Finite Group Databasebeta

← #237790

Subgroups — #237790

Order 512.

1806 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1806 conjugacy classes of subgroups — too many to draw. Only its 194 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#35020C211yesmaximal, characteristic, complemented
2#20601C212yesmaximal
3#13851C211yesmaximal, characteristic
4#10345C214yesmaximal, complemented
5#10344C214yesmaximal, complemented
6#8019C212yesmaximal, complemented
7#7911C211yesmaximal, characteristic, complemented
8C23×#41C2211yescharacteristic
9C2×#417—24no—
10C2×#407C2212yes—
11#2851—22no—
12#2646C2214yes—
13#2569—24no—
14#2442—24no—
15#2315—22no—
16C2×#380C2212yes—
17C2×C4×C22⋊C4—22no—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#1788—24no—
21#1787C2214yes—
22#1786C2214yes—
23#1446C2212yes—
24#1438—22no—
25#1438C2212yes—
26#1294C2218yes—
27#1280C2214yes—
28C22×#41—22no—
29C22×#41—22no—
30C22×#41—22no—
31C22×#41—24no—
32C22×#41—48no—
33C22×#41Q814yes—
34C23×Q8C2311yescharacteristic
35C23×D4—21no—
36C23×D4C2311yescharacteristic
37C24×C4—21no—
38C24×C4C2311yescharacteristic
39C22×M16—22no—
40C2×#125—22no—
41C22×C4⋊C4D414yes—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—24no—
45C22×C22⋊C4D412yes—
46C22×C42—21no—
47C22×C42D412yes—
48#417—416no—
49C2×#103—28no—
50C2×#102—28no—
51#407—24no—
52#407—24no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8C2314yes—
55#400—28no—
56#398—28no—
57#389—24no—
58#387—28no—
59#385—28no—
60#385—28no—
61#382—22no—
62#380—28no—
63#380—22no—
64#380—22no—
65#380D414yes—
66#380C2314yes—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4—24no—
70C4×C22⋊C4—48no—
71C2×#95—22no—
72C2×#95—24no—
73C2×#95C2312yes—
74C2×#95C2312yes—
75C25—21no—
76C25C2×D411yescharacteristic
77C2×#41—216no—
78C2×#41—28no—
79C2×#41—28no—
80C2×#41—416no—
81C2×#41—48no—
82C2×#41—416no—
83C2×#41—48no—
84C2×#41—48no—
85C2×#41—432no—
86C2×#41—44no—
87C2×#41—44no—
88C22×Q8—21no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—44no—
92C22×Q8C2×Q812yes—
93C22×D4—24no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—21no—
97C22×D4—44no—
98C22×D4—44no—
99C22×D4—44no—
100C22×D4—42no—
101C22×D4—42no—
102C22×D4C2×Q812yes—
103C23×C4—28no—
104C23×C4—21no—
105C23×C4—24no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—44no—
109C23×C4—44no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—42no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D414yes—
116C23×C4C2×D412yes—
117C23×C4C2×Q812yes—
118C23×C4#4114yes—
119C23×C4#4112yes—
120C23×C4C2411yescharacteristic
121C2×M16—48no—
122C2×M16—44no—
123C22×C8—24no—
124#125—84no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—48no—
129C2×C4⋊C4C2×D418yes—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—48no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—48no—
138C2×C22⋊C4—48no—
139C2×C22⋊C4—48no—
140C2×C22⋊C4—42no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—44no—
144C2×C42—44no—
145C2×C42—42no—
146#103—416no—
147#102—416no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C22⋊C8—44no—
151#95—28no—
152#95—28no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—48no—
158C24—24no—
159C24—24no—
160C24—22no—
161C24—42no—
162C24—42no—
163C24—44no—
164C24—42no—
165C24—41no—
166C24—41no—
167C24—44no—
168C24#12012yes—
169C24C22⋊Q812yes—
170C24#12412yes—
171C24C22×D411yescharacteristic
172#41—232no—
173#41—432no—
174#41—416no—
175#41—416no—
176#41—416no—
177#41—432no—
178C2×Q8—28no—
179C2×Q8—24no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—48no—
184C2×Q8—42no—
185C2×D4—28no—
186C2×D4—24no—
187C2×D4—44no—
188C2×D4—48no—
189C2×D4—44no—
190C2×D4—48no—
191C2×D4—44no—
192C2×D4—48no—
193C2×D4—416no—
194C2×D4—416no—
195C2×D4—48no—
196C2×D4—42no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—216no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—28no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—44no—
208C22×C4—48no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—48no—
213C22×C4—416no—
214C22×C4—48no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—48no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—48no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4#12011yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4C22⋊Q814yes—
233C22×C4C22⋊Q812yes—
234C22×C4C22⋊Q814yes—
235C22×C4#12414yes—
236C22×C4#12412yes—
237C22×C4#12414yes—
238C22×C4C22×D412yes—
239C22×C4C22×Q811yescharacteristic
240C22×C4C2×#4112yes—
241C22×C4C2×#4111yescharacteristic
242M16—88no—
243C2×C8—48no—
244C2×C8—44no—
245C4⋊C4—48no—
246C4⋊C4—48no—
247C4⋊C4—88no—
248C4⋊C4—88no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—48no—
252C22⋊C4—48no—
253C22⋊C4—432no—
254C22⋊C4—48no—
255C42—44no—
256C42—82no—
257C42—84no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—24no—
263C23—24no—
264C23—24no—
265C23—24no—
266C23—24no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—48no—
272C23—42no—
273C23—44no—
274C23—44no—
275C23—48no—
276C23—44no—
277C23—42no—
278C23—44no—
279C23—84no—
280C23—84no—
281C23—82no—
282C23—84no—
283C23—84no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23#39314yes—
288C23C2×#12011yescharacteristic
289C23C2×C22⋊Q811yescharacteristic
290C23C2×#12411yescharacteristic
291C23#57514yes—
292Q8—28no—
293Q8—48no—
294Q8—44no—
295D4—28no—
296D4—48no—
297D4—44no—
298D4—432no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—28no—
303C2×C4—24no—
304C2×C4—28no—
305C2×C4—24no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—416no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—416no—
317C2×C4—44no—
318C2×C4—48no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—48no—
322C2×C4—44no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—168no—
331C2×C4#39312yes—
332C2×C4#39312yes—
333C2×C4B(4,𝔽2)14yes—
334C2×C4C2×C22⋊Q812yes—
335C2×C4C2×#12412yes—
336C8—84no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—24no—
342C22—24no—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—48no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—84no—
351C22—84no—
352C22—88no—
353C22—84no—
354C22—82no—
355C22—82no—
356C22—88no—
357C22C2×#39311yescharacteristic
358C22#300512yes—
359C22#301812yes—
360C22C2×B(4,𝔽2)12yes—
361C22#308012yes—
362C22#314212yes—
363C4—22no—
364C4—22no—
365C4—22no—
366C4—44no—
367C4—84no—
368C4—84no—
369C4—88no—
370C4#202914yes—
371C2—22no—
372C2—22no—
373C2—44no—
374C2—84no—
375C2#1816814yes—
376C2#2345311yescharacteristic
377C2#2346111yescharacteristic
378C2C2×#202911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
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
1. #35020C2semidirect
4. #10345C2semidirect
5. #10344C2semidirect
6. #8019C2semidirect
7. #7911C2semidirect

Maximal subgroups

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

subgroupindex
#350202
#206012
#138512
#103452
#103442
#80192
#79112