Finite Group Databasebeta

← #406724

Subgroups — #406724

Order 512.

1528 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35090C211yesmaximal, characteristic
2#14684C212yesmaximal
3#13756C211yesmaximal, characteristic
4#11390C212yesmaximal
5#11287C212yesmaximal
6#11287C214yesmaximal
7#10736C212yesmaximal
8#10726C211yesmaximal, characteristic
9C2×#543C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C42C2211yescharacteristic
13#2855—22no—
14#2627—22no—
15#2438—28no—
16#2315C2212yes—
17#2315C2211yescharacteristic
18C2×#381C2212yes—
19C2×#381C2212yes—
20C2×#342C2214yes—
21C2×#342C2212yes—
22C2×#342C2212yes—
23C2×#342C2214yes—
24C4⋊#95—22no—
25#1434—28no—
26C22⋊#95C2214yes—
27C22⋊#95C2212yes—
28C22⋊#95C2214yes—
29C22⋊#95C2212yes—
30C4×#95—22no—
31C24×C4C2312yes—
32C24×C4C2311yescharacteristic
33#543—24no—
34#543—42no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—24no—
37C2×#117—24no—
38C2×#117—24no—
39C2×#117—24no—
40C2×#117—24no—
41C2×#117—42no—
42C2×#117—42no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D412yes—
45C22×C22⋊C4C2314yes—
46C22×C22⋊C4C2311yescharacteristic
47C22×C22⋊C4C2312yes—
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2311yescharacteristic
51C22×C42—21no—
52C22×C42—24no—
53C22×C42—21no—
54C22×C42—21no—
55C22×C42—24no—
56C22×C42—22no—
57C22×C42D412yes—
58C22×C42Q814yescomplemented
59#400—24no—
60#398—28no—
61#395—24no—
62#391—24no—
63#389—24no—
64#387—28no—
65#385—24no—
66#385—24no—
67#385—24no—
68#381—22no—
69#381—22no—
70#381—24no—
71#381—24no—
72#381—24no—
73#380—22no—
74#380—24no—
75C2×#95—24no—
76C2×#95—24no—
77C2×#95—28no—
78C2×#95—28no—
79C2×#95—24no—
80C2×#95D418yes—
81#342—28no—
82#342—24no—
83#342—44no—
84#342—42no—
85C25C2411yescharacteristic
86C22×Q8—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—24no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—48no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D414yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D414yes—
103C23×C4C2×D414yes—
104C23×C4C2×D412yes—
105C23×C4C2×Q812yes—
106C23×C4C2×Q814yes—
107C23×C4#4114yes—
108C23×C4#4112yes—
109C22⋊Q8—416no—
110C22⋊Q8—44no—
111C22⋊Q8—44no—
112#117—416no—
113#117—416no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4#4114yes—
147C2×C22⋊C4#4114yes—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—216no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—42no—
155C2×C42—42no—
156C2×C42—42no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42—48no—
160C2×C42—42no—
161C2×C42—42no—
162#95—24no—
163#95—28no—
164#95—28no—
165#95—28no—
166#95—48no—
167#95—48no—
168#95—416no—
169#95—416no—
170#95—48no—
171C24—44no—
172C24C22×D411yescharacteristic
173C24C22×D412yes—
174C24C22×Q811yescharacteristic
175C24C2×#4111yescharacteristic
176C24C2×#4112yes—
177C24C2×#4112yes—
178C24C2×#4112yes—
179C24H(5,𝔽2)12yes—
180C242−1+412yes—
181C2×Q8—44no—
182C2×Q8—84no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—28no—
196C22×C4—28no—
197C22×C4—28no—
198C22×C4—28no—
199C22×C4—28no—
200C22×C4—216no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—28no—
212C22×C4—28no—
213C22×C4—48no—
214C22×C4—48no—
215C22×C4—416no—
216C22×C4—44no—
217C22×C4—48no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—48no—
228C22×C4—48no—
229C22×C4—48no—
230C22×C4—48no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4#12012yes—
235C22×C4#12012yes—
236C22×C4C22⋊Q812yes—
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q818yes—
239C22×C4#12314yes—
240C22×C4#12418yes—
241C22×C4#12412yes—
242C22×C4#12412yes—
243C4⋊C4—48no—
244C4⋊C4—48no—
245C4⋊C4—48no—
246C4⋊C4—44no—
247C4⋊C4—84no—
248C4⋊C4—88no—
249C22⋊C4—28no—
250C22⋊C4—28no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C42—48no—
264C42—48no—
265C42—48no—
266C42—48no—
267C23—24no—
268C23—28no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23—44no—
273C23—44no—
274C23—44no—
275C23—44no—
276C23C2×#12011yescharacteristic
277C23C2×C22⋊Q811yescharacteristic
278C23C2×C22⋊Q812yes—
279C23C2×#12311yescharacteristic
280C23#52511yescharacteristic
281C23C2×#12411yescharacteristic
282C23C2×#12412yes—
283C23#52912yes—
284C23#53312yes—
285C23#53411yescharacteristic
286C23#53611yescharacteristic
287C23#53912yes—
288C23#54412yes—
289C23#54614yes—
290C23#54714yes—
291C23#55114yes—
292C23#55214yes—
293Q8—164no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—216no—
305C2×C4—216no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—28no—
309C2×C4—216no—
310C2×C4—24no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—48no—
314C2×C4—416no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—48no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—48no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—84no—
332C2×C4—88no—
333C2×C4—82no—
334C2×C4—88no—
335C2×C4—88no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—88no—
339C2×C4—84no—
340C2×C4#39314yes—
341C2×C4#39314yes—
342C22—22no—
343C22—28no—
344C22—22no—
345C22—22no—
346C22—24no—
347C22—22no—
348C22—24no—
349C22—22no—
350C22—22no—
351C22—28no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—44no—
357C22—44no—
358C22C2×#39311yescharacteristic
359C22#240511yescharacteristic
360C22#240611yescharacteristic
361C22#243814yes—
362C22#248814yes—
363C22#250311yescharacteristic
364C22#256111yescharacteristic
365C22#262711yescharacteristic
366C22#285511yescharacteristic
367C22#302514yes—
368C4—24no—
369C4—24no—
370C4—48no—
371C4—44no—
372C4—84no—
373C4—84no—
374C4—82no—
375C4—84no—
376C4—164no—
377C2—21no—
378C2—24no—
379C2—21no—
380C2—21no—
381C2—24no—
382C2—21no—
383C2#1493411yescharacteristic
384C2#2346712yes—
385C2#2405712yes—
386C2#2411312yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ 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
58. C22×C42Q8semidirect

Maximal subgroups

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

subgroupindex
#350902
#146842
#137562
#113902
#112872
#107362
#107262