Finite Group Databasebeta

← #4423047

Subgroups — #4423047

Order 512.

3148 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic, complemented
2C2×#2409C211yesmaximal, characteristic
3C2×#2638C211yesmaximal, characteristic
4C2×#2357C211yesmaximal, characteristic
5#16805C212yesmaximal, complemented
6#16565C212yesmaximal, complemented
7#16506C212yesmaximal, complemented
8#16262C212yesmaximal, complemented
9#15398C214yesmaximal, complemented
10#15300C214yesmaximal
11#15128C212yesmaximal, complemented
12#15097C212yesmaximal, complemented
13#14872C212yesmaximal, complemented
14#14763C212yesmaximal, complemented
15#14660C211yesmaximal, characteristic, complemented
16#14355C211yesmaximal, characteristic, complemented
17#14336C211yesmaximal, characteristic, complemented
18C24×Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2638—28no—
24#2434—28no—
25#2409—28no—
26C2×#398C2211yescharacteristic
27C2×#397C2212yes—
28C2×#397C2211yescharacteristic
29C2×#397C2214yes—
30C2×#395C2211yescharacteristic
31C2×#395C2212yes—
32C2×#395C2212yes—
33C2×#395C2212yes—
34C2×#394C2212yes—
35C2×#393C2211yescharacteristic
36C2×#393C2212yes—
37#2357—28no—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#390C2212yes—
41C2×#389C2214yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#387C2212yes—
45C2×#386C2212yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#385C2214yes—
49C2×#382C2211yescharacteristic
50C2×#381C2211yescharacteristic
51C2×#381C2211yescharacteristic
52C2×#381C2211yescharacteristic
53C2×#381C2214yes—
54C2×#380C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2211yescharacteristic
59#1446C2212yes—
60#1446C2212yes—
61#1446C2212yes—
62#1446C2212yes—
63#1446C2212yes—
64#1445C2214yes—
65#1445C2212yes—
66#1444C2212yes—
67#1444C2214yes—
68#1443C2214yes—
69#1442C2212yes—
70#1442C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2212yes—
74#1441C2214yes—
75#1441C2212yes—
76#1441C2214yes—
77#1440C2214yes—
78#1440C2214yes—
79#1439C2214yes—
80#1439C2214yes—
81#1439C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84C4⋊#95C2212yes—
85C4⋊#95C2214yes—
86#1434C2212yes—
87C23×Q8—24no—
88C23×Q8—24no—
89C23×Q8—24no—
90C23×Q8—48no—
91C23×Q8C2312yes—
92C23×Q8C2312yes—
93C23×Q8C2311yescharacteristic
94C23×D4C2311yescharacteristic
95C24×C4C2312yes—
96C24×C4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—48no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—48no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4D418yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2312yes—
136#398—24no—
137#397—24no—
138#397—28no—
139#397—48no—
140#395—24no—
141#395—28no—
142#395—28no—
143#395—28no—
144#394—28no—
145#393—24no—
146#393—28no—
147#391—28no—
148#391—28no—
149#390—28no—
150#389—48no—
151#388—28no—
152#388—28no—
153#387—28no—
154#386—28no—
155#385—28no—
156#385—28no—
157#385—216no—
158#382—24no—
159#381—24no—
160#381—24no—
161#381—24no—
162#381—48no—
163#380—24no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—24no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4D418yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C25C2411yescharacteristic
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—28no—
210C22×Q8—28no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8—432no—
215C22×Q8—432no—
216C22×Q8—432no—
217C22×Q8—416no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—41no—
223C22×D4—41no—
224C22×D4C2×D414yes—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2414yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C4⋊Q8—432no—
254C22⋊Q8—432no—
255C22⋊Q8—416no—
256C4×D4—232no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D414yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
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—24no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×D414yes—
318C2×C22⋊C4C2×D414yes—
319C2×C22⋊C4C2×D418yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—44no—
326C2×C42—42no—
327C2×C42—44no—
328C2×C42C2×D414yes—
329C2×C42C2×D418yes—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95C2×D414yes—
359#95C2×D414yes—
360C24—24no—
361C24—24no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—24no—
369C24—24no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24C22×D412yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C2×#4112yes—
378C24C2511yescharacteristic
379C2×Q8—216no—
380C2×Q8—216no—
381C2×Q8—416no—
382C2×Q8—416no—
383C2×Q8—416no—
384C2×Q8—416no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—4128no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—84no—
393C2×Q8—84no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—216no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4#121116yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)18yes—
490C22×C42−1+418yes—
491C22×C42−1+412yes—
492C22×C42−1+412yes—
493C4⋊C4—28no—
494C4⋊C4—48no—
495C4⋊C4—416no—
496C4⋊C4—416no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—84no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—216no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C42—28no—
518C42—48no—
519C42—48no—
520C42—84no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23C2×#12114yes—
555C23C23×D411yescharacteristic
556C23C23×D411yescharacteristic
557C23C22×#4111yescharacteristic
558C23C2×H(5,𝔽2)14yes—
559C23C2×2−1+411yescharacteristic
560C23C2×2−1+411yescharacteristic
561C23C2×2−1+414yes—
562C23#58511yescharacteristic
563C23#58511yescharacteristic
564Q8—432no—
565Q8—432no—
566Q8—88no—
567D4—416no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—416no—
595C2×C4—432no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—88no—
646C2×C4C2×#12114yes—
647C2×C4C2×#12114yes—
648C2×C4#53618yes—
649C2×C4#53614yes—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22C22×#12111yescharacteristic
669C22C2×#53612yes—
670C22C2×#53611yescharacteristic
671C22#344911yescharacteristic
672C22#348412yes—
673C22#348412yes—
674C22#348911yescharacteristic
675C22#348911yescharacteristic
676C22#349012yes—
677C22#349511yescharacteristic
678C22#349511yescharacteristic
679C22#349912yes—
680C22#350912yes—
681C22#354114yes—
682C22#355014yes—
683C22#355214yes—
684C22#355414yes—
685C4—24no—
686C4—24no—
687C4—28no—
688C4—24no—
689C4—48no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—82no—
694C4—84no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C2—44no—
701C2—82no—
702C2#3530611yescharacteristic
703C2#3531611yescharacteristic
704C2#3534811yescharacteristic
705C2#4050312yes—
706C2#4059112yes—
707C2#4242712yes—
708C2#4248112yes—
709C2#5133912yes—
710C2#5134812yes—
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
1. C2×#2434C2semidirect
5. #16805C2semidirect
6. #16565C2semidirect
7. #16506C2semidirect
8. #16262C2semidirect
9. #15398C2semidirect
11. #15128C2semidirect
12. #15097C2semidirect
13. #14872C2semidirect
14. #14763C2semidirect
15. #14660C2semidirect
16. #14355C2semidirect
17. #14336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#24092
C2×#26382
C2×#23572
#168052
#165652
#165062
#162622
#153982
#153002
#151282
#150972
#148722
#147632
#146602
#143552
#143362