Finite Group Databasebeta

← #345320

Subgroups — #345320

Order 512.

667 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 148 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#26234C212yesmaximal, complemented
2#24811C212yesmaximal
3#10602C212yesmaximal, complemented
4#10485C212yesmaximal, complemented
5#10076C212yesmaximal, complemented
6#9831C211yesmaximal, characteristic, complemented
7#9663C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic, complemented
9#8894C212yesmaximal, complemented
10C2×#500C2211yescharacteristic
11C2×#483—22no—
12C2×#474C2212yes—
13C2×C4×D8C2211yescharacteristic
14#2542C2212yes—
15D4×C4⋊C4C2212yes—
16#2083—22no—
17#2067—22no—
18#2035—22no—
19#2035—22no—
20#1990C2212yes—
21#1981C2212yes—
22#1956C2212yes—
23#1954C2212yes—
24#1948C2212yes—
25#1939C2212yes—
26#1879—22no—
27#1872—22no—
28#1834C2211yescharacteristic
29#1802—22no—
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×D4—22no—
44C2×C4×D4C2312yes—
45C22×C4⋊C4—22no—
46#500—22no—
47#500—41no—
48#483—48no—
49#474—42no—
50#474—42no—
51C4×D8—41no—
52C4×D8—41no—
53C2×#107—22no—
54C2×#107—41no—
55C2×#107D411yescharacteristic
56C2×#106—22no—
57C2×#106D411yescharacteristic
58C2×#106D411yescharacteristic
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102D412yes—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#399—42no—
75#390—22no—
76#389—22no—
77#385—42no—
78#384—22no—
79#384C2312yes—
80#380—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—
97C23×C4—42no—
98C2×D8—24no—
99C2×D8—41no—
100C22×C8—21no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C22⋊Q8—44no—
112C22⋊Q8—44no—
113C22⋊Q8—82no—
114C22⋊Q8—82no—
115C4×D4—22no—
116C4×D4—22no—
117C4×D4—22no—
118C4×D4—22no—
119C4×D4—44no—
120C4×D4—82no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C42—22no—
139C2×C42C2411yescharacteristic
140#107—41no—
141#107—44no—
142#107—44no—
143#106—41no—
144#106—44no—
145#106D814yescomplemented
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—82no—
151#103—44no—
152#103—44no—
153#103—82no—
154#103—82no—
155#102—42no—
156#102—44no—
157#102—82no—
158C22⋊C8—44no—
159C22⋊C8—82no—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8#4112yes—
166#95—22no—
167#95—42no—
168C24—22no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4#13714yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—84no—
193C22×C4—84no—
194C22×C4—162no—
195C22×C4C22×D411yescharacteristic
196C22×C4C22×D411yescharacteristic
197C22×C4H(5,𝔽2)11yescharacteristic
198D8—44no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—22no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8C2×D812yes—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C4⋊C4C2×D814yes—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—84no—
238C22⋊C4—84no—
239C22⋊C4—84no—
240C42—42no—
241C42—42no—
242C42C2×#4111yescharacteristic
243C42H(5,𝔽2)11yescharacteristic
244C42H(5,𝔽2)11yescharacteristic
245C422−1+411yescharacteristic
246C23—24no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23D4211yescharacteristic
254Q8—82no—
255Q8—82no—
256Q8—82no—
257Q8—82no—
258D4—24no—
259D4—24no—
260D4—42no—
261D4—42no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—168no—
283C2×C4#53411yescharacteristic
284C2×C4#53711yescharacteristic
285C2×C4#53811yescharacteristic
286C2×C4#54711yescharacteristic
287C2×C4#55011yescharacteristic
288C2×C4#56211yescharacteristic
289C2×C4C22×D811yescharacteristic
290C2×C4C2×#13712yes—
291C2×C4#57511yescharacteristic
292C2×C4#57611yescharacteristic
293C2×C4#57611yescharacteristic
294C2×C4#57712yes—
295C8—22no—
296C8—42no—
297C8—41no—
298C8—41no—
299C8—41no—
300C8—82no—
301C8—82no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22#268211yescharacteristic
310C22#326612yes—
311C22#327311yescharacteristic
312C22D4×D811yescharacteristic
313C22#328912yes—
314C4—21no—
315C4—21no—
316C4—44no—
317C4—82no—
318C4—82no—
319C4—162no—
320C4—162no—
321C4#319511yescharacteristic
322C4#320411yescharacteristic
323C4#324511yescharacteristic
324C4#325911yescharacteristic
325C4#334412yes—
326C4#339111yescharacteristic
327C4#340311yescharacteristic
328C2—84no—
329C2#1975712yes—
330C2#2017812yes—
331C2#3120311yescharacteristic
332C2#3120811yescharacteristic
333C2#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. #26234C2semidirect
3. #10602C2semidirect
4. #10485C2semidirect
5. #10076C2semidirect
6. #9831C2semidirect
8. #9596C2semidirect
9. #8894C2semidirect
145. #106D8semidirect

Maximal subgroups

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

subgroupindex
#262342
#248112
#106022
#104852
#100762
#98312
#96632
#95962
#88942