Finite Group Databasebeta

← #32997

Subgroups — #32997

Order 256.

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

normal subgroups only

This group has 663 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#2504C211yesmaximal, characteristic, complemented
5#1883C214yesmaximal, complemented
6#1860C214yesmaximal, complemented
7#1826C211yesmaximal, characteristic, complemented
8#1821C211yesmaximal, characteristic, complemented
9#1789C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C2×AGL(1,ℤ/8)C2212yes—
12C2×#121C2211yescharacteristic
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#482—22no—
25#481—21no—
26#480—21no—
27#449—22no—
28#448—21no—
29#447—21no—
30#428C2212yescomplemented
31C2×#104C2214yes—
32#418C2212yescomplemented
33#417C2212yes—
34#407C2211yescharacteristic
35#407C2211yescharacteristic
36#397—22no—
37#390C2212yescomplemented
38#386—22no—
39#382C2212yescomplemented
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#121—21no—
63#121—21no—
64#121—21no—
65#121—21no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×D4—28no—
69C4×D4—24no—
70C4×D4—24no—
71C4×D4—28no—
72#117—24no—
73#117—21no—
74#117—21no—
75#117—24no—
76#117D412yes—
77#117C2312yes—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—24no—
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×C22⋊C4—24no—
95C2×C42—22no—
96C2×C42—21no—
97C2×C42C2312yes—
98#107—22no—
99#106—22no—
100#104—28no—
101#103—22no—
102#102—22no—
103#102—22no—
104#102—22no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109#95—21no—
110#95—24no—
111C24—21no—
112C24—21no—
113C24C2×D411yescharacteristic
114#41—28no—
115#41—28no—
116#41—24no—
117#41—24no—
118#41—24no—
119#41#4118yes—
120C2×Q8—24no—
121C2×Q8—21no—
122C2×Q8C2×D412yes—
123C2×D4—28no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—24no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—24no—
131C2×D4—44no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4C2×D412yes—
135C22×C4—28no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—28no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—82no—
148C22×C4—84no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D414yes—
155C22×C4C2411yescharacteristic
156SD16—44no—
157D8—44no—
158M16—24no—
159M16—24no—
160C2×C8—22no—
161C2×C8—22no—
162C4⋊C4—28no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4C2×D412yes—
172C4⋊C4C2×D412yes—
173C22⋊C4—28no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—42no—
177C22⋊C4—41no—
178C22⋊C4—41no—
179C22⋊C4—84no—
180C42—28no—
181C42—22no—
182C42—22no—
183C42—24no—
184C23—24no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—41no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—41no—
194C23—42no—
195C23#12011yescharacteristic
196C23#12011yescharacteristic
197C23C22×D411yescharacteristic
198Q8—24no—
199Q8—22no—
200Q8#12314yes—
201D4—28no—
202D4—24no—
203D4—28no—
204D4—22no—
205D4—42no—
206D4—42no—
207D4—82no—
208D4—82no—
209D4#12314yes—
210C2×C4—24no—
211C2×C4—28no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—28no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—88no—
233C2×C4#12011yescharacteristic
234C2×C4#12011yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4C2×#4114yes—
238C8—42no—
239C8—42no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—24no—
246C22—21no—
247C22—21no—
248C22—44no—
249C22—48no—
250C22—42no—
251C22—42no—
252C22—84no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22#45412yes—
257C22C2×#12011yescharacteristic
258C22#52512yes—
259C4—21no—
260C4—24no—
261C4—21no—
262C4—42no—
263C4—44no—
264C4—42no—
265C4—82no—
266C4—81no—
267C4C2×#12312yes—
268C4#52512yes—
269C2—22no—
270C2—22no—
271C2—41no—
272C2—44no—
273C2—161no—
274C2#284911yescharacteristic
275C2C2×#45411yescharacteristic
276C2#301911yescharacteristic
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. #2504C2semidirect
5. #1883C2semidirect
6. #1860C2semidirect
7. #1826C2semidirect
8. #1821C2semidirect
9. #1789C2semidirect
30. #428C22semidirect
32. #418C22semidirect
37. #390C22semidirect
39. #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
#30942
#30062
#25042
#18832
#18602
#18262
#18212
#17892