Finite Group Databasebeta

← #329599

Subgroups — #329599

Order 512.

736 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 154 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#32906C212yesmaximal
2#29876C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic
4#10623C212yesmaximal, complemented
5#10478C214yesmaximal, complemented
6#10265C212yesmaximal, complemented
7#9130C212yesmaximal
8#8599C211yesmaximal, characteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#500C2211yescharacteristic
11C2×#462C2214yes—
12C2×#451—22no—
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2857C2212yes—
16#2086—22no—
17#2075—22no—
18#1990C2212yes—
19#1984—22no—
20#1984C2212yes—
21#1984C2212yes—
22#1878—24no—
23#1872—22no—
24#1834C2211yescharacteristic
25#1817—22no—
26#1803C2214yes—
27#1797C2212yes—
28C2×#335C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31#1277C2212yes—
32C23×Q8D411yescharacteristic
33C22×Q16—24no—
34C2×C4⋊Q8C2312yes—
35C2×C22⋊Q8—22no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—42no—
39C2×C4×Q8—42no—
40C2×C4×Q8D412yes—
41C2×C4×Q8C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#500—42no—
47#462—44no—
48#462—44no—
49#451—48no—
50#431—42no—
51#431—42no—
52C2×#107—22no—
53C2×C4⋊C8D412yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2314yes—
56C2×#103—24no—
57C2×#103—24no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8D412yes—
61C2×C4×C8—21no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#391—22no—
65#389—22no—
66#389—22no—
67#384C2312yes—
68#380—22no—
69#380—22no—
70#340—22no—
71#340—24no—
72#340—24no—
73#336—24no—
74#336—24no—
75#336—22no—
76#335—42no—
77#332—44no—
78C22×Q8—21no—
79C22×Q8—22no—
80C22×Q8—22no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8D814yes—
84C22×Q8C2×D412yes—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4#4111yescharacteristic
90C2×Q16—48no—
91C2×Q16—48no—
92C2×Q16—48no—
93C22×C8—24no—
94C22×C8—22no—
95C22×C8C2×D412yes—
96C22×C8C2×D412yes—
97C4⋊Q8—24no—
98C4⋊Q8—24no—
99C22⋊Q8—44no—
100C22⋊Q8—44no—
101C4×Q8—24no—
102C4×Q8—24no—
103C4×Q8—44no—
104C4×Q8—42no—
105C4×Q8—42no—
106C4×Q8—44no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C2×D412yes—
119C2×C22⋊C4—42no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42C2411yescharacteristic
130#107—44no—
131C4⋊C8—24no—
132C4⋊C8—24no—
133C4⋊C8—24no—
134C4⋊C8—42no—
135#103—48no—
136#103—48no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139C22⋊C8—44no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8—42no—
143#95—42no—
144#95—42no—
145C24#12111yescharacteristic
146C2×Q8—22no—
147C2×Q8—24no—
148C2×Q8—24no—
149C2×Q8—24no—
150C2×Q8—48no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—48no—
154C2×Q8—44no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—44no—
158C2×Q8C2×D814yes—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—82no—
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4C2×D812yes—
179C22×C4#13512yes—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D411yescharacteristic
182C22×C4C2×#4111yescharacteristic
183Q16—816no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8—44no—
189C2×C8—42no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—44no—
193C4⋊C4—24no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—44no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—82no—
204C4⋊C4—84no—
205C22⋊C4—84no—
206C42—21no—
207C42—22no—
208C42—21no—
209C42—22no—
210C42—44no—
211C42—42no—
212C42—42no—
213C42H(5,𝔽2)12yes—
214C42H(5,𝔽2)12yes—
215C23—21no—
216C23—22no—
217C23—21no—
218C23—22no—
219C23#46612yes—
220C23C2×#12111yescharacteristic
221Q8—28no—
222Q8—48no—
223Q8—44no—
224Q8—84no—
225Q8—84no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—48no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—82no—
244C2×C4—84no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4#46612yes—
252C2×C4#53412yes—
253C2×C4#53411yescharacteristic
254C2×C4#53511yescharacteristic
255C2×C4#53812yes—
256C2×C4C22×D811yescharacteristic
257C2×C4C2×#13511yescharacteristic
258C2×C4#57512yes—
259C2×C4#57814yes—
260C8—42no—
261C8—42no—
262C8—84no—
263C8—82no—
264C22—22no—
265C22—24no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22#261711yescharacteristic
270C22#304412yes—
271C22C2×#46611yescharacteristic
272C22#305411yescharacteristic
273C22#306912yes—
274C4—22no—
275C4—22no—
276C4—41no—
277C4—44no—
278C4—41no—
279C4—82no—
280C4—162no—
281C4—162no—
282C4#318612yes—
283C4#319112yes—
284C4#320212yes—
285C4#322312yes—
286C2—42no—
287C2#1940412yes—
288C2#1944312yes—
289C2#2972411yescharacteristic
290C2#2973011yescharacteristic
291C2#2975211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10623C2semidirect
5. #10478C2semidirect
6. #10265C2semidirect

Maximal subgroups

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

subgroupindex
#329062
#298762
#218182
#106232
#104782
#102652
#91302
#85992