Finite Group Databasebeta

← #192134

Subgroups — #192134

Order 512.

1002 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34059C211yesmaximal, characteristic
2#21983C212yesmaximal
3C2×#1461C211yesmaximal, characteristic
4#14346C211yesmaximal, characteristic
5#8857C212yesmaximal
6#8856C212yesmaximal
7#8855C214yesmaximal
8#8025C212yesmaximal
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2212yes—
11#2320—22no—
12#2320—22no—
13#2316C2212yes—
14#2313—21no—
15#2313C2212yes—
16#2310—21no—
17#2295—24no—
18#2283—21no—
19#2283C2212yes—
20C2×#383C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C4×#117—21no—
23#1949—24no—
24#1853C2214yes—
25#1852C2214yes—
26#1846—22no—
27#1835—22no—
28#1461C418yescomplemented
29C2×#324C2212yes—
30C2×#324C2211yescharacteristic
31C2×#324C2211yescharacteristic
32#1446—21no—
33#1445—21no—
34#1441C2212yes—
35#1441C2212yes—
36#1434—21no—
37#1434—21no—
38#1294C2214yes—
39#1294C2214yes—
40C23×Q8D411yescharacteristic
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—24no—
44C2×C4×Q8C2×C414yes—
45C2×C4×Q8Q814yes—
46C2×#117—22no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42C2311yescharacteristic
54#431—24no—
55#431C2×C418yes—
56C2×#107—24no—
57C2×#106—24no—
58C2×C4⋊C8—22no—
59C2×C22⋊C8C2312yes—
60C2×C22⋊C8C2312yes—
61C2×C4×C8—22no—
62#391—24no—
63#391—24no—
64#389—24no—
65#389—24no—
66#387—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#383—22no—
71#383—22no—
72#383—24no—
73#382—21no—
74#382—21no—
75#382—22no—
76#382—22no—
77#382D412yes—
78#382D412yes—
79#381—22no—
80#381—22no—
81#380—22no—
82#380C2314yes—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4D414yes—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90#376—22no—
91#376—22no—
92#376—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95C2312yes—
102C43—22no—
103#340—24no—
104#340—24no—
105#324C2×C418yes—
106#324C2×C414yes—
107#324C2×C414yes—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—24no—
111C22×Q8—42no—
112C22×Q8C4⋊C414yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C4×Q8—28no—
125C4×Q8—28no—
126C4×Q8—28no—
127C4×Q8—28no—
128#117—44no—
129#117—44no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4C4⋊C414yes—
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C412yes—
145C2×C4⋊C4C2×D414yes—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×Q812yes—
148C2×C4⋊C4C2×Q812yes—
149C2×C22⋊C4—24no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42C4⋊C414yes—
161C2×C42C4⋊C414yes—
162C2×C42C22×C412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×Q812yes—
166C2×C42#4114yes—
167#107—84no—
168#106—84no—
169C4⋊C8—28no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8C22×C414yes—
173C22⋊C8C22×C414yes—
174C4×C8—28no—
175#95—24no—
176#95—24no—
177#95—22no—
178#95—22no—
179#95—24no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—44no—
188#95—41no—
189#95—42no—
190#95—41no—
191#95—42no—
192#95—42no—
193#95—44no—
194#95—82no—
195#95—82no—
196C24C22×D411yescharacteristic
197C2×Q8—28no—
198C2×Q8—28no—
199C2×Q8—44no—
200C2×Q8—48no—
201C2×Q8—48no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C2×C4⋊C412yes—
239C22×C4C2×C4⋊C412yes—
240C22×C4C2×C4⋊C412yes—
241C22×C4C2×C4⋊C412yes—
242C22×C4C2×C4⋊C412yes—
243C22×C4C2×C4⋊C412yes—
244C22×C4#12011yescharacteristic
245C22×C4#12012yes—
246C22×C4#12011yescharacteristic
247C22×C4#12312yes—
248C22×C4#12312yes—
249C22×C4#13514yes—
250C22×C4C23×C411yescharacteristic
251C22×C4C22×D412yes—
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4112yes—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—28no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C42—24no—
276C42—24no—
277C42—24no—
278C42—24no—
279C42—48no—
280C42—48no—
281C42—82no—
282C42—82no—
283C42C4×D414yes—
284C42C4×D414yes—
285C42C22⋊Q814yes—
286C42C22⋊Q814yes—
287C23—22no—
288C23—21no—
289C23—21no—
290C23C22×C4⋊C411yescharacteristic
291C23C2×#12011yescharacteristic
292C23C2×#12311yescharacteristic
293C23C2×#13512yes—
294C23#57512yes—
295Q8—416no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
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—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4#38512yes—
349C2×C4#38512yes—
350C2×C4#38514yes—
351C2×C4#42714yes—
352C2×C4C2×C4×D412yes—
353C2×C4C2×C22⋊Q812yes—
354C8—44no—
355C8—44no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—21no—
363C22—21no—
364C22C2×#38511yescharacteristic
365C22C2×#42711yescharacteristic
366C22#291111yescharacteristic
367C22#300212yes—
368C22#302012yes—
369C22#302711yescharacteristic
370C22#302811yescharacteristic
371C22#309112yes—
372C4—24no—
373C4—24no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—84no—
382C4—84no—
383C4—162no—
384C4—162no—
385C2—22no—
386C2—21no—
387C2—21no—
388C2#1830712yes—
389C2#1831512yes—
390C2#2084311yescharacteristic
391C2#2085711yescharacteristic
392C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#340592
#219832
C2×#14612
#143462
#88572
#88562
#88552
#80252