Finite Group Databasebeta

← #309649

Subgroups — #309649

Order 512.

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

normal subgroups only

This group has 622 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#24967C212yesmaximal
2#22969C212yesmaximal, complemented
3#10676C211yesmaximal, characteristic, complemented
4#10481C212yesmaximal, complemented
5#10235C211yesmaximal, characteristic
6#9619C212yesmaximal, complemented
7#9545C212yesmaximal, complemented
8#8928C211yesmaximal, characteristic, complemented
9#8882C212yesmaximal, complemented
10C2×#476C2212yes—
11#2537C2211yescharacteristic
12#2369C2211yescharacteristic
13D4×C4⋊C4C2212yes—
14#2077—22no—
15#2034—22no—
16#1990C2212yes—
17#1956C2211yescharacteristic
18#1954C2212yes—
19#1948C2212yes—
20#1947C2212yes—
21#1944—22no—
22#1920—22no—
23#1874—22no—
24#1862—22no—
25#1845C2211yescharacteristic
26#1834C2212yes—
27#1824—22no—
28#1802C2212yes—
29#1770C2212yes—
30#1762—22no—
31C4×#102C2212yes—
32C2×#335C2212yes—
33C2×#333C2211yescharacteristic
34C2×D4⋊C8C2212yes—
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8C2311yescharacteristic
40C2×#121—22no—
41C2×C4×D4—22no—
42C2×C4×D4C2312yes—
43C22×C4⋊C4—22no—
44#476—24no—
45#476—24no—
46C2×#107—22no—
47C2×#107—41no—
48C2×#106—22no—
49C2×#106—22no—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×#103—41no—
53C2×#102—22no—
54C2×#102—22no—
55C2×#102D412yes—
56C2×#102D412yes—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×#97—22no—
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2312yes—
62#394—41no—
63#391—41no—
64#389—22no—
65#389—22no—
66#385—42no—
67#384C2312yes—
68#384C2311yescharacteristic
69#380—22no—
70#380—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#335—24no—
84#333—22no—
85D4⋊C8—24no—
86C22×Q8—21no—
87C22×D4—41no—
88C22×D4C2×D412yes—
89C23×C4—22no—
90C23×C4—42no—
91C22×C8—22no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8C2×D412yes—
95C22×C8C2×D412yes—
96C4⋊Q8—22no—
97C4⋊Q8—21no—
98C4⋊Q8#4112yes—
99#121—44no—
100#121—82no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—44no—
106C4×D4—82no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4C2×D412yes—
118C2×C4⋊C4C2×D412yes—
119C2×C4⋊C4C2×D412yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C42—22no—
124C2×C42—41no—
125C2×C42C2411yescharacteristic
126#107—82no—
127#107—82no—
128#106—28no—
129#106—82no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8#4114yes—
134#103—82no—
135#102—28no—
136#102—24no—
137#102—42no—
138#102—82no—
139C22⋊C8—82no—
140C22⋊C8—82no—
141#97—82no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145#95—22no—
146#95—42no—
147#95—41no—
148#95—42no—
149C24—22no—
150C2×Q8—22no—
151C2×Q8—41no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—84no—
160C2×D4—161no—
161C2×D4#13714yes—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—84no—
175C22×C4—84no—
176C22×C4—162no—
177C22×C4C22×D412yes—
178C22×C4H(5,𝔽2)11yescharacteristic
179C2×C8—22no—
180C2×C8—24no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C4⋊C4—24no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—81no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C4⋊C4—161no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—84no—
215C22⋊C4—84no—
216C42—42no—
217C42—42no—
218C42—161no—
219C42C2×#4111yescharacteristic
220C42C2×#4112yes—
221C422−1+411yescharacteristic
222C23—24no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23D4211yescharacteristic
230Q8—82no—
231Q8—161no—
232D4—24no—
233D4—24no—
234D4—42no—
235D4—42no—
236D4—164no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—168no—
261C2×C4#52512yes—
262C2×C4#53911yescharacteristic
263C2×C4#54712yes—
264C2×C4#55911yescharacteristic
265C2×C4C2×#13712yes—
266C2×C4#57512yes—
267C2×C4#57612yes—
268C2×C4#57712yes—
269C8—42no—
270C8—42no—
271C8—42no—
272C8—44no—
273C8—82no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22#245511yescharacteristic
282C22#326612yes—
283C22#327312yes—
284C22#328911yescharacteristic
285C22#329211yescharacteristic
286C4—21no—
287C4—21no—
288C4—82no—
289C4—82no—
290C4—82no—
291C4—82no—
292C4—161no—
293C4—162no—
294C4#311712yes—
295C4#311912yes—
296C4#334612yes—
297C4#334812yes—
298C2—84no—
299C2#2004011yescharacteristic
300C2#2017812yes—
301C2#2030211yescharacteristic
302C2#2656712yes—
303C2#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. #22969C2semidirect
3. #10676C2semidirect
4. #10481C2semidirect
6. #9619C2semidirect
7. #9545C2semidirect
8. #8928C2semidirect
9. #8882C2semidirect

Maximal subgroups

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

subgroupindex
#249672
#229692
#106762
#104812
#102352
#96192
#95452
#89282
#88822