Finite Group Databasebeta

← #244001

Subgroups — #244001

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic
2#21835C211yesmaximal, characteristic
3#21409C211yesmaximal, characteristic
4#21400C211yesmaximal, characteristic
5#20824C211yesmaximal, characteristic
6#20797C211yesmaximal, characteristic
7#10685C211yesmaximal, characteristic, complemented
8#10684C211yesmaximal, characteristic, complemented
9#10684C211yesmaximal, characteristic, complemented
10#10681C211yesmaximal, characteristic, complemented
11#10679C211yesmaximal, characteristic, complemented
12#10675C211yesmaximal, characteristic, complemented
13#10675C211yesmaximal, characteristic, complemented
14#10674C211yesmaximal, characteristic, complemented
15#7848C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#427—22no—
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1989—22no—
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1949—22no—
38#1894—22no—
39#1892—22no—
40#1854—22no—
41#1846—22no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1844C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1825—22no—
55#1809—22no—
56#1776—22no—
57#1763—22no—
58C4×#103—22no—
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×M16D412yes—
64C23×C8D412yes—
65#533—22no—
66#533—22no—
67#533—22no—
68#533Q814yes—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×#117—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83#427—48no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×C4⋊C8D412yes—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×C22⋊C8D412yes—
98C4×M16—22no—
99C4×M16—22no—
100C2×#97D412yes—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8D412yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#382—21no—
117#382—21no—
118#382—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143C22×Q8—21no—
144C22×Q8—21no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—24no—
149C2×M16—24no—
150C2×M16—42no—
151C22×C8—24no—
152C22×C8—24no—
153C22×C8—24no—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8C2×Q812yes—
167C4⋊Q8C2×Q812yes—
168#125—24no—
169#125—42no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—42no—
177C4×Q8—42no—
178#117—24no—
179#117—42no—
180#117—44no—
181#117—44no—
182#117—82no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C42—41no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223#107—44no—
224#107—44no—
225#107—44no—
226#107—82no—
227#106—44no—
228#106—44no—
229#106—44no—
230#106—82no—
231C4⋊C8—24no—
232#103—44no—
233#103—44no—
234#103—44no—
235#103—44no—
236C22⋊C8—24no—
237#97—24no—
238C4×C8—24no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257C24#12711yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4C22⋊Q812yes—
281C22×C4C22⋊Q812yes—
282C22×C4C22⋊Q812yes—
283C22×C4#12312yes—
284C22×C4#12312yes—
285C22×C4#12312yes—
286C22×C4#12711yescharacteristic
287C22×C4#12711yescharacteristic
288C22×C4#12711yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—48no—
293C2×C8—216no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8#13514yes—
310C2×C8AGL(1,ℤ/8)14yes—
311C2×C8#13714yes—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—44no—
345C42—82no—
346C42—82no—
347C42—82no—
348C42C22⋊Q812yes—
349C42C22⋊Q812yes—
350C42C22⋊Q812yes—
351C42#12312yes—
352C42#12312yes—
353C42#12312yes—
354C42C22×Q811yescharacteristic
355C42C2×#4111yescharacteristic
356C42C2×#4111yescharacteristic
357C42C2×#4111yescharacteristic
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23#39912yes—
365C23C2×#12711yescharacteristic
366Q8—44no—
367Q8—44no—
368Q8—82no—
369Q8—82no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4#39912yes—
407C2×C4#39912yes—
408C2×C4#39912yes—
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×#12311yescharacteristic
413C2×C4C2×#12311yescharacteristic
414C2×C4C2×#12311yescharacteristic
415C2×C4C2×#13512yes—
416C2×C4C2×AGL(1,ℤ/8)12yes—
417C2×C4C2×#13712yes—
418C2×C4#57512yes—
419C2×C4#57712yes—
420C2×C4#57812yes—
421C8—24no—
422C8—24no—
423C8—24no—
424C8—42no—
425C8—42no—
426C8—42no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22C2×#39911yescharacteristic
435C22#314711yescharacteristic
436C22#315011yescharacteristic
437C22#315111yescharacteristic
438C22#315211yescharacteristic
439C22#315411yescharacteristic
440C22#315611yescharacteristic
441C4—22no—
442C4—22no—
443C4—22no—
444C4—82no—
445C4—82no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4#307612yes—
453C4#307912yes—
454C4#308212yes—
455C4#309112yes—
456C4#309212yes—
457C4#309712yes—
458C2—42no—
459C2#1964111yescharacteristic
460C2#1964811yescharacteristic
461C2#1968711yescharacteristic
462C2#1969311yescharacteristic
463C2#2373011yescharacteristic
464C2#2373411yescharacteristic
465C2#2373611yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10685C2semidirect
8. #10684C2semidirect
9. #10684C2semidirect
10. #10681C2semidirect
11. #10679C2semidirect
12. #10675C2semidirect
13. #10675C2semidirect
14. #10674C2semidirect

Maximal subgroups

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

subgroupindex
#219842
#218352
#214092
#214002
#208242
#207972
#106852
#106842
#106812
#106792
#106752
#106742
#78482