Finite Group Databasebeta

← #325348

Subgroups — #325348

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33256C211yesmaximal, characteristic, complemented
2#29122C212yesmaximal
3C8×#124C211yesmaximal, characteristic
4#10624C211yesmaximal, characteristic, complemented
5#10623C211yesmaximal, characteristic, complemented
6#10622C211yesmaximal, characteristic
7#10621C211yesmaximal, characteristic
8#9901C212yesmaximal, complemented
9#8643C212yesmaximal, complemented
10#8642C212yesmaximal
11C4×#334C211yesmaximal, characteristic, complemented
12C2×C8×Q8C2211yescharacteristic
13C2×C8×D4C2211yescharacteristic
14#2859C2212yes—
15C4×#124C2211yescharacteristic
16#2101—22no—
17#2042—22no—
18#1984C2211yescharacteristic
19#1954C2211yescharacteristic
20#1941—22no—
21#1941—22no—
22#1934—22no—
23#1933—22no—
24#1923—22no—
25#1844C2212yes—
26#1834C2211yescharacteristic
27#1834C2211yescharacteristic
28#1831C2211yescharacteristic
29#1803C2212yes—
30#1803C2212yes—
31#1802C2212yes—
32#1802C2212yes—
33C4×C4⋊C8C2212yes—
34C8×C22⋊C4—22no—
35C42×C8C2211yescharacteristic
36C2×#334C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2212yes—
40#1296C2212yes—
41#1277C2211yescharacteristic
42#1277C2211yescharacteristic
43C23×C8—21no—
44C2×#125—22no—
45C2×#124D411yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47C2×C4×D4C2311yescharacteristic
48C8×Q8—24no—
49C8×D4—24no—
50C2×#106—22no—
51C2×#106—22no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2311yescharacteristic
56C2×C4⋊C8C2311yescharacteristic
57C2×#103—22no—
58C2×#103—22no—
59C2×#102—22no—
60C2×#102—22no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—22no—
63C2×C4×C8—21no—
64C2×C4×C8—22no—
65C2×C4×C8D412yes—
66C2×C4×C8D412yes—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#391D411yescharacteristic
70#390D411yescharacteristic
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#388—22no—
76#384C2312yes—
77#384C2312yes—
78#382—22no—
79#382D411yescharacteristic
80C4×C22⋊C4—22no—
81C43C2311yescharacteristic
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#336—22no—
89#336—22no—
90#336—22no—
91#334—24no—
92C22×Q8C2×D411yescharacteristic
93C22×D4C2×D411yescharacteristic
94C23×C4—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—42no—
100C22×C8—44no—
101C22×C8C2×D412yes—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104#125—84no—
105#124—22no—
106#124—22no—
107C4×Q8—22no—
108C4×Q8—22no—
109C4×D4—22no—
110C4×D4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4C2×D411yescharacteristic
121C2×C4⋊C4C2×D411yescharacteristic
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—21no—
125C2×C42—21no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42#4112yes—
130C2×C42C2411yescharacteristic
131#106—44no—
132#106—44no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—42no—
140#103—44no—
141#103—82no—
142#102—44no—
143#102—82no—
144C22⋊C8—44no—
145C22⋊C8—42no—
146C4×C8—24no—
147C4×C8—22no—
148C4×C8—42no—
149C4×C8—42no—
150C4×C8—82no—
151C4×C8#4114yes—
152#95—22no—
153#95—22no—
154#95—42no—
155#95—42no—
156C24—21no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8#13512yes—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4#13512yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4#12112yes—
175C22×C4#12112yes—
176C22×C4C22×D411yescharacteristic
177C22×C4C22×D411yescharacteristic
178C22×C4C2×#4111yescharacteristic
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—84no—
194C2×C8#12412yes—
195C2×C8#12412yes—
196C2×C8#13514yes—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C4⋊C4—84no—
212C22⋊C4—28no—
213C22⋊C4—42no—
214C22⋊C4—44no—
215C42—22no—
216C42—21no—
217C42—21no—
218C42—22no—
219C42—22no—
220C42—24no—
221C42—41no—
222C42—41no—
223C42C2×#4112yes—
224C422−1+412yes—
225C23—22no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23C2×#12111yescharacteristic
231Q8—24no—
232Q8—42no—
233D4—24no—
234D4—42no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4#47114yes—
258C2×C4C2×#12411yescharacteristic
259C2×C4#53611yescharacteristic
260C2×C4#54712yes—
261C2×C4#55612yes—
262C2×C4C2×#13511yescharacteristic
263C2×C4C2×#13511yescharacteristic
264C2×C4C2×#13512yes—
265C2×C4#57512yes—
266C2×C4#57712yes—
267C8—22no—
268C8—42no—
269C8—82no—
270C8—82no—
271C8—81no—
272C8—81no—
273C8#48814yes—
274C22—42no—
275C22—44no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22#254711yescharacteristic
280C22#303512yes—
281C22#305011yescharacteristic
282C22#306011yescharacteristic
283C22C2×#47112yes—
284C4—24no—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—162no—
291C4—162no—
292C4C2×#48811yescharacteristic
293C4#314211yescharacteristic
294C4#323312yes—
295C4#333712yes—
296C4#333712yes—
297C2—82no—
298C2#1912712yes—
299C2#1922112yes—
300C2#2870711yescharacteristic
301C2#2872111yescharacteristic
302C2#2873211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #33256C2semidirect
4. #10624C2semidirect
5. #10623C2semidirect
8. #9901C2semidirect
9. #8643C2semidirect
11. C4×#334C2semidirect

Maximal subgroups

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

subgroupindex
#332562
#291222
C8×#1242
#106242
#106232
#106222
#106212
#99012
#86432
#86422
C4×#3342