Finite Group Databasebeta

← #309543

Subgroups — #309543

Order 512.

594 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 142 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#24968C212yesmaximal
2#23061C212yesmaximal, complemented
3#10676C211yesmaximal, characteristic, complemented
4#10480C212yesmaximal, complemented
5#10235C211yesmaximal, characteristic
6#9613C212yesmaximal, complemented
7#9542C212yesmaximal, complemented
8#8929C211yesmaximal, characteristic, complemented
9#8885C212yesmaximal, complemented
10C2×#476C2212yes—
11#2537C2211yescharacteristic
12#2369C2211yescharacteristic
13#2354C2212yes—
14#2076—22no—
15#2033—22no—
16#1990C2212yes—
17#1956C2211yescharacteristic
18#1954C2212yes—
19#1948C2212yes—
20#1947C2212yes—
21#1934—22no—
22#1926—22no—
23#1920—22no—
24#1875—22no—
25#1845C2211yescharacteristic
26#1834C2212yes—
27#1825—22no—
28#1802C2212yes—
29C4×#106—22no—
30C8×C4⋊C4C2212yes—
31C4×#102C2212yes—
32C2×#334C2212yes—
33C2×#333C2211yescharacteristic
34C2×D4⋊C8C2212yes—
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8C2311yescharacteristic
40C2×#121—22no—
41C2×C4×D4C2312yes—
42C2×#117—22no—
43#476—24no—
44#476—24no—
45C2×#107—22no—
46C2×#107—41no—
47C2×#106—22no—
48C2×#106—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#103—41no—
52C2×#102—22no—
53C2×#102—22no—
54C2×#102D412yes—
55C2×#102D412yes—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C4×C8—22no—
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2312yes—
61#394—41no—
62#391—41no—
63#390—22no—
64#389—22no—
65#388—42no—
66#384C2312yes—
67#384C2311yescharacteristic
68#380—22no—
69#380—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2312yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340D412yes—
81#340D412yes—
82#336—22no—
83#334—24no—
84#333—22no—
85D4⋊C8—24no—
86C22×Q8—21no—
87C22×D4—41no—
88C22×D4C2×D412yes—
89C23×C4—22no—
90C22×C8—22no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8C2×D412yes—
94C22×C8C2×D412yes—
95C4⋊Q8—22no—
96C4⋊Q8—21no—
97C4⋊Q8#4112yes—
98#121—44no—
99#121—82no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104#117—44no—
105#117—82no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C42—22no—
121C2×C42—42no—
122C2×C42—41no—
123C2×C42C2411yescharacteristic
124#107—82no—
125#107—82no—
126#106—28no—
127#106—44no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8#4114yes—
132#103—82no—
133#102—28no—
134#102—24no—
135#102—42no—
136#102—82no—
137C22⋊C8—82no—
138C22⋊C8—82no—
139C4×C8—22no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8—44no—
143#95—22no—
144#95—42no—
145#95—42no—
146#95—41no—
147C24—22no—
148C2×Q8—22no—
149C2×Q8—41no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—84no—
159C2×D4—161no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4C22×D412yes—
173C22×C4H(5,𝔽2)11yescharacteristic
174C2×C8—22no—
175C2×C8—24no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C4⋊C4—22no—
190C4⋊C4—24no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—81no—
203C4⋊C4—82no—
204C4⋊C4—161no—
205C4⋊C4#13514yes—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—84no—
209C22⋊C4—84no—
210C22⋊C4—162no—
211C42—42no—
212C42—42no—
213C42—84no—
214C42—161no—
215C42C2×#4111yescharacteristic
216C42C2×#4112yes—
217C422−1+411yescharacteristic
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23D4211yescharacteristic
226Q8—82no—
227Q8—161no—
228D4—42no—
229D4—42no—
230D4—42no—
231D4—42no—
232D4—164no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4#52512yes—
257C2×C4#53911yescharacteristic
258C2×C4#54712yes—
259C2×C4#55911yescharacteristic
260C2×C4C2×#13512yes—
261C2×C4#57512yes—
262C2×C4#57712yes—
263C2×C4#57712yes—
264C8—42no—
265C8—42no—
266C8—42no—
267C8—44no—
268C8—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22#245511yescharacteristic
277C22#326712yes—
278C22#328212yes—
279C22#328911yescharacteristic
280C22#329211yescharacteristic
281C4—21no—
282C4—21no—
283C4—44no—
284C4—82no—
285C4—82no—
286C4—82no—
287C4—162no—
288C4—161no—
289C4#310212yes—
290C4#312612yes—
291C4#334012yes—
292C4#335112yes—
293C2—162no—
294C2#1982412yes—
295C2#2000311yescharacteristic
296C2#2013111yescharacteristic
297C2#2655312yes—
298C2#2657811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radical#2537C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23061C2semidirect
3. #10676C2semidirect
4. #10480C2semidirect
6. #9613C2semidirect
7. #9542C2semidirect
8. #8929C2semidirect
9. #8885C2semidirect

Maximal subgroups

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

subgroupindex
#249682
#230612
#106762
#104802
#102352
#96132
#95422
#89292
#88852