Finite Group Databasebeta

← #219604

Subgroups — #219604

Order 512.

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

normal subgroups only

This group has 542 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#22275C211yesmaximal, characteristic
2#22258C211yesmaximal, characteristic, complemented
3#19651C211yesmaximal, characteristic
4#9777C211yesmaximal, characteristic
5#9759C211yesmaximal, characteristic, complemented
6#9296C212yesmaximal, complemented
7#9295C212yesmaximal
8#9294C212yesmaximal
9#9292C212yesmaximal, complemented
10#8791C211yesmaximal, characteristic, complemented
11#8142C211yesmaximal, characteristic, complemented
12#3229C2212yes—
13#3208C2212yes—
14#3160C2211yescharacteristic
15C2×#492C2211yescharacteristic
16#2961C2212yes—
17#2328C2211yescharacteristic
18#2327C2211yescharacteristic
19#1995C2212yes—
20#1961C2211yescharacteristic
21#1960C2211yescharacteristic
22#1922—22no—
23#1922—22no—
24#1880—22no—
25#1877—22no—
26#1837C2211yescharacteristic
27#1835C2211yescharacteristic
28#1746C2211yescharacteristic
29#1714C414yescomplemented
30#1708C412yescomplemented
31#1705C412yescomplemented
32#1546C2212yes—
33#1545C2212yes—
34#1518C2212yes—
35#1518C2212yes—
36#1452C2211yescharacteristic
37#1450C2211yescharacteristic
38#1286C2212yes—
39#1286C2212yes—
40#1275C2212yes—
41#1274C2212yes—
42C22×SD16—22no—
43#533C2312yes—
44C2×C4⋊Q8C2311yescharacteristic
45C2×#127C2311yescharacteristic
46C2×C4×Q8—21no—
47C2×C4×D4—21no—
48#500C2×C412yescomplemented
49#499—22no—
50#498C2×C411yescharacteristic, complemented
51#498C2×C411yescharacteristic, complemented
52#498C2×C414yescomplemented
53#492—22no—
54#492C2×C412yes—
55#492C2×C412yes—
56#492C2×C414yes—
57#487C2×C414yescomplemented
58#486C2×C414yes—
59#483—42no—
60#475—42no—
61#460—42no—
62#450—42no—
63#446—42no—
64#446—42no—
65#431C2312yes—
66#431C2312yes—
67#427—22no—
68#424—22no—
69#423C2312yes—
70C2×#103—22no—
71C2×#102—22no—
72#405—22no—
73C2×#97C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—21no—
76#390—21no—
77#387—42no—
78#386—42no—
79#382C2311yescharacteristic
80#382C2311yescharacteristic
81#376C2311yescharacteristic
82#341—22no—
83#340—22no—
84#339—22no—
85#339—22no—
86#336D412yes—
87#336D412yes—
88#332—42no—
89#331—42no—
90#328—42no—
91#328—42no—
92#322—22no—
93#322C2×C412yescomplemented
94C82C2×C412yescomplemented
95C22×Q8—21no—
96C22×D4—21no—
97C22×D4—22no—
98C23×C4—41no—
99C2×SD16—24no—
100C2×SD16—24no—
101C2×SD16—44no—
102C2×SD16—82no—
103C2×SD16—82no—
104C2×M16—22no—
105C22×C8C2×D412yes—
106C22×C8C2×D412yes—
107C4⋊Q8—21no—
108C4⋊Q8—22no—
109C4⋊Q8C22×C412yes—
110C4⋊Q8C22×C412yes—
111C4⋊Q8C22×C412yes—
112#127—21no—
113#127—22no—
114#127C22×C412yes—
115#125—22no—
116C22⋊Q8—22no—
117C22⋊Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—82no—
121C4×D4—82no—
122#117—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—42no—
129C2×C42—41no—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2411yescharacteristic
133#107—24no—
134#106—22no—
135#106—22no—
136#106—44no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—44no—
142#103—44no—
143#103—82no—
144#102—44no—
145#102—82no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148#97—22no—
149#97—42no—
150C4×C8—22no—
151C4×C8—42no—
152C4×C8C22×C414yes—
153C4×C8C22×C411yescharacteristic
154C4×C8C22×C411yescharacteristic
155C4×C8#4112yes—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—42no—
160C24—41no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—41no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×D4—22no—
168C2×D4—24no—
169C2×D4—41no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—162no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—82no—
180C22×C4—162no—
181C22×C4C22×D411yescharacteristic
182C22×C4H(5,𝔽2)12yes—
183SD16—84no—
184SD16—84no—
185SD16—164no—
186M16—42no—
187M16—42no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—24no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8C4×D412yes—
197C2×C8C4×D412yes—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—82no—
209C22⋊C4—42no—
210C22⋊C4—82no—
211C22⋊C4—162no—
212C42—22no—
213C42—22no—
214C42—42no—
215C42—82no—
216C42C23×C411yescharacteristic
217C42C2×#4111yescharacteristic
218C42H(5,𝔽2)12yes—
219C23—42no—
220C23—82no—
221C23—82no—
222C23—81no—
223C23—81no—
224C23#53511yescharacteristic
225Q8—44no—
226Q8—82no—
227Q8—82no—
228Q8—82no—
229D4—44no—
230D4—84no—
231D4—82no—
232D4—162no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—164no—
248C2×C4C2×C4×D411yescharacteristic
249C2×C4#51812yes—
250C2×C4#53511yescharacteristic
251C2×C4#54012yes—
252C2×C4#57712yes—
253C2×C4#57712yes—
254C8—22no—
255C8—22no—
256C8—44no—
257C8—42no—
258C8—82no—
259C8—82no—
260C22—22no—
261C22—82no—
262C22—84no—
263C22—161no—
264C22—161no—
265C22—162no—
266C22#232811yescharacteristic
267C22#322211yescharacteristic
268C22#322211yescharacteristic
269C4—22no—
270C4—22no—
271C4—81no—
272C4—81no—
273C4—82no—
274C4—82no—
275C4—82no—
276C4—161no—
277C4—161no—
278C4#294912yes—
279C4#322112yes—
280C2—21no—
281C2—21no—
282C2—162no—
283C2#2227511yescharacteristic
284C2#2228412yes—
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
2. #22258C2semidirect
5. #9759C2semidirect
6. #9296C2semidirect
9. #9292C2semidirect
10. #8791C2semidirect
11. #8142C2semidirect
29. #1714C4semidirect
30. #1708C4semidirect
31. #1705C4semidirect
48. #500C2×C4semidirect
50. #498C2×C4semidirect
51. #498C2×C4semidirect
52. #498C2×C4semidirect
57. #487C2×C4semidirect
93. #322C2×C4semidirect
94. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222752
#222582
#196512
#97772
#97592
#92962
#92952
#92942
#92922
#87912
#81422