Finite Group Databasebeta

← #24711

Subgroups — #24711

Order 256.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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×#477C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic
3#2857C211yesmaximal, characteristic
4C4×C22⋊Q8C211yesmaximal, characteristic
5#2075C212yesmaximal
6#2035C212yesmaximal
7#1987C211yesmaximal, characteristic, complemented
8#1948C211yesmaximal, characteristic, complemented
9#1944C212yesmaximal, complemented
10#1935C212yesmaximal, complemented
11#1803C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C22×C42C2211yescharacteristic
17#477—24no—
18#432—24no—
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2212yes—
26C2×#97C2212yes—
27#391—21no—
28#391C2211yescharacteristic
29#389—21no—
30#389C2212yes—
31#389C2211yescharacteristic
32#387—21no—
33#385—21no—
34#384C2211yescharacteristic
35#380—21no—
36#380C2211yescharacteristic
37#380C2212yes—
38C4×C4⋊C4—21no—
39C4×C4⋊C4C2211yescharacteristic
40C4×C22⋊C4—21no—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C22⋊Q8—22no—
55C22⋊Q8—22no—
56C22⋊Q8—44no—
57C4×Q8—22no—
58C4×Q8—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42D411yescharacteristic
79C2×C42D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#107—24no—
83#106—24no—
84C4⋊C8—22no—
85C4⋊C8D414yes—
86#103—24no—
87#103—24no—
88C22⋊C8—24no—
89#97—24no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—22no—
96#95—22no—
97C24C2×D411yescharacteristic
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×Q8#4112yes—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—41no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2411yescharacteristic
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8C2×D414yes—
137C4⋊C4—24no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4#4112yes—
154C22⋊C4—24no—
155C22⋊C4—42no—
156C22⋊C4—44no—
157C42—22no—
158C42—22no—
159C42—21no—
160C42—21no—
161C42—42no—
162C42—42no—
163C42C2×D412yes—
164C42#4112yes—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23AGL(1,ℤ/8)12yes—
172C23C22×D411yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—41no—
176Q8—41no—
177Q8—84no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—44no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—44no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4#12112yes—
214C2×C4#12112yes—
215C2×C4#13712yes—
216C2×C4#13712yes—
217C2×C4C22×D411yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C4C2×#4111yescharacteristic
220C2×C4C2×#4111yescharacteristic
221C2×C42−1+411yescharacteristic
222C2×C42−1+411yescharacteristic
223C8—24no—
224C8—42no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22C2×#12111yescharacteristic
233C22#52511yescharacteristic
234C22#53611yescharacteristic
235C22C2×AGL(1,ℤ/8)11yescharacteristic
236C22C2×#13711yescharacteristic
237C22C2×#13711yescharacteristic
238C22#57511yescharacteristic
239C4—22no—
240C4—42no—
241C4—41no—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—82no—
246C4—81no—
247C4—81no—
248C4—82no—
249C4#47014yes—
250C4#54711yescharacteristic
251C4#54811yescharacteristic
252C4#55211yescharacteristic
253C4#55611yescharacteristic
254C2—42no—
255C2#241511yescharacteristic
256C2C2×#47011yescharacteristic
257C2#305511yescharacteristic
258C2#311211yescharacteristic
259C2#311611yescharacteristic
260C2#323711yescharacteristic
261C2#323811yescharacteristic
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
1. C2×#477C2semidirect
7. #1987C2semidirect
8. #1948C2semidirect
9. #1944C2semidirect
10. #1935C2semidirect
11. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4322
#28572
C4×C22⋊Q82
#20752
#20352
#19872
#19482
#19442
#19352
#18032