Finite Group Databasebeta

← #237834

Subgroups — #237834

Order 512.

1290 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 226 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#21758C212yesmaximal
3#14152C211yesmaximal, characteristic
4#10346C214yesmaximal, complemented
5#10344C214yesmaximal, complemented
6C2×#1294C212yesmaximal, complemented
7#7907C211yesmaximal, characteristic, complemented
8C22×#121C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2569—24no—
13#2442—24no—
14#2315—22no—
15C2×#386—22no—
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18C22×#95C2211yescharacteristic
19#1893—24no—
20#1874—24no—
21#1823C2214yes—
22#1786C2214yes—
23#1445C2212yes—
24#1438C2212yes—
25#1437—22no—
26#1294C2218yes—
27#1280C2214yes—
28C23×D4—21no—
29C23×D4C2311yescharacteristic
30C24×C4—21no—
31C24×C4C2311yescharacteristic
32C23×C8—22no—
33C2×#125—22no—
34C2×#121—22no—
35C2×#121—22no—
36C2×#121—22no—
37C2×#121—24no—
38C2×#121Q814yes—
39C22×C4⋊C4D414yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—22no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—24no—
44C22×C22⋊C4D412yes—
45C22×C42D412yes—
46C2×#102—28no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8C2314yes—
51#400—28no—
52#398—28no—
53#389—24no—
54#387—28no—
55#386—48no—
56#385—28no—
57#385—28no—
58#382—22no—
59#380—28no—
60#380—22no—
61#380—22no—
62#380D414yes—
63#380C2314yes—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C2×#95—22no—
67C2×#95—24no—
68C2×#95—21no—
69C2×#95—24no—
70C2×#95C2312yes—
71C2×#95C2312yes—
72#340—28no—
73C25—21no—
74C25C2×D411yescharacteristic
75C22×D4—24no—
76C22×D4—22no—
77C22×D4—22no—
78C22×D4—21no—
79C22×D4—42no—
80C22×D4—42no—
81C22×D4—44no—
82C22×D4—44no—
83C22×D4—82no—
84C22×D4C2×Q812yes—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—24no—
88C23×C4—44no—
89C23×C4—42no—
90C23×C4—44no—
91C23×C4—82no—
92C23×C4—82no—
93C23×C4C2×D414yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×Q812yes—
98C23×C4#4114yes—
99C23×C4#4112yes—
100C23×C4C2411yescharacteristic
101C22×C8—24no—
102C22×C8—44no—
103C22×C8—84no—
104#125—84no—
105#121—28no—
106#121—28no—
107#121—44no—
108#121—44no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—84no—
116C2×C4⋊C4C2×D418yes—
117C2×C4⋊C4C2×Q812yes—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—44no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—48no—
131C2×C22⋊C4—48no—
132C2×C22⋊C4—84no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—82no—
136#102—88no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139C22⋊C8—44no—
140#95—28no—
141#95—28no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—48no—
147#95—42no—
148C24—22no—
149C24—42no—
150C24—42no—
151C24—44no—
152C24—44no—
153C24—41no—
154C24—41no—
155C24—42no—
156C24—42no—
157C24#12012yes—
158C24C22⋊Q812yes—
159C24#12412yes—
160C24#13514yes—
161C24AGL(1,ℤ/8)14yes—
162C24C22×D411yescharacteristic
163C2×D4—24no—
164C2×D4—48no—
165C2×D4—48no—
166C2×D4—416no—
167C2×D4—42no—
168C2×D4—82no—
169C2×D4—84no—
170C2×D4—82no—
171C2×D4—84no—
172C2×D4—84no—
173C2×D4—84no—
174C2×D4—84no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—28no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—48no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4—82no—
198C22×C4—82no—
199C22×C4—84no—
200C22×C4—84no—
201C22×C4—84no—
202C22×C4—816no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4#12011yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4C22⋊Q814yes—
208C22×C4C22⋊Q814yes—
209C22×C4C22⋊Q812yes—
210C22×C4#12414yes—
211C22×C4#12414yes—
212C22×C4#12412yes—
213C22×C4C22×D412yes—
214C22×C4C22×Q811yescharacteristic
215C22×C4C2×#4112yes—
216C22×C4C2×#4111yescharacteristic
217C2×C8—48no—
218C2×C8—44no—
219C2×C8—88no—
220C4⋊C4—24no—
221C4⋊C4—48no—
222C4⋊C4—48no—
223C4⋊C4—42no—
224C4⋊C4—88no—
225C4⋊C4—88no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—48no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—816no—
234C42—44no—
235C42—82no—
236C23—22no—
237C23—24no—
238C23—22no—
239C23—24no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—48no—
244C23—42no—
245C23—48no—
246C23—44no—
247C23—44no—
248C23—44no—
249C23—42no—
250C23—84no—
251C23—82no—
252C23—82no—
253C23—84no—
254C23—88no—
255C23—81no—
256C23—81no—
257C23—82no—
258C23—82no—
259C23—82no—
260C23#39314yes—
261C23#44914yes—
262C23#44914yes—
263C23#47818yes—
264C23#48814yes—
265C23#49014yes—
266C23C2×#12011yescharacteristic
267C23C2×C22⋊Q811yescharacteristic
268C23C2×#12411yescharacteristic
269C23C2×#13512yes—
270C23C2×AGL(1,ℤ/8)12yes—
271D4—48no—
272D4—84no—
273D4—816no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4—48no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—816no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—84no—
305C2×C4—168no—
306C2×C4#39312yes—
307C2×C4#39312yes—
308C2×C4C2×C22⋊Q812yes—
309C2×C4C2×#12412yes—
310C8—84no—
311C22—21no—
312C22—21no—
313C22—24no—
314C22—21no—
315C22—21no—
316C22—24no—
317C22—24no—
318C22—24no—
319C22—28no—
320C22—28no—
321C22—44no—
322C22—48no—
323C22—44no—
324C22—88no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—162no—
329C22—162no—
330C22—164no—
331C22#202518yes—
332C22C2×#39311yescharacteristic
333C22C2×#44912yes—
334C22#301812yes—
335C22#302612yes—
336C22C2×#47812yes—
337C22C2×#48811yescharacteristic
338C22C2×#49011yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—84no—
344C4—82no—
345C4—84no—
346C4—88no—
347C2—21no—
348C2—22no—
349C2—21no—
350C2—28no—
351C2—162no—
352C2#1838414yes—
353C2C2×#202511yescharacteristic
354C2#2346111yescharacteristic
355C2#2346811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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. #10346C2semidirect
5. #10344C2semidirect
6. C2×#1294C2semidirect
7. #7907C2semidirect

Maximal subgroups

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

subgroupindex
#350202
#217582
#141522
#103462
#103442
C2×#12942
#79072