Finite Group Databasebeta

← #244004

Subgroups — #244004

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 462 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#20760C211yesmaximal, characteristic
7#10684C211yesmaximal, characteristic, complemented
8#10682C211yesmaximal, characteristic, complemented
9#10682C211yesmaximal, characteristic, complemented
10#10681C211yesmaximal, characteristic, complemented
11#10681C211yesmaximal, characteristic, complemented
12#10679C211yesmaximal, characteristic, complemented
13#10679C211yesmaximal, characteristic, complemented
14#10675C211yesmaximal, characteristic, complemented
15#7848C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2285C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1989—22no—
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1949—22no—
37#1894—22no—
38#1892—22no—
39#1854—22no—
40#1845C2211yescharacteristic
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1844C2211yescharacteristic
44#1835—22no—
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1825—22no—
54#1809—22no—
55#1776—22no—
56#1763—22no—
57C4×#103—22no—
58#1735—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—
83C2×#107—22no—
84C2×#107—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×C4⋊C8D412yes—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×C22⋊C8D412yes—
93C4×M16—22no—
94C4×M16—22no—
95C2×#97D412yes—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8D412yes—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#382—21no—
112#382—21no—
113#382—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
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—
142C22×Q8—21no—
143C22×Q8—21no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—42no—
150C22×C8—24no—
151C22×C8—42no—
152C22×C8—42no—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8C2×Q812yes—
166C4⋊Q8C2×Q812yes—
167#125—24no—
168#125—42no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—42no—
176C4×Q8—42no—
177#117—24no—
178#117—42no—
179#117—44no—
180#117—44no—
181#117—82no—
182C2×C4⋊C4—21no—
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—41no—
205C2×C4⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42C2411yescharacteristic
222#107—44no—
223#107—82no—
224#106—44no—
225#106—82no—
226C4⋊C8—24no—
227#103—44no—
228#103—44no—
229#103—44no—
230#103—44no—
231C22⋊C8—24no—
232#97—24no—
233C4×C8—24no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24#12711yescharacteristic
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4C22⋊Q812yes—
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4#12312yes—
279C22×C4#12312yes—
280C22×C4#12312yes—
281C22×C4#12711yescharacteristic
282C22×C4#12711yescharacteristic
283C22×C4#12711yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—48no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
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—48no—
306C2×C8#13514yes—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
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—
335C22⋊C4—44no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C42—44no—
340C42—82no—
341C42—82no—
342C42—82no—
343C42C22⋊Q812yes—
344C42C22⋊Q812yes—
345C42C22⋊Q812yes—
346C42#12312yes—
347C42#12312yes—
348C42#12312yes—
349C42C22×Q811yescharacteristic
350C42C2×#4111yescharacteristic
351C42C2×#4111yescharacteristic
352C42C2×#4111yescharacteristic
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23#39912yes—
360C23C2×#12711yescharacteristic
361Q8—44no—
362Q8—44no—
363Q8—82no—
364Q8—82no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
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#39912yes—
402C2×C4#39912yes—
403C2×C4#39912yes—
404C2×C4C2×C22⋊Q811yescharacteristic
405C2×C4C2×C22⋊Q811yescharacteristic
406C2×C4C2×C22⋊Q811yescharacteristic
407C2×C4C2×#12311yescharacteristic
408C2×C4C2×#12311yescharacteristic
409C2×C4C2×#12311yescharacteristic
410C2×C4C2×#13512yes—
411C2×C4#57512yes—
412C2×C4#57512yes—
413C2×C4#57512yes—
414C2×C4#57712yes—
415C2×C4#57812yes—
416C8—24no—
417C8—42no—
418C8—42no—
419C8—42no—
420C8—42no—
421C8—42no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22C2×#39911yescharacteristic
430C22#314711yescharacteristic
431C22#315011yescharacteristic
432C22#315311yescharacteristic
433C22#315311yescharacteristic
434C22#315411yescharacteristic
435C22#315611yescharacteristic
436C4—22no—
437C4—22no—
438C4—22no—
439C4—82no—
440C4—82no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4—161no—
447C4#307612yes—
448C4#308012yes—
449C4#308212yes—
450C4#309112yes—
451C4#309312yes—
452C4#309712yes—
453C2—42no—
454C2#1964311yescharacteristic
455C2#1964911yescharacteristic
456C2#1969211yescharacteristic
457C2#1969611yescharacteristic
458C2#2373011yescharacteristic
459C2#2373311yescharacteristic
460C2#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. #10684C2semidirect
8. #10682C2semidirect
9. #10682C2semidirect
10. #10681C2semidirect
11. #10681C2semidirect
12. #10679C2semidirect
13. #10679C2semidirect
14. #10675C2semidirect

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
#207602
#106842
#106822
#106812
#106792
#106752
#78482