Finite Group Databasebeta

← #354136

Subgroups — #354136

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#433C211yesmaximal, characteristic
2C2×#1750C211yesmaximal, characteristic
3C2×#1746C211yesmaximal, characteristic, complemented
4C2×C42×C8C211yesmaximal, characteristic, complemented
5#13761C211yesmaximal, characteristic
6#10921C214yesmaximal
7#10876C214yesmaximal, complemented
8#10694C211yesmaximal, characteristic, complemented
9#10693C211yesmaximal, characteristic, complemented
10C22×#117C2211yescharacteristic
11C2×#433—24no—
12C2×#433—24no—
13C2×#433C418yescomplemented
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C22×C4×C8C2214yes—
20C22×C4×C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×#376C2211yescharacteristic
24C2×C43C2211yescharacteristic
25#1750C418yescomplemented
26#1746—24no—
27C2×#336C2214yes—
28C2×#336C2214yes—
29C2×#336C2214yes—
30C2×#336C2212yes—
31C42×C8—24no—
32#1437C414yes—
33#1436C414yes—
34C4×#95C2212yes—
35C24×C4C2311yescharacteristic
36C23×C8C2312yes—
37C23×C8C2312yes—
38C23×C8C2312yes—
39C23×C8C2312yes—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—22no—
44C2×#117C2×C414yes—
45C2×#117C2×C414yes—
46C22×C4⋊C4C2×C412yes—
47C22×C22⋊C4C2×C414yes—
48C22×C22⋊C4C2×C412yes—
49C22×C42C2×C414yes—
50C22×C42C2312yes—
51C22×C42C2311yescharacteristic
52C22×C42C2312yes—
53C22×C42C2311yescharacteristic
54#433—232no—
55C2×C4⋊C8—24no—
56C2×C4⋊C8—24no—
57C2×C4⋊C8C2×C418yes—
58C2×C22⋊C8—24no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8—28no—
61C2×C22⋊C8C2×C418yes—
62C2×C22⋊C8C2×C418yes—
63C2×#97—22no—
64C2×#97—22no—
65C2×#97—22no—
66C2×#97—22no—
67C2×#97C2×C414yes—
68C2×#97C2×C414yes—
69C2×C4×C8—28no—
70C2×C4×C8—28no—
71C2×C4×C8—22no—
72C2×C4×C8—28no—
73C2×C4×C8—22no—
74C2×C4×C8C2×C414yes—
75C2×C4×C8C2×C414yes—
76C2×C4×C8D414yes—
77C2×C4×C8D414yes—
78C4×C22⋊C4—28no—
79#376—22no—
80#376—22no—
81C2×#95C2×C418yes—
82C2×#95C2×C414yes—
83C2×#95C2×C414yes—
84C43—22no—
85C43—22no—
86#336—28no—
87#336—28no—
88#336—28no—
89#336C2×C418yes—
90C25C22×C411yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4C22×C414yes—
98C23×C4C22×C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C412yes—
101C23×C4C22×C412yes—
102C23×C4C22×C412yes—
103C23×C4C22×C414yes—
104C23×C4C2411yescharacteristic
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—24no—
108C22×C8—24no—
109C22×C8—24no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—24no—
114C22×C8—24no—
115C22×C8—24no—
116C22×C8—22no—
117C22×C8—24no—
118C22×C8C22×C414yes—
119C22×C8C22×C414yes—
120C22×C8C22×C414yes—
121C22×C8C22×C414yes—
122C22×C8C2×D414yes—
123C22×C8C2×D414yes—
124C22×C8#4118yes—
125#117—28no—
126#117—28no—
127#117—28no—
128#117—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4C4218yes—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—216no—
136C2×C22⋊C4C4218yes—
137C2×C42—28no—
138C2×C42—216no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—22no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42—24no—
146C2×C42—22no—
147C2×C42—24no—
148C2×C42—22no—
149C2×C42—24no—
150C2×C42C22×C412yes—
151C2×C42C22×C412yes—
152C2×C42C22×C412yes—
153C2×C42C22×C412yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42#4118yes—
157C2×C42#4112yes—
158C2×C42#4112yes—
159C4⋊C8—216no—
160C22⋊C8—216no—
