Finite Group Databasebeta

← #219642

Subgroups — #219642

Order 512.

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

normal subgroups only

This group has 533 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#22277C211yesmaximal, characteristic, complemented
2#22236C211yesmaximal, characteristic
3#19699C211yesmaximal, characteristic
4#9772C211yesmaximal, characteristic, complemented
5#9757C211yesmaximal, characteristic
6#9300C212yesmaximal
7#9296C212yesmaximal, complemented
8#9292C212yesmaximal, complemented
9#9291C212yesmaximal
10#8794C211yesmaximal, characteristic, complemented
11#8141C211yesmaximal, characteristic, complemented
12#3229C2212yes—
13#3226C2212yes—
14C2×#494C2211yescharacteristic
15#3131C2211yescharacteristic
16#2961C2211yescharacteristic
17#2932C2211yescharacteristic
18#2327C2212yes—
19#1995C2211yescharacteristic
20#1963C2211yescharacteristic
21#1960C2212yes—
22#1929—22no—
23#1878—22no—
24#1878—22no—
25#1866—22no—
26#1848C2211yescharacteristic
27#1835C2211yescharacteristic
28#1746C2211yescharacteristic
29#1723C414yescomplemented
30#1720C414yescomplemented
31#1546C2212yes—
32#1546C2212yes—
33#1518C2212yes—
34#1517C2212yes—
35#1452C2211yescharacteristic
36#1450C2211yescharacteristic
37#1286C2212yes—
38#1286C2212yes—
39#1275C2212yes—
40#1274C2212yes—
41C22×Q16—22no—
42C2×#131—21no—
43#533C2311yescharacteristic
44#532C2311yescharacteristic
45C2×C4⋊Q8C2312yes—
46C2×C4×Q8—22no—
47#500C2×C414yes—
48#498C2×C414yes—
49#494—21no—
50#494—21no—
51#494C2×C414yes—
52#488—21no—
53#488—21no—
54#487C2×C412yes—
55#487C2×C414yes—
56#486C2×C412yes—
57#486C2×C414yes—
58#483—42no—
59#480—42no—
60#475—42no—
61#474—42no—
62#446—42no—
63#435—42no—
64#431C2312yes—
65#431C2312yes—
66C2×#108—22no—
67#424—21no—
68#423C2311yescharacteristic
69#423C2311yescharacteristic
70C2×#104—22no—
71#416—22no—
72C2×#103—22no—
73C2×#103—22no—
74#408—21no—
75#405—21no—
76C2×#97C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—22no—
79#387—42no—
80#387—42no—
81#382C2312yes—
82#376C2311yescharacteristic
83#340—22no—
84#339—22no—
85#336D412yes—
86#336D412yes—
87#332—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—24no—
97C2×Q16—24no—
98C2×Q16—82no—
99C2×Q16—82no—
100#131—82no—
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—41no—
104C22×C8—41no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D412yes—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8C22×C414yes—
112C4⋊Q8C22×C411yescharacteristic
113C4⋊Q8C22×C412yes—
114#127C22×C411yescharacteristic
115#125—21no—
116#124—21no—
117C22⋊Q8—22no—
118#121—22no—
119C4×Q8—22no—
120C4×Q8—82no—
121C4×Q8—82no—
122C4×D4—22no—
123#117—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C42—42no—
129C2×C42C2×D412yes—
130C2×C42C2411yescharacteristic
131#108—82no—
132#107—44no—
133#106—44no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—42no—
139#104—82no—
140#103—22no—
141#103—22no—
142#103—44no—
143#103—82no—
144#103—82no—
145#102—22no—
146#102—22no—
147#102—44no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—42no—
151#97—22no—
152#97—41no—
153C4×C8—22no—
154C4×C8—41no—
155C4×C8C22×C414yes—
156C4×C8C22×C412yes—
157C4×C8#4111yescharacteristic
158C4×C8#4111yescharacteristic
159#95—22no—
160#95—22no—
161#95—22no—
162#95—44no—
163#41—42no—
164#41—42no—
165C2×Q8—21no—
166C2×Q8—24no—
167C2×Q8—22no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×D4—21no—
174C2×D4—22no—
175C2×D4—41no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4C22×D411yescharacteristic
181C22×C4H(5,𝔽2)12yes—
182Q16—84no—
183Q16—84no—
184M16—42no—
185M16—42no—
186M16—82no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—24no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—82no—
198C2×C8C4×D412yes—
199C2×C8C4×D412yes—
200C4⋊C4—22no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—84no—
211C22⋊C4—41no—
212C22⋊C4—42no—
213C42—22no—
214C42—22no—
215C42—41no—
216C42—84no—
217C42C23×C411yescharacteristic
218C42C2×#4111yescharacteristic
219C42H(5,𝔽2)12yes—
220C23—41no—
221C23#53511yescharacteristic
222Q8—44no—
223Q8—44no—
224Q8—42no—
225Q8—82no—
226Q8—82no—
227Q8—82no—
228D4—42no—
229D4—82no—
230D4—82no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—84no—
242C2×C4—82no—
243C2×C4C2×C4×D411yescharacteristic
244C2×C4#51812yes—
245C2×C4#53511yescharacteristic
246C2×C4#54012yes—
247C2×C4#57612yes—
248C2×C4#57712yes—
249C8—22no—
250C8—22no—
251C8—44no—
252C8—41no—
253C8—41no—
254C8—82no—
255C8—81no—
256C8—81no—
257C22—22no—
258C22—81no—
259C22—82no—
260C22#232811yescharacteristic
261C22#322412yes—
262C4—22no—
263C4—22no—
264C4—82no—
265C4—81no—
266C4—84no—
267C4—82no—
268C4—162no—
269C4#294612yes—
270C4#322011yescharacteristic
271C4#322111yescharacteristic
272C2—22no—
273C2—161no—
274C2#2227911yescharacteristic
275C2#2228512yes—
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. #22277C2semidirect
4. #9772C2semidirect
7. #9296C2semidirect
8. #9292C2semidirect
10. #8794C2semidirect
11. #8141C2semidirect
29. #1723C4semidirect
30. #1720C4semidirect

Maximal subgroups

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

subgroupindex
#222772
#222362
#196992
#97722
#97572
#93002
#92962
#92922
#92912
#87942
#81412