Finite Group Databasebeta

← #325378

Subgroups — #325378

Order 512.

552 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33291C211yesmaximal, characteristic, complemented
2#29131C212yesmaximal
3C8×#124C211yesmaximal, characteristic
4#10624C211yesmaximal, characteristic, complemented
5#10623C211yesmaximal, characteristic, complemented
6#10622C211yesmaximal, characteristic
7#10621C211yesmaximal, characteristic
8#9907C212yesmaximal, complemented
9#8647C212yesmaximal, complemented
10#8646C212yesmaximal
11#8273C211yesmaximal, characteristic, complemented
12C2×C8×Q8C2211yescharacteristic
13C2×C8×D4C2211yescharacteristic
14#2859C2212yes—
15C4×#124C2211yescharacteristic
16#2101—22no—
17#2042—22no—
18#1991—22no—
19#1991—22no—
20#1984C2211yescharacteristic
21#1954C2211yescharacteristic
22#1936—22no—
23#1935—22no—
24#1923—22no—
25#1834C2211yescharacteristic
26#1834C2211yescharacteristic
27#1834C2212yes—
28#1831C2211yescharacteristic
29#1803C2212yes—
30#1803C2212yes—
31#1802C2212yes—
32#1802C2212yes—
33#1767C2212yes—
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×C4⋊C8—21no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2311yescharacteristic
54C2×C4⋊C8C2311yescharacteristic
55C2×#103—22no—
56C2×#103—22no—
57C2×#102—22no—
58C2×#102—22no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—22no—
61C2×#97—22no—
62C2×C4×C8—21no—
63C2×C4×C8D412yes—
64C2×C4×C8D412yes—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#391D411yescharacteristic
68#390D411yescharacteristic
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#388—22no—
74#384C2312yes—
75#384C2312yes—
76#382—22no—
77#382D411yescharacteristic
78C4×C22⋊C4—22no—
79C43C2311yescharacteristic
80#340—22no—
81#340—22no—
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—44no—
100C22×C8—42no—
101C22×C8C2×D412yes—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104#125—44no—
105#125—44no—
106#124—22no—
107#124—22no—
108C4×Q8—22no—
109C4×Q8—22no—
110C4×D4—22no—
111C4×D4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4C2×D411yescharacteristic
122C2×C4⋊C4C2×D411yescharacteristic
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—21no—
126C2×C42—21no—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D412yes—
129C2×C42C2×D411yescharacteristic
130C2×C42#4112yes—
131C2×C42C2411yescharacteristic
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—42no—
139#103—44no—
140#103—82no—
141#102—44no—
142#102—82no—
143C22⋊C8—44no—
144C22⋊C8—42no—
145#97—82no—
146C4×C8—22no—
147C4×C8—42no—
148C4×C8—42no—
149C4×C8—42no—
150C4×C8#4114yes—
151#95—22no—
152#95—22no—
153#95—42no—
154#95—42no—
155C24—21no—
156C2×Q8—21no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8#13512yes—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4#13512yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4#12112yes—
174C22×C4#12112yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C2×#4111yescharacteristic
178C2×C8—22no—
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—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
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—82no—
208C4⋊C4—82no—
209C4⋊C4—84no—
210C4⋊C4—84no—
211C22⋊C4—28no—
212C22⋊C4—44no—
213C22⋊C4—42no—
214C42—21no—
215C42—21no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42—42no—
220C42—41no—
221C42—41no—
222C42C2×#4112yes—
223C422−1+412yes—
224C23—22no—
225C23—42no—
226C23—41no—
227C23—41no—
228C23—42no—
229C23C2×#12111yescharacteristic
230Q8—24no—
231Q8—42no—
232D4—24no—
233D4—42no—
234C2×C4—22no—
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×C4C2×#12411yescharacteristic
258C2×C4#53611yescharacteristic
259C2×C4#54712yes—
260C2×C4#55612yes—
261C2×C4C2×#13511yescharacteristic
262C2×C4C2×#13511yescharacteristic
263C2×C4#57512yes—
264C2×C4#57512yes—
265C2×C4#57712yes—
266C8—22no—
267C8—22no—
268C8—42no—
269C8—82no—
270C8—82no—
271C8—81no—
272C8—81no—
273C22—42no—
274C22—44no—
275C22—81no—
276C22—81no—
277C22—82no—
278C22#254711yescharacteristic
279C22#304012yes—
280C22#305011yescharacteristic
281C22#306011yescharacteristic
282C22#306712yes—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—81no—
288C4—81no—
289C4—162no—
290C4—162no—
291C4#313511yescharacteristic
292C4#314011yescharacteristic
293C4#324012yes—
294C4#333712yes—
295C4#333712yes—
296C2—82no—
297C2#1927812yes—
298C2#1936612yes—
299C2#2870711yescharacteristic
300C2#2872711yescharacteristic
301C2#2873411yescharacteristic
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. #33291C2semidirect
4. #10624C2semidirect
5. #10623C2semidirect
8. #9907C2semidirect
9. #8647C2semidirect
11. #8273C2semidirect

Maximal subgroups

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

subgroupindex
#332912
#291312
C8×#1242
#106242
#106232
#106222
#106212
#99072
#86472
#86462
#82732