Finite Group Databasebeta

← #726484

Subgroups — #726484

Order 512.

3808 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3C2×#2547C211yesmaximal, characteristic
4C2×#2660C211yesmaximal, characteristic, complemented
5#16673C214yesmaximal, complemented
6#15849C214yesmaximal, complemented
7#15262C214yesmaximal
8#15236C212yesmaximal, complemented
9#15182C212yesmaximal, complemented
10#14907C214yesmaximal, complemented
11#14160C211yesmaximal, characteristic, complemented
12#13880C212yesmaximal, complemented
13#13879C212yesmaximal, complemented
14#13847C211yesmaximal, characteristic
15#13786C211yesmaximal, characteristic, complemented
16C2×#552—28no—
17C2×#552—28no—
18C2×#552—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#126C2212yes—
21C22×#125C2212yes—
22C22×#125C2211yescharacteristic
23C22×#124C2212yes—
24C22×#123C2212yes—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C22×#117C2212yes—
31C23×C4⋊C4C2211yescharacteristic
32#2660—28no—
33#2567—28no—
34#2547—28no—
35C2×#398C2214yes—
36C2×#397C2212yes—
37C2×#397C2212yes—
38C2×#395C2214yes—
39C2×#395C2214yes—
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#389C2214yes—
44C2×#385C2218yes—
45C2×#384C2212yes—
46C2×#384C2212yes—
47C2×#382C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×#376C2211yescharacteristic
55#1445C2212yes—
56#1444C2218yes—
57#1442C2218yes—
58#1441C2214yes—
59#1441C2218yes—
60#1440C2218yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2218yes—
64#1439C2218yes—
65#1438C2212yes—
66#1436C2218yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79#552—2128no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×#126—28no—
84C2×#126—28no—
85C2×#126—28no—
86C2×#125—28no—
87C2×#125—28no—
88C2×#125—28no—
89C2×#125—24no—
90C2×#125—24no—
91C2×#125D418yes—
92C2×#124—28no—
93C2×#124—48no—
94C2×#123—28no—
95C2×#123—28no—
96C2×#123—28no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8D418yes—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4D418yes—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117—24no—
112C2×#117—28no—
113C2×#117—28no—
114C2×#117—28no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2318yes—
133C22×C4⋊C4C2318yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2311yescharacteristic
141C22×C22⋊C4C2311yescharacteristic
142C22×C22⋊C4C2314yes—
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2312yes—
146C22×C42C2312yes—
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149#398—216no—
150#397—28no—
151#397—28no—
152#395—216no—
153#395—216no—
154#391—28no—
155#391—28no—
156#391—28no—
157#389—216no—
158#385—416no—
159#384—28no—
160#384—28no—
161#382—28no—
162#380—28no—
163#380—28no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—24no—
166C4×C22⋊C4—28no—
167C4×C22⋊C4—28no—
168#376—24no—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2318yes—
178C2×#95C2318yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2318yes—
182C2×#95C2318yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8C2×D414yes—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—42no—
199C22×D4C2×D414yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C4⋊Q8—416no—
229#126—264no—
230#125—264no—
231#125—232no—
232#124—432no—
233#123—264no—
234C22⋊Q8—232no—
235C22⋊Q8—232no—
236C4×Q8—232no—
237C4×D4—232no—
238#117—232no—
239#117—264no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—216no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D418yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—216no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—48no—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42C2×D418yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42#4118yes—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—216no—
337#95—216no—
338#95—28no—
339#95—28no—
340#95—216no—
341#95—216no—
342#95—48no—
343#95—48no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24C22×D412yes—
358C24C22×D412yes—
359C24H(5,𝔽2)14yes—
360C24C2511yescharacteristic
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—88no—
370C2×Q8—88no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—44no—
374C2×D4—48no—
375C2×D4—48no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—216no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4118yes—
455C22×C4C2×#4114yes—
456C22×C42−1+418yes—
457C22×C42−1+414yes—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—816no—
473C4⋊C4—816no—
474C22⋊C4—232no—
475C22⋊C4—216no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—28no—
479C22⋊C4—28no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C42—216no—
484C42—216no—
485C42—216no—
486C42—48no—
487C42—48no—
488C42#120116yes—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—24no—
496C23—28no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23#53514yes—
521C23C23×D411yescharacteristic
522C23C23×D411yescharacteristic
523C23C23×D411yescharacteristic
524C23C22×#4112yes—
525C23C2×H(5,𝔽2)12yes—
526C23C2×2−1+412yes—
527C23C2×2−1+414yes—
528C23#58512yes—
529Q8—416no—
530Q8—816no—
531D4—416no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—432no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
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—416no—
584C2×C4—416no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4C2×#12014yes—
596C2×C4C2×#12014yes—
597C2×C4C2×#12014yes—
598C2×C4#53214yes—
599C2×C4#53614yes—
600C2×C4#54718yes—
601C2×C4#54818yes—
602C22—24no—
603C22—28no—
604C22—44no—
605C22—42no—
606C22—44no—
607C22—42no—
608C22—48no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—82no—
616C22—82no—
617C22—84no—
618C22C22×#12011yescharacteristic
619C22C2×#53211yescharacteristic
620C22C2×#53511yescharacteristic
621C22C2×#53611yescharacteristic
622C22#344911yescharacteristic
623C22C2×#54712yes—
624C22C2×#54812yes—
625C22#348612yes—
626C22#348812yes—
627C22#348812yes—
628C22#349214yes—
629C22#349514yes—
630C22#349912yes—
631C22#349912yes—
632C22#354414yes—
633C22#355514yes—
634C4—24no—
635C4—24no—
636C4—44no—
637C4—48no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—48no—
642C4—44no—
643C4—44no—
644C4—84no—
645C4—84no—
646C4#243218yes—
647C2—44no—
648C2—82no—
649C2C2×#243211yescharacteristic
650C2#3525011yescharacteristic
651C2#3566611yescharacteristic
652C2#3592612yes—
653C2#3786312yes—
654C2#3786812yes—
655C2#4281312yes—
656C2#4497114yes—
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×#2567C2semidirect
4. C2×#2660C2semidirect
5. #16673C2semidirect
6. #15849C2semidirect
8. #15236C2semidirect
9. #15182C2semidirect
10. #14907C2semidirect
11. #14160C2semidirect
12. #13880C2semidirect
13. #13879C2semidirect
15. #13786C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#25672
C2×#25472
C2×#26602
#166732
#158492
#152622
#152362
#151822
#149072
#141602
#138802
#138792
#138472
#137862