Finite Group Databasebeta

← #4397596

Subgroups — #4397596

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2816C211yesmaximal, characteristic
2C2×#2504C211yesmaximal, characteristic
3C2×#2503C211yesmaximal, characteristic, complemented
4C2×#2333C211yesmaximal, characteristic
5#16805C212yesmaximal, complemented
6#16701C212yesmaximal, complemented
7#15738C214yesmaximal, complemented
8#15702C214yesmaximal, complemented
9#15538C214yesmaximal, complemented
10#15495C214yesmaximal, complemented
11#15050C212yesmaximal, complemented
12#15016C212yesmaximal, complemented
13#14342C211yesmaximal, characteristic, complemented
14#14292C211yesmaximal, characteristic, complemented
15#13953C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2816—28no—
20#2504—28no—
21#2503—28no—
22C2×#400C2214yes—
23C2×#399C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#393C2212yes—
30#2333—28no—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#388C2212yes—
36C2×#387C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#382C2211yescharacteristic
41C2×#382C2212yes—
42C2×#382C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×#381C2211yescharacteristic
45C2×#381C2214yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#1446C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1445C2212yes—
53#1445C2212yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2212yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1440C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2214yes—
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×#121—24no—
89C2×#121—24no—
90C2×#121D418yes—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#400—216no—
122#399—28no—
123#398—28no—
124#397—28no—
125#397—28no—
126#394—28no—
127#394—216no—
128#393—28no—
129#391—24no—
130#391—24no—
131#390—28no—
132#390—28no—
133#388—28no—
134#387—28no—
135#386—216no—
136#386—28no—
137#386—28no—
138#382—28no—
139#382—24no—
140#382—24no—
141#381—24no—
142#381—24no—
143#381—216no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4D418yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4C2×D418yes—
189C22×D4C2×D414yes—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4#4114yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C22⋊Q8—416no—
219#121—232no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C2×D414yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4C2×D418yes—
283C2×C22⋊C4C2×D418yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95C2×D418yes—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24C22×D414yes—
335C24C22×D412yes—
336C24C22×D412yes—
337C24C2×#4112yes—
338C24C2511yescharacteristic
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×D4—216no—
345C2×D4—28no—
346C2×D4—48no—
347C2×D4—416no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—88no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—88no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4#121116yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C4⋊C4—28no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—88no—
445C4⋊C4—84no—
446C4⋊C4—88no—
447C22⋊C4—216no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—82no—
455C22⋊C4—84no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C22⋊C4—82no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C42—44no—
462C42—44no—
463C42—48no—
464C42—84no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—24no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23C2×#12114yes—
504C23C23×D411yescharacteristic
505C23C23×D411yescharacteristic
506C23C22×#4111yescharacteristic
507C23#58512yes—
508C23#58512yes—
509C23#58512yes—
510C23#58512yes—
511C23#58514yes—
512Q8—88no—
513D4—88no—
514D4—816no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—416no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—816no—
584C2×C4—816no—
585C2×C4C2×#12114yes—
586C2×C4C2×#12114yes—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—48no—
595C22—44no—
596C22—44no—
597C22—48no—
598C22—48no—
599C22—44no—
600C22—84no—
601C22—84no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22C22×#12111yescharacteristic
607C22#344912yes—
608C22#348512yes—
609C22#348511yescharacteristic
610C22#348511yescharacteristic
611C22#348512yes—
612C22#349112yes—
613C22#350012yes—
614C22#350012yes—
615C22#351011yescharacteristic
616C22#351012yes—
617C22#351011yescharacteristic
618C22#356614yes—
619C22#356614yes—
620C22#356614yes—
621C22#356714yes—
622C4—24no—
623C4—24no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—84no—
632C4—84no—
633C4—82no—
634C4—82no—
635C2—44no—
636C2—84no—
637C2#3532711yescharacteristic
638C2#3544711yescharacteristic
639C2#3561711yescharacteristic
640C2#4138712yes—
641C2#4215112yes—
642C2#4685712yes—
643C2#4722612yes—
644C2#4882514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2503C2semidirect
5. #16805C2semidirect
6. #16701C2semidirect
7. #15738C2semidirect
8. #15702C2semidirect
9. #15538C2semidirect
10. #15495C2semidirect
11. #15050C2semidirect
12. #15016C2semidirect
13. #14342C2semidirect
14. #14292C2semidirect
15. #13953C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28162
C2×#25042
C2×#25032
C2×#23332
#168052
#167012
#157382
#157022
#155382
#154952
#150502
#150162
#143422
#142922
#139532