Finite Group Databasebeta

← #241188

Subgroups — #241188

Order 512.

1052 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1052 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#13966C211yesmaximal, characteristic
4#10571C214yesmaximal, complemented
5#10569C212yesmaximal, complemented
6#10568C212yesmaximal, 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—
17#2280C2212yes—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1854—24no—
22#1853C2214yes—
23#1852C2214yes—
24#1808—22no—
25#1807—22no—
26#1442C2212yes—
27#1442C2212yes—
28#1434—22no—
29C22⋊#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⋊C4C2312yes—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C42—21no—
46C2×#107—24no—
47C2×#106—24no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8Q818yescomplemented
51C2×C22⋊C8C2312yes—
52C2×C22⋊C8C2312yes—
53#400—24no—
54#398—24no—
55#398—24no—
56#395—24no—
57#391—24no—
58#389—24no—
59#387—24no—
60#387—24no—
61#385—24no—
62#385—24no—
63#385—24no—
64#381—22no—
65#380—22no—
66#380—22no—
67#380—22no—
68#380—22no—
69#380—22no—
70#380D412yes—
71#380D412yes—
72#380C2314yes—
73#379—22no—
74C4×C22⋊C4—22no—
75C4×C22⋊C4—22no—
76C4×C22⋊C4—22no—
77C4×C22⋊C4Q814yescomplemented
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95D412yes—
86C2×#95C2311yescharacteristic
87C2×#95C2311yescharacteristic
88C2×#95C2312yes—
89#340—24no—
90#340—24no—
91C25C2×D411yescharacteristic
92C22×Q8—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—42no—
99C23×C4—44no—
100C23×C4—82no—
101C23×C4—82no—
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×Q814yes—
108C23×C4C2×Q812yes—
109C23×C4#4112yes—
110C23×C4C2411yescharacteristic
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—28no—
114C22×C8—84no—
115C22⋊Q8—44no—
116C22⋊Q8—44no—
117C22⋊Q8—84no—
118C22⋊Q8—84no—
119#117—84no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4C2×D414yes—
129C2×C4⋊C4C2×D414yes—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C42—24no—
141C2×C42—24no—
142C2×C42—82no—
143#107—84no—
144#106—84no—
145C22⋊C8—28no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8C2×Q814yes—
149C22⋊C8C2×Q814yes—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—22no—
155#95—22no—
156#95—22no—
157#95—22no—
158#95—24no—
159#95—22no—
160#95—44no—
161#95—42no—
162#95—44no—
163#95—82no—
164#95—82no—
165#95C2×Q814yes—
166C24—22no—
167C24—22no—
168C24—22no—
169C24—22no—
170C24—42no—
171C24#12011yescharacteristic
172C24#12011yescharacteristic
173C24C22⋊Q812yes—
174C24C4⋊Q812yes—
175C24C22×D411yescharacteristic
176C2×Q8—44no—
177C2×Q8—82no—
178C2×Q8—82no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—82no—
213C22×C4—84no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—88no—
218C22×C4—88no—
219C22×C4#12012yes—
220C22×C4C22⋊Q814yes—
221C22×C4C22⋊Q814yes—
222C22×C4C22⋊Q812yes—
223C22×C4C4⋊Q814yes—
224C22×C4C4⋊Q814yes—
225C22×C4C4⋊Q812yes—
226C22×C4#13514yes—
227C22×C4C22×D412yes—
228C22×C4C22×Q812yes—
229C22×C4C22×Q811yescharacteristic
230C22×C4C2×#4111yescharacteristic
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—88no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—48no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C22⋊C4—48no—
247C22⋊C4—48no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—84no—
255C42—84no—
256C23—22no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—24no—
263C23—24no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23—21no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23—44no—
273C23—44no—
274C23—44no—
275C23—44no—
276C23#39512yes—
277C23#39512yes—
278C23#45412yes—
279C23#45512yes—
280C23C2×#12011yescharacteristic
281C23C2×C22⋊Q811yescharacteristic
282C23C2×C4⋊Q811yescharacteristic
283C23C2×#13512yes—
284C23#57512yes—
285Q8—164no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—82no—
315C2×C4—88no—
316C2×C4—88no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#39514yes—
330C2×C4C2×C22⋊Q812yes—
331C2×C4C2×C4⋊Q812yes—
332C8—44no—
333C8—44no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—24no—
342C22—22no—
343C22—22no—
344C22—24no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—48no—
350C22—44no—
351C22—44no—
352C22—42no—
353C22—42no—
354C22#206114yes—
355C22C2×#39511yescharacteristic
356C22#300212yes—
357C22C2×#45511yescharacteristic
358C22C2×#45411yescharacteristic
359C22#302612yes—
360C22#307612yes—
361C22#316011yescharacteristic
362C22#316211yescharacteristic
363C4—22no—
364C4—22no—
365C4—24no—
366C4—44no—
367C4—84no—
368C4—82no—
369C4—84no—
370C4—84no—
371C4—162no—
372C2—22no—
373C2—24no—
374C2—21no—
375C2—21no—
376C2—42no—
377C2#1837712yes—
378C2#1837812yes—
379C2#2355111yescharacteristic
380C2C2×#206111yescharacteristic
381C2#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. #10571C2semidirect
5. #10569C2semidirect
6. #10568C2semidirect
50. C2×C22⋊C8Q8semidirect
77. 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
#139662
#105712
#105692
#105682
#80122
C2×#12802