Finite Group Databasebeta

← #393506

Subgroups — #393506

Order 512.

1848 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2#33996C211yesmaximal, characteristic
3C2×#1820C211yesmaximal, characteristic
4#13870C212yesmaximal
5#10936C214yesmaximal
6#10918C214yesmaximal
7#10726C212yesmaximal
8C22×#117C2212yes—
9C22×#117C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11#2484—24no—
12#2484C418yescomplemented
13C2×#400—21no—
14C2×#398—21no—
15C2×#387—22no—
16C2×#385C2212yes—
17C2×#381C2214yes—
18C2×#381C2212yes—
19C2×#381C2212yes—
20C2×#381C2211yescharacteristic
21#2274—24no—
22#2274—21no—
23#2274—21no—
24#2273C2214yes—
25C4×#117—24no—
26C4×#117—22no—
27#1820—24no—
28C2×#342C2214yes—
29C2×#342C2218yes—
30#1441—24no—
31#1436—24no—
32C22⋊#95C2214yes—
33C24×C4C2311yescharacteristic
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—24no—
43C2×#117—216no—
44C2×#117C2×C414yes—
45C2×#117D414yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4D414yes—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C22⋊C4C2314yes—
58C22×C22⋊C4C2312yes—
59C22×C22⋊C4C2312yes—
60C22×C22⋊C4C2314yes—
61C22×C42D414yes—
62C22×C42D412yes—
63#400—28no—
64#398—28no—
65#387—216no—
66#385—24no—
67#385C2×C418yes—
68#381—22no—
69#381—24no—
70#381—24no—
71#381—24no—
72#381—216no—
73#381C2×C414yes—
74#381C2×C418yes—
75C4×C4⋊C4—28no—
76C4×C4⋊C4—28no—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—24no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—216no—
82#376—24no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—28no—
89C2×#95—22no—
90C2×#95—28no—
91C2×#95—22no—
92C43—24no—
93#342—28no—
94#342—216no—
95C25C2411yescharacteristic
96C23×C4—28no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—28no—
107C23×C4—24no—
108C23×C4—44no—
109C23×C4C22×C412yes—
110C23×C4C22×C412yes—
111C23×C4C22×C412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D418yes—
120C23×C4#4114yes—
121#117—216no—
122#117—216no—
123#117—216no—
124#117—216no—
125#117—28no—
126#117—28no—
127#117—28no—
128#117—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—216no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—48no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C22⋊C414yes—
150C2×C4⋊C4C22⋊C414yes—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4C22×C414yes—
163C2×C22⋊C4C22×C414yes—
164C2×C22⋊C4C2×D414yes—
165C2×C22⋊C4#41116yes—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42—216no—
171C2×C42—24no—
172C2×C42—28no—
173C2×C42—28no—
174C2×C42C22⋊C418yes—
175#95—28no—
176#95—28no—
177#95—24no—
178#95—28no—
179#95—28no—
180#95—24no—
181#95—28no—
182#95—416no—
183#95—44no—
184#95—44no—
185#95—416no—
186C24—42no—
187C24—42no—
188C24C23×C411yescharacteristic
189C24C22×D411yescharacteristic
190C24C22×D412yes—
191C24C22×D411yescharacteristic
192C24C2×#4112yes—
193C24C2×#4118yes—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—216no—
202C22×C4—28no—
203C22×C4—28no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—28no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—48no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—48no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—48no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—41no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—48no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C414yes—
248C22×C4C2×C22⋊C414yes—
249C22×C4C4×D414yes—
250C22×C4C4×D414yes—
251C22×C4#12011yescharacteristic
252C22×C4#12012yes—
253C22×C4#12011yescharacteristic
254C22×C4#12114yes—
255C22×C4#12114yes—
256C22×C4#12312yes—
257C22×C4#12312yes—
258C22×C4#123116yes—
259C4⋊C4—216no—
260C4⋊C4—216no—
261C4⋊C4—28no—
262C4⋊C4—28no—
263C4⋊C4—416no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—48no—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4—216no—
270C22⋊C4—216no—
271C22⋊C4—28no—
272C22⋊C4—44no—
273C22⋊C4—48no—
274C22⋊C4—48no—
275C42—216no—
276C42—216no—
277C42—28no—
278C42—28no—
279C42—88no—
280C42—88no—
281C23—22no—
282C23—24no—
283C23—22no—
284C23—24no—
285C23—41no—
286C23—41no—
287C23—42no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—416no—
292C23C22×C22⋊C411yescharacteristic
293C23C2×C4×D412yes—
294C23C4×#4114yes—
295C23C2×#12011yescharacteristic
296C23C2×#12112yes—
297C23C2×#12314yes—
298C23C2×#12311yescharacteristic
299C23#52514yes—
300C23#52514yes—
301C23#52918yes—
302C23#53214yes—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—28no—
310C2×C4—28no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—216no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—416no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—88no—
342C2×C4—88no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—88no—
349C2×C4—88no—
350C2×C4—84no—
351C2×C4#38612yes—
352C2×C4#38612yes—
353C2×C4#38614yes—
354C22—21no—
355C22—22no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—21no—
361C22—22no—
362C22—28no—
363C22—28no—
364C22—28no—
365C22—44no—
366C22—48no—
367C22—44no—
368C22C2×#38611yescharacteristic
369C22#229712yes—
370C22C4×#12314yes—
371C22#284912yes—
372C22#285312yes—
373C22#285414yes—
374C22#302512yes—
375C22#302812yes—
376C4—22no—
377C4—24no—
378C4—22no—
379C4—44no—
380C4—44no—
381C4—44no—
382C4—416no—
383C4—84no—
384C4—82no—
385C4—164no—
386C2—22no—
387C2—24no—
388C2—22no—
389C2—24no—
390C2#1470611yescharacteristic
391C2#2095911yescharacteristic
392C2#2096211yescharacteristic
393C2#3302314yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2484C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
#339962
C2×#18202
#138702
#109362
#109182
#107262