21no—
187C42C2×D412yes—
188C42C2×D412yes—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23—22no—
195C23—41no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23C22×D411yescharacteristic
200Q8—24no—
201Q8—42no—
202Q8—42no—
203Q8—42no—
204Q8—42no—
205Q8—42no—
206Q8—84no—
207D4—24no—
208D4—42no—
209D4—84no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4#12012yes—
236C2×C4#12012yes—
237C2×C4C22×D411yescharacteristic
238C2×C4C22×D411yescharacteristic
239C2×C4C2×#4112yes—
240C2×C4H(5,𝔽2)12yes—
241C8—24no—
242C8—42no—
243C22—42no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—82no—
252C22—82no—
253C22C2×#12011yescharacteristic
254C22#53211yescharacteristic
255C22#53511yescharacteristic
256C22#57712yes—
257C22#57812yes—
258C4—22no—
259C4—22no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—42no—
264C4—42no—
265C4—41no—
266C4—82no—
267C4—82no—
268C4#54612yes—
269C4#55012yes—
270C2—81no—
271C2—82no—
272C2#243111yescharacteristic
273C2#300911yescharacteristic
274C2#301111yescharacteristic
275C2#317812yes—
276C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#497C2semidirect
3. C2×#487C2semidirect
4. #2621C2semidirect
5. #2001C2semidirect
7. #1993C2semidirect
8. #1958C2semidirect
9. #1878C2semidirect
10. #1877C2semidirect
11. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4972
C2×#4872
#26212
#20012
#20002
#19932
#19582
#18782
#18772
#17502