Finite Group Databasebeta

← #181552

Subgroups — #181552

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3C2×#2351C211yesmaximal, characteristic
4#8026C212yesmaximal
5#8021C214yesmaximal
6#8017C214yesmaximal
7#8014C212yesmaximal
8C22×C4×Q8—21no—
9C22×C4×Q8C2211yescharacteristic
10#2983—24no—
11#2983C418yescomplemented
12C2×#446—24no—
13C2×#432C2214yes—
14C2×#431C2212yes—
15#2356—24no—
16#2355C418yescomplemented
17#2351—24no—
18#2351C418yescomplemented
19#2313—22no—
20#2310—21no—
21C2×#389—22no—
22C2×#389—21no—
23#2285C2212yes—
24#2283C2214yes—
25C2×#382C2212yes—
26C2×#382C2211yescharacteristic
27#2274—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30#1294C2214yes—
31#1294C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C2×C4×Q8—28no—
35C2×C4×Q8—48no—
36C2×C4×Q8C2×C414yes—
37C2×C4×Q8C2×C414yes—
38C2×C4×Q8C2×C414yes—
39C2×C4×Q8D414yes—
40C2×C4×Q8D414yes—
41C2×C4×Q8Q814yes—
42C2×#117—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4D411yescharacteristic
49C22×C42—22no—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52C22×C42C2312yes—
53#446—232no—
54#432—28no—
55#432C2×C4116yes—
56#431—24no—
57#431C2×C418yes—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—24no—
60C2×C22⋊C8C2314yes—
61C2×#97—24no—
62C2×C4×C8—22no—
63#391—28no—
64#389—216no—
65#389—28no—
66#387—216no—
67#384C2×C414yes—
68#383—24no—
69#382—22no—
70#382—24no—
71#382—22no—
72#382C2×C418yes—
73#382C2×C418yes—
74#382C2×C414yes—
75#381—24no—
76#380—22no—
77#380C2314yes—
78C4×C4⋊C4—28no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4C2×C418yes—
83C4×C4⋊C4C2×C414yes—
84C4×C4⋊C4C2×C414yes—
85C4×C4⋊C4C2×C418yes—
86C4×C22⋊C4—24no—
87#376—24no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95C2311yescharacteristic
91C2×#95C2311yescharacteristic
92C2×#95C2312yes—
93C22×Q8—24no—
94C22×Q8—24no—
95C22×Q8C22⋊C414yes—
96C22×Q8C22⋊C414yes—
97C22×Q8C4⋊C414yes—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2411yescharacteristic
106C22×C8—24no—
107C4×Q8—432no—
108C4×Q8C42116yes—
109C4×Q8C22⋊C4116yes—
110C4×Q8C4⋊C4116yes—
111C4×Q8C4⋊C4116yes—
112#117—44no—
113#117—44no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—48no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—48no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4C22⋊C414yes—
125C2×C4⋊C4C22⋊C418yes—
126C2×C4⋊C4C22⋊C418yes—
127C2×C4⋊C4C22⋊C414yes—
128C2×C4⋊C4C4⋊C418yes—
129C2×C4⋊C4C4⋊C414yes—
130C2×C4⋊C4C22×C418yes—
131C2×C22⋊C4—24no—
132C2×C42—24no—
133C2×C42—28no—
134C2×C42—24no—
135C2×C42—28no—
136C2×C42—48no—
137C2×C42—44no—
138C2×C42C22×C412yes—
139C2×C42C22×C414yes—
140C2×C42C22×C414yes—
141C2×C42C22×C414yes—
142C2×C42C22×C412yes—
143C2×C42C22×C412yes—
144C2×C42C2×D414yes—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D414yes—
148C2×C42C2×Q814yes—
149C2×C42C2×Q812yes—
150C4⋊C8—28no—
151C4⋊C8—216no—
152C22⋊C8—24no—
153C22⋊C8C22×C418yes—
154#97—216no—
155C4×C8—28no—
156#95—24no—
157#95—28no—
158#95—28no—
159#95—28no—
160#95—24no—
161#95—28no—
162#95—21no—
163#95—21no—
164#95—22no—
165#95—24no—
166#95C22×C412yes—
167#95C22×C412yes—
168#95C22×C414yes—
169C24C22×D411yescharacteristic
170C2×Q8—28no—
171C2×Q8—28no—
172C2×Q8—28no—
173C2×Q8—28no—
174C2×Q8—216no—
175C2×Q8#95116yes—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—28no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4C2×C22⋊C414yes—
195C22×C4C2×C22⋊C414yes—
196C22×C4C2×C22⋊C414yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C414yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C4⋊C414yes—
203C22×C4C2×C4⋊C412yes—
204C22×C4C2×C4⋊C414yes—
205C22×C4C2×C4⋊C412yes—
206C22×C4AGL(1,ℤ/8)14yes—
207C22×C4#13714yes—
208C22×C4C23×C411yescharacteristic
209C22×C4C23×C411yescharacteristic
210C22×C4C23×C411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C22×Q811yescharacteristic
214C2×C8—28no—
215C2×C8—48no—
216C4⋊C4—48no—
217C4⋊C4—416no—
218C4⋊C4—48no—
219C4⋊C4—416no—
220C4⋊C4—48no—
221C4⋊C4—48no—
222C4⋊C4—84no—
223C4⋊C4#95132yes—
224C4⋊C4#95116yes—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C42—416no—
228C42—48no—
229C42—48no—
230C42—88no—
231C42C2×C4218yes—
232C42C2×C4214yes—
233C42C2×C22⋊C414yes—
234C42C2×C22⋊C418yes—
235C42C2×C4⋊C418yes—
236C42C2×C4⋊C418yes—
237C42C2×C4⋊C414yes—
238C42C2×C4⋊C414yes—
239C23—22no—
240C23—22no—
241C23C22×C22⋊C411yescharacteristic
242C23C22×C22⋊C411yescharacteristic
243C23C22×C4⋊C411yescharacteristic
244C23C2×AGL(1,ℤ/8)12yes—
245C23C2×#13712yes—
246Q8—232no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—28no—
252C2×C4—28no—
253C2×C4—24no—
254C2×C4—28no—
255C2×C4—24no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—48no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—48no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—48no—
275C2×C4—48no—
276C2×C4—88no—
277C2×C4—84no—
278C2×C4C2×#9514yes—
279C2×C4C2×#9518yes—
280C2×C4C2×#9518yes—
281C2×C4C2×#9514yes—
282C2×C4#41714yes—
283C2×C4#41714yes—
284C2×C4#41814yes—
285C2×C4#41914yes—
286C2×C4#42818yes—
287C2×C4C22×C4211yescharacteristic
288C2×C4C22×C22⋊C411yescharacteristic
289C2×C4C22×C4⋊C411yescharacteristic
290C2×C4C22×C4⋊C411yescharacteristic
291C8—48no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—24no—
297C22—24no—
298C22C22×#9511yescharacteristic
299C22#288412yes—
300C22C2×#41811yescharacteristic
301C22C2×#41911yescharacteristic
302C22C2×#41712yes—
303C22#290112yes—
304C22C2×#42812yes—
305C4—28no—
306C4—24no—
307C4—28no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—44no—
313C4—84no—
314C4—82no—
315C4—88no—
316C4#173618yes—
317C2—21no—
318C2—21no—
319C2—22no—
320C2#1762814yes—
321C2#2034311yescharacteristic
322C2C2×#173611yescharacteristic
323C2#2035511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2983C4semidirect
16. #2355C4semidirect
18. #2351C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#341872
C2×#23512
#80262
#80212
#80172
#80142