Finite Group Databasebeta

← #207015

Subgroups — #207015

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2#20524C211yesmaximal, characteristic
3#13966C211yesmaximal, characteristic
4#9889C212yesmaximal, complemented
5#9888C212yesmaximal, complemented
6#9886C214yesmaximal, complemented
7#8020C212yesmaximal
8#8016C212yesmaximal
9C2×#407C2212yes—
10#2857C2212yes—
11#2851C2212yes—
12#2626—22no—
13#2608—22no—
14#2607—22no—
15#2441C414yescomplemented
16#2439C414yescomplemented
17#2329—22no—
18#2280C2212yes—
19C2×#380C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21C22×#95C2211yescharacteristic
22#1753—22no—
23#1753—22no—
24#1749C2211yescharacteristic
25#1749C2211yescharacteristic
26C4×C22⋊C8C2212yes—
27#1442—22no—
28#1434C2212yes—
29#1434C2212yes—
30C22⋊#95—22no—
31#1294C2214yes—
32#1294C2214yes—
33#1294C2214yes—
34#1294C2214yes—
35C24×C4—21no—
36C24×C4C2311yescharacteristic
37C22×M16—22no—
38C2×C4⋊Q8C2×C412yescomplemented
39C2×#125C2×C412yescomplemented
40C2×C22⋊Q8—22no—
41C2×#117C2×C412yes—
42C2×#117C2×C412yes—
43C22×C4⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4D412yes—
46C22×C42C2311yescharacteristic
47C22×C42C2311yescharacteristic
48#407—24no—
49#407Q818yescomplemented
50C2×C22⋊C8—22no—
51C2×C22⋊C8C2312yes—
52C2×C22⋊C8C2312yes—
53C2×#97—21no—
54C2×#97—21no—
55C2×C4×C8—22no—
56#399—24no—
57#396—24no—
58#396—24no—
59#393—24no—
60#391—22no—
61#391—24no—
62#389—22no—
63#389—24no—
64#389C2×C414yescomplemented
65#389C2×C414yescomplemented
66#387—24no—
67#387—24no—
68#387—24no—
69#385—24no—
70#384C2×C412yescomplemented
71#382C2×C412yescomplemented
72#381—22no—
73#380—22no—
74#380—24no—
75#380C2×C414yes—
76#380C2314yes—
77#379—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4C2×C414yes—
81C4×C22⋊C4D412yes—
82C4×C22⋊C4D412yes—
83C2×#95—24no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D412yes—
90C2×#95C2312yes—
91C2×#95C2312yes—
92#336—22no—
93#336—21no—
94#336—21no—
95C25C2×D411yescharacteristic
96C22×Q8—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—44no—
100C23×C4—42no—
101C23×C4—44no—
102C23×C4—81no—
103C23×C4—81no—
104C23×C4—81no—
105C23×C4—81no—
106C23×C4C22×C412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×Q814yes—
112C23×C4#4114yes—
113C23×C4C2411yescharacteristic
114C2×M16—28no—
115C2×M16—82no—
116C2×M16—82no—
117C22×C8—22no—
118C22×C8—22no—
119C4⋊Q8—82no—
120#125—82no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123#117—82no—
124#117—82no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—81no—
133C2×C4⋊C4—81no—
134C2×C4⋊C4C22×C414yes—
135C2×C4⋊C4C22×C414yes—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4C22⋊C414yes—
145C2×C22⋊C4C22⋊C414yes—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—81no—
149C2×C42—81no—
150C2×C42C22×C412yes—
151C2×C42C22×C412yes—
152C2×C42C2×D412yes—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C22⋊C8—28no—
156C22⋊C8—22no—
157C22⋊C8—22no—
158C22⋊C8C2×Q814yes—
159C22⋊C8C2×Q814yes—
160#97—81no—
161#97—81no—
162C4×C8—82no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—44no—
171#95—44no—
172#95—42no—
173#95—82no—
174#95—82no—
175#95C22⋊C414yes—
176#95C22⋊C414yes—
177C24—22no—
178C24—22no—
179C24—42no—
180C24—41no—
181C24—41no—
182C24C2×C22⋊C412yes—
183C24#12412yes—
184C24C4⋊Q812yes—
185C24C22×D411yescharacteristic
186C2×Q8—44no—
187C2×Q8—82no—
188C2×Q8—82no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—48no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4—88no—
225C22×C4—88no—
226C22×C4—84no—
227C22×C4—84no—
228C22×C4C2×C22⋊C414yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C4×Q818yes—
233C22×C4C22⋊Q814yes—
234C22×C4C22⋊Q814yes—
235C22×C4#12412yes—
236C22×C4C4⋊Q812yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C22×Q812yes—
240C22×C4C2×#4112yes—
241M16—88no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—48no—
251C4⋊C4—44no—
252C4⋊C4—82no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—82no—
256C4⋊C4—84no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—84no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C22⋊C4—84no—
265C42—42no—
266C42—41no—
267C42—41no—
268C42—42no—
269C42—42no—
270C42—82no—
271C42—82no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—24no—
280C23—24no—
281C23—24no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23#39114yes—
291C23C22×C22⋊C411yescharacteristic
292C23C2×#12411yescharacteristic
293C23C2×C4⋊Q811yescharacteristic
294C23#57512yes—
295C23#57512yes—
296Q8—164no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—816no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4#39114yes—
345C2×C4C2×C4×Q812yes—
346C2×C4C2×C22⋊Q812yes—
347C8—44no—
348C8—44no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—24no—
358C22—24no—
359C22—24no—
360C22—44no—
361C22—44no—
362C22—48no—
363C22—44no—
364C22—44no—
365C22C2×#39111yescharacteristic
366C22#288311yescharacteristic
367C22#288311yescharacteristic
368C22#289612yes—
369C22#290112yes—
370C22#314011yescharacteristic
371C22#314011yescharacteristic
372C22#316612yes—
373C4—22no—
374C4—22no—
375C4—24no—
376C4—42no—
377C4—42no—
378C4—44no—
379C4—84no—
380C4—84no—
381C4—84no—
382C4—162no—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—44no—
387C2#1762312yes—
388C2#1762912yes—
389C2#2130711yescharacteristic
390C2#2132311yescharacteristic
391C2#2133311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9889C2semidirect
5. #9888C2semidirect
6. #9886C2semidirect
15. #2441C4semidirect
16. #2439C4semidirect
38. C2×C4⋊Q8C2×C4semidirect
39. C2×#125C2×C4semidirect
49. #407Q8semidirect
64. #389C2×C4semidirect
65. #389C2×C4semidirect
70. #384C2×C4semidirect
71. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350332
#205242
#139662
#98892
#98882
#98862
#80202
#80162