Finite Group Databasebeta

← #416012

Subgroups — #416012

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2#14685C212yesmaximal
3C4×#381C211yesmaximal, characteristic
4#11287C214yesmaximal
5#11070C214yesmaximal
6#10854C212yesmaximal
7#10846C211yesmaximal, characteristic
8C23×C42C2211yescharacteristic
9#2851—22no—
10#2646—22no—
11#2569—24no—
12#2442—24no—
13#2315C2214yes—
14C2×#381C2212yes—
15C2×#381C2214yes—
16C2×#380C2212yes—
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20C2×#342C2214yes—
21C2×#342C2214yes—
22C2×#342C2214yes—
23C4⋊#95—24no—
24#1434—24no—
25#1434—22no—
26C22⋊#95C2214yes—
27C22⋊#95C2212yes—
28C4×#95—22no—
29C24×C4C2311yescharacteristic
30C24×C4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C2×#125—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4D414yes—
35C22×C22⋊C4C2314yes—
36C22×C22⋊C4C2312yes—
37C22×C22⋊C4C2314yes—
38C22×C22⋊C4C2312yes—
39C22×C42—21no—
40C22×C42—22no—
41C22×C42—21no—
42C22×C42—22no—
43C22×C42—24no—
44C22×C42—22no—
45C22×C42—21no—
46C22×C42D412yes—
47C22×C42Q814yes—
48#400—28no—
49#398—28no—
50#389—24no—
51#387—28no—
52#385—28no—
53#385—28no—
54#382—22no—
55#381—24no—
56#381—24no—
57#381—28no—
58#381—44no—
59#380—22no—
60#380—28no—
61#380—22no—
62#380—22no—
63#380—22no—
64#380—24no—
65#380—24no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—24no—
69C2×#95—24no—
70C2×#95—24no—
71C2×#95—28no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—28no—
75C2×#95D414yes—
76C2×#95D412yes—
77#342—28no—
78#342—44no—
79#342—44no—
80C25C2411yescharacteristic
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—44no—
95C23×C4—44no—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D418yes—
99C23×C4C2×D414yes—
100C23×C4C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×Q812yes—
103C23×C4C2×Q812yes—
104C23×C4C2×Q812yes—
105C23×C4#4114yes—
106C23×C4#4114yes—
107C23×C4#4112yes—
108#125—84no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—48no—
114C2×C4⋊C4—48no—
115C2×C4⋊C4—44no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4#4114yes—
140C2×C42—22no—
141C2×C42—24no—
142C2×C42—28no—
143C2×C42—24no—
144C2×C42—28no—
145C2×C42—22no—
146C2×C42—41no—
147C2×C42—42no—
148C2×C42—41no—
149C2×C42—42no—
150C2×C42—44no—
151C2×C42—42no—
152C2×C42—44no—
153C2×C42—44no—
154C2×C42—44no—
155C2×C42—42no—
156#95—216no—
157#95—28no—
158#95—28no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—416no—
163#95—48no—
164#95—416no—
165#95—42no—
166#95—44no—
167C24—42no—
168C24—42no—
169C24C22×D411yescharacteristic
170C24C22×D412yes—
171C24C22×Q811yescharacteristic
172C24C2×#4111yescharacteristic
173C24C2×#4112yes—
174C24C2×#4112yes—
175C24C2×#4112yes—
176C24H(5,𝔽2)12yes—
177C242−1+412yes—
178C22×C4—24no—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—28no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—28no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—28no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—48no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—48no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4#12011yescharacteristic
238C22×C4#12012yes—
239C22×C4#12011yescharacteristic
240C22×C4C22⋊Q812yes—
241C22×C4C22⋊Q814yes—
242C22×C4C22⋊Q812yes—
243C22×C4C22⋊Q814yes—
244C22×C4#12412yes—
245C22×C4#12414yes—
246C22×C4#12412yes—
247C22×C4#12414yes—
248C22×C4#12514yescomplemented
249C4⋊C4—48no—
250C4⋊C4—84no—
251C4⋊C4—88no—
252C4⋊C4—88no—
253C4⋊C4—88no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C42—44no—
271C42—48no—
272C42—44no—
273C42—44no—
274C42—82no—
275C42—84no—
276C23—22no—
277C23—24no—
278C23—22no—
279C23—24no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—42no—
284C23—41no—
285C23—42no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23C2×#12011yescharacteristic
291C23C2×C22⋊Q811yescharacteristic
292C23C2×C22⋊Q812yes—
293C23#52512yes—
294C23C2×#12411yescharacteristic
295C23C2×#12412yes—
296C23C2×#12511yescharacteristic
297C23#52912yes—
298C23#53311yescharacteristic
299C23#53712yes—
300C23#54011yescharacteristic
301C23#54111yescharacteristic
302C23#54412yes—
303C23#54612yes—
304C23#54712yes—
305C23#54812yes—
306C23#55012yes—
307C23#55212yes—
308C23#55212yes—
309C23#55312yes—
310C23#55312yes—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—28no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—24no—
320C2×C4—28no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—82no—
348C2×C4—84no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4—88no—
352C2×C4—88no—
353C2×C4—84no—
354C2×C4—88no—
355C2×C4—88no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4#39312yes—
359C2×C4#39314yes—
360C2×C4#39312yes—
361C2×C4#45814yes—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—21no—
369C22—22no—
370C22—21no—
371C22—22no—
372C22—21no—
373C22—22no—
374C22—21no—
375C22—22no—
376C22—41no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22C2×#39311yescharacteristic
387C22#240712yes—
388C22#244112yes—
389C22#244212yes—
390C22#248912yes—
391C22#249012yes—
392C22#256212yes—
393C22#262911yescharacteristic
394C22#285111yescharacteristic
395C22C2×#45812yes—
396C22#302512yes—
397C4—22no—
398C4—24no—
399C4—44no—
400C4—44no—
401C4—84no—
402C4—84no—
403C4—88no—
404C4—164no—
405C4#203014yes—
406C2—22no—
407C2—22no—
408C2—21no—
409C2—21no—
410C2—22no—
411C2—22no—
412C2—22no—
413C2#1521611yescharacteristic
414C2C2×#203011yescharacteristic
415C2#2346711yescharacteristic
416C2#2414114yes—
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
248. C22×C4#125semidirect

Maximal subgroups

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

subgroupindex
#350202
#146852
C4×#3812
#112872
#110702
#108542
#108462