Finite Group Databasebeta

← #381123

Subgroups — #381123

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35448C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic
3#20392C211yesmaximal, characteristic
4#11205C212yesmaximal, complemented
5#11169C212yesmaximal
6#11070C212yesmaximal, complemented
7#10966C212yesmaximal
8#10936C212yesmaximal, complemented
9#10915C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C2×#518C2211yescharacteristic
12C22×#99C2212yes—
13C22×#99C2212yes—
14#2484C414yescomplemented
15#2483C414yescomplemented
16#2419—24no—
17#2419—24no—
18#2315C2211yescharacteristic
19#2315C2211yescharacteristic
20#2315C2212yes—
21#2308C2211yescharacteristic
22C2×#381C2211yescharacteristic
23C2×#381C2211yescharacteristic
24C2×#381C2214yes—
25#2273C2212yes—
26#1799—24no—
27C4×#99—24no—
28C2×#342C2214yes—
29C2×#342C2212yes—
30C2×#342C2212yes—
31C2×#342C2214yes—
32C2×#342C2214yes—
33C23×D4C2311yescharacteristic
34C24×C4C2311yescharacteristic
35C2×#123—22no—
36C2×#123—22no—
37C2×#123—44no—
38#518—44no—
39C2×C4×D4D414yes—
40C2×C4×D4D414yes—
41C2×#117C2×C412yes—
42C22×C4⋊C4C2×C412yes—
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C412yes—
45C22×C22⋊C4C2314yes—
46C22×C22⋊C4C2311yescharacteristic
47C22×C22⋊C4C2311yescharacteristic
48C22×C22⋊C4C2311yescharacteristic
49C22×C22⋊C4C2314yes—
50C22×C22⋊C4C2311yescharacteristic
51C22×C22⋊C4C2311yescharacteristic
52C2×#99—24no—
53C2×#99—28no—
54C2×#99—24no—
55C2×#99—44no—
56#400—24no—
57#398—24no—
58#397—24no—
59#397—24no—
60#387—24no—
61#387—24no—
62#386—22no—
63#386—22no—
64#385—24no—
65#385—22no—
66#385—22no—
67#385—44no—
68#385C2×C414yes—
69#385C2×C414yes—
70#381—24no—
71#381—22no—
72#381—22no—
73#381—28no—
74#381—44no—
75#381C2×C412yes—
76#381C2×C414yes—
77#381C2×C414yes—
78#380—22no—
79#380—22no—
80#380—24no—
81#379C2×C412yes—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—44no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—44no—
87#342—24no—
88#342—24no—
89#342—28no—
90#342—44no—
91#342—44no—
92C25C22×C411yescharacteristic
93C25C2411yescharacteristic
94C22×D4—24no—
95C22×D4—24no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4C2×D412yes—
99C22×D4C2×D412yes—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4C22⋊C414yes—
111C23×C4C22×C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116#123—48no—
117#123—48no—
118C4×D4—48no—
119C4×D4—48no—
120#117—44no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4C22⋊C414yes—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4C22⋊C414yes—
153C2×C22⋊C4C22×C412yes—
154C2×C22⋊C4C22×C412yes—
155C2×C22⋊C4C22×C412yes—
156C2×C22⋊C4C22×C412yes—
157C2×C22⋊C4C2×D412yes—
158C2×C22⋊C4C2×D412yes—
159C2×C22⋊C4#4118yes—
160C2×C42—44no—
161C2×C42C22⋊C414yes—
162#99—216no—
163#99—48no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—416no—
172#95—44no—
173#95—44no—
174C24—22no—
175C24—22no—
176C24—22no—
177C24—82no—
178C24—82no—
179C24C2×C22⋊C412yes—
180C24C23×C411yescharacteristic
181C24C22×D411yescharacteristic
182C24C22×D411yescharacteristic
183C24C2×#4114yes—
184C24H(5,𝔽2)14yes—
185C24H(5,𝔽2)12yes—
186C242−1+412yes—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—24no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—44no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—84no—
237C22×C4—88no—
238C22×C4—88no—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C414yes—
242C4⋊C4—44no—
243C4⋊C4—48no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C4⋊C4—164no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—44no—
256C22⋊C4—48no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—84no—
269C42—44no—
270C42—164no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—48no—
277C23—42no—
278C23—44no—
279C23—44no—
280C23—84no—
281C23—84no—
282C23—82no—
283C23—82no—
284C23—162no—
285C23—162no—
286C23C22×C22⋊C411yescharacteristic
287C23C4×#4112yes—
288C23#51812yes—
289C23#52012yes—
290C23#52512yes—
291C23#53212yes—
292C23#53412yes—
293C23#53412yes—
294C23#53511yescharacteristic
295C23#53611yescharacteristic
296C23#53712yes—
297C23#53814yes—
298C23#53914yes—
299C23#54014yes—
300C23#54214yes—
301D4—48no—
302D4—44no—
303D4—44no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—88no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—88no—
331C2×C4—88no—
332C2×C4—88no—
333C2×C4—84no—
334C2×C4—164no—
335C2×C4—162no—
336C2×C4—162no—
337C2×C4—164no—
338C2×C4—164no—
339C22—22no—
340C22—24no—
341C22—24no—
342C22—24no—
343C22—44no—
344C22—42no—
345C22—42no—
346C22—44no—
347C22—44no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—42no—
356C22—82no—
357C22—82no—
358C22—164no—
359C22#229711yescharacteristic
360C22#230811yescharacteristic
361C22#231111yescharacteristic
362C22#233514yes—
363C22#263014yes—
364C22#266112yes—
365C22#266312yes—
366C4—44no—
367C4—44no—
368C4—42no—
369C4—82no—
370C4—82no—
371C4—88no—
372C4—82no—
373C4—82no—
374C4—164no—
375C2—21no—
376C2—21no—
377C2—24no—
378C2—22no—
379C2—24no—
380C2—22no—
381C2—82no—
382C2#1428711yescharacteristic
383C2#2177712yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11205C2semidirect
6. #11070C2semidirect
8. #10936C2semidirect
9. #10915C2semidirect
14. #2484C4semidirect
15. #2483C4semidirect

Maximal subgroups

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

subgroupindex
#354482
#206332
#203922
#112052
#111692
#110702
#109662
#109362
#109152