Finite Group Databasebeta

← #345332

Subgroups — #345332

Order 512.

631 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 144 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#26233C212yesmaximal, complemented
2#24676C212yesmaximal
3#10594C212yesmaximal, complemented
4#10485C212yesmaximal, complemented
5#10076C212yesmaximal, complemented
6#9831C211yesmaximal, characteristic, complemented
7#9663C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic, complemented
9#8897C212yesmaximal, complemented
10C2×#500C2211yescharacteristic
11C2×#485—22no—
12C2×#474C2212yes—
13C2×C4×D8C2211yescharacteristic
14#2542C2212yes—
15#2354C2212yes—
16#2071—22no—
17#2054—22no—
18#2036—22no—
19#2035—22no—
20#1990C2212yes—
21#1981C2212yes—
22#1956C2212yes—
23#1954C2212yes—
24#1947—22no—
25#1947C2212yes—
26#1939C2212yes—
27#1925—22no—
28#1873—22no—
29#1834C2211yescharacteristic
30#1802C2212yes—
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#332C2212yes—
35C2×D4⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×D8D411yescharacteristic
40C2×C4⋊Q8C2312yes—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C4×D4C2312yes—
44C2×#117—22no—
45#500—22no—
46#500—41no—
47#485—48no—
48#474—42no—
49#474—42no—
50C4×D8—41no—
51C4×D8—41no—
52C2×#107—22no—
53C2×#107—22no—
54C2×#107—41no—
55C2×#107D411yescharacteristic
56C2×#106D411yescharacteristic
57C2×#106D411yescharacteristic
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—22no—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102D412yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#399—42no—
74#390—22no—
75#390—22no—
76#388—42no—
77#384—22no—
78#384C2312yes—
79#380—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2312yes—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—41no—
87#340D412yes—
88#336—22no—
89#335—41no—
90#335—41no—
91#332—42no—
92D4⋊C8—24no—
93#322—41no—
94C22×Q8—22no—
95C22×D4C2×D412yes—
96C23×C4—22no—
97C2×D8—42no—
98C2×D8—41no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C22⋊Q8—44no—
111C22⋊Q8—44no—
112C22⋊Q8—82no—
113C22⋊Q8—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⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C42—22no—
136C2×C42—42no—
137C2×C42C2411yescharacteristic
138#107—41no—
139#107—44no—
140#107—44no—
141#107—44no—
142#106—41no—
143#106D814yescomplemented
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—44no—
149#103—44no—
150#103—44no—
151#103—82no—
152#103—82no—
153#102—42no—
154#102—44no—
155#102—44no—
156C22⋊C8—44no—
157C22⋊C8—82no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8#4112yes—
164#95—22no—
165#95—42no—
166C24—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4H(5,𝔽2)11yescharacteristic
193D8—44no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—22no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8C2×D812yes—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—84no—
226C4⋊C4—82no—
227C4⋊C4—84no—
228C4⋊C4C2×D814yes—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—84no—
232C22⋊C4—84no—
233C22⋊C4—84no—
234C22⋊C4—162no—
235C42—42no—
236C42—42no—
237C42—84no—
238C42C2×#4111yescharacteristic
239C42H(5,𝔽2)11yescharacteristic
240C42H(5,𝔽2)11yescharacteristic
241C422−1+411yescharacteristic
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23D4211yescharacteristic
250Q8—82no—
251Q8—82no—
252Q8—82no—
253Q8—82no—
254D4—42no—
255D4—42no—
256D4—42no—
257D4—42no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4#53411yescharacteristic
279C2×C4#53711yescharacteristic
280C2×C4#53811yescharacteristic
281C2×C4#54711yescharacteristic
282C2×C4#55011yescharacteristic
283C2×C4#56211yescharacteristic
284C2×C4C22×D811yescharacteristic
285C2×C4#57512yes—
286C2×C4#57511yescharacteristic
287C2×C4#57611yescharacteristic
288C2×C4#57611yescharacteristic
289C2×C4#57812yes—
290C8—22no—
291C8—42no—
292C8—41no—
293C8—41no—
294C8—41no—
295C8—82no—
296C8—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22#268211yescharacteristic
305C22#326912yes—
306C22#327311yescharacteristic
307C22D4×D811yescharacteristic
308C22#328912yes—
309C4—21no—
310C4—21no—
311C4—44no—
312C4—82no—
313C4—82no—
314C4—162no—
315C4—162no—
316C4#319711yescharacteristic
317C4#320211yescharacteristic
318C4#324511yescharacteristic
319C4#325911yescharacteristic
320C4#334912yes—
321C4#339111yescharacteristic
322C4#340311yescharacteristic
323C2—162no—
324C2#1976012yes—
325C2#2022312yes—
326C2#3120511yescharacteristic
327C2#3120811yescharacteristic
328C2#3123311yescharacteristic
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. #26233C2semidirect
3. #10594C2semidirect
4. #10485C2semidirect
5. #10076C2semidirect
6. #9831C2semidirect
8. #9596C2semidirect
9. #8897C2semidirect
143. #106D8semidirect

Maximal subgroups

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

subgroupindex
#262332
#246762
#105942
#104852
#100762
#98312
#96632
#95962
#88972