Finite Group Databasebeta

← #24130

Subgroups — #24130

Order 256.

592 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 135 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, complemented
2C2×#474C211yesmaximal, characteristic, complemented
3C2×#440C211yesmaximal, characteristic, complemented
4#2535C211yesmaximal, characteristic
5#2036C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1874C212yesmaximal, complemented
10#1852C212yesmaximal, complemented
11#1805C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13#554—24no—
14#554—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—21no—
17C2×C22⋊Q8—22no—
18C2×C22⋊Q8C2212yes—
19C2×#121C2212yes—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C4⋊C4C2212yes—
25#474—24no—
26#440—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2212yes—
32C2×#102C2211yescharacteristic
33C2×#102C2212yes—
34C2×C22⋊C8C2212yes—
35C2×C22⋊C8C2212yes—
36C2×#97C2211yescharacteristic
37#397—21no—
38#392—21no—
39#386—22no—
40#382C2211yescharacteristic
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×D4—22no—
48C22×D4D411yescharacteristic
49C22×D4C2311yescharacteristic
50C23×C4—22no—
51C23×C4—21no—
52C23×C4C2312yes—
53C2×SD16—22no—
54C2×SD16—22no—
55C2×SD16—22no—
56C22×C8C2311yescharacteristic
57C22×C8C2312yes—
58C22×C8C2311yescharacteristic
59C4⋊Q8—22no—
60C4⋊Q8D414yescomplemented
61#125—28no—
62C22⋊Q8—216no—
63C22⋊Q8—24no—
64C22⋊Q8—24no—
65#121—24no—
66#121—42no—
67C4×Q8—22no—
68C4×Q8D414yescomplemented
69C4×D4—28no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4D412yes—
90C2×C42—21no—
91C2×C42C2311yescharacteristic
92#107—22no—
93#107—24no—
94C4⋊C8—22no—
95#103—22no—
96#103—42no—
97#102—22no—
98#102—24no—
99C22⋊C8—24no—
100C22⋊C8—42no—
101#97—22no—
102#95—22no—
103#95—21no—
104#95—21no—
105C24—21no—
106C24C2×D412yes—
107C2×Q8—24no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—82no—
114C2×Q8C2×D412yes—
115C2×D4—24no—
116C2×D4—22no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—44no—
123C2×D4—44no—
124C2×D4—82no—
125C2×D4#4112yes—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—28no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4C2×D412yes—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2411yescharacteristic
146SD16—28no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8—21no—
155C2×C8#4112yes—
156C4⋊C4—28no—
157C4⋊C4—28no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D414yes—
169C22⋊C4—28no—
170C22⋊C4—24no—
171C22⋊C4—44no—
172C22⋊C4—42no—
173C22⋊C4—82no—
174C42—24no—
175C42C2×D412yes—
176C42C2×D412yes—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23#13714yes—
189C23C22×D411yescharacteristic
190Q8—22no—
191Q8—22no—
192Q8—41no—
193Q8—41no—
194Q8—84no—
195D4—28no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—41no—
200D4—88no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—84no—
224C2×C4—88no—
225C2×C4#12012yes—
226C2×C4#12012yes—
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C2×#4111yescharacteristic
230C2×C4C2×#4111yescharacteristic
231C2×C4H(5,𝔽2)11yescharacteristic
232C2×C42−1+411yescharacteristic
233C8—22no—
234C8—41no—
235C8—42no—
236C22—24no—
237C22—24no—
238C22—24no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—44no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22C2×#12011yescharacteristic
249C22#54611yescharacteristic
250C22#54711yescharacteristic
251C22C2×#13712yes—
252C22#57612yes—
253C4—22no—
254C4—22no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—44no—
259C4—81no—
260C4—82no—
261C4#52511yescharacteristic
262C4#53711yescharacteristic
263C4#54811yescharacteristic
264C4#55011yescharacteristic
265C2—44no—
266C2—82no—
267C2#240711yescharacteristic
268C2#300411yescharacteristic
269C2#300811yescharacteristic
270C2#331412yes—
271C2#334612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#554C2semidirect
2. C2×#474C2semidirect
3. C2×#440C2semidirect
5. #2036C2semidirect
6. #2000C2semidirect
7. #1993C2semidirect
8. #1955C2semidirect
9. #1874C2semidirect
10. #1852C2semidirect
11. #1805C2semidirect
60. C4⋊Q8D4semidirect
68. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4742
C2×#4402
#25352
#20362
#20002
#19932
#19552
#18742
#18522
#18052