Finite Group Databasebeta

← #497097

Subgroups — #497097

Order 512.

500 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2255C211yesmaximal, characteristic
2C4×#499C211yesmaximal, characteristic
3C4×#500C211yesmaximal, characteristic
4#13284C211yesmaximal, characteristic
5#13283C211yesmaximal, characteristic
6#13112C212yesmaximal
7#13111C212yesmaximal
8#13111C212yesmaximal
9#13109C212yesmaximal
10#12890C211yesmaximal, characteristic
11#12591C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#499C2211yescharacteristic
14C4×C4⋊Q8—21no—
15C4×#125—21no—
16#2255C418yescomplemented
17C2×#367C2212yes—
18C2×#366C2212yes—
19C2×#346C2211yescharacteristic
20C2×#346C2211yescharacteristic
21#1991—21no—
22#1990—21no—
23#1942—21no—
24#1942C2211yescharacteristic
25#1942C2211yescharacteristic
26#1942C2212yes—
27#1941—21no—
28#1846C2211yescharacteristic
29#1845C2211yescharacteristic
30C4×#107—21no—
31C4×#107C2212yes—
32C4×#107C2211yescharacteristic
33C4×#107C2211yescharacteristic
34C4×#106—21no—
35C42×C8C2211yescharacteristic
36#1380C2212yes—
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2212yes—
43#1374C2212yes—
44C2×C4⋊Q8—21no—
45C2×#125—21no—
46C2×C4×Q8—41no—
47C22×C16C2312yes—
48C22×C16C2312yes—
49#500—22no—
50#500C2×C414yes—
51#499—22no—
52#499C2×C414yes—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107C2312yes—
56C2×#107C2311yescharacteristic
57C2×#107C2311yescharacteristic
58C2×#107C2311yescharacteristic
59C2×#107C2311yescharacteristic
60C2×#107C2312yes—
61C2×#106—21no—
62C2×#106—21no—
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#391—41no—
70#389—21no—
71#389—21no—
72#389—22no—
73#389—41no—
74#387—41no—
75#387—41no—
76#384—21no—
77#382—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—41no—
82C4×C4⋊C4—41no—
83C4×C4⋊C4—41no—
84C43D411yescharacteristic
85#367C2×C418yes—
86#366C2×C418yes—
87#346—22no—
88#346C2×C414yes—
89#340—21no—
90#340—21no—
91#340—22no—
92#340—22no—
93#340—21no—
94#340—21no—
95C22×Q8—41no—
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C22×C8C2411yescharacteristic
100C4⋊Q8—24no—
101C4⋊Q8—42no—
102#125—24no—
103#125—42no—
104C4×Q8—84no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C42—42no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42D811yescharacteristic
123C2×C42D811yescharacteristic
124C2×C42D811yescharacteristic
125C2×C42D811yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C16—22no—
130C2×C16—22no—
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—22no—
134C2×C16C22×C414yes—
135#107—24no—
136#107—21no—
137#107—21no—
138#107—22no—
139#107—24no—
140#107—22no—
141#107—22no—
142#107—42no—
143#107C22×C412yes—
144#107C22×C414yes—
145#107C22×C412yes—
146#106—24no—
147#106—42no—
148C4×C8—22no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8C22×C412yes—
154C4×C8#4112yes—
155#95—41no—
156#95—41no—
157#95—42no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—42no—
163C2×Q8—44no—
164C2×Q8—82no—
165C2×Q8—82no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4C2×D811yescharacteristic
173C22×C4C2×D811yescharacteristic
174C22×C4C2×D811yescharacteristic
175C22×C4C2×D811yescharacteristic
176C22×C4C2×D811yescharacteristic
177C22×C4C2×D811yescharacteristic
178C22×C4C22×D411yescharacteristic
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8C4×D412yes—
186C2×C8#13712yes—
187C2×C8#13712yes—
188C2×C8C23×C411yescharacteristic
189C2×C8C2×#4111yescharacteristic
190C2×C82−1+412yes—
191C2×C82−1+412yes—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C4⋊C4—82no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—22no—
217C42—22no—
218C42—41no—
219C42—41no—
220C42—82no—
221C42—84no—
222C42—82no—
223C42C4×D412yes—
224C42#13512yes—
225C16—24no—
226C16—42no—
227C23C22×D811yescharacteristic
228Q8—88no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4C4×D812yes—
262C2×C4C4×D812yes—
263C2×C4#50912yes—
264C2×C4#51012yes—
265C2×C4C2×C4×D411yescharacteristic
266C2×C4#53611yescharacteristic
267C2×C4#53611yescharacteristic
268C2×C4C2×#13511yescharacteristic
269C2×C4C2×#13711yescharacteristic
270C2×C4C2×#13711yescharacteristic
271C8—41no—
272C8—41no—
273C8#44114yes—
274C8#51912yes—
275C8#54112yes—
276C22C2×C4×D811yescharacteristic
277C22#322611yescharacteristic
278C22#322611yescharacteristic
279C22C2×#50911yescharacteristic
280C22C2×#51011yescharacteristic
281C22#341411yescharacteristic
282C22#341411yescharacteristic
283C4—22no—
284C4—22no—
285C4—22no—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—82no—
290C4—84no—
291C4—82no—
292C4—162no—
293C4—161no—
294C4—161no—
295C4#232711yescharacteristic
296C4C2×#44111yescharacteristic
297C4#323111yescharacteristic
298C4#323111yescharacteristic
299C2#2224711yescharacteristic
300C2C4×#50911yescharacteristic
301C2C4×#51011yescharacteristic
302C2#3380111yescharacteristic
303C2#3380211yescharacteristic
304C2#3380411yescharacteristic
305C2#3380611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2255C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22552
C4×#4992
C4×#5002
#132842
#132832
#131122
#131112
#131092
#128902
#125912