Finite Group Databasebeta

← #403331

Subgroups — #403331

Order 512.

2438 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 2438 conjugacy classes of subgroups — too many to draw. Only its 290 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
1C2×#3458C211yesmaximal, characteristic
2C2×#2028C214yesmaximal, complemented
3#20911C212yesmaximal
4#14138C211yesmaximal, characteristic
5#11392C214yesmaximal, complemented
6#10733C212yesmaximal
7#10728C211yesmaximal, characteristic
8#3458—216no—
9C2×#544—24no—
10C2×#544C2214yes—
11C2×#543—22no—
12C2×#533—22no—
13C2×#533—24no—
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2212yes—
16C22×#117C2211yescharacteristic
17C2×#419C2214yes—
18C2×#407C2212yes—
19C2×#391C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#2028—216no—
22#1873—24no—
23#1872—24no—
24C2×#343C2214yes—
25#1736C2214yes—
26C2×#339C2218yes—
27#1440—22no—
28#1436—22no—
29C4⋊#95C2212yes—
30C23×Q8C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C22×M16C2314yes—
33#544—264no—
34#544—216no—
35#544—28no—
36#544D4116yes—
37#543—232no—
38#533—264no—
39#533—232no—
40C2×C4⋊Q8—24no—
41C2×C4⋊Q8—28no—
42C2×#125—24no—
43C2×#125—28no—
44C2×C22⋊Q8—216no—
45C2×C22⋊Q8—28no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8Q814yescomplemented
51C2×C4×Q8—28no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—216no—
57C2×#117—28no—
58C2×#117D414yes—
59C2×#117C2314yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D414yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4D412yes—
68C22×C42—22no—
69C22×C42D412yes—
70C22×C42C2312yes—
71#419—216no—
72C2×#103—28no—
73#407—24no—
74#407—24no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77#391—28no—
78C4×C22⋊C4—24no—
79C2×#95—24no—
80C2×#95—21no—
81C2×#95—24no—
82#343—28no—
83#340—28no—
84#339—216no—
85C25C2×D411yescharacteristic
86C22×Q8—28no—
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8—24no—
90C22×Q8—22no—
91C22×Q8—24no—
92C22×Q8C2×Q812yes—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D414yes—
103C23×C4C2×Q812yes—
104C23×C4#4114yes—
105C23×C4#4112yes—
106C23×C4C2411yescharacteristic
107C2×M16—28no—
108C2×M16—24no—
109C2×M16—24no—
110C2×M16—28no—
111C22×C8—24no—
112C4⋊Q8—232no—
113C4⋊Q8—232no—
114C4⋊Q8—232no—
115#125—232no—
116#125—232no—
117#125—232no—
118C22⋊Q8—2128no—
119C22⋊Q8—232no—
120C22⋊Q8—232no—
121C22⋊Q8—28no—
122C22⋊Q8—28no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125C4×Q8—264no—
126#117—216no—
127#117—216no—
128#117—216no—
129#117—216no—
130#117—28no—
131#117—216no—
132#117C2×D4116yes—
133C2×C4⋊C4—216no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—216no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C2×D418yes—
147C2×C4⋊C4C2×D418yes—
148C2×C4⋊C4C2×Q812yes—
149C2×C22⋊C4—216no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—28no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C42—216no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—28no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—24no—
168#103—416no—
169C22⋊C8—44no—
170C22⋊C8—44no—
171#95—48no—
172#95—48no—
173#95—42no—
174C24—22no—
175C24—22no—
176C24—42no—
177C24—42no—
178C24#12012yes—
179C24C22⋊Q812yes—
180C24#12412yes—
181C24C22×D411yescharacteristic
182C2×Q8—232no—
183C2×Q8—24no—
184C2×Q8—28no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—48no—
190C2×Q8—88no—
191C2×Q8—88no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—216no—
200C22×C4—28no—
201C22×C4—24no—
202C22×C4—28no—
203C22×C4—216no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—84no—
224C22×C4—84no—
225C22×C4#12011yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12018yes—
228C22×C4C22⋊Q814yes—
229C22×C4C22⋊Q814yes—
230C22×C4C22⋊Q812yes—
231C22×C4#12414yes—
232C22×C4#12414yes—
233C22×C4#12412yes—
234C22×C4C22×D412yes—
235C22×C4C22×D414yes—
236C22×C4C22×Q811yescharacteristic
237C22×C4C2×#4112yes—
238C22×C4C2×#4111yescharacteristic
239C22×C42−1+414yes—
240M16—48no—
241M16—48no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C4⋊C4—216no—
246C4⋊C4—264no—
247C4⋊C4—232no—
248C4⋊C4—216no—
249C4⋊C4—28no—
250C4⋊C4—24no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4#120116yes—
255C22⋊C4—216no—
256C22⋊C4—216no—
257C22⋊C4—216no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C42—28no—
261C42—216no—
262C42—28no—
263C42—216no—
264C42—48no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—24no—
270C23—24no—
271C23—24no—
272C23—24no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—82no—
278C23—82no—
279C23—82no—
280C23—82no—
281C23#39314yes—
282C23C2×#12012yes—
283C23C2×#12011yescharacteristic
284C23C2×C22⋊Q811yescharacteristic
285C23C2×#12411yescharacteristic
286C23#53612yes—
287Q8—48no—
288Q8—832no—
289Q8—84no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—28no—
301C2×C4—216no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—84no—
315C2×C4—816no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#39312yes—
326C2×C4#39312yes—
327C2×C4C2×#12014yes—
328C2×C4C2×C22⋊Q812yes—
329C2×C4C2×#12412yes—
330C2×C4#53614yes—
331C2×C4#54818yes—
332C2×C4#54814yes—
333C2×C4D4×Q814yes—
334C8—84no—
335C22—22no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—44no—
347C22—44no—
348C22—84no—
349C22—84no—
350C22C2×#39311yescharacteristic
351C22#240912yes—
352C22#243612yes—
353C22#261411yescharacteristic
354C22#284611yescharacteristic
355C22#301812yes—
356C22#302212yes—
357C4—22no—
358C4—22no—
359C4—48no—
360C4—416no—
361C4—84no—
362C4—82no—
363C4—82no—
364C4#240914yes—
365C4#243614yes—
366C2—22no—
367C2—22no—
368C2—44no—
369C2#1487311yescharacteristic
370C2#2346111yescharacteristic
371C2#2346411yescharacteristic
372C2#2965512yes—
373C2#3289312yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2028C2semidirect
5. #11392C2semidirect
50. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
C2×#20282
#209112
#141382
#113922
#107332
#107282