Finite Group Databasebeta

← #726434

Subgroups — #726434

Order 512.

3886 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3C2×#2536C211yesmaximal, characteristic
4C2×#2327C211yesmaximal, characteristic, complemented
5#16683C214yesmaximal, complemented
6#15985C218yesmaximal, complemented
7#15236C212yesmaximal, complemented
8#15178C212yesmaximal, complemented
9#14860C214yesmaximal
10#14282C211yesmaximal, characteristic, complemented
11#14139C212yesmaximal, complemented
12#13942C211yesmaximal, characteristic, complemented
13#13881C212yesmaximal, complemented
14#13847C211yesmaximal, characteristic
15C2×#554—28no—
16C2×#554—28no—
17C2×#554—28no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2214yes—
23C22×C22⋊Q8C2212yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4C2212yes—
29#2568—28no—
30#2536—28no—
31C2×#400C2214yes—
32C2×#398C2212yes—
33C2×#394C2212yes—
34#2327—28no—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2214yes—
39C2×#391C2212yes—
40C2×#388C2214yes—
41C2×#387C2214yes—
42C2×#387C2214yes—
43C2×#382C2211yescharacteristic
44C2×#382C2212yes—
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×#376C2211yescharacteristic
51#1446C2214yes—
52#1445C2214yes—
53#1445C2212yes—
54#1444C2218yes—
55#1442C2218yes—
56#1442C2218yes—
57#1441C2218yes—
58#1441C2214yes—
59#1440C2218yes—
60#1439C2218yes—
61#1439C2214yes—
62#1438C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67C22⋊#95C2218yes—
68C22⋊#95C2218yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2314yes—
75#554—2128no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8D418yes—
82C2×#125—28no—
83C2×#125—28no—
84C2×#125—28no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—44no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8D418yes—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4D418yes—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#400—216no—
135#398—28no—
136#394—28no—
137#391—216no—
138#391—24no—
139#391—28no—
140#391—24no—
141#391—44no—
142#388—216no—
143#387—216no—
144#387—216no—
145#382—28no—
146#382—24no—
147#382—24no—
148#380—28no—
149C4×C4⋊C4—24no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—44no—
152#376—24no—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C2×#95C2314yes—
170C25C2412yes—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—48no—
186C22×Q8C2×D414yes—
187C22×Q8C2×D418yes—
188C22×Q8#4114yes—
189C22×D4—28no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4C2×D414yes—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—48no—
203C23×C4—48no—
204C23×C4C2×D418yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4⋊Q8—232no—
219C4⋊Q8—232no—
220#125—264no—
221C22⋊Q8—264no—
222C22⋊Q8—2128no—
223C22⋊Q8—432no—
224C4×Q8—232no—
225C4×Q8—416no—
226C4×D4—264no—
227C4×D4—232no—
228C2×C4⋊C4—264no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D418yes—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4C2×D418yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42C2×D418yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42#4114yes—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—216no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—216no—
313#95—28no—
314#95—48no—
315#95—48no—
316#95—48no—
317#95—44no—
318#95—48no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24C22×D414yes—
329C242−1+418yes—
330C24C2511yescharacteristic
331C2×Q8—28no—
332C2×Q8—216no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—416no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—88no—
348C2×Q8—88no—
349C2×D4—216no—
350C2×D4—216no—
351C2×D4—216no—
352C2×D4—28no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—232no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—816no—
414C22×C4—816no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4C22×D414yes—
418C22×C4C22×D418yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+412yes—
436C4⋊C4—232no—
437C4⋊C4—232no—
438C4⋊C4—232no—
439C4⋊C4—28no—
440C4⋊C4—216no—
441C4⋊C4—28no—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C22⋊C4—232no—
450C22⋊C4—216no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C42—216no—
456C42—28no—
457C42—48no—
458C42—44no—
459C42—48no—
460C42#120116yes—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23#53618yes—
484C23C23×D411yescharacteristic
485C23C23×D411yescharacteristic
486C23C23×D411yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×2−1+414yes—
491C23C2×2−1+411yescharacteristic
492C23#58514yes—
493Q8—416no—
494Q8—416no—
495Q8—816no—
496D4—232no—
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—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—216no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—432no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—832no—
561C2×C4C2×#12014yes—
562C2×C4C2×#12014yes—
563C2×C4C2×#12014yes—
564C2×C4#52514yes—
565C2×C4#53714yes—
566C2×C4#54614yes—
567C2×C4#54714yes—
568C2×C4#54814yes—
569C2×C4#55014yes—
570C22—28no—
571C22—28no—
572C22—28no—
573C22—44no—
574C22—48no—
575C22—48no—
576C22—48no—
577C22—44no—
578C22—48no—
579C22—44no—
580C22—44no—
581C22C22×#12011yescharacteristic
582C22C2×#52511yescharacteristic
583C22C2×#53612yes—
584C22C2×#53711yescharacteristic
585C22C2×#54611yescharacteristic
586C22C2×#54711yescharacteristic
587C22C2×#54811yescharacteristic
588C22C2×#55011yescharacteristic
589C22#348512yes—
590C22#348612yes—
591C22#348612yes—
592C22#349514yes—
593C22#350012yes—
594C22#350012yes—
595C22#351614yes—
596C22#354114yes—
597C22#356314yes—
598C4—24no—
599C4—24no—
600C4—44no—
601C4—48no—
602C4—44no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—82no—
607C4—84no—
608C4—82no—
609C4—84no—
610C4#240718yes—
611C2—48no—
612C2C2×#240711yescharacteristic
613C2#3524211yescharacteristic
614C2#3561611yescharacteristic
615C2#3573812yes—
616C2#3786512yes—
617C2#3808312yes—
618C2#4279012yes—
619C2#4513812yes—
620C2#4514312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2568C2semidirect
4. C2×#2327C2semidirect
5. #16683C2semidirect
6. #15985C2semidirect
7. #15236C2semidirect
8. #15178C2semidirect
10. #14282C2semidirect
11. #14139C2semidirect
12. #13942C2semidirect
13. #13881C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#25682
C2×#25362
C2×#23272
#166832
#159852
#152362
#151782
#148602
#142822
#141392
#139422
#138812
#138472