161C22⋊C8—216no—
162#97—28no—
163#97—28no—
164C4×C8—232no—
165C4×C8—28no—
166C4×C8C22⋊C4116yescomplemented
167#95—28no—
168#95—28no—
169#95—216no—
170C24—22no—
171C24—21no—
172C24—21no—
173C24—22no—
174C24—28no—
175C24C2×C4212yes—
176C24C23×C411yescharacteristic
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—28no—
180C22×C4—28no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—28no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—28no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4C2×C4214yes—
206C22×C4C2×C4212yes—
207C22×C4C2×C4214yes—
208C22×C4#11718yes—
209C22×C4C4×D414yes—
210C22×C4C4×D414yes—
211C22×C4#131116yes—
212C22×C4C23×C411yescharacteristic
213C22×C4C23×C411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C2×#4111yescharacteristic
217C22×C4C2×#4111yescharacteristic
218C22×C4C2×#4114yes—
219C22×C4C2×#4114yes—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8—24no—
223C2×C8—24no—
224C2×C8—28no—
225C2×C8—28no—
226C2×C8—28no—
227C2×C8—28no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8C2×C22⋊C418yes—
239C2×C8#11718yes—
240C4⋊C4—216no—
241C22⋊C4—216no—
242C22⋊C4—416no—
243C42—232no—
244C42—22no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—28no—
249C42—28no—
250C42—44no—
251C42—44no—
252C42C2×C22⋊C414yes—
253C42#11714yes—
254C42C4×D414yes—
255C42C4×D414yes—
256C23—28no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—44no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23—48no—
265C23—42no—
266C23—42no—
267C23C22×C4211yescharacteristic
268C23C2×#11712yes—
269C23C2×C4×D411yescharacteristic
270C23C2×C4×D411yescharacteristic
271C23C4×#4112yes—
272C23C2×#13118yes—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—28no—
276C2×C4—216no—
277C2×C4—28no—
278C2×C4—216no—
279C2×C4—28no—
280C2×C4—28no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—48no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4C4×C22⋊C418yes—
299C2×C4#433116yescomplemented
300C2×C4C22×C22⋊C411yescharacteristic
301C2×C4C2×#11712yes—
302C2×C4C2×#11711yescharacteristic
303C2×C4C2×C4×D411yescharacteristic
304C2×C4C2×C4×D411yescharacteristic
305C2×C4C4×#4112yes—
306C2×C4C4×#4112yes—
307C2×C4C4×#4112yes—
308C2×C4#52514yes—
309C2×C4#52914yes—
310C2×C4#52914yes—
311C2×C4#53214yes—
312C8—28no—
313C8—28no—
314C8—44no—
315C8—44no—
316C22—44no—
317C22—42no—
318C22—44no—
319C22—44no—
320C22—84no—
321C22—84no—
322C22C2×C4×C22⋊C411yescharacteristic
323C22C4×#11711yescharacteristic
324C22C4×#11711yescharacteristic
325C22#229012yes—
326C22#229312yes—
327C22C4×#13114yes—
328C22C2×#43314yes—
329C22#292214yes—
330C22#292914yes—
331C22#293114yes—
332C22#296618yes—
333C4—216no—
334C4—28no—
335C4—44no—
336C4—42no—
337C4—84no—
338C4#229012yes—
339C4#229312yes—
340C4#229712yes—
341C4#229812yes—
342C2—82no—
343C2#1376111yescharacteristic
344C2#2038312yes—
345C2C4×#43312yes—
346C2#2047412yes—
347C2#2110314yes—
348C2#2126614yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1746C2semidirect
4. C2×C42×C8C2semidirect
7. #10876C2semidirect
8. #10694C2semidirect
9. #10693C2semidirect
13. C2×#433C4semidirect
25. #1750C4semidirect
166. C4×C8C22⋊C4semidirect
299. C2×C4#433semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
C2×#17502
C2×#17462
C2×C42×C82
#137612
#109212
#108762
#106942
#106932