Finite Group Databasebeta

← #4308982

Subgroups — #4308982

Order 512.

5317 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic
3C2×#2614C211yesmaximal, characteristic, complemented
4C2×#2623C211yesmaximal, characteristic, complemented
5#16632C214yesmaximal, complemented
6#16606C214yesmaximal, complemented
7#16601C214yesmaximal, complemented
8#16600C214yesmaximal, complemented
9#15073C212yesmaximal, complemented
10#15069C212yesmaximal, complemented
11#14918C212yesmaximal, complemented
12#14911C212yesmaximal
13#14681C211yesmaximal, characteristic, complemented
14#14342C211yesmaximal, characteristic, complemented
15#14336C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#123C2211yescharacteristic
20C22×#123C2212yes—
21C22×C22⋊Q8—28no—
22C22×C22⋊Q8—28no—
23C22×C22⋊Q8—28no—
24C22×C22⋊Q8C2214yes—
25C22×C22⋊Q8C2214yes—
26C22×#120C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2211yescharacteristic
29C23×C22⋊C4C2212yes—
30#2849—28no—
31#2623—28no—
32#2614—28no—
33C2×#400C2212yescomplemented
34C2×#399C2212yescomplemented
35C2×#398C2212yescomplemented
36C2×#397C2212yescomplemented
37C2×#395C2212yescomplemented
38C2×#395C2212yescomplemented
39C2×#393C2212yescomplemented
40C2×#393C2212yescomplemented
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#381C2211yescharacteristic
48C2×#381C2211yescharacteristic
49C2×#381C2211yescharacteristic
50C2×#381C2211yescharacteristic
51C2×#381C2212yes—
52C2×#381C2212yes—
53C2×#381C2212yes—
54C2×#381C2212yes—
55C2×#379C2211yescharacteristic
56C2×#379C2212yes—
57#1446C2214yescomplemented
58#1446C2214yescomplemented
59#1445C2214yescomplemented
60#1445C2214yescomplemented
61#1444C2218yescomplemented
62#1442C2218yescomplemented
63#1441C2218yescomplemented
64#1440C2218yes—
65#1439C2218yes—
66#1439C2218yes—
67#1439C2218yescomplemented
68#1438C2218yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C26C2311yescharacteristic
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8—44no—
77C23×Q8—44no—
78C23×Q8D418yescomplemented
79C23×Q8C2312yes—
80C23×Q8C2312yes—
81C23×D4C2311yescharacteristic
82C23×D4C2312yes—
83C24×C4—24no—
84C24×C4—24no—
85C24×C4—44no—
86C24×C4—44no—
87C24×C4D418yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2312yes—
93C24×C4C2312yes—
94C2×#123—24no—
95C2×#123—24no—
96C2×#123—28no—
97C2×#123—24no—
98C2×#123—24no—
99C2×#123—44no—
100C2×#123—44no—
101C2×#123—44no—
102C2×C22⋊Q8—2128no—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—216no—
105C2×C22⋊Q8—216no—
106C2×C22⋊Q8—216no—
107C2×C22⋊Q8—216no—
108C2×C22⋊Q8—216no—
109C2×#120—24no—
110C2×#120—44no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4—28no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4D418yescomplemented
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2318yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4—28no—
126C22×C22⋊C4—28no—
127C22×C22⋊C4—28no—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2318yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2311yescharacteristic
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2318yes—
142C22×C22⋊C4C2312yes—
143#400—28no—
144#399—28no—
145#398—28no—
146#397—28no—
147#395—28no—
148#395—28no—
149#393—28no—
150#393—28no—
151#386—28no—
152#386—28no—
153#386—28no—
154#385—28no—
155#385—28no—
156#385—28no—
157#381—24no—
158#381—24no—
159#381—28no—
160#381—24no—
161#381—24no—
162#381—44no—
163#381—44no—
164#381—44no—
165#379—24no—
166#379—44no—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2318yes—
180C2×#95C2318yes—
181C2×#95C2318yes—
182C2×#95C2318yes—
183C2×#95C2318yes—
184C25—22no—
185C25—22no—
186C25—28no—
187C25—24no—
188C25—24no—
189C25—44no—
190C25C2×D414yes—
191C25C2411yescharacteristic
192C25C2412yes—
193C25C2411yescharacteristic
194C22×Q8—232no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—432no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8—416no—
205C22×Q8—416no—
206C22×Q8—432no—
207C22×Q8C2×D418yes—
208C22×Q8C2×D418yes—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—28no—
216C22×D4—44no—
217C22×D4—44no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—232no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—416no—
238C23×C4—416no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—48no—
244C23×C4—432no—
245C23×C4—432no—
246C23×C4C2×D418yes—
247C23×C4C2×D414yes—
248C23×C4C2×D414yes—
249C23×C4C2×D414yes—
250C23×C4C2×D414yes—
251C23×C4C2×D418yes—
252C23×C4C2×D418yes—
253C23×C4C2×D418yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2414yes—
257C23×C4C2414yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C23×C4C2412yes—
261C23×C4C2414yes—
