Finite Group Databasebeta

← #6497996

Subgroups — #6497996

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2#36635C211yesmaximal, characteristic
3#33296C212yesmaximal, complemented
4#33295C212yesmaximal, complemented
5#30231C214yesmaximal, complemented
6#29226C212yesmaximal, complemented
7#29225C212yesmaximal, complemented
8#29224C214yesmaximal, complemented
9#27968C214yesmaximal
10#27967C214yesmaximal
11#21837C214yesmaximal
12#20463C211yesmaximal, characteristic
13C2×#533C2212yes—
14C2×#431C2212yes—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2864C2212yes—
18#2863—22no—
19#2862C2211yescharacteristic
20#2862C2212yes—
21#2859C2211yescharacteristic
22#2859C2212yes—
23#2859C2212yes—
24#2857—22no—
25#2857—22no—
26#2857C2212yes—
27#2855—22no—
28#2851C2212yes—
29C4×C4⋊Q8—22no—
30C4×C4⋊Q8—22no—
31C4×C4⋊Q8C2212yes—
32C4×#125C2212yes—
33C4×C22⋊Q8—22no—
34#2295—22no—
35#2285C2212yes—
36#2285C2212yes—
37#2283C2212yes—
38C2×C43C2211yescharacteristic
39#2101C22116yes—
40#2100C2218yes—
41#2099C2218yes—
42#1993C2218yes—
43#1992C2214yes—
44#1992C2214yes—
45#1991C2218yes—
46#1991C2218yes—
47#1988C2218yes—
48#1987C2218yes—
49#1836C2218yes—
50#1836C2214yes—
51#1836C2214yes—
52#1825C22116yes—
53#1767C2212yes—
54#1767C2212yes—
55#1749C2212yes—
56#1749C2212yes—
57#1744C2212yes—
58C4×#97C2212yes—
59#533—216no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2312yes—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#125C2312yes—
66C2×#125C2314yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×#117C2314yes—
74C22×C42D412yes—
75C22×C42D412yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#431—28no—
80C2×C4⋊C8C2314yes—
81C2×C22⋊C8C2314yes—
82C2×#97C2314yes—
83C2×#97C2314yes—
84C2×C4×C8C2314yes—
85#391—24no—
86#391—24no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391—22no—
91#389—24no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389C2314yes—
100#389C2314yes—
101#387—24no—
102#387—24no—
103#387—24no—
104#387—24no—
105#385—24no—
106#384—22no—
107#384—22no—
108#384C2312yes—
109#384C2312yes—
110#384C2312yes—
111#384C2312yes—
112#384C2312yes—
113#383—22no—
114#382C2312yes—
115#382C2314yes—
116#382C2314yes—
117#382C2312yes—
118#382C2312yes—
119#381—22no—
120#380—22no—
121#380—24no—
122#380—24no—
123#380C2314yes—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2314yes—
129C4×C22⋊C4—22no—
130#376—22no—
131C43D412yes—
132C43D412yes—
133C43C2311yescharacteristic
134C43C2311yescharacteristic
135C43C2311yescharacteristic
136C43C2311yescharacteristic
137#340C23116yes—
138#340C23116yes—
139#340C23116yes—
140#340C23116yes—
141#336C2314yes—
142#336C2314yes—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8C2418yes—
152C4⋊Q8—28no—
153C4⋊Q8—28no—
154C4⋊Q8—28no—
155C4⋊Q8—48no—
156C4⋊Q8—48no—
157#125—216no—
158#125—28no—
159C22⋊Q8—48no—
160C22⋊Q8—48no—
161C22⋊Q8—48no—
162C4×Q8—48no—
163C4×Q8—48no—
164C4×Q8—48no—
165#117—216no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C2414yes—
181C2×C4⋊C4C2414yes—
182C2×C4⋊C4C2414yes—
183C2×C4⋊C4C2414yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—44no—
191C2×C42—44no—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4118yes—
205C2×C42C2411yescharacteristic
206C2×C42C2411yescharacteristic
207C2×C42C2411yescharacteristic
208C2×C42C2411yescharacteristic
209C2×C42C2411yescharacteristic
210C2×C42C2411yescharacteristic
211C4⋊C8—28no—
212C22⋊C8—28no—
213#97—28no—
214#97—28no—
215C4×C8—28no—
216#95—24no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—28no—
222#95—28no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227C24C22×D411yescharacteristic
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—48no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—48no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4#12414yes—
250C22×C4#12414yes—
251C22×C4C22×D412yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D412yes—
257C22×C4C22×D412yes—
258C22×C4C22×D412yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C2×#4114yes—
261C22×C4C2511yescharacteristic
262C2×C8—28no—
263C2×C8—28no—
264C2×C8—28no—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—44no—
270C4⋊C4—48no—
271C4⋊C4—44no—
272C4⋊C4—48no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—48no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—48no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C22⋊C4—48no—
284C22⋊C4—48no—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—24no—
290C42—24no—
291C42—22no—
292C42—22no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42—42no—
297C42—48no—
298C42—48no—
299C42#12414yes—
300C42#12414yes—
301C42C2×#4114yes—
302C42C2×#4114yes—
303C422−1+414yes—
304C422−1+414yes—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23C2×#12412yes—
310C23C23×D411yescharacteristic
311Q8—88no—
312Q8—88no—
313Q8—88no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—44no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4C2×#12412yes—
344C2×C4C2×#12412yes—
345C2×C4C2×#12412yes—
346C2×C4C2×#12412yes—
347C2×C4C2×#12412yes—
348C2×C4#53614yes—
349C2×C4#53614yes—
350C2×C4#57514yes—
351C2×C4#57514yes—
352C2×C4C22×#4112yes—
353C2×C4C2×2−1+412yes—
354C2×C4C2×2−1+412yes—
355C8—48no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—42no—
360C22—42no—
361C22C22×#12411yescharacteristic
362C22C2×#53611yescharacteristic
363C22C2×#53611yescharacteristic
364C22C2×#57512yes—
365C22C2×#57512yes—
366C4—24no—
367C4—22no—
368C4—24no—
369C4—24no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—84no—
375C4—84no—
376C4—84no—
377C4—84no—
378C4#313514yes—
379C4#350914yes—
380C2—42no—
381C2#3532011yescharacteristic
382C2C2×#313511yescharacteristic
383C2#6061811yescharacteristic
384C2#6086312yes—
385C2#6086612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33296C2semidirect
4. #33295C2semidirect
5. #30231C2semidirect
6. #29226C2semidirect
7. #29225C2semidirect
8. #29224C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366352
#332962
#332952
#302312
#292262
#292252
#292242
#279682
#279672
#218372
#204632