Finite Group Databasebeta

← #305778

Subgroups — #305778

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28829C211yesmaximal, characteristic, complemented
2#26021C212yesmaximal
3#22657C211yesmaximal, characteristic
4#10656C212yesmaximal, complemented
5#10579C212yesmaximal, complemented
6#9612C211yesmaximal, characteristic, complemented
7#9593C211yesmaximal, characteristic, complemented
8#8885C212yesmaximal
9#8814C211yesmaximal, characteristic, complemented
10#8642C211yesmaximal, characteristic
11C8×#102C211yesmaximal, characteristic
12C2×C8×D4C2211yescharacteristic
13C2×#427—21no—
14#2552C2212yes—
15#2354C2212yes—
16#2084—22no—
17#2065—22no—
18#2025—22no—
19#1989—22no—
20#1948C2212yes—
21#1948C2212yes—
22#1947C2212yes—
23#1941—21no—
24#1939C2212yes—
25#1919—22no—
26#1854—22no—
27#1845C2212yes—
28#1844—21no—
29#1844C2211yescharacteristic
30#1844C2211yescharacteristic
31#1834C2212yes—
32#1807—21no—
33#1802C2212yes—
34C4×#106C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C4×#102C2211yescharacteristic
37C2×#334C2211yescharacteristic
38C2×D4⋊C8C2211yescharacteristic
39C2×C82C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C23×C8—21no—
46C2×#125—22no—
47C2×C4×D4C2311yescharacteristic
48C2×#117—22no—
49C8×D4—22no—
50C8×D4Q814yes—
51#427—44no—
52C2×#107—22no—
53C2×#107—42no—
54C2×#106—21no—
55C2×#106—21no—
56C2×#106—22no—
57C2×#106—21no—
58C2×#106—21no—
59C2×#106—22no—
60C2×#106D411yescharacteristic
61C2×#106D411yescharacteristic
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×#102—22no—
67C2×#102D412yes—
68C2×C22⋊C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#398—42no—
74#390—22no—
75#389—22no—
76#389—22no—
77#388—42no—
78#384—22no—
79#384C2312yes—
80#384C2312yes—
81#380—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4C2312yes—
84C4×C4⋊C4C2311yescharacteristic
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340D412yes—
92#340D412yes—
93#336—21no—
94#336—22no—
95#334—22no—
96D4⋊C8Q814yes—
97C82—22no—
98C22×D4C2×D411yescharacteristic
99C23×C4—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—24no—
104C22×C8—22no—
105C22×C8—42no—
106C22×C8—42no—
107C22×C8C2×D412yes—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110#125—44no—
111#125—82no—
112C4×D4—21no—
113C4×D4—21no—
114C4×D4C2×Q812yes—
115C4×D4C2×Q812yes—
116#117—44no—
117#117—82no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C4⋊C4C2×D412yes—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C42—22no—
135C2×C42—21no—
136C2×C42—42no—
137C2×C42C2411yescharacteristic
138#107—44no—
139#107—84no—
140#106—22no—
141#106—22no—
142#106—42no—
143#106—42no—
144#106—44no—
145#106—82no—
146#106—81no—
147#106—81no—
148C4⋊C8—21no—
149C4⋊C8—22no—
150C4⋊C8—21no—
151C4⋊C8—22no—
152C4⋊C8—82no—
153C4⋊C8C2×Q812yes—
154C4⋊C8C2×Q812yes—
155#102—24no—
156#102—82no—
157C22⋊C8—24no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—42no—
162C4×C8C2×Q812yes—
163C4×C8C2×Q812yes—
164C4×C8#4112yes—
165#95—22no—
166#95—21no—
167#95—42no—
168#95—42no—
169C24—21no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4C4⋊Q812yes—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4H(5,𝔽2)11yescharacteristic
189C2×C8—24no—
190C2×C8—22no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—48no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8C22⋊Q812yes—
207C2×C8C22⋊Q812yes—
208C2×C8#13512yes—
209C2×C8AGL(1,ℤ/8)12yes—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—84no—
225C4⋊C4—84no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—162no—
229C4⋊C4C4⋊Q812yes—
230C4⋊C4#13514yes—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—84no—
234C22⋊C4—162no—
235C42—41no—
236C42—42no—
237C42—42no—
238C42—41no—
239C42—84no—
240C42C22×Q811yescharacteristic
241C42C22×Q811yescharacteristic
242C42C2×#4111yescharacteristic
243C422−1+411yescharacteristic
244C23—22no—
245C23—41no—
246C23—42no—
247C23—41no—
248C23—42no—
249C23D4211yescharacteristic
250D4—22no—
251D4—22no—
252D4—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
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—42no—
269C2×C4—42no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4C2×C22⋊Q811yescharacteristic
277C2×C4C2×C4⋊Q811yescharacteristic
278C2×C4#54411yescharacteristic
279C2×C4#54711yescharacteristic
280C2×C4D4×Q811yescharacteristic
281C2×C4#55411yescharacteristic
282C2×C4C2×#13511yescharacteristic
283C2×C4C2×#13512yes—
284C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
285C2×C4#57611yescharacteristic
286C2×C4#57712yes—
287C2×C4#57811yescharacteristic
288C8—24no—
289C8—22no—
290C8—42no—
291C8—42no—
292C8—42no—
293C8—82no—
294C8#47814yes—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—82no—
299C22—82no—
300C22#244011yescharacteristic
301C22#327411yescharacteristic
302C22#328212yes—
303C22#328311yescharacteristic
304C22#329112yes—
305C4—21no—
306C4—21no—
307C4—44no—
308C4—42no—
309C4—82no—
310C4—82no—
311C4—162no—
312C4—162no—
313C4C2×#47811yescharacteristic
314C4#308311yescharacteristic
315C4#316011yescharacteristic
316C4#316311yescharacteristic
317C4#334012yes—
318C4#338011yescharacteristic
319C4#338611yescharacteristic
320C2—82no—
321C2#1992212yes—
322C2#1999712yes—
323C2#2604111yescharacteristic
324C2#2606111yescharacteristic
325C2#2608111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #28829C2semidirect
4. #10656C2semidirect
5. #10579C2semidirect
6. #9612C2semidirect
7. #9593C2semidirect
9. #8814C2semidirect

Maximal subgroups

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

subgroupindex
#288292
#260212
#226572
#106562
#105792
#96122
#95932
#88852
#88142
#86422
C8×#1022