Finite Group Databasebeta

← #393523

Subgroups — #393523

Order 512.

1889 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2623C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3C2×#1899C212yesmaximal
4#14342C211yesmaximal, characteristic, complemented
5#10960C214yesmaximal
6#10912C214yesmaximal, complemented
7#10736C212yesmaximal, complemented
8C22×#123C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C22×#99C2214yes—
12#2623—24no—
13#2623—24no—
14C2×#400C412yescomplemented
15C2×#399C412yescomplemented
16C2×#398C412yescomplemented
17C2×#397C412yescomplemented
18C2×#386C2212yes—
19C2×#385C2212yes—
20C2×#381C2212yes—
21C2×#381C2211yescharacteristic
22#2274—28no—
23C2×C4×C4⋊C4—24no—
24C2×C4×C22⋊C4C2214yes—
25#1899—28no—
26C2×#342C2214yes—
27C2×#342C2218yes—
28#1445—24no—
29C22⋊#95C2214yes—
30C23×D4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C24×C4C2312yes—
33C2×#123—22no—
34C2×#123—22no—
35C2×#123—24no—
36C2×#123—24no—
37C2×#117—24no—
38C2×#117—216no—
39C2×#117D418yes—
40C22×C4⋊C4—24no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C414yes—
48C22×C4⋊C4D414yes—
49C22×C4⋊C4D414yes—
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C414yes—
53C22×C22⋊C4C2314yes—
54C22×C22⋊C4C2311yescharacteristic
55C22×C22⋊C4C2312yes—
56C22×C22⋊C4C2314yes—
57C22×C42—22no—
58C22×C42—24no—
59C2×#99—28no—
60C2×#99—216no—
61#400—28no—
62#399—28no—
63#398—28no—
64#397—28no—
65#386—24no—
66#386—24no—
67#385—24no—
68#385—24no—
69#381—22no—
70#381—22no—
71#381—24no—
72#381—24no—
73C4×C4⋊C4—232no—
74C4×C22⋊C4—216no—
75C2×#95—24no—
76C2×#95—24no—
77C2×#95—28no—
78C2×#95C2×C412yes—
79C2×#95C2×C414yes—
80C2×#95C2×C414yes—
81C2×#95C2×C412yes—
82#342—28no—
83#342—216no—
84C25C22×C411yescharacteristic
85C25C2411yescharacteristic
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—28no—
91C23×C4—28no—
92C23×C4—24no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—84no—
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C414yes—
103C23×C4C22×C414yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D414yes—
107C23×C4C2×D414yes—
108C23×C4C2×D414yes—
109C23×C4#4114yes—
110#123—44no—
111#123—48no—
112#123—44no—
113#117—232no—
114#117—232no—
115C2×C4⋊C4—216no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—216no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—84no—
128C2×C4⋊C4C22⋊C418yes—
129C2×C4⋊C4C22⋊C418yes—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—28no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—84no—
149C2×C22⋊C4C2×D418yes—
150C2×C22⋊C4#41116yes—
151C2×C42—216no—
152C2×C42—28no—
153C2×C42—216no—
154C2×C42—88no—
155#99—416no—
156#95—216no—
157#95—24no—
158#95—24no—
159#95—28no—
160#95—28no—
161#95—48no—
162#95—88no—
163C24—21no—
164C24—22no—
165C24—24no—
166C24—42no—
167C24—42no—
168C24—42no—
169C24—42no—
170C24C23×C411yescharacteristic
171C24C22×D411yescharacteristic
172C24C22×D411yescharacteristic
173C24C22×D412yes—
174C24C2×#4112yes—
175C24C2×#4118yes—
176C2×D4—41no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—41no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—48no—
183C2×D4—44no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—28no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—28no—
190C22×C4—216no—
191C22×C4—28no—
192C22×C4—28no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—216no—
196C22×C4—28no—
197C22×C4—28no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—28no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—48no—
212C22×C4—48no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—44no—
220C22×C4—48no—
221C22×C4—88no—
222C22×C4—88no—
223C22×C4C2×C22⋊C418yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C4×D414yes—
227C22×C4C4×D414yes—
228C22×C4#12014yes—
229C22×C4#12118yes—
230C22×C4#12314yes—
231C4⋊C4—232no—
232C4⋊C4—216no—
233C4⋊C4—216no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—88no—
237C22⋊C4—28no—
238C22⋊C4—216no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C22⋊C4—41no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—48no—
249C22⋊C4—88no—
250C22⋊C4#123116yes—
251C42—232no—
252C42—816no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—416no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—41no—
267C23—42no—
268C23—44no—
269C23—88no—
270C23—82no—
271C23—82no—
272C23C22×C22⋊C411yescharacteristic
273C23C2×C4×D412yes—
274C23C4×#4114yes—
275C23C2×#12011yescharacteristic
276C23C2×#12112yes—
277C23C2×#12311yescharacteristic
278C23C2×#12314yes—
279C23#52514yes—
280C23#52514yes—
281C23#52918yes—
282C23#53214yes—
283D4—82no—
284D4—82no—
285D4—84no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—28no—
289C2×C4—216no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—216no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—432no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—88no—
314C2×C4—88no—
315C2×C4—88no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—816no—
319C2×C4—84no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4#38614yes—
323C2×C4#38614yes—
324C22—28no—
325C22—28no—
326C22—28no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—22no—
331C22—21no—
332C22—22no—
333C22—22no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—48no—
338C22—81no—
339C22—84no—
340C22—81no—
341C22—81no—
342C22—164no—
343C22C2×#38611yescharacteristic
344C22#229712yes—
345C22C4×#12314yes—
346C22#284912yes—
347C22#285312yes—
348C22#285414yes—
349C22#288312yes—
350C22#288412yes—
351C4—24no—
352C4—24no—
353C4—416no—
354C4—48no—
355C4—81no—
356C4—84no—
357C4—81no—
358C4—81no—
359C4—164no—
360C2—22no—
361C2—24no—
362C2—24no—
363C2—22no—
364C2—161no—
365C2#1470611yescharacteristic
366C2#2098511yescharacteristic
367C2#2098611yescharacteristic
368C2#2142314yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2274C2semidirect
4. #14342C2semidirect
6. #10912C2semidirect
7. #10736C2semidirect
14. C2×#400C4semidirect
15. C2×#399C4semidirect
16. C2×#398C4semidirect
17. C2×#397C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
C2×#22742
C2×#18992
#143422
#109602
#109122
#107362