Finite Group Databasebeta

← #780288

Subgroups — #780288

Order 512.

3834 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#2447C212yesmaximal
3C2×#2349C211yesmaximal, characteristic
4#16262C212yesmaximal, complemented
5#16024C214yesmaximal, complemented
6#15911C214yesmaximal, complemented
7#15202C211yesmaximal, characteristic, complemented
8#15188C211yesmaximal, characteristic, complemented
9#15180C211yesmaximal, characteristic, complemented
10#14934C212yesmaximal, complemented
11#14892C212yesmaximal, complemented
12#14128C212yesmaximal, complemented
13#14027C212yesmaximal
14#14017C212yesmaximal, complemented
15#13978C212yesmaximal, complemented
16#13916C211yesmaximal, characteristic
17C4×#395C211yesmaximal, characteristic, complemented
18C2×#556—216no—
19C2×#556—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2214yes—
22C22×#124C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2212yes—
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C22×C4×D4C2211yescharacteristic
29C22×#117C2212yes—
30#2447—216no—
31C2×#397C2214yes—
32C2×#397C2214yes—
33C2×#395C2211yescharacteristic
34#2349—28no—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#390C2212yes—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#385C2212yes—
44C2×#385C2214yes—
45C2×#385C2212yes—
46C2×#382C2211yescharacteristic
47C2×#381C2212yes—
48C2×#381C2214yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C22×#95C2212yes—
53#1446C2211yescharacteristic
54#1446C2212yes—
55#1444C2211yescharacteristic
56#1444C2212yes—
57#1444C2214yes—
58#1444C2214yes—
59#1444C2212yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2211yescharacteristic
69#1441C2214yes—
70#1441C2212yes—
71#1441C2212yes—
72#1440C2214yes—
73#1440C2214yes—
74#1440C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C22⋊#95C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2312yes—
93#556—2128no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—24no—
96C2×#126—216no—
97C2×#126—216no—
98C2×#126—216no—
99C2×#124—28no—
100C2×#124—28no—
101C2×#124D4116yes—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—216no—
104C2×C4×Q8—28no—
105C2×C4×Q8—28no—
106C2×C4×Q8—28no—
107C2×C4×Q8—28no—
108C2×C4×Q8—24no—
109C2×C4×D4—24no—
110C2×C4×D4—28no—
111C2×C4×D4—24no—
112C2×C4×D4—28no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4D418yes—
116C2×#117—28no—
117C2×#117—216no—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
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⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C42C2314yes—
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144C22×C42C2311yescharacteristic
145C22×C42C2312yes—
146C22×C42C2311yescharacteristic
147#397—216no—
148#397—216no—
149#395—24no—
150#391—28no—
151#391—28no—
152#391—28no—
153#391—28no—
154#390—28no—
155#389—24no—
156#389—24no—
157#389—28no—
158#385—28no—
159#385—28no—
160#385—216no—
161#382—24no—
162#381—216no—
163#381—28no—
164#380—28no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—28no—
167C2×#95—28no—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2311yescharacteristic
198C25C2412yes—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—42no—
210C22×Q8C2×D418yes—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4C2×D414yes—
215C22×D4C2×D418yes—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—28no—
224C23×C4—24no—
225C23×C4—44no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2×D418yes—
231C23×C4C2411yescharacteristic
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C4⋊Q8—232no—
248#126—2128no—
249#124—264no—
250C22⋊Q8—264no—
251C4×Q8—264no—
252C4×Q8—232no—
253C4×D4—232no—
254C4×D4—232no—
255C4×D4—416no—
256#117—264no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4#4114yes—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D4116yes—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D4116yes—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42C2×D414yes—
