Finite Group Databasebeta

← #725982

Subgroups — #725982

Order 512.

2682 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#2627C211yesmaximal, characteristic
3C22×#398C211yesmaximal, characteristic
4C2×#2368C211yesmaximal, characteristic
5#16464C214yesmaximal, complemented
6#15885C214yesmaximal, complemented
7#15500C214yesmaximal, complemented
8#15073C212yesmaximal
9#14935C212yesmaximal
10#14871C214yesmaximal, complemented
11#13992C212yesmaximal
12#13908C211yesmaximal, characteristic
13#13904C211yesmaximal, characteristic
14#13904C212yesmaximal
15#13776C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20#2627—28no—
21#2503—28no—
22C2×#400C2212yes—
23C2×#400C2214yes—
24C2×#398—28no—
25C2×#398C2214yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#398C2212yes—
29C2×#395C2212yes—
30#2368—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#382C2212yes—
38C2×#381C2212yes—
39C2×#381C2211yescharacteristic
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#381C2211yescharacteristic
44C2×#381C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C22×#95C2212yes—
52C22×#95C2211yescharacteristic
53#1445C2214yes—
54#1445C2214yes—
55#1445C2214yes—
56#1444C2212yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2214yes—
73C4⋊#95C2214yes—
74C4⋊#95C2214yes—
75#1434C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8D418yes—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120#400—28no—
121#400—216no—
122#398—216no—
123#398—28no—
124#398—28no—
125#398—216no—
126#395—28no—
127#391—28no—
128#391—28no—
129#387—28no—
130#387—28no—
131#385—28no—
132#385—28no—
133#382—28no—
134#381—24no—
135#381—28no—
136#381—28no—
137#381—28no—
138#381—24no—
139#381—28no—
140#381—44no—
141#380—24no—
142#380—44no—
143C4×C4⋊C4—28no—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147C2×#95—28no—
148C2×#95D418yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C25C2411yescharacteristic
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×Q8C2×D414yes—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—28no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D418yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4#4114yes—
210C23×C4#4114yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C22⋊Q8—232no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—432no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D418yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4C2×D418yes—
282C2×C22⋊C4#4114yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95C2×D414yes—
319#95C2×D414yes—
320#95C2×D418yes—
321C24—24no—
322C24—24no—
323C24C22×D412yes—
324C24C22×D412yes—
325C24C2×#4112yes—
326C24C2×#4112yes—
327C24C2×#4112yes—
328C24H(5,𝔽2)12yes—
329C242−1+412yes—
330C24C2511yescharacteristic
331C2×Q8—28no—
332C2×Q8—48no—
333C2×Q8—84no—
334C2×Q8—84no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—216no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—216no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4#121116yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D414yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C42−1+418yes—
436C4⋊C4—28no—
437C4⋊C4—216no—
438C4⋊C4—416no—
439C4⋊C4—416no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—84no—
446C4⋊C4—88no—
447C4⋊C4—84no—
448C4⋊C4—88no—
449C22⋊C4—216no—
450C22⋊C4—28no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—48no—
455C42—48no—
456C42—48no—
457C42—84no—
458C42—84no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23C2×#12114yes—
470C23#52914yes—
471C23#54214yes—
472C23#54614yes—
473C23#54814yes—
474C23#54814yes—
475C23#55014yes—
476C23C23×D411yescharacteristic
477C23C23×D411yescharacteristic
478C23C22×#4111yescharacteristic
479C23C22×#4111yescharacteristic
480C23C22×#4111yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×2−1+414yes—
483C23C2×2−1+411yescharacteristic
484C23#58514yes—
485Q8—88no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—416no—
509C2×C4—432no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—416no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—88no—
561C2×C4—816no—
562C2×C4C2×#12114yes—
563C2×C4C2×#12114yes—
564C2×C4#53618yes—
565C22—24no—
566C22—28no—
567C22—28no—
568C22—28no—
569C22—28no—
570C22—28no—
571C22—28no—
572C22—24no—
573C22#242618yes—
574C22C22×#12111yescharacteristic
575C22C2×#52911yescharacteristic
576C22C2×#53612yes—
577C22C2×#54211yescharacteristic
578C22C2×#54611yescharacteristic
579C22C2×#54811yescharacteristic
580C22C2×#54811yescharacteristic
581C22C2×#55011yescharacteristic
582C22#348512yes—
583C22#348512yes—
584C22#349112yes—
585C22#349514yes—
586C22#349912yes—
587C22#350912yes—
588C22#351514yes—
589C22#353514yes—
590C22#356314yes—
591C4—24no—
592C4—24no—
593C4—28no—
594C4—44no—
595C4—44no—
596C4—44no—
597C4—44no—
598C4—44no—
599C4—82no—
600C4—84no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—84no—
605C2—28no—
606C2C2×#242611yescharacteristic
607C2#3531611yescharacteristic
608C2#3544711yescharacteristic
609C2#3602312yes—
610C2#3783512yes—
611C2#3912212yes—
612C2#4139312yes—
613C2#4349712yes—
614C2#4444812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #16464C2semidirect
6. #15885C2semidirect
7. #15500C2semidirect
10. #14871C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#26272
C22×#3982
C2×#23682
#164642
#158852
#155002
#150732
#149352
#148712
#139922
#139082
#139042
#137762