Finite Group Databasebeta

← #6374829

Subgroups — #6374829

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34026C211yesmaximal, characteristic
2#21925C218yesmaximal, complemented
3#21917C212yesmaximal, complemented
4#21913C212yesmaximal, complemented
5#21907C212yesmaximal
6#21903C212yesmaximal
7#21435C218yesmaximal, complemented
8#20811C212yesmaximal, complemented
9#20719C212yesmaximal
10C4×#432C212yesmaximal, complemented
11C2×C4×#41C2211yescharacteristic
12C2×#432C2214yes—
13#2298C2214yes—
14#2297C2211yescharacteristic
15#2297C2211yescharacteristic
16#2296C2211yescharacteristic
17#2296C2211yescharacteristic
18#2295C414yes—
19#2293C414yes—
20#2290C418yes—
21C4×#117C2214yes—
22C4×#117C2211yescharacteristic
23C2×C43C2211yescharacteristic
24#1924C22116yes—
25#1923C22116yes—
26#1917C22116yes—
27#1916C22116yes—
28#1843C2214yes—
29#1832C2214yes—
30#1818C2214yes—
31#1798C2214yes—
32#1798C2214yes—
33#1798C2214yes—
34#1798C2214yes—
35#1780C2214yes—
36#1767C2214yes—
37#1767C2214yes—
38#1767C2214yes—
39C4×C4⋊C8C2214yes—
40#1749C2214yes—
41#1749C2214yes—
42#1749C2214yes—
43C4×C22⋊C8C2214yes—
44#1745C2214yes—
45C4×#97C2214yes—
46C22×#41C2311yescharacteristic
47C4×#41—28no—
48C2×C4×Q8C2×C412yes—
49C2×C4×D4C2×C414yes—
50C2×C4×D4C2×C412yes—
51C2×#117C2×C418yes—
52C2×#117C2312yes—
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C42C2×C414yes—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61#432—216no—
62C2×C4⋊C8C2318yes—
63C2×C4⋊C8C2318yes—
64C2×C4⋊C8C2318yes—
65C2×C22⋊C8C2318yes—
66C2×C22⋊C8C2318yes—
67C2×C22⋊C8C2318yes—
68C2×#97C2318yes—
69C2×#97C2318yes—
70C2×C4×C8C2318yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#390C2×C412yes—
74#390C2×C412yes—
75#389C2×C414yes—
76#389C2×C414yes—
77#388C2×C418yes—
78#388C2×C418yes—
79#387C2×C418yes—
80#387C2×C418yes—
81#386C2×C414yes—
82#386C2×C414yes—
83#385C2×C414yes—
84#385C2×C414yes—
85C4×C4⋊C4C2×C418yes—
86C4×C4⋊C4C2×C414yes—
87C4×C4⋊C4C2314yes—
88C4×C4⋊C4C2314yes—
89C4×C22⋊C4C2×C414yes—
90C4×C22⋊C4C2×C418yes—
91C4×C22⋊C4C2314yes—
92C4×C22⋊C4C2314yes—
93#376C2×C418yes—
94#376C2312yes—
95C43C2×C414yes—
96C43C2312yes—
97C43C2312yes—
98C43C2312yes—
99#336C2318yes—
100#336C2318yes—
101#336C2318yes—
102#336C2318yes—
103#336C2318yes—
104#336C2318yes—
105C2×#41—24no—
106C2×#41—42no—
107C2×#41—42no—
108C22×Q8C22×C411yescharacteristic
109C22×D4C22×C412yes—
110C22×D4C22×C411yescharacteristic
111C23×C4C22×C412yes—
112C23×C4C22×C414yes—
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C2412yes—
116C23×C4C2411yescharacteristic
117C22×C8C24116yes—
118C4×Q8—28no—
119C4×D4—216no—
120C4×D4—28no—
121#117—24no—
122#117—28no—
123#117—42no—
124#117—42no—
125#117—416no—
126C2×C4⋊C4C22×C418yes—
127C2×C4⋊C4C22×C414yes—
128C2×C4⋊C4C22×C412yes—
129C2×C4⋊C4C22×C418yes—
130C2×C4⋊C4C22×C412yes—
131C2×C4⋊C4C22×C412yes—
132C2×C22⋊C4C22×C418yes—
133C2×C22⋊C4C22×C412yes—
134C2×C22⋊C4C22×C414yes—
135C2×C22⋊C4C22×C418yes—
136C2×C22⋊C4C22×C412yes—
137C2×C22⋊C4C22×C412yes—
138C2×C42—28no—
139C2×C42—48no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42C22×C414yes—