331C2×C42C2×D418yes—
332C2×C42#4114yes—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—22no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—44no—
358#95—42no—
359#95—44no—
360#95—44no—
361#95—42no—
362#95#4114yes—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24C22×D418yes—
373C24C22×D414yes—
374C24C2511yescharacteristic
375C2×Q8—28no—
376C2×Q8—216no—
377C2×Q8—216no—
378C2×Q8—216no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—216no—
390C2×D4—216no—
391C2×D4#120116yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—216no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—216no—
446C22×C4—216no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—48no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—48no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4—88no—
482C22×C4—84no—
483C22×C4C22×D418yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D414yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C42−1+412yes—
499C22×C42−1+412yes—
500C22×C42−1+414yes—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—232no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—88no—
517C22⋊C4—216no—
518C22⋊C4—232no—
519C22⋊C4—216no—
520C22⋊C4—232no—
521C22⋊C4—416no—
522C22⋊C4—48no—
523C22⋊C4—88no—
524C42—232no—
525C42—28no—
526C42—216no—
527C42—28no—
528C42—44no—
529C42—48no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—24no—
537C23—28no—
538C23—24no—
539C23—28no—
540C23—28no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23C2×#12018yes—
553C23C23×D411yescharacteristic
554C23C23×D412yes—
555C23C22×#4111yescharacteristic
556C23C22×#4111yescharacteristic
557C23C22×#4111yescharacteristic
558C23C2×2−1+412yes—
559C23C2×2−1+411yescharacteristic
560C23C2×2−1+411yescharacteristic
561C23#58512yes—
562C23#58511yescharacteristic
563C23#58512yes—
564Q8—416no—
565Q8—432no—
566D4—232no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—432no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—816no—
657C2×C4—816no—
658C2×C4C2×#12014yes—
659C2×C4#52514yes—
660C2×C4#53614yes—
661C2×C4#54714yes—
662C2×C4#54718yes—
663C2×C4#54718yes—
664C22—28no—
665C22—28no—
666C22—28no—
667C22—44no—
668C22—44no—
669C22—48no—
670C22—48no—
671C22—44no—
672C22—48no—
673C22—48no—
674C22—44no—
675C22C22×#12011yescharacteristic
676C22C2×#52511yescharacteristic
677C22C2×#53611yescharacteristic
678C22C2×#54711yescharacteristic
679C22C2×#54712yes—
680C22C2×#54712yes—
681C22D4×#4112yes—
682C22#348011yescharacteristic
683C22#348211yescharacteristic
684C22#348511yescharacteristic
685C22#348911yescharacteristic
686C22#348912yes—
687C22#348912yes—
688C22#350011yescharacteristic
689C22#350012yes—
690C22#350712yes—
691C22#350911yescharacteristic
692C22#351012yes—
693C22#351811yescharacteristic
694C22#352112yes—
695C22#355512yes—
696C22#355512yes—
697C22#355712yes—
698C4—24no—
699C4—42no—
700C4—44no—
701C4—44no—
702C4—48no—
703C4—44no—
704C4—44no—
705C4—42no—
706C4—44no—
707C4—42no—
708C4—42no—
709C4—81no—
710C4—82no—
711C4—84no—
712C4—82no—
713C4#240618yes—
714C2—48no—
715C2C2×#240611yescharacteristic
716C2#3603311yescharacteristic
717C2#3607811yescharacteristic
718C2#3621311yescharacteristic
719C2#3640511yescharacteristic
720C2#3949011yescharacteristic
721C2#4267611yescharacteristic
722C2#4522712yes—
723C2#4542912yes—
724C2#5906912yes—
725C2#5982912yes—
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
4. #16262C2semidirect
5. #16024C2semidirect
6. #15911C2semidirect
7. #15202C2semidirect
8. #15188C2semidirect
9. #15180C2semidirect
10. #14934C2semidirect
11. #14892C2semidirect
12. #14128C2semidirect
14. #14017C2semidirect
15. #13978C2semidirect
17. C4×#395C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#24472
C2×#23492
#162622
#160242
#159112
#152022
#151882
#151802
#149342
#148922
#141282
#140272
#140172
#139782
#139162
C4×#3952