Finite Group Databasebeta

← #32999

Subgroups — #32999

Order 256.

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

normal subgroups only

This group has 642 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×#520C211yesmaximal, characteristic
2#3095C211yesmaximal, characteristic, complemented
3#3007C211yesmaximal, characteristic
4#2503C211yesmaximal, characteristic, complemented
5#1882C214yesmaximal, complemented
6#1860C214yesmaximal, complemented
7#1826C211yesmaximal, characteristic, complemented
8#1821C211yesmaximal, characteristic, complemented
9#1789C211yesmaximal, characteristic
10C22×#41C2211yescharacteristic
11C2×#137C2212yes—
12C2×C22⋊Q8C2211yescharacteristic
13#520—22no—
14#520—22no—
15#520—22no—
16#520C2214yes—
17C2×C4×Q8—21no—
18C2×C4×D4—21no—
19C2×C4×D4—22no—
20C2×#117—21no—
21C2×#117C2211yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4C2211yescharacteristic
24#485—22no—
25#484—21no—
26#483—21no—
27#452—22no—
28#451—21no—
29#450—21no—
30#428C2212yescomplemented
31C2×#104C2214yes—
32#419C2212yescomplemented
33#417C2212yes—
34#407C2211yescharacteristic
35#407C2211yescharacteristic
36#400—22no—
37#391C2212yescomplemented
38#382C2212yescomplemented
39#381—22no—
40C4×C22⋊C4C2211yescharacteristic
41#343C2214yescomplemented
42#339C2214yescomplemented
43C2×#41—28no—
44C2×#41—21no—
45C2×#41—21no—
46C2×#41D412yes—
47C2×#41C2312yes—
48C22×Q8D411yescharacteristic
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4D411yescharacteristic
52C23×C4—21no—
53C23×C4—22no—
54C23×C4D412yes—
55C23×C4C2311yescharacteristic
56#137—28no—
57C2×Q16—22no—
58C2×SD16—22no—
59C2×M16C2312yes—
60C2×M16C2312yes—
61C22⋊Q8—21no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C22⋊Q8—21no—
65C4×Q8—24no—
66C4×Q8—24no—
67C4×D4—28no—
68C4×D4—24no—
69C4×D4—24no—
70C4×D4—28no—
71#117—24no—
72#117—21no—
73#117—21no—
74#117—24no—
75#117D412yes—
76#117C2312yes—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D412yes—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2312yes—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42C2312yes—
97#107—22no—
98#106—22no—
99#104—28no—
100#103—22no—
101#103—22no—
102#103—22no—
103#102—22no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108#95—21no—
109#95—24no—
110#95—24no—
111C24—21no—
112C24C2×D411yescharacteristic
113#41—28no—
114#41—28no—
115#41—24no—
116#41—24no—
117#41—24no—
118#41#4118yes—
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—24no—
124C2×Q8C2×D412yes—
125C2×D4—28no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—24no—
129C2×D4—21no—
130C2×D4C2×D412yes—
131C22×C4—28no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—28no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—44no—
145C22×C4—84no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D414yes—
152C22×C4C2411yescharacteristic
153Q16—44no—
154SD16—44no—
155M16—24no—
156M16—24no—
157C2×C8—22no—
158C2×C8—22no—
159C4⋊C4—28no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4C2×D412yes—
171C4⋊C4C2×D412yes—
172C22⋊C4—28no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—42no—
176C22⋊C4—41no—
177C22⋊C4—41no—
178C42—28no—
179C42—22no—
180C42—22no—
181C42—24no—
182C23—24no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—41no—
187C23—42no—
188C23—42no—
189C23#12011yescharacteristic
190C23#12011yescharacteristic
191C23C22×D411yescharacteristic
192Q8—24no—
193Q8—22no—
194Q8—42no—
195Q8—42no—
196Q8#12314yes—
197D4—28no—
198D4—24no—
199D4—28no—
200D4—22no—
201D4#12314yes—
202C2×C4—24no—
203C2×C4—28no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—82no—
226C2×C4—88no—
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4C22×D411yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C2×#4114yes—
232C8—42no—
233C8—42no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—24no—
240C22—21no—
241C22—21no—
242C22—44no—
243C22—48no—
244C22—42no—
245C22—42no—
246C22#45512yes—
247C22C2×#12011yescharacteristic
248C22#52512yes—
249C4—21no—
250C4—24no—
251C4—21no—
252C4—44no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—81no—
257C4—81no—
258C4C2×#12312yes—
259C4#52512yes—
260C2—22no—
261C2—22no—
262C2—41no—
263C2—44no—
264C2#284911yescharacteristic
265C2C2×#45511yescharacteristic
266C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3095C2semidirect
4. #2503C2semidirect
5. #1882C2semidirect
6. #1860C2semidirect
7. #1826C2semidirect
8. #1821C2semidirect
30. #428C22semidirect
32. #419C22semidirect
37. #391C22semidirect
38. #382C22semidirect
41. #343C22semidirect
42. #339C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
#30952
#30072
#25032
#18822
#18602
#18262
#18212
#17892