Finite Group Databasebeta

← #237758

Subgroups — #237758

Order 512.

1288 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1288 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
1C2×#2646C211yesmaximal, characteristic, complemented
2#21758C212yesmaximal
3#14271C211yesmaximal, characteristic
4#10346C214yesmaximal, complemented
5#10344C214yesmaximal, complemented
6#8023C212yesmaximal, complemented
7C2×#1280C211yesmaximal, characteristic, complemented
8C22×#121C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10#2646—24no—
11#2646—22no—
12#2646C2214yes—
13C2×#400—22no—
14C2×#398—22no—
15C2×#386—22no—
16C2×#385—22no—
17C2×#380C2211yescharacteristic
18C2×#380C2212yes—
19C22×#95C2211yescharacteristic
20#1893—24no—
21#1874—22no—
22#1874—22no—
23#1823C2214yes—
24#1786C2214yes—
25#1445C2212yes—
26#1439—22no—
27#1438C2212yes—
28#1294C2218yes—
29#1280C2214yes—
30C23×D4—21no—
31C23×D4C2311yescharacteristic
32C24×C4D412yes—
33C24×C4C2311yescharacteristic
34C23×C8—22no—
35C2×#121—22no—
36C2×#121—22no—
37C2×#121—22no—
38C2×#121—24no—
39C2×#121Q814yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4C2311yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—24no—
50C2×#102—24no—
51C2×#102—24no—
52C2×C22⋊C8—24no—
53C2×C22⋊C8—24no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8C2314yes—
56#400—28no—
57#400—28no—
58#398—28no—
59#398—28no—
60#386—48no—
61#385—28no—
62#385—28no—
63#380—22no—
64#380—22no—
65#380—24no—
66#380D414yes—
67#380C2314yes—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—24no—
74C2×#95D412yes—
75C2×#95C2312yes—
76C2×#95C2312yes—
77#340—24no—
78#340—24no—
79C25—21no—
80C25C2×D411yescharacteristic
81C22×D4—24no—
82C22×D4—22no—
83C22×D4—22no—
84C22×D4—21no—
85C22×D4—41no—
86C22×D4—41no—
87C22×D4—42no—
88C22×D4—44no—
89C22×D4—44no—
90C22×D4—82no—
91C22×D4C2×Q812yes—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—82no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4C2×D414yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×Q812yes—
108C23×C4#4114yes—
109C23×C4#4112yes—
110C23×C4C2411yescharacteristic
111C22×C8—24no—
112C22×C8—44no—
113C22×C8—82no—
114C22×C8—82no—
115#121—28no—
116#121—28no—
117#121—44no—
118#121—44no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4—82no—
129C2×C4⋊C4—84no—
130C2×C4⋊C4—84no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×Q812yes—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—48no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—84no—
146C2×C22⋊C4—84no—
147#102—84no—
148#102—84no—
149C22⋊C8—24no—
150C22⋊C8—24no—
151C22⋊C8—44no—
152#95—28no—
153#95—28no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—48no—
160C24—22no—
161C24—41no—
162C24—42no—
163C24—41no—
164C24—42no—
165C24—44no—
166C24—42no—
167C24—41no—
168C24—41no—
169C24—42no—
170C24—42no—
171C24#12011yescharacteristic
172C24#12011yescharacteristic
173C24C22⋊Q812yes—
174C24#12412yes—
175C24#13514yes—
176C24AGL(1,ℤ/8)14yes—
177C24C22×D411yescharacteristic
178C2×D4—24no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—416no—
184C2×D4—42no—
185C2×D4—82no—
186C2×D4—84no—
187C2×D4—82no—
188C2×D4—84no—
189C2×D4—84no—
190C2×D4—84no—
191C2×D4—84no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—82no—
226C22×C4—84no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—88no—
231C22×C4—88no—
232C22×C4#12012yes—
233C22×C4C22⋊Q814yes—
234C22×C4C22⋊Q814yes—
235C22×C4C22⋊Q812yes—
236C22×C4#12414yes—
237C22×C4#12414yes—
238C22×C4#12412yes—
239C22×C4C22×D412yes—
240C22×C4C22×Q811yescharacteristic
241C22×C4C2×#4112yes—
242C22×C4C2×#4111yescharacteristic
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—88no—
247C4⋊C4—24no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—88no—
254C4⋊C4—88no—
255C4⋊C4—84no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C22⋊C4—816no—
261C22⋊C4—84no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—44no—
274C23—44no—
275C23—42no—
276C23—42no—
277C23—44no—
278C23—44no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—42no—
283C23—84no—
284C23—82no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23—82no—
289C23—81no—
290C23—81no—
291C23—82no—
292C23—81no—
293C23—81no—
294C23—82no—
295C23#39312yes—
296C23#39312yes—
297C23#44914yes—
298C23#44914yes—
299C23#47818yes—
300C23#48814yes—
301C23#49014yes—
302C23C2×#12011yescharacteristic
303C23C2×C22⋊Q811yescharacteristic
304C23C2×#12411yescharacteristic
305C23C2×#13512yes—
306C23C2×AGL(1,ℤ/8)12yes—
307D4—48no—
308D4—84no—
309D4—88no—
310D4—88no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—88no—
337C2×C4—88no—
338C2×C4—88no—
339C2×C4—88no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—82no—
346C2×C4—84no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#39314yes—
350C2×C4C2×C22⋊Q812yes—
351C2×C4C2×#12412yes—
352C8—84no—
353C22—22no—
354C22—24no—
355C22—22no—
356C22—24no—
357C22—22no—
358C22—24no—
359C22—22no—
360C22—24no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—22no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—22no—
369C22—48no—
370C22—44no—
371C22—44no—
372C22—88no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—162no—
377C22—162no—
378C22—162no—
379C22—162no—
380C22#202514yes—
381C22#202514yes—
382C22C2×#39311yescharacteristic
383C22C2×#44912yes—
384C22#301911yescharacteristic
385C22#302011yescharacteristic
386C22#302612yes—
387C22C2×#47812yes—
388C22C2×#48811yescharacteristic
389C22C2×#49011yescharacteristic
390C4—22no—
391C4—22no—
392C4—24no—
393C4—84no—
394C4—84no—
395C4—84no—
396C4—82no—
397C4—162no—
398C2—21no—
399C2—22no—
400C2—21no—
401C2—24no—
402C2—24no—
403C2—162no—
404C2#1828812yes—
405C2#1829612yes—
406C2C2×#202511yescharacteristic
407C2#2346211yescharacteristic
408C2#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. C2×#2646C2semidirect
4. #10346C2semidirect
5. #10344C2semidirect
6. #8023C2semidirect
7. C2×#1280C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
#217582
#142712
#103462
#103442
#80232
C2×#12802