Finite Group Databasebeta

← #345515

Subgroups — #345515

Order 512.

604 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 134 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#26301C212yesmaximal, complemented
2#24627C212yesmaximal
3#10588C212yesmaximal, complemented
4#10476C212yesmaximal, complemented
5#10066C212yesmaximal, complemented
6#9836C211yesmaximal, characteristic, complemented
7#9665C211yesmaximal, characteristic
8#9600C211yesmaximal, characteristic, complemented
9#8889C212yesmaximal, complemented
10C2×#500C2211yescharacteristic
11C2×#461C2212yes—
12C2×C4×D8C2211yescharacteristic
13#2547C2212yes—
14#2354C2212yes—
15#2077—22no—
16#2076—22no—
17#2055—22no—
18#2012—22no—
19#1990C2212yes—
20#1988C2212yes—
21#1983—22no—
22#1956C2212yes—
23#1954C2212yes—
24#1947C2212yes—
25#1939C2212yes—
26#1928—22no—
27#1854—22no—
28#1844C2211yescharacteristic
29#1825—22no—
30#1802C2212yes—
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#334C2211yescharacteristic
34C2×#332C2212yes—
35C2×D4⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×SD16—22no—
40C22×D8D411yescharacteristic
41C2×C4⋊Q8C2312yes—
42C2×#124—22no—
43C2×C4×D4C2312yes—
44C2×#117—22no—
45#500—22no—
46#500—41no—
47#461—42no—
48#461—42no—
49C4×D8—41no—
50C4×D8—41no—
51C2×#107—22no—
52C2×#107—41no—
53C2×#107D411yescharacteristic
54C2×#106—41no—
55C2×#106D412yes—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#102—22no—
63C2×#102D412yes—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#397—42no—
70#391—22no—
71#390—22no—
72#388—42no—
73#384C2312yes—
74#380—22no—
75#380—22no—
76#380—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4C2312yes—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340D411yescharacteristic
86#340D411yescharacteristic
87#336—22no—
88#335—41no—
89#334—41no—
90#332—42no—
91D4⋊C8—24no—
92#322—41no—
93C22×Q8—22no—
94C22×D4C2×D412yes—
95C23×C4—22no—
96C2×SD16—44no—
97C2×SD16—82no—
98C2×SD16—82no—
99C2×D8—42no—
100C2×D8—41no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112#124—44no—
113#124—82no—
114C4×D4—22no—
115C4×D4—22no—
116C4×D4—22no—
117C4×D4—22no—
118#117—44no—
119#117—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4C2×D412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C42—22no—
134C2×C42—42no—
135C2×C42C2411yescharacteristic
136#107—41no—
137#107—44no—
138#107—82no—
139#106—42no—
140#106—82no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—82no—
146#103—44no—
147#103—44no—
148#103—82no—
149#102—42no—
150#102—82no—
151C22⋊C8—44no—
152C22⋊C8—82no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8#4112yes—
159#95—22no—
160#95—42no—
161#95—42no—
162#95—42no—
163C24—22no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4C22×D411yescharacteristic
188C22×C4C22×D411yescharacteristic
189C22×C4H(5,𝔽2)11yescharacteristic
190SD16—88no—
191D8—44no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—84no—
227C22⋊C4—84no—
228C22⋊C4—84no—
229C22⋊C4—162no—
230C42—42no—
231C42—42no—
232C42—84no—
233C42C2×#4111yescharacteristic
234C42H(5,𝔽2)11yescharacteristic
235C42H(5,𝔽2)11yescharacteristic
236C422−1+411yescharacteristic
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23D4211yescharacteristic
245Q8—82no—
246Q8—82no—
247Q8—82no—
248Q8—82no—
249D4—42no—
250D4—42no—
251D4—42no—
252D4—42no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4#53411yescharacteristic
275C2×C4#53711yescharacteristic
276C2×C4#53811yescharacteristic
277C2×C4#54711yescharacteristic
278C2×C4#55011yescharacteristic
279C2×C4#56211yescharacteristic
280C2×C4#57512yes—
281C2×C4#57512yes—
282C2×C4#57612yes—
283C2×C4#57712yes—
284C8—24no—
285C8—41no—
286C8—41no—
287C8—41no—
288C8—41no—
289C8—82no—
290C8—82no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22#268211yescharacteristic
299C22#326712yes—
300C22#326812yes—
301C22#329111yescharacteristic
302C22#329311yescharacteristic
303C4—21no—
304C4—21no—
305C4—82no—
306C4—82no—
307C4—82no—
308C4—162no—
309C4—162no—
310C4#319711yescharacteristic
311C4#319911yescharacteristic
312C4#325611yescharacteristic
313C4#326311yescharacteristic
314C4#334712yes—
315C4#340112yes—
316C2—162no—
317C2#2013212yes—
318C2#2014412yes—
319C2#3120711yescharacteristic
320C2#3122311yescharacteristic
321C2#3123011yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
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
1. #26301C2semidirect
3. #10588C2semidirect
4. #10476C2semidirect
5. #10066C2semidirect
6. #9836C2semidirect
8. #9600C2semidirect
9. #8889C2semidirect

Maximal subgroups

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

subgroupindex
#263012
#246272
#105882
#104762
#100662
#98362
#96652
#96002
#88892