143C2×C42C22×C414yes—
144C2×C42C22×C414yes—
145C2×C42C22×C418yes—
146C2×C42C22×C412yes—
147C2×C42C22×C412yes—
148C2×C42C22×C414yes—
149C2×C42C22×C414yes—
150C2×C42C22×C412yes—
151C2×C42#4118yes—
152C2×C42C2412yes—
153C2×C42C2412yes—
154C2×C42C2412yes—
155C2×C42C2412yes—
156C2×C42C2412yes—
157C2×C42C2411yescharacteristic
158C2×C42C2411yescharacteristic
159C4⋊C8—216no—
160C4⋊C8—48no—
161C4⋊C8—48no—
162C22⋊C8—216no—
163C22⋊C8—48no—
164C22⋊C8—48no—
165#97—216no—
166#97—48no—
167C4×C8—48no—
168#95C22×C418yes—
169#95C22×C418yes—
170#95C22×C418yes—
171#95C22×C418yes—
172#95C22×C418yes—
173C24C23×C412yes—
174C24C23×C411yescharacteristic
175#41—416no—
176C2×Q8—22no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×D4—24no—
180C2×D4—22no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—42no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—28no—
188C22×C4—28no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4#11718yes—
198C22×C4C23×C412yes—
199C22×C4C23×C414yes—
200C22×C4C23×C414yes—
201C22×C4C23×C414yes—
202C22×C4C23×C414yes—
203C22×C4C23×C412yes—
204C22×C4C23×C412yes—
205C22×C4C23×C411yescharacteristic
206C22×C4C23×C414yes—
207C22×C4C2×#4114yes—
208C22×C4C2511yescharacteristic
209C2×C8—216no—
210C2×C8—216no—
211C2×C8—216no—
212C4⋊C4—24no—
213C4⋊C4—28no—
214C4⋊C4—216no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—48no—
218C22⋊C4—24no—
219C22⋊C4—28no—
220C22⋊C4—216no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—48no—
224C42—28no—
225C42—28no—
226C42—28no—
227C42—24no—
228C42—24no—
229C42—24no—
230C42—24no—
231C42—24no—
232C42—42no—
233C42—42no—
234C42—44no—
235C42—44no—
236C42—48no—
237C42#11718yes—
238C42C2×#4114yes—
239C42C2×#4114yes—
240C23—21no—
241C23—21no—
242C23—24no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—28no—
247C23C2×#11712yes—
248C23C24×C411yescharacteristic
249Q8—44no—
250D4—44no—
251D4—48no—
252C2×C4—24no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—28no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—28no—
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—416no—
274C2×C4—48no—
275C2×C4—48no—
276C2×C4C2×#11714yes—
277C2×C4C2×#11712yes—
278C2×C4C2×#11714yes—
279C2×C4#56818yes—
280C2×C4C22×#4112yes—
281C2×C4#58512yes—
282C2×C4#58512yes—
283C8—416no—
284C22—22no—
285C22—24no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—44no—
290C22—44no—
291C22—44no—
292C22C22×#11711yescharacteristic
293C22#342811yescharacteristic
294C22#342811yescharacteristic
295C22C2×#56814yes—
296C4—22no—
297C4—28no—
298C4—24no—
299C4—24no—
300C4—42no—
301C4—44no—
302C4—44no—
303C4—44no—
304C4—44no—
305C4#345514yes—
306C2—42no—
307C2—82no—
308C2#3404811yescharacteristic
309C2#6012312yes—
310C2#6028612yes—
311C2#6028712yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC22×#41C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21925C2semidirect
3. #21917C2semidirect
4. #21913C2semidirect
7. #21435C2semidirect
8. #20811C2semidirect
10. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#340262
#219252
#219172
#219132
#219072
#219032
#214352
#208112
#207192
C4×#4322