Finite Group Databasebeta

← #315040

Subgroups — #315040

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28792C211yesmaximal, characteristic, complemented
2#28600C211yesmaximal, characteristic
3#27748C211yesmaximal, characteristic
4#27727C211yesmaximal, characteristic
5#10421C211yesmaximal, characteristic
6#10416C211yesmaximal, characteristic
7#10357C211yesmaximal, characteristic, complemented
8#10353C211yesmaximal, characteristic, complemented
9#9821C211yesmaximal, characteristic, complemented
10#9736C211yesmaximal, characteristic, complemented
11#9566C211yesmaximal, characteristic, complemented
12#9561C211yesmaximal, characteristic, complemented
13#8646C211yesmaximal, characteristic
14C8×#106C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic
16C2×#487C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18#2571C2211yescharacteristic
19#2552C2211yescharacteristic
20#2547C2211yescharacteristic
21#2542C2211yescharacteristic
22#2099—21no—
23#2085—21no—
24#2074—21no—
25#2071—21no—
26#2059—21no—
27#2042—21no—
28#2035—21no—
29#2035—21no—
30#1991—21no—
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1988C2211yescharacteristic
35#1988C2211yescharacteristic
36#1981C2211yescharacteristic
37#1981C2211yescharacteristic
38#1958—21no—
39#1956C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2211yescharacteristic
42#1942—21no—
43#1934—21no—
44#1933—21no—
45#1892—21no—
46#1892—21no—
47#1834—21no—
48#1834C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51C4×#106C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×#124—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C4×D4C2311yescharacteristic
75#487—22no—
76#487—22no—
77C8×D4—22no—
78C8×D4—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106D411yescharacteristic
82C2×#106D411yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#103—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102D411yescharacteristic
98C2×#102D411yescharacteristic
99C2×C22⋊C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#400—41no—
107#399—41no—
108#398—41no—
109#397—41no—
110#391—21no—
111#390—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#384—21no—
117#384—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121#382—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—41no—
140#340—41no—
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#334—22no—
149#332—22no—
150#332—22no—
151D4⋊C8—22no—
152C82—22no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×D4C2×D411yescharacteristic
156C23×C4—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—42no—
163C22×C8—81no—
164C22×C8—81no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177#125—42no—
178#125—81no—
179#124—42no—
180#124—81no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—81no—
184C22⋊Q8—81no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—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—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C2411yescharacteristic
221#107—42no—
222#107—81no—
223#106—22no—
224#106—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8#4112yes—
233#103—42no—
234#103—42no—
235#103—42no—
236#103—81no—
237#103—81no—
238#103—81no—
239#103—81no—
240#103—81no—
241#102—22no—
242#102—22no—
243#102—42no—
244#102—81no—
245C22⋊C8—42no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—42no—
251C4×C8—81no—
252C4×C8—81no—
253C4×C8#4112yes—
254C4×C8#4112yes—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264C24—21no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4#13512yes—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—84no—
328C2×C8#12412yes—
329C2×C8#12412yes—
330C2×C8#13512yes—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—161no—
366C4⋊C4—161no—
367C4⋊C4#13512yes—
368C4⋊C4#13512yes—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C422−1+411yescharacteristic
385C23—22no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23D4211yescharacteristic
393Q8—81no—
394Q8—81no—
395Q8—81no—
396Q8—81no—
397Q8—81no—
398Q8—81no—
399Q8—81no—
400Q8—81no—
401D4—22no—
402D4—22no—
403D4—41no—
404D4—41no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4C2×#12411yescharacteristic
445C2×C4#53211yescharacteristic
446C2×C4#54211yescharacteristic
447C2×C4#54711yescharacteristic
448C2×C4#54811yescharacteristic
449C2×C4#55511yescharacteristic
450C2×C4C2×#13511yescharacteristic
451C2×C4C2×#13511yescharacteristic
452C2×C4C2×#13511yescharacteristic
453C2×C4C2×#13511yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C8—22no—
459C8—22no—
460C8—22no—
461C8—42no—
462C8—41no—
463C8—42no—
464C8—81no—
465C8—81no—
466C8—81no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22#249111yescharacteristic
475C22#328011yescharacteristic
476C22#328211yescharacteristic
477C22#328211yescharacteristic
478C22#328411yescharacteristic
479C22#329011yescharacteristic
480C22#329311yescharacteristic
481C4—21no—
482C4—21no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4#313111yescharacteristic
495C4#313611yescharacteristic
496C4#316811yescharacteristic
497C4#317411yescharacteristic
498C4#333711yescharacteristic
499C4#333711yescharacteristic
500C4#336111yescharacteristic
501C4#336511yescharacteristic
502C2—82no—
503C2#1999011yescharacteristic
504C2#1999111yescharacteristic
505C2#1999811yescharacteristic
506C2#1999911yescharacteristic
507C2#2777111yescharacteristic
508C2#2778811yescharacteristic
509C2#2780811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28792C2semidirect
7. #10357C2semidirect
8. #10353C2semidirect
9. #9821C2semidirect
10. #9736C2semidirect
11. #9566C2semidirect
12. #9561C2semidirect
14. C8×#106C2semidirect

Maximal subgroups

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

subgroupindex
#287922
#286002
#277482
#277272
#104212
#104162
#103572
#103532
#98212
#97362
#95662
#95612
#86462
C8×#1062
C8×#1022