Finite Group Databasebeta

← #207014

Subgroups — #207014

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2#20524C211yesmaximal, characteristic
3#13965C211yesmaximal, characteristic
4#9887C214yesmaximal, complemented
5#9885C212yesmaximal, complemented
6#9884C212yesmaximal, 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—
18C2×#385—22no—
19#2280C2212yes—
20C2×#380C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C22×#95C2211yescharacteristic
23#1753—22no—
24#1753—22no—
25#1749C2212yes—
26C4×C22⋊C8C2211yescharacteristic
27C4×C22⋊C8C2211yescharacteristic
28#1444—22no—
29C4⋊#95C2212yes—
30C4⋊#95C2212yes—
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—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4D412yes—
48C22×C42C2311yescharacteristic
49C22×C42C2311yescharacteristic
50#407—24no—
51#407Q818yescomplemented
52C2×C22⋊C8—22no—
53C2×C22⋊C8C2312yes—
54C2×C22⋊C8C2312yes—
55C2×#97—22no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#399—24no—
59#396—24no—
60#396—24no—
61#393—24no—
62#391—22no—
63#391—24no—
64#389—22no—
65#389—24no—
66#389C2×C414yescomplemented
67#389C2×C414yescomplemented
68#387—24no—
69#387—24no—
70#387—24no—
71#385—24no—
72#385—48no—
73#384C2×C412yescomplemented
74#382C2×C412yescomplemented
75#381—22no—
76#380—22no—
77#380—24no—
78#380C2×C414yes—
79#380C2314yes—
80#379—22no—
81C4×C22⋊C4—22no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4C2×C414yes—
84C4×C22⋊C4D412yes—
85C4×C22⋊C4D412yes—
86C2×#95—24no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95D412yes—
91C2×#95C2312yes—
92C2×#95C2312yes—
93#336—21no—
94#336—21no—
95#336—22no—
96C25C2×D411yescharacteristic
97C22×Q8—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—44no—
101C23×C4—42no—
102C23×C4—44no—
103C23×C4—81no—
104C23×C4—81no—
105C23×C4—81no—
106C23×C4—81no—
107C23×C4C22×C412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×Q814yes—
113C23×C4#4114yes—
114C23×C4C2411yescharacteristic
115C2×M16—28no—
116C2×M16—82no—
117C2×M16—82no—
118C22×C8—22no—
119C22×C8—22no—
120C4⋊Q8—82no—
121#125—82no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124#117—82no—
125#117—82no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—82no—
138C2×C4⋊C4—82no—
139C2×C4⋊C4—81no—
140C2×C4⋊C4—81no—
141C2×C4⋊C4C22×C414yes—
142C2×C4⋊C4C22×C414yes—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4C22⋊C414yes—
152C2×C22⋊C4C22⋊C414yes—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—81no—
156C2×C42—81no—
157C2×C42C22×C412yes—
158C2×C42C22×C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C22⋊C8—28no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8C2×Q814yes—
166C22⋊C8C2×Q814yes—
167#97—82no—
168C4×C8—81no—
169C4×C8—81no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—24no—
176#95—24no—
177#95—44no—
178#95—42no—
179#95—44no—
180#95C22⋊C414yes—
181#95C22⋊C414yes—
182C24—22no—
183C24—22no—
184C24—42no—
185C24—41no—
186C24—41no—
187C24C2×C22⋊C412yes—
188C24#12412yes—
189C24C4⋊Q812yes—
190C24C22×D411yescharacteristic
191C2×Q8—44no—
192C2×Q8—82no—
193C2×Q8—82no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—48no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4—88no—
228C22×C4—88no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4C2×C22⋊C414yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C4×Q818yes—
236C22×C4C22⋊Q814yes—
237C22×C4C22⋊Q814yes—
238C22×C4#12412yes—
239C22×C4C4⋊Q812yes—
240C22×C4AGL(1,ℤ/8)14yes—
241C22×C4#13714yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×Q812yes—
245C22×C4C2×#4112yes—
246M16—88no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—48no—
257C4⋊C4—82no—
258C4⋊C4—84no—
259C4⋊C4—84no—
260C4⋊C4—82no—
261C4⋊C4—88no—
262C4⋊C4—88no—
263C4⋊C4—84no—
264C22⋊C4—28no—
265C22⋊C4—28no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—84no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C22⋊C4—84no—
272C42—41no—
273C42—41no—
274C42—42no—
275C42—42no—
276C42—42no—
277C42—82no—
278C42—82no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—24no—
287C23—24no—
288C23—24no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—44no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23#39114yes—
298C23C22×C22⋊C411yescharacteristic
299C23C2×#12411yescharacteristic
300C23C2×C4⋊Q811yescharacteristic
301C23C2×AGL(1,ℤ/8)12yes—
302C23C2×#13712yes—
303Q8—164no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
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—44no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—816no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#39114yes—
350C2×C4#41714yes—
351C2×C4#41714yes—
352C2×C4C2×C4×Q812yes—
353C2×C4C2×C22⋊Q812yes—
354C8—44no—
355C8—44no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—24no—
365C22—24no—
366C22—24no—
367C22—44no—
368C22—44no—
369C22—48no—
370C22—44no—
371C22—44no—
372C22C2×#39111yescharacteristic
373C22#288311yescharacteristic
374C22#288311yescharacteristic
375C22C2×#41712yes—
376C22#290112yes—
377C22#313811yescharacteristic
378C22#313911yescharacteristic
379C22#316512yes—
380C4—22no—
381C4—22no—
382C4—24no—
383C4—24no—
384C4—24no—
385C4—44no—
386C4—84no—
387C4—84no—
388C4—84no—
389C4—162no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—44no—
394C2#1762212yes—
395C2#1763012yes—
396C2#2130711yescharacteristic
397C2#2132211yescharacteristic
398C2#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. #9887C2semidirect
5. #9885C2semidirect
6. #9884C2semidirect
15. #2441C4semidirect
16. #2439C4semidirect
38. C2×C4⋊Q8C2×C4semidirect
39. C2×#125C2×C4semidirect
51. #407Q8semidirect
66. #389C2×C4semidirect
67. #389C2×C4semidirect
73. #384C2×C4semidirect
74. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350332
#205242
#139652
#98872
#98852
#98842
#80202
#80162