Finite Group Databasebeta

← #241187

Subgroups — #241187

Order 512.

1124 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 214 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35052C211yesmaximal, characteristic
2#21941C212yesmaximal
3#13965C211yesmaximal, characteristic
4#10572C214yesmaximal, complemented
5#10570C212yesmaximal, complemented
6#10567C212yesmaximal, complemented
7#8012C212yesmaximal
8C2×#1280C211yesmaximal, characteristic
9C22×C22⋊C8C2212yes—
10#2848—22no—
11#2659C2212yes—
12#2646C2212yes—
13#2569—22no—
14#2549—22no—
15#2438—24no—
16#2329—22no—
17C2×#385—22no—
18#2280C2212yes—
19C2×#380C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21C22×#95C2211yescharacteristic
22#1854—24no—
23#1853C2214yes—
24#1852C2214yes—
25#1808—22no—
26#1807—22no—
27#1444C2212yes—
28#1444C2212yes—
29C4⋊#95—22no—
30#1294C2214yes—
31#1294C2214yes—
32#1280C2211yescharacteristic
33#1280C2211yescharacteristic
34#1280C2212yes—
35C24×C4D412yes—
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×#117D414yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2312yes—
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C42—21no—
48C2×#107—24no—
49C2×#106—24no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—24no—
52C2×C22⋊C8Q818yescomplemented
53C2×C22⋊C8C2312yes—
54C2×C22⋊C8C2312yes—
55#400—24no—
56#398—24no—
57#398—24no—
58#395—24no—
59#391—24no—
60#389—24no—
61#387—24no—
62#387—24no—
63#385—24no—
64#385—24no—
65#385—24no—
66#385—48no—
67#381—22no—
68#380—22no—
69#380—22no—
70#380—22no—
71#380—22no—
72#380—22no—
73#380D412yes—
74#380D412yes—
75#380C2314yes—
76#379—22no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4Q814yescomplemented
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D412yes—
87C2×#95C2311yescharacteristic
88C2×#95C2311yescharacteristic
89C2×#95C2312yes—
90#340—24no—
91#340—24no—
92C25C2×D411yescharacteristic
93C22×Q8—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—22no—
99C23×C4—42no—
100C23×C4—44no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×Q814yes—
109C23×C4C2×Q812yes—
110C23×C4#4112yes—
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—28no—
115C22×C8—84no—
116C22⋊Q8—44no—
117C22⋊Q8—44no—
118C22⋊Q8—84no—
119C22⋊Q8—84no—
120#117—84no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4C2×D414yes—
136C2×C4⋊C4C2×D414yes—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—82no—
150#107—84no—
151#106—84no—
152C22⋊C8—28no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155C22⋊C8C2×Q814yes—
156C22⋊C8C2×Q814yes—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—24no—
166#95—22no—
167#95—44no—
168#95—44no—
169#95—42no—
170#95C2×Q814yes—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—22no—
175C24—42no—
176C24#12011yescharacteristic
177C24#12011yescharacteristic
178C24C22⋊Q812yes—
179C24C4⋊Q812yes—
180C24C22×D411yescharacteristic
181C2×Q8—44no—
182C2×Q8—82no—
183C2×Q8—82no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—82no—
218C22×C4—84no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—88no—
223C22×C4—88no—
224C22×C4#12012yes—
225C22×C4C22⋊Q814yes—
226C22×C4C22⋊Q814yes—
227C22×C4C22⋊Q812yes—
228C22×C4C4⋊Q814yes—
229C22×C4C4⋊Q814yes—
230C22×C4C4⋊Q812yes—
231C22×C4#13514yes—
232C22×C4C22×D412yes—
233C22×C4C22×Q812yes—
234C22×C4C22×Q811yescharacteristic
235C22×C4C2×#4111yescharacteristic
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—88no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—48no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C4⋊C4—88no—
252C4⋊C4—88no—
253C22⋊C4—216no—
254C22⋊C4—48no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C42—84no—
262C42—84no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—22no—
268C23—24no—
269C23—24no—
270C23—24no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23#39512yes—
284C23#39512yes—
285C23#45312yes—
286C23#45612yes—
287C23C2×#12011yescharacteristic
288C23C2×C22⋊Q811yescharacteristic
289C23C2×C4⋊Q811yescharacteristic
290C23C2×#13512yes—
291C23#57512yes—
292Q8—164no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—28no—
296C2×C4—24no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—88no—
327C2×C4—88no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4#39514yes—
337C2×C4C2×C22⋊Q812yes—
338C2×C4C2×C4⋊Q812yes—
339C8—44no—
340C8—44no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—24no—
349C22—22no—
350C22—22no—
351C22—24no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—48no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22#206014yes—
362C22C2×#39511yescharacteristic
363C22#300212yes—
364C22C2×#45311yescharacteristic
365C22C2×#45611yescharacteristic
366C22#302612yes—
367C22#307612yes—
368C22#316011yescharacteristic
369C22#316211yescharacteristic
370C4—22no—
371C4—22no—
372C4—24no—
373C4—48no—
374C4—44no—
375C4—82no—
376C4—84no—
377C4—84no—
378C4—162no—
379C2—22no—
380C2—24no—
381C2—21no—
382C2—21no—
383C2—42no—
384C2#1837612yes—
385C2#1837912yes—
386C2#2355111yescharacteristic
387C2C2×#206011yescharacteristic
388C2#2356511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10572C2semidirect
5. #10570C2semidirect
6. #10567C2semidirect
52. C2×C22⋊C8Q8semidirect
80. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350522
#219412
#139652
#105722
#105702
#105672
#80122
C2×#12802