Finite Group Databasebeta

← #237814

Subgroups — #237814

Order 512.

1142 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1142 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#35181C211yesmaximal, characteristic, complemented
2#20732C212yesmaximal
3#13905C211yesmaximal, characteristic
4#10346C214yesmaximal, complemented
5#10344C214yesmaximal
6#8026C212yesmaximal, complemented
7#7908C211yesmaximal, characteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C2×#431C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2571—22no—
13#2570—24no—
14#2552—22no—
15#2317—22no—
16#2285C2212yes—
17C2×#382C2211yescharacteristic
18C2×#380—22no—
19C2×C4×C4⋊C4C2211yescharacteristic
20#1823C2214yes—
21#1802—22no—
22#1802—22no—
23#1786C2214yes—
24C4×#102—24no—
25#1445C2212yes—
26#1438C2212yes—
27#1437—22no—
28#1294C2218yes—
29#1280C2214yes—
30C23×D4C2311yescharacteristic
31C24×C4—21no—
32C2×#125—22no—
33C2×C4×D4—22no—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4—24no—
37C2×C4×D4Q814yes—
38C2×#117—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4D412yes—
41C22×C4⋊C4C2311yescharacteristic
42C22×C22⋊C4—21no—
43C22×C22⋊C4—24no—
44C22×C42D412yes—
45C22×C42C2311yescharacteristic
46#431—24no—
47#431—24no—
48C2×C4⋊C8—22no—
49C2×#102—24no—
50C2×#102—24no—
51C2×C22⋊C8C2314yes—
52C2×C4×C8—22no—
53#400—28no—
54#398—28no—
55#389—24no—
56#389—24no—
57#389—24no—
58#387—28no—
59#385—28no—
60#384—24no—
61#384D412yes—
62#384D412yes—
63#382—24no—
64#382—22no—
65#382—22no—
66#382—24no—
67#380—22no—
68#380—48no—
69#380C2314yes—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4D414yes—
74C2×#95—21no—
75C2×#95—24no—
76C2×#95C2312yes—
77C2×#95C2312yes—
78#340—24no—
79#340—24no—
80C25—21no—
81C22×D4—22no—
82C22×D4—22no—
83C22×D4—21no—
84C22×D4—44no—
85C22×D4C2×Q812yes—
86C23×C4—21no—
87C23×C4—24no—
88C23×C4—22no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—44no—
92C23×C4—44no—
93C23×C4C2×D414yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2411yescharacteristic
97C22×C8—24no—
98#125—84no—
99C4×D4—28no—
100C4×D4—28no—
101C4×D4—44no—
102C4×D4—44no—
103#117—44no—
104#117—44no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D414yes—
122C2×C4⋊C4C2×Q812yes—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—48no—
129C2×C22⋊C4—48no—
130C2×C22⋊C4—84no—
131C2×C42—24no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—82no—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C2×C42C2×Q812yes—
138C2×C42#4114yes—
139C2×C42#4112yes—
140C4⋊C8—44no—
141#102—48no—
142#102—48no—
143C22⋊C8—24no—
144C22⋊C8—24no—
145C4×C8—44no—
146#95—28no—
147#95—28no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—48no—
153#95—42no—
154C24—24no—
155C24—22no—
156C24—42no—
157C24—42no—
158C24—42no—
159C24—41no—
160C24—41no—
161C24—42no—
162C24—42no—
163C24C22×D411yescharacteristic
164C2×D4—24no—
165C2×D4—24no—
166C2×D4—24no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4—48no—
171C2×D4—44no—
172C2×D4—42no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—48no—
189C22×C4—416no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—84no—
196C22×C4—84no—
197C22×C4—84no—
198C22×C4—84no—
199C22×C4#12011yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12012yes—
202C22×C4C22⋊Q812yes—
203C22×C4C22⋊Q812yes—
204C22×C4#12412yes—
205C22×C4#12412yes—
206C22×C4#13514yes—
207C22×C4AGL(1,ℤ/8)14yes—
208C22×C4C22×D412yes—
209C22×C4C22×Q811yescharacteristic
210C22×C4C2×#4112yes—
211C22×C4C2×#4111yescharacteristic
212C2×C8—44no—
213C2×C8—44no—
214C2×C8—44no—
215C4⋊C4—24no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—88no—
226C4⋊C4—88no—
227C4⋊C4—84no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—816no—
233C42—44no—
234C42—48no—
235C42—82no—
236C42C22⋊Q814yes—
237C42C22⋊Q814yes—
238C42#12414yes—
239C42#12414yes—
240C23—21no—
241C23—21no—
242C23—22no—
243C23—24no—
244C23—44no—
245C23—44no—
246C23—42no—
247C23—44no—
248C23—44no—
249C23—44no—
250C23—44no—
251C23—44no—
252C23—42no—
253C23—44no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23—81no—
260C23—81no—
261C23—82no—
262C23—81no—
263C23—81no—
264C23C2×#12011yescharacteristic
265C23C2×C22⋊Q811yescharacteristic
266C23C2×#12411yescharacteristic
267C23C2×#13512yes—
268C23C2×AGL(1,ℤ/8)12yes—
269D4—28no—
270D4—44no—
271D4—44no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—816no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#39312yes—
307C2×C4#39312yes—
308C2×C4#39314yes—
309C2×C4#44914yes—
310C2×C4#44914yes—
311C2×C4#47818yes—
312C2×C4#48814yes—
313C2×C4#49014yes—
314C2×C4C2×C22⋊Q812yes—
315C2×C4C2×#12412yes—
316C8—84no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—48no—
326C22—44no—
327C22—84no—
328C22—84no—
329C22—84no—
330C22—84no—
331C22—84no—
332C22—82no—
333C22—82no—
334C22C2×#39311yescharacteristic
335C22C2×#44912yes—
336C22#301911yescharacteristic
337C22#302011yescharacteristic
338C22#302512yes—
339C22C2×#47812yes—
340C22C2×#48811yescharacteristic
341C22C2×#49011yescharacteristic
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—84no—
347C4—84no—
348C4—84no—
349C4—82no—
350C4—162no—
351C4#202518yes—
352C2—21no—
353C2—22no—
354C2—21no—
355C2—84no—
356C2#1821412yes—
357C2#1822212yes—
358C2C2×#202511yescharacteristic
359C2#2346211yescharacteristic
360C2#2346711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
1. #35181C2semidirect
4. #10346C2semidirect
6. #8026C2semidirect
7. #7908C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#207322
#139052
#103462
#103442
#80262
#79082