Finite Group Databasebeta

← #192085

Subgroups — #192085

Order 512.

1160 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34177C211yesmaximal, characteristic
2#21363C212yesmaximal, complemented
3C2×#1464C211yesmaximal, characteristic
4#13852C211yesmaximal, characteristic
5#8855C214yesmaximal, complemented
6#8854C212yesmaximal
7#8853C212yesmaximal
8#8021C212yesmaximal, complemented
9C22×#117C2211yescharacteristic
10C2×#432C2212yes—
11#2343—24no—
12#2343—24no—
13#2315C2211yescharacteristic
14#2315C2211yescharacteristic
15#2313—21no—
16#2313C2212yes—
17#2283—21no—
18#2283C2212yes—
19C2×#381—21no—
20#2273—21no—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4—21no—
24#1949—24no—
25#1853C2214yes—
26#1852C2214yes—
27#1776—24no—
28#1464C418yescomplemented
29C2×#324C2212yes—
30C2×#323C2212yes—
31#1441—21no—
32#1441C2212yes—
33#1439—21no—
34#1439C2212yes—
35C4×#95—21no—
36#1294C2214yes—
37#1294C2214yes—
38C24×C4D411yescharacteristic
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—24no—
42C2×#117C2×C414yes—
43C2×#117Q814yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D411yescharacteristic
52C22×C22⋊C4C2311yescharacteristic
53C22×C42—21no—
54C22×C42—21no—
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57#432—24no—
58#432C2×C418yescomplemented
59C2×#107—24no—
60C2×#106—24no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8C2314yes—
63C2×#97—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#382—22no—
73#382D414yes—
74#381—22no—
75#381—22no—
76#381—22no—
77#381—22no—
78#380—21no—
79#380—21no—
80#380C2312yes—
81#380C2312yes—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4D414yes—
87C4×C22⋊C4—24no—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—44no—
92#376—22no—
93#376—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95C2312yes—
101#340—24no—
102#340—24no—
103#324C2×C418yescomplemented
104#323C2×C418yes—
105C25C2×D411yescharacteristic
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4C4⋊C414yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125#117—28no—
126#117—28no—
127#117—28no—
128#117—28no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—42no—
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—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—44no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4C4⋊C414yes—
160C2×C22⋊C4C22×C412yes—
161C2×C22⋊C4C2×Q812yes—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—42no—
174C2×C42C4⋊C414yes—
175C2×C42C22×C412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×Q812yes—
179C2×C42#4114yes—
180#107—48no—
181#106—48no—
182C4⋊C8—28no—
183C22⋊C8—24no—
184C22⋊C8C22×C418yes—
185#97—28no—
186#95—24no—
187#95—24no—
188#95—22no—
189#95—22no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—24no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—42no—
201C24—21no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—42no—
206C24—42no—
207C24C2×C4⋊C412yes—
208C24AGL(1,ℤ/8)12yes—
209C24#13712yes—
210C24C22×D411yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—28no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4C2×C4⋊C412yes—
262C22×C4C2×C4⋊C412yes—
263C22×C4C2×C4⋊C412yes—
264C22×C4C2×C4⋊C412yes—
265C22×C4C2×C4⋊C412yes—
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12012yes—
269C22×C4#12312yes—
270C22×C4#12312yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D412yes—
273C22×C4C22×Q811yescharacteristic
274C22×C4C2×#4112yes—
275C2×C8—28no—
276C2×C8—44no—
277C2×C8—44no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—28no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—416no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C42—24no—
299C42—24no—
300C42—84no—
301C42—84no—
302C42—82no—
303C42—82no—
304C42—84no—
305C42C4×D414yes—
306C42C4×D414yes—
307C42C22⋊Q814yes—
308C42C22⋊Q814yes—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—44no—
318C23—42no—
319C23—42no—
320C23—44no—
321C23—42no—
322C23—42no—
323C23—82no—
324C23—82no—
325C23—81no—
326C23—81no—
327C23#42814yes—
328C23C22×C4⋊C411yescharacteristic
329C23C2×#12011yescharacteristic
330C23C2×#12311yescharacteristic
331C23C2×AGL(1,ℤ/8)11yescharacteristic
332C23C2×#13711yescharacteristic
333C23#57511yescharacteristic
334C23#57511yescharacteristic
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—48no—
356C2×C4—48no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—164no—
388C2×C4—164no—
389C2×C4#38512yes—
390C2×C4#38512yes—
391C2×C4#38514yes—
392C2×C4C2×C4×D412yes—
393C2×C4C2×C22⋊Q812yes—
394C8—48no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—24no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22C2×#38511yescharacteristic
413C22C2×#42811yescharacteristic
414C22#291111yescharacteristic
415C22#300611yescharacteristic
416C22#300711yescharacteristic
417C22#301812yes—
418C22#302512yes—
419C22#309411yescharacteristic
420C22#309511yescharacteristic
421C4—22no—
422C4—22no—
423C4—24no—
424C4—22no—
425C4—22no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—84no—
430C4—84no—
431C4—84no—
432C4—84no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—44no—
438C2#1817612yes—
439C2#1817712yes—
440C2#2084611yescharacteristic
441C2#2085511yescharacteristic
442C2#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. #21363C2semidirect
5. #8855C2semidirect
8. #8021C2semidirect
28. #1464C4semidirect
58. #432C2×C4semidirect
103. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341772
#213632
C2×#14642
#138522
#88552
#88542
#88532
#80212