Finite Group Databasebeta

← #6497995

Subgroups — #6497995

Order 512.

1482 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#36635C211yesmaximal, characteristic
2C2×#2859C211yesmaximal, characteristic
3#33296C212yesmaximal, complemented
4#33295C212yesmaximal, complemented
5#30231C214yesmaximal, complemented
6#29227C214yesmaximal, complemented
7#29223C214yesmaximal, complemented
8#27969C214yesmaximal
9#27966C212yesmaximal
10#27965C212yesmaximal
11#21836C214yesmaximal
12#20463C211yesmaximal, characteristic
13C22×#125C2212yes—
14C2×#431C2212yes—
15#2864C2212yes—
16#2863—22no—
17#2862C2211yescharacteristic
18#2859—22no—
19#2859—22no—
20#2859C2211yescharacteristic
21#2859C2212yes—
22#2859C2212yes—
23#2859C2212yes—
24#2859C2212yes—
25#2857C2212yes—
26#2855—22no—
27#2851C2212yes—
28C4×C4⋊Q8C2212yes—
29C4×#125C2212yes—
30C4×C22⋊Q8—22no—
31#2295—22no—
32C2×#389—22no—
33C2×#389—22no—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36#2283C2212yes—
37C2×#382C2212yes—
38C2×C43C2211yescharacteristic
39#2101C22116yes—
40#2100C2218yes—
41#2099C2218yes—
42#1993C2218yes—
43#1992C2214yes—
44#1992C2214yes—
45#1991C2218yes—
46#1991C2218yes—
47#1989C22116yes—
48#1836C2218yes—
49#1836C2214yes—
50#1836C2214yes—
51#1824C2218yes—
52#1823C2218yes—
53#1767C2212yes—
54#1767C2212yes—
55#1749C2212yes—
56#1749C2212yes—
57#1744C2212yes—
58C4×#97C2212yes—
59C2×C4⋊Q8C2312yes—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—216no—
63C2×#125C2312yes—
64C2×#125C2314yes—
65C2×#125C2314yes—
66C2×C22⋊Q8—22no—
67C2×C4×Q8—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
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—22no—
86#391—22no—
87#391—22no—
88#391—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—24no—
99#389—24no—
100#389—22no—
101#389C2314yes—
102#389C2314yes—
103#387—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#385—24no—
108#384—22no—
109#384—22no—
110#384C2312yes—
111#384C2312yes—
112#384C2312yes—
113#384C2312yes—
114#384C2312yes—
115#383—22no—
116#382—24no—
117#382C2312yes—
118#382C2314yes—
119#382C2314yes—
120#382C2312yes—
121#382C2312yes—
122#381—22no—
123#380—22no—
124#380C2314yes—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2314yes—
128C4×C22⋊C4—22no—
129#376—22no—
130C2×#95—24no—
131C43D412yes—
132C43D412yes—
133C43C2311yescharacteristic
134C43C2311yescharacteristic
135C43C2311yescharacteristic
136C43C2311yescharacteristic
137#340C23116yes—
138#340C23116yes—
139#340C23116yes—
140#340C23116yes—
141#336C2314yes—
142#336C2314yes—
143C22×Q8—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8C2418yes—
152C4⋊Q8—28no—
153#125—216no—
154#125—216no—
155#125—28no—
156#125—48no—
157#125—48no—
158C22⋊Q8—48no—
159C4×Q8—48no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4C2414yes—
183C2×C4⋊C4C2414yes—
184C2×C4⋊C4C2414yes—
185C2×C4⋊C4C2414yes—
186C2×C22⋊C4—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—28no—
217#95—28no—
218#95—24no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227C24C22×D411yescharacteristic
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—44no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—48no—
249C22×C4#12414yes—
250C22×C4#12414yes—
251C22×C4C22×D412yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D412yes—
258C22×C4C22×D412yes—
259C22×C4C22×D412yes—
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—48no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—48no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—48no—
285C22⋊C4—48no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—24no—
290C42—22no—
291C42—22no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42—42no—
297C42—42no—
298C42—48no—
299C42#12414yes—
300C42#12414yes—
301C42C2×#4114yes—
302C42C2×#4114yes—
303C422−1+414yes—
304C422−1+414yes—
305C23—21no—
306C23—21no—
307C23—22no—
308C23C2×#12412yes—
309C23#57514yes—
310C23C23×D411yescharacteristic
311Q8—88no—
312C2×C4—216no—
313C2×C4—28no—
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—22no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—44no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—48no—
341C2×C4—48no—
342C2×C4—88no—
343C2×C4—88no—
344C2×C4C2×#12412yes—
345C2×C4C2×#12412yes—
346C2×C4C2×#12412yes—
347C2×C4C2×#12412yes—
348C2×C4C2×#12412yes—
349C2×C4#53614yes—
350C2×C4#53614yes—
351C2×C4#57514yes—
352C2×C4C22×#4112yes—
353C2×C4C2×2−1+412yes—
354C2×C4C2×2−1+412yes—
355C8—48no—
356C22—24no—
357C22—24no—
358C22—22no—
359C22—22no—
360C22#313514yes—
361C22C22×#12411yescharacteristic
362C22C2×#53611yescharacteristic
363C22C2×#53611yescharacteristic
364C22C2×#57512yes—
365C22C2×#57512yes—
366C4—24no—
367C4—24no—
368C4—22no—
369C4—22no—
370C4—24no—
371C4—22no—
372C4—22no—
373C4—22no—
374C4—42no—
375C4—84no—
376C4—84no—
377C4—84no—
378C4—84no—
379C4#350914yes—
380C2—24no—
381C2#3532011yescharacteristic
382C2C2×#313511yescharacteristic
383C2#6061811yescharacteristic
384C2#6086112yes—
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. #29227C2semidirect
7. #29223C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×#28592
#332962
#332952
#302312
#292272
#292232
#279692
#279662
#279652
#218362
#204632