Finite Group Databasebeta

← #237772

Subgroups — #237772

Order 512.

868 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 190 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#34595C211yesmaximal, characteristic
2#21836C212yesmaximal
3#14289C211yesmaximal, characteristic
4#10347C212yesmaximal
5#10346C212yesmaximal
6#10343C212yesmaximal
7#10342C212yesmaximal
8#8025C212yesmaximal
9#7908C211yesmaximal, characteristic
10C22×#125C2211yescharacteristic
11C2×#431C2212yes—
12#2666—24no—
13#2659C2212yes—
14#2646C2212yes—
15#2627—22no—
16#2402—22no—
17#2399—22no—
18C4×C22⋊Q8—22no—
19#2283C2212yes—
20C2×#383C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1991—24no—
23#1989—22no—
24#1989—22no—
25#1824C2214yes—
26#1823C2214yes—
27#1445C2211yescharacteristic
28#1445C2211yescharacteristic
29#1445C2211yescharacteristic
30#1445C2211yescharacteristic
31#1441—22no—
32#1436—22no—
33#1294C2214yes—
34#1294C2214yes—
35#1280C2212yes—
36#1280C2212yes—
37C2×#125—22no—
38C2×#125—22no—
39C2×#125—22no—
40C2×#125—24no—
41C2×#125Q814yescomplemented
42C2×C22⋊Q8—22no—
43C2×C4×Q8—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C42D412yes—
50C22×C42C2311yescharacteristic
51#431—24no—
52#431—24no—
53C2×C4⋊C8—22no—
54C2×C22⋊C8C2314yes—
55C2×C4×C8—22no—
56#400—24no—
57#400—24no—
58#398—24no—
59#398—24no—
60#398—24no—
61#398—24no—
62#395—24no—
63#395—24no—
64#391—22no—
65#389—22no—
66#387—24no—
67#385—24no—
68#383—22no—
69#383—24no—
70#382D414yes—
71#381—22no—
72#380—22no—
73#380C2314yes—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4D412yes—
79C4×C4⋊C4D412yes—
80C4×C22⋊C4—22no—
81#376—22no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95C2311yescharacteristic
89C2×#95C2311yescharacteristic
90C2×#95C2311yescharacteristic
91C2×#95C2311yescharacteristic
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96C22×Q8—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2411yescharacteristic
106C22×C8—24no—
107#125—28no—
108#125—28no—
109#125—44no—
110#125—44no—
111C22⋊Q8—44no—
112C22⋊Q8—44no—
113C4×Q8—84no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4C2×D414yes—
138C2×C4⋊C4C2×Q812yes—
139C2×C4⋊C4C2×Q812yes—
140C2×C22⋊C4—24no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—82no—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42C2×Q812yes—
149C2×C42#4114yes—
150C2×C42#4112yes—
151C4⋊C8—44no—
152C22⋊C8—24no—
153C22⋊C8—24no—
154C4×C8—44no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—24no—
165#95—22no—
166#95—22no—
167#95—44no—
168#95—42no—
169#95—42no—
170#95—44no—
171#95—82no—
172#95—82no—
173C24C22×D411yescharacteristic
174C2×Q8—44no—
175C2×Q8—82no—
176C2×Q8—82no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4#12011yescharacteristic
214C22×C4#12012yes—
215C22×C4#12011yescharacteristic
216C22×C4C22⋊Q812yes—
217C22×C4C22⋊Q812yes—
218C22×C4#12412yes—
219C22×C4#12412yes—
220C22×C4#13514yes—
221C22×C4C22×D412yes—
222C22×C4C22×Q811yescharacteristic
223C22×C4C2×#4112yes—
224C22×C4C2×#4111yescharacteristic
225C2×C8—48no—
226C2×C8—44no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—48no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—48no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—82no—
240C4⋊C4—84no—
241C4⋊C4—82no—
242C4⋊C4—84no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C42—44no—
246C42—44no—
247C42—44no—
248C42—82no—
249C42C22⋊Q814yes—
250C42C22⋊Q814yes—
251C42#12414yes—
252C42#12414yes—
253C23—21no—
254C23—21no—
255C23—22no—
256C23C2×#12011yescharacteristic
257C23C2×C22⋊Q811yescharacteristic
258C23C2×#12411yescharacteristic
259C23C2×#13512yes—
260C23#57512yes—
261Q8—88no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4#39314yes—
313C2×C4#39312yes—
314C2×C4#39312yes—
315C2×C4C2×C22⋊Q812yes—
316C2×C4C2×#12412yes—
317C8—84no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22C2×#39311yescharacteristic
327C22#300212yes—
328C22#301911yescharacteristic
329C22#302011yescharacteristic
330C22#302812yes—
331C22#307612yes—
332C22#313111yescharacteristic
333C22#313511yescharacteristic
334C4—24no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—44no—
340C4—82no—
341C4—84no—
342C4—84no—
343C4—82no—
344C4—82no—
345C4—161no—
346C4—161no—
347C2—21no—
348C2—21no—
349C2—22no—
350C2#1831412yes—
351C2#1831512yes—
352C2#2345111yescharacteristic
353C2#2346211yescharacteristic
354C2#2347011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
41. C2×#125Q8semidirect

Maximal subgroups

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

subgroupindex
#345952
#218362
#142892
#103472
#103462
#103432
#103422
#80252
#79082