Finite Group Databasebeta

← #192096

Subgroups — #192096

Order 512.

1112 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34177C211yesmaximal, characteristic
2#21361C212yesmaximal, complemented
3C2×#1458C211yesmaximal, characteristic
4#13938C211yesmaximal, characteristic
5#8855C214yesmaximal, complemented
6#8854C212yesmaximal
7#8853C212yesmaximal
8#8025C212yesmaximal, complemented
9C22×#117C2211yescharacteristic
10C2×#431C2212yes—
11#2343—24no—
12#2343—22no—
13#2343—22no—
14#2315C2212yes—
15#2313—21no—
16#2313C2212yes—
17#2283—21no—
18#2283C2212yes—
19C2×#383C2211yescharacteristic
20C2×#381—21no—
21#2273—21no—
22C2×C4×C4⋊C4C2211yescharacteristic
23#1949—24no—
24#1853C2214yes—
25#1852C2214yes—
26C4×#107—22no—
27C4×#106—22no—
28#1458C418yescomplemented
29C2×#324C2212yes—
30C2×#323C2211yescharacteristic
31C2×#323C2211yescharacteristic
32#1441C2212yes—
33#1439C2212yes—
34#1437—21no—
35#1436—21no—
36#1434—21no—
37C22⋊#95—21no—
38#1294C2214yes—
39#1294C2214yes—
40C24×C4D411yescharacteristic
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—24no—
44C2×#117C2×C414yes—
45C2×#117Q814yes—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D411yescharacteristic
52C22×C22⋊C4C2311yescharacteristic
53C22×C42—21no—
54C22×C42D411yescharacteristic
55C22×C42C2311yescharacteristic
56#431—24no—
57#431C2×C418yescomplemented
58C2×#107—24no—
59C2×#106—24no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2312yes—
62C2×C22⋊C8C2312yes—
63C2×C4×C8—22no—
64#387—24no—
65#387—24no—
66#387—24no—
67#387—24no—
68#385—24no—
69#385—24no—
70#383—22no—
71#383—22no—
72#383—24no—
73#382—21no—
74#382—21no—
75#382D414yes—
76#381—22no—
77#381—22no—
78#381—22no—
79#381—22no—
80#380—22no—
81#380C2314yes—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D412yes—
86C4×C4⋊C4D412yes—
87C4×C22⋊C4—24no—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91#376—22no—
92#376—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95C2312yes—
103#340—24no—
104#340—24no—
105#324C2×C418yescomplemented
106#323C2×C414yes—
107#323C2×C414yes—
108C25C2×D411yescharacteristic
109C23×C4—24no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—42no—
118C23×C4C4⋊C414yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127#117—28no—
128#117—28no—
129#117—28no—
130#117—28no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4C4⋊C414yes—
140C2×C4⋊C4C22×C412yes—
141C2×C4⋊C4C2×D414yes—
142C2×C4⋊C4C2×D414yes—
143C2×C4⋊C4C2×Q812yes—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—82no—
155C2×C22⋊C4C4⋊C414yes—
156C2×C22⋊C4C22×C412yes—
157C2×C22⋊C4C2×Q812yes—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—24no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—42no—
165C2×C42—82no—
166C2×C42C4⋊C414yes—
167C2×C42C22×C412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×Q812yes—
171C2×C42#4114yes—
172#107—48no—
173#106—48no—
174C4⋊C8—28no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8C22×C414yes—
178C22⋊C8C22×C414yes—
179C4×C8—28no—
180#95—22no—
181#95—24no—
182#95—22no—
183#95—24no—
184#95—22no—
185#95—22no—
186#95—24no—
187#95—22no—
188#95—22no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—44no—
193#95—44no—
194#95—41no—
195#95—41no—
196#95—42no—
197#95—42no—
198#95—82no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—42no—
203C24—41no—
204C24—41no—
205C24C2×C4⋊C412yes—
206C24#13514yes—
207C24C22×D411yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—28no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—41no—
228C22×C4—48no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4C2×C4⋊C412yes—
254C22×C4C2×C4⋊C412yes—
255C22×C4C2×C4⋊C412yes—
256C22×C4C2×C4⋊C412yes—
257C22×C4C2×C4⋊C412yes—
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12012yes—
261C22×C4#12312yes—
262C22×C4#12312yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D412yes—
265C22×C4C22×Q811yescharacteristic
266C22×C4C2×#4112yes—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—44no—
270C2×C8—44no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—28no—
283C22⋊C4—48no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—88no—
287C42—24no—
288C42—24no—
289C42—82no—
290C42—82no—
291C42—84no—
292C42—84no—
293C42C4×D414yes—
294C42C4×D414yes—
295C42C22⋊Q814yes—
296C42C22⋊Q814yes—
297C23—21no—
298C23—21no—
299C23—22no—
300C23—24no—
301C23—22no—
302C23—24no—
303C23—22no—
304C23—44no—
305C23—42no—
306C23—42no—
307C23—44no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—82no—
312C23—82no—
313C23—81no—
314C23—81no—
315C23#42714yes—
316C23C22×C4⋊C411yescharacteristic
317C23C2×#12011yescharacteristic
318C23C2×#12311yescharacteristic
319C23C2×#13512yes—
320C23#57512yes—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—164no—
371C2×C4—164no—
372C2×C4#38512yes—
373C2×C4#38512yes—
374C2×C4#38514yes—
375C2×C4C2×C4×D412yes—
376C2×C4C2×C22⋊Q812yes—
377C8—44no—
378C8—44no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—42no—
392C22—42no—
393C22—84no—
394C22—82no—
395C22C2×#38511yescharacteristic
396C22C2×#42711yescharacteristic
397C22#291111yescharacteristic
398C22#300212yes—
399C22#302111yescharacteristic
400C22#302211yescharacteristic
401C22#302512yes—
402C22#309112yes—
403C4—22no—
404C4—22no—
405C4—24no—
406C4—22no—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—84no—
412C4—84no—
413C4—84no—
414C4—162no—
415C2—22no—
416C2—21no—
417C2—21no—
418C2—44no—
419C2#1820812yes—
420C2#1820912yes—
421C2#2084311yescharacteristic
422C2#2085811yescharacteristic
423C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×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
2. #21361C2semidirect
5. #8855C2semidirect
8. #8025C2semidirect
28. #1458C4semidirect
57. #431C2×C4semidirect
105. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341772
#213612
C2×#14582
#139382
#88552
#88542
#88532
#80252