Finite Group Databasebeta

← #128024

Subgroups — #128024

Order 512.

1376 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1376 conjugacy classes of subgroups — too many to draw. Only its 144 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#20591C211yesmaximal, characteristic
2#20584C212yesmaximal
3#7316C214yesmaximal, complemented
4C23×#41C2211yescharacteristic
5C2×#413—22no—
6C2×#410—22no—
7C2×#410—22no—
8C2×#407C412yescomplemented
9#2280C414yescomplemented
10#1785—22no—
11#1784C412yescomplemented
12#1783C2212yes—
13#1782—22no—
14#1782—22no—
15#1781C414yescomplemented
16#1781C2214yes—
17C22×#41—44no—
18C22×#41—44no—
19C22×#41—42no—
20C22×#41—42no—
21C22×#41D411yescharacteristic
22C22×#41D411yescharacteristic
23C22×#41D412yes—
24C22×#41D412yes—
25C22×#41Q811yescharacteristic
26C22×#41Q811yescharacteristic
27C23×Q8C2×C411yescharacteristic
28C23×D4C2×C412yes—
29C23×D4C2311yescharacteristic
30C24×C4C2×C412yes—
31C24×C4C2×C411yescharacteristic
32C22×M16—22no—
33C2×#117—24no—
34#413—44no—
35#413—44no—
36C2×#101—24no—
37C2×#100—24no—
38#410—44no—
39#410—44no—
40#410—44no—
41#410—44no—
42C2×#99—24no—
43C2×#99—24no—
44C2×#99—24no—
45C2×#99—24no—
46#407—22no—
47#407—42no—
48#407C2×C414yes—
49C2×C22⋊C8—42no—
50#381—24no—
51#380—44no—
52#379C2×C418yes—
53C4×C22⋊C4—44no—
54C25C4211yescharacteristic
55C25C22×C412yes—
56C2×#41—24no—
57C2×#41—24no—
58C2×#41—24no—
59C2×#41—24no—
60C2×#41—24no—
61C2×#41—24no—
62C2×#41—41no—
63C2×#41—41no—
64C2×#41—42no—
65C2×#41—42no—
66C2×#41—42no—
67C2×#41—42no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41—42no—
71C2×#41—44no—
72C2×#41—44no—
73C2×#41—42no—
74C2×#41—44no—
75C2×#41—44no—
76C2×#41—44no—
77C2×#41—432no—
78C2×#41—416no—
79C2×#41—416no—
80C22×Q8—44no—
81C22×Q8—42no—
82C22×Q8C22⋊C412yes—
83C22×Q8C4⋊C411yescharacteristic
84C22×Q8C4⋊C411yescharacteristic
85C22×D4—44no—
86C22×D4—42no—
87C22×D4—44no—
88C22×D4—44no—
89C22×D4—44no—
90C22×D4C22⋊C412yes—
91C22×D4C22⋊C412yes—
92C22×D4C4⋊C412yes—
93C22×D4C4⋊C412yes—
94C22×D4C2×D412yes—
95C22×D4C2×D411yescharacteristic
96C22×D4C2×Q811yescharacteristic
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4C4211yescharacteristic
108C23×C4C4212yes—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C411yescharacteristic
113C23×C4C4⋊C412yes—
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C412yes—
116C23×C4C4⋊C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C2×M16—24no—
119C2×M16—42no—
120C2×M16—44no—
121C2×M16—44no—
122C22×C8—42no—
123#117—44no—
124#117—44no—
125#117—44no—
126#117—44no—
127C2×C4⋊C4—44no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—48no—
130C2×C42—44no—
131#101—48no—
132#100—48no—
133#99—48no—
134#99—48no—
135#99—48no—
136#99—48no—
137C22⋊C8—44no—
138C22⋊C8—44no—
139#95—48no—
140C24—22no—
141C24—24no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—82no—
146C24—84no—
147C24#9511yescharacteristic
148C24#9511yescharacteristic
149C24C2×C4211yescharacteristic
150C24C2×C22⋊C412yes—
151C24C2×C4⋊C412yes—
152#41—48no—
153#41—48no—
154#41—44no—
155#41—48no—
156#41—48no—
157#41—44no—
158#41—48no—
159#41—432no—
160#41—416no—
161#41—416no—
162#41—48no—
163#41—44no—
164#41—44no—
165C2×Q8—24no—
166C2×Q8—24no—
167C2×Q8—24no—
168C2×Q8—42no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—41no—
172C2×Q8—416no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—42no—
192C2×D4—432no—
193C2×D4—416no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—48no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4#9511yescharacteristic
238C22×C4#9512yes—
239C22×C4#9512yes—
240C22×C4#9511yescharacteristic
241C22×C4#9914yes—
242C22×C4#9914yes—
243C22×C4#9912yes—
244C22×C4#9912yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C4⋊C411yescharacteristic
247M16—48no—
248M16—84no—
249C2×C8—44no—
250C2×C8—82no—
251C2×C8—82no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C22⋊C4—48no—
255C22⋊C4—48no—
256C22⋊C4—816no—
257C42—84no—
258C42—84no—
259C23—22no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—42no—
270C23—48no—
271C23—48no—
272C23—44no—
273C23—44no—
274C23—84no—
275C23—84no—
276C23—84no—
277C23—81no—
278C23—81no—
279C23—82no—
280C23—82no—
281C23—84no—
282C23—82no—
283C23—84no—
284C23—84no—
285C23—84no—
286C23#34212yes—
287C23C2×#9511yescharacteristic
288C23C2×#9914yes—
289Q8—48no—
290Q8—44no—
291Q8—44no—
292D4—48no—
293D4—48no—
294D4—48no—
295D4—44no—
296D4—48no—
297D4—48no—
298D4—44no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—44no—
317C2×C4—48no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—88no—
333C2×C4—816no—
334C2×C4#34214yes—
335C2×C4#34212yes—
336C2×C4#34211yescharacteristic
337C2×C4#34212yes—
338C2×C4#34211yescharacteristic
339C2×C4C2×#9912yes—
340C8—84no—
341C22—24no—
342C22—22no—
343C22—24no—
344C22—22no—
345C22—41no—
346C22—44no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—42no—
351C22—44no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—816no—
362C22—88no—
363C22C2×#34212yes—
364C22C2×#34211yescharacteristic
365C4—22no—
366C4—24no—
367C4—41no—
368C4—42no—
369C4—41no—
370C4—168no—
371C4#130412yes—
372C4#130411yescharacteristic
373C4#130411yescharacteristic
374C2—22no—
375C2—21no—
376C2—44no—
377C2—42no—
378C2—42no—
379C2—81no—
380C2—82no—
381C2C2×#130411yescharacteristic
subgroupG/H
centerC4#1304
commutatorC24C2×C42
Frattini, cosocleC23×D4C23
socleC2C2×#1304
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7316C2semidirect
8. C2×#407C4semidirect
9. #2280C4semidirect
11. #1784C4semidirect
15. #1781C4semidirect

Maximal subgroups

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

subgroupindex
#205912
#205842
#73162