Finite Group Databasebeta

← #207013

Subgroups — #207013

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2#20526C211yesmaximal, characteristic
3#13953C211yesmaximal, characteristic
4#9889C212yesmaximal, complemented
5#9888C212yesmaximal, complemented
6#9887C214yesmaximal, complemented
7#8024C212yesmaximal
8#8012C212yesmaximal
9C22×C22⋊C8C2212yes—
10#2857C2212yes—
11#2851C2212yes—
12#2626—22no—
13#2608—22no—
14#2607—22no—
15#2441C414yescomplemented
16#2439C414yescomplemented
17#2329—22no—
18#2280C2212yes—
19C2×#381C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22#1752—22no—
23#1752—22no—
24#1749C2212yes—
25#1749C2211yescharacteristic
26#1749C2211yescharacteristic
27#1439—22no—
28#1436—22no—
29C4⋊#95C2212yes—
30#1434C2212yes—
31#1294C2214yes—
32#1294C2214yes—
33#1294C2214yes—
34#1294C2214yes—
35C24×C4C2311yescharacteristic
36C23×C8—22no—
37C2×C4⋊Q8C2×C412yes—
38C2×#125C2×C412yes—
39C2×C22⋊Q8—22no—
40C2×#117C2×C412yes—
41C2×#117C2×C412yes—
42C22×C4⋊C4—22no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4D412yes—
46C22×C42—21no—
47C22×C42C2311yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C22⋊C8—22no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8Q818yescomplemented
52C2×C22⋊C8C2312yes—
53C2×C22⋊C8C2312yes—
54C2×#97—22no—
55C2×#97—21no—
56C2×#97—21no—
57#399—24no—
58#396—24no—
59#396—24no—
60#393—24no—
61#391—22no—
62#391—24no—
63#389—22no—
64#389—24no—
65#389C2×C414yes—
66#389C2×C414yes—
67#387—24no—
68#387—24no—
69#387—24no—
70#385—24no—
71#384C2×C412yes—
72#382C2×C412yes—
73#381—22no—
74#381—24no—
75#380—22no—
76#380C2×C414yes—
77#380C2314yes—
78#379—22no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4C2×C414yes—
83C4×C22⋊C4D412yes—
84C4×C22⋊C4D412yes—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
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—81no—
100C23×C4—81no—
101C23×C4C22×C412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2×Q814yes—
107C23×C4#4114yes—
108C23×C4C2411yescharacteristic
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—28no—
112C22×C8—82no—
113C22×C8—82no—
114C4⋊Q8—82no—
115#125—82no—
116C22⋊Q8—44no—
117C22⋊Q8—44no—
118#117—82no—
119#117—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—82no—
129C2×C4⋊C4C22×C414yes—
130C2×C4⋊C4C22×C414yes—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—81no—
141C2×C22⋊C4—81no—
142C2×C22⋊C4C22⋊C414yes—
143C2×C22⋊C4C22⋊C414yes—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—42no—
147C2×C42—81no—
148C2×C42—81no—
149C2×C42—81no—
150C2×C42—81no—
151C2×C42C22×C412yes—
152C2×C42C22×C412yes—
153C2×C42C2×D412yes—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C22⋊C8—28no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8C2×Q814yes—
160C22⋊C8C2×Q814yes—
161#97—82no—
162#97—81no—
163#97—81no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—44no—
172#95—44no—
173#95—42no—
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—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
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—42no—
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—84no—
222C22×C4—84no—
223C22×C4C2×C22⋊C414yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C4×Q818yes—
228C22×C4C22⋊Q814yes—
229C22×C4C22⋊Q814yes—
230C22×C4#12412yes—
231C22×C4C4⋊Q812yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×Q812yes—
235C22×C4C2×#4112yes—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—88no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—48no—
246C4⋊C4—44no—
247C4⋊C4—82no—
248C4⋊C4—84no—
249C4⋊C4—84no—
250C4⋊C4—82no—
251C4⋊C4—88no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—48no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—84no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—84no—
261C42—42no—
262C42—41no—
263C42—41no—
264C42—42no—
265C42—42no—
266C42—82no—
267C42—82no—
268C42—84no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23—22no—
278C23—22no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—82no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23#39114yes—
288C23C22×C22⋊C411yescharacteristic
289C23C2×#12411yescharacteristic
290C23C2×C4⋊Q811yescharacteristic
291C23#57512yes—
292C23#57512yes—
293Q8—164no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
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—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4#39114yes—
341C2×C4C2×C4×Q812yes—
342C2×C4C2×C22⋊Q812yes—
343C8—44no—
344C8—44no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—24no—
354C22—24no—
355C22—24no—
356C22—44no—
357C22—44no—
358C22—84no—
359C22—82no—
360C22—82no—
361C22C2×#39111yescharacteristic
362C22#288311yescharacteristic
363C22#288311yescharacteristic
364C22#289612yes—
365C22#290112yes—
366C22#313512yes—
367C22#316211yescharacteristic
368C22#316211yescharacteristic
369C4—22no—
370C4—22no—
371C4—24no—
372C4—42no—
373C4—42no—
374C4—44no—
375C4—84no—
376C4—84no—
377C4—84no—
378C4—162no—
379C2—22no—
380C2—21no—
381C2—21no—
382C2—44no—
383C2#1762512yes—
384C2#1762912yes—
385C2#2130711yescharacteristic
386C2#2132111yescharacteristic
387C2#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. #9887C2semidirect
15. #2441C4semidirect
16. #2439C4semidirect
51. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
#205262
#139532
#98892
#98882
#98872
#80242
#80122