Finite Group Databasebeta

← #32996

Subgroups — #32996

Order 256.

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

normal subgroups only

This group has 691 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, complemented
2#3094C211yesmaximal, characteristic, complemented
3#3006C211yesmaximal, characteristic
4#2501C211yesmaximal, characteristic, complemented
5#1883C214yesmaximal, complemented
6#1859C214yesmaximal, complemented
7#1826C211yesmaximal, characteristic, complemented
8#1821C211yesmaximal, characteristic, complemented
9#1789C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C2×AGL(1,ℤ/8)C2212yes—
12C2×#123—22no—
13C2×#121C2211yescharacteristic
14#520—22no—
15#520—22no—
16#520—22no—
17#520C2214yes—
18C2×C4×Q8—21no—
19C2×C4×D4—21no—
20C2×C4×D4—22no—
21C2×#117—21no—
22C2×#117C2211yescharacteristic
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2211yescharacteristic
25#482—22no—
26#481—21no—
27#480—21no—
28#449—22no—
29#448—21no—
30#447—21no—
31#428C2212yescomplemented
32C2×#104C2214yes—
33#418C2212yescomplemented
34#417C2212yes—
35#407C2211yescharacteristic
36#407C2211yescharacteristic
37#399—22no—
38#390C2212yescomplemented
39#384C2212yescomplemented
40C4×C22⋊C4C2211yescharacteristic
41#343C2214yescomplemented
42#339C2214yescomplemented
43C2×#41—28no—
44C2×#41—21no—
45C2×#41—21no—
46C2×#41D412yes—
47C2×#41C2312yes—
48C22×Q8D411yescharacteristic
49C22×D4—21no—
50C22×D4—21no—
51C22×D4D411yescharacteristic
52C22×D4C2311yescharacteristic
53C23×C4—21no—
54C23×C4—22no—
55C23×C4D412yes—
56C23×C4C2311yescharacteristic
57AGL(1,ℤ/8)—28no—
58C2×SD16—22no—
59C2×D8—22no—
60C2×M16C2312yes—
61C2×M16C2312yes—
62#123—48no—
63#121—21no—
64#121—21no—
65#121—21no—
66#121—21no—
67C4×Q8—24no—
68C4×Q8—24no—
69C4×D4—28no—
70C4×D4—24no—
71C4×D4—24no—
72C4×D4—28no—
73#117—24no—
74#117—21no—
75#117—21no—
76#117—24no—
77#117D412yes—
78#117C2312yes—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4D412yes—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2312yes—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—24no—
97C2×C42—22no—
98C2×C42—21no—
99C2×C42C2312yes—
100#107—22no—
101#106—22no—
102#104—28no—
103#103—22no—
104#102—22no—
105#102—22no—
106#102—22no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111#95—21no—
112C24—21no—
113C24—21no—
114C24C2×D411yescharacteristic
115#41—28no—
116#41—28no—
117#41—24no—
118#41—24no—
119#41—24no—
120#41#4118yes—
121C2×Q8—24no—
122C2×Q8—21no—
123C2×Q8C2×D412yes—
124C2×D4—28no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—24no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—24no—
132C2×D4—44no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4C2×D412yes—
136C22×C4—28no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—28no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—44no—
149C22×C4—84no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D414yes—
156C22×C4C2411yescharacteristic
157SD16—44no—
158D8—44no—
159M16—24no—
160M16—24no—
161C2×C8—22no—
162C2×C8—22no—
163C4⋊C4—28no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—48no—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—28no—
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—48no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C42—28no—
183C42—22no—
184C42—22no—
185C42—24no—
186C23—24no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—41no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—41no—
196C23—42no—
197C23#12011yescharacteristic
198C23#12011yescharacteristic
199C23C22×D411yescharacteristic
200Q8—24no—
201Q8—22no—
202Q8#12314yescomplemented
203D4—28no—
204D4—24no—
205D4—28no—
206D4—22no—
207D4—42no—
208D4—42no—
209D4—82no—
210D4—82no—
211D4#12314yescomplemented
212C2×C4—24no—
213C2×C4—28no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—28no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4—88no—
235C2×C4#12011yescharacteristic
236C2×C4#12011yescharacteristic
237C2×C4C22×D411yescharacteristic
238C2×C4C22×D411yescharacteristic
239C2×C4C2×#4114yes—
240C8—42no—
241C8—42no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—24no—
248C22—21no—
249C22—21no—
250C22—44no—
251C22—48no—
252C22—42no—
253C22—42no—
254C22—84no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#45312yes—
259C22C2×#12011yescharacteristic
260C22#52512yes—
261C4—21no—
262C4—24no—
263C4—21no—
264C4—44no—
265C4—44no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4C2×#12312yes—
270C4#52512yes—
271C2—22no—
272C2—22no—
273C2—41no—
274C2—44no—
275C2—161no—
276C2#284911yescharacteristic
277C2C2×#45311yescharacteristic
278C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3006C2
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
1. C2×#520C2semidirect
2. #3094C2semidirect
4. #2501C2semidirect
5. #1883C2semidirect
6. #1859C2semidirect
7. #1826C2semidirect
8. #1821C2semidirect
9. #1789C2semidirect
31. #428C22semidirect
33. #418C22semidirect
38. #390C22semidirect
39. #384C22semidirect
41. #343C22semidirect
42. #339C22semidirect
202. Q8#123semidirect
211. D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
#30942
#30062
#25012
#18832
#18592
#18262
#18212
#17892