Finite Group Databasebeta

← #237812

Subgroups — #237812

Order 512.

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

normal subgroups only

This group has 930 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#35120C211yesmaximal, characteristic
2#20735C212yesmaximal
3#13961C211yesmaximal, characteristic
4#10346C212yesmaximal
5#10345C212yesmaximal
6#10342C212yesmaximal
7#10341C212yesmaximal
8#8025C212yesmaximal
9#7906C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#431C2212yes—
12#2855—22no—
13#2659C2212yes—
14#2646C2212yes—
15#2612—24no—
16#2606—22no—
17#2605—22no—
18#2329—22no—
19#2283C2212yes—
20C2×#383C2211yescharacteristic
21C2×#376C2211yescharacteristic
22#1823C2214yes—
23#1805—24no—
24#1787C2214yes—
25C4×#103—22no—
26C4×#103—22no—
27#1446C2211yescharacteristic
28#1446C2211yescharacteristic
29#1445C2211yescharacteristic
30#1445C2211yescharacteristic
31#1436—22no—
32#1434—22no—
33#1294C2214yes—
34#1294C2214yes—
35#1280C2212yes—
36#1280C2212yes—
37C23×Q8C2311yescharacteristic
38C2×C22⋊Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—24no—
43C2×C4×Q8Q814yescomplemented
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2311yescharacteristic
46C22×C42—21no—
47C22×C42D412yes—
48C22×C42C2311yescharacteristic
49#431—24no—
50#431—24no—
51C2×C4⋊C8—22no—
52C2×#103—24no—
53C2×#103—24no—
54C2×C22⋊C8C2314yes—
55C2×C4×C8—22no—
56#400—24no—
57#398—24no—
58#398—24no—
59#395—24no—
60#391—24no—
61#391—22no—
62#391—22no—
63#389—24no—
64#389—22no—
65#389—22no—
66#387—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#385—24no—
71#383—22no—
72#383—24no—
73#382D414yes—
74#381—22no—
75#380—22no—
76#380C2314yes—
77C4×C4⋊C4D412yes—
78C4×C4⋊C4D412yes—
79C4×C22⋊C4—22no—
80#376—22no—
81#376—22no—
82#376—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95D412yes—
90C2×#95C2311yescharacteristic
91C2×#95C2311yescharacteristic
92C2×#95C2311yescharacteristic
93C2×#95C2311yescharacteristic
94#340—24no—
95#340—24no—
96C22×Q8—22no—
97C22×Q8—21no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—44no—
101C22×Q8C2×Q812yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2411yescharacteristic
111C22×C8—24no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114C4×Q8—28no—
115C4×Q8—28no—
116C4×Q8—44no—
117C4×Q8—44no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×Q812yes—
134C2×C22⋊C4—24no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×Q812yes—
147C2×C42#4114yes—
148C2×C42#4112yes—
149C4⋊C8—44no—
150#103—48no—
151#103—48no—
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—24no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95—24no—
166#95—22no—
167#95—22no—
168#95—44no—
169#95—44no—
170#95—44no—
171#95—42no—
172#95—42no—
173#95—44no—
174C24C22×D411yescharacteristic
175C2×Q8—28no—
176C2×Q8—24no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×Q8—48no—
182C2×Q8—42no—
183C2×Q8—82no—
184C2×Q8—82no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4#12012yes—
222C22×C4#12011yescharacteristic
223C22×C4#12011yescharacteristic
224C22×C4C22⋊Q812yes—
225C22×C4C22⋊Q812yes—
226C22×C4#12412yes—
227C22×C4#12412yes—
228C22×C4#13514yes—
229C22×C4C22×D412yes—
230C22×C4C22×Q811yescharacteristic
231C22×C4C2×#4112yes—
232C22×C4C2×#4111yescharacteristic
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—44no—
236C4⋊C4—24no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—48no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C42—44no—
251C42—44no—
252C42—48no—
253C42—82no—
254C42—82no—
255C42C22⋊Q814yes—
256C42C22⋊Q814yes—
257C42#12414yes—
258C42#12414yes—
259C23—21no—
260C23—21no—
261C23—22no—
262C23C2×#12011yescharacteristic
263C23C2×C22⋊Q811yescharacteristic
264C23C2×#12411yescharacteristic
265C23C2×#13512yes—
266C23#57512yes—
267Q8—28no—
268Q8—48no—
269Q8—44no—
270Q8—164no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4#39314yes—
321C2×C4#39312yes—
322C2×C4#39312yes—
323C2×C4C2×C22⋊Q812yes—
324C2×C4C2×#12412yes—
325C8—84no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22C2×#39311yescharacteristic
335C22#300212yes—
336C22#302211yescharacteristic
337C22#302211yescharacteristic
338C22#302512yes—
339C22#307612yes—
340C22#313111yescharacteristic
341C22#313511yescharacteristic
342C4—22no—
343C4—22no—
344C4—24no—
345C4—22no—
346C4—22no—
347C4—44no—
348C4—44no—
349C4—84no—
350C4—84no—
351C4—82no—
352C4—82no—
353C4—161no—
354C4—161no—
355C2—21no—
356C2—21no—
357C2—22no—
358C2#1820912yes—
359C2#1821712yes—
360C2#2345111yescharacteristic
361C2#2346411yescharacteristic
362C2#2346711yescharacteristic
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
43. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351202
#207352
#139612
#103462
#103452
#103422
#103412
#80252
#79062