Finite Group Databasebeta

← #219646

Subgroups — #219646

Order 512.

501 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 162 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
1#22253C211yesmaximal, characteristic, complemented
2#22236C211yesmaximal, characteristic
3#19703C211yesmaximal, characteristic
4#9767C211yesmaximal, characteristic, complemented
5#9757C211yesmaximal, characteristic
6#9302C212yesmaximal
7#9297C212yesmaximal, complemented
8#9290C212yesmaximal, complemented
9#9287C212yesmaximal
10#8794C211yesmaximal, characteristic, complemented
11#8141C211yesmaximal, characteristic, complemented
12#3227C2212yes—
13#3209C2212yes—
14C2×#498C2211yescharacteristic
15#3131C2211yescharacteristic
16#2961C2211yescharacteristic
17#2932C2211yescharacteristic
18#2331C2212yes—
19#1995C2211yescharacteristic
20#1993C2212yes—
21#1963C2211yescharacteristic
22#1929—22no—
23#1920—22no—
24#1919—22no—
25#1866—22no—
26#1848C2211yescharacteristic
27#1835C2211yescharacteristic
28#1746C2211yescharacteristic
29#1720C414yescomplemented
30#1716C414yescomplemented
31#1549C2212yes—
32#1548C2212yes—
33#1518C2212yes—
34#1517C2212yes—
35#1452C2211yescharacteristic
36#1450C2211yescharacteristic
37#1286C2212yes—
38#1286C2212yes—
39#1275C2212yes—
40#1274C2212yes—
41C2×#131—21no—
42#533C2311yescharacteristic
43#532C2311yescharacteristic
44C2×C4⋊Q8C2312yes—
45#498—21no—
46#498—21no—
47#498C2×C414yes—
48#498C2×C414yes—
49#494C2×C414yes—
50#488—21no—
51#488—21no—
52#487C2×C412yes—
53#487C2×C414yes—
54#486C2×C412yes—
55#486C2×C414yes—
56#481—42no—
57#476—42no—
58#462—42no—
59#451—42no—
60#446—42no—
61#435—42no—
62#431C2312yes—
63#431C2312yes—
64C2×#108—22no—
65C2×#106—22no—
66#424—21no—
67#423C2311yescharacteristic
68#423C2311yescharacteristic
69C2×#104—22no—
70#416—22no—
71#408—21no—
72#405—21no—
73C2×#97C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—42no—
76#389—22no—
77#387—42no—
78#382C2312yes—
79C4×C4⋊C4—22no—
80#376C2311yescharacteristic
81#340—22no—
82#340—22no—
83#340—22no—
84#339—22no—
85#336D412yes—
86#336D412yes—
87#333—42no—
88#332—42no—
89#328—42no—
90#327—42no—
91#322—21no—
92#322C2×C414yes—
93C82—21no—
94C2×#41—21no—
95C22×Q8—22no—
96C2×Q16—44no—
97#131—82no—
98C2×M16—21no—
99C2×M16—21no—
100C2×M16—41no—
101C22×C8—41no—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D412yes—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8C22×C414yes—
109C4⋊Q8C22×C411yescharacteristic
110C4⋊Q8C22×C412yes—
111#127C22×C411yescharacteristic
112#125—21no—
113#124—21no—
114C22⋊Q8—22no—
115#121—22no—
116C4×Q8—22no—
117C4×D4—22no—
118#117—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—42no—
123C2×C42—42no—
124C2×C42C2×D412yes—
125C2×C42C2411yescharacteristic
126#108—82no—
127#106—24no—
128#106—24no—
129#106—44no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—42no—
135#104—82no—
136#103—22no—
137#103—22no—
138#103—44no—
139#102—22no—
140#102—22no—
141#102—44no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—42no—
145#97—22no—
146#97—41no—
147C4×C8—22no—
148C4×C8—41no—
149C4×C8C22×C414yes—
150C4×C8C22×C412yes—
151C4×C8#4111yescharacteristic
152C4×C8#4111yescharacteristic
153#95—22no—
154#95—22no—
155#95—22no—
156#95—42no—
157#95—42no—
158#41—42no—
159#41—42no—
160C2×Q8—21no—
161C2×Q8—24no—
162C2×Q8—22no—
163C2×Q8—42no—
164C2×Q8—162no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—41no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4C22×D411yescharacteristic
173C22×C4H(5,𝔽2)12yes—
174Q16—164no—
175M16—42no—
176M16—42no—
177M16—82no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—24no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—42no—
188C2×C8—82no—
189C2×C8C4×D412yes—
190C2×C8C4×D412yes—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—162no—
202C22⋊C4—41no—
203C22⋊C4—42no—
204C42—22no—
205C42—22no—
206C42—41no—
207C42—162no—
208C42C23×C411yescharacteristic
209C42C2×#4111yescharacteristic
210C42H(5,𝔽2)12yes—
211C23—41no—
212C23#53511yescharacteristic
213Q8—42no—
214Q8—82no—
215Q8—84no—
216Q8—162no—
217D4—42no—
218D4—82no—
219D4—82no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4C2×C4×D411yescharacteristic
235C2×C4#51812yes—
236C2×C4#53511yescharacteristic
237C2×C4#54012yes—
238C2×C4#57712yes—
239C2×C4#57812yes—
240C8—22no—
241C8—22no—
242C8—44no—
243C8—41no—
244C8—41no—
245C8—82no—
246C8—81no—
247C8—81no—
248C22—22no—
249C22—81no—
250C22—82no—
251C22#232811yescharacteristic
252C22#322512yes—
253C4—22no—
254C4—22no—
255C4—82no—
256C4—81no—
257C4—84no—
258C4—82no—
259C4—162no—
260C4#294712yes—
261C4#322211yescharacteristic
262C4#322311yescharacteristic
263C2—22no—
264C2—161no—
265C2#2227811yescharacteristic
266C2#2228412yes—
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#532C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22253C2semidirect
4. #9767C2semidirect
7. #9297C2semidirect
8. #9290C2semidirect
10. #8794C2semidirect
11. #8141C2semidirect
29. #1720C4semidirect
30. #1716C4semidirect

Maximal subgroups

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

subgroupindex
#222532
#222362
#197032
#97672
#97572
#93022
#92972
#92902
#92872
#87942
#81412