262C23×C4C2414yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266#123—416no—
267#123—432no—
268#123—416no—
269C22⋊Q8—2128no—
270C22⋊Q8—2128no—
271#120—416no—
272C2×C4⋊C4—264no—
273C2×C4⋊C4—232no—
274C2×C4⋊C4—216no—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—216no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4C2×D418yes—
295C2×C4⋊C4C2×D418yes—
296C2×C22⋊C4—264no—
297C2×C22⋊C4—216no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—216no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—216no—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—48no—
333C2×C22⋊C4—48no—
334C2×C22⋊C4—48no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—48no—
348#95—48no—
349#95—48no—
350#95—48no—
351#95—48no—
352C24—24no—
353C24—24no—
354C24—216no—
355C24—216no—
356C24—28no—
357C24—216no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—28no—
366C24—28no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—24no—
374C24—48no—
375C24—432no—
376C24—432no—
377C24—48no—
378C24—48no—
379C24—48no—
380C24—48no—
381C24C22×D414yes—
382C24C22×D412yes—
383C24C22×D412yes—
384C24C22×D412yes—
385C24H(5,𝔽2)14yes—
386C24H(5,𝔽2)14yes—
387C242−1+418yes—
388C24C2511yescharacteristic
389C2×Q8—216no—
390C2×Q8—216no—
391C2×Q8—416no—
392C2×Q8—416no—
393C2×Q8—4128no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—416no—
399C2×Q8—416no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—816no—
409C22×C4—216no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—216no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—216no—
422C22×C4—28no—
423C22×C4—216no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—24no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—4128no—
477C22×C4—416no—
478C22×C4—416no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4#120116yes—
490C22×C4C22×D418yes—
491C22×C4C22×D418yes—
492C22×C4C22×D414yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C4⋊C4—264no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—48no—
503C4⋊C4—416no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—88no—
507C22⋊C4—264no—
508C22⋊C4—48no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—88no—
521C22⋊C4—816no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—216no—
529C23—216no—
530C23—232no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—28no—
538C23—28no—
539C23—216no—
540C23—216no—
541C23—432no—
542C23—432no—
543C23—464no—
544C23—416no—
545C23—464no—
546C23—416no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—48no—
552C23—48no—
553C23—41no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—48no—
565C23—48no—
566C23C2×#12014yes—
567C23#53418yes—
568C23#53414yes—
569C23#53414yes—
570C23#53518yes—
571C23#53618yes—
572C23C23×D411yescharacteristic
573C23C23×D411yescharacteristic
574C23C23×D411yescharacteristic
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×2−1+414yes—
578C23#58514yes—
579Q8—432no—
580Q8—432no—
581D4—816no—
582D4—88no—
583C2×C4—232no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—432no—
602C2×C4—432no—
603C2×C4—432no—
604C2×C4—416no—
605C2×C4—432no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4—48no—
644C2×C4—48no—
645C2×C4C2×#12014yes—
646C2×C4C2×#12014yes—
647C22—28no—
648C22—216no—
649C22—28no—
650C22—24no—
651C22—24no—
652C22—216no—
653C22—28no—
654C22—28no—
655C22—28no—
656C22—464no—
657C22—44no—
658C22—416no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—416no—
663C22—82no—
664C22—84no—
665C22—82no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—84no—
670C22#261318yes—
671C22C22×#12011yescharacteristic
672C22C2×#53411yescharacteristic
673C22C2×#53412yes—
674C22C2×#53411yescharacteristic
675C22C2×#53512yes—
676C22C2×#53612yes—
677C22#344912yes—
678C22#348512yes—
679C22#348612yes—
680C22#348612yes—
681C22#348712yes—
682C22#353518yes—
683C22#353518yes—
684C4—24no—
685C4—24no—
686C4—48no—
687C4—44no—
688C4—48no—
689C4—82no—
690C4—84no—
691C4—82no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—84no—
696C2—28no—
697C2—48no—
698C2—161no—
699C2C2×#261311yescharacteristic
700C2#3514911yescharacteristic
701C2#3524211yescharacteristic
702C2#3711814yes—
703C2#3718614yes—
704C2#3744112yes—
705C2#3744712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊Q8C2semidirect
3. C2×#2614C2semidirect
4. C2×#2623C2semidirect
5. #16632C2semidirect
6. #16606C2semidirect
7. #16601C2semidirect
8. #16600C2semidirect
9. #15073C2semidirect
10. #15069C2semidirect
11. #14918C2semidirect
13. #14681C2semidirect
14. #14342C2semidirect
15. #14336C2semidirect
33. C2×#400C22semidirect
34. C2×#399C22semidirect
35. C2×#398C22semidirect
36. C2×#397C22semidirect
37. C2×#395C22semidirect
38. C2×#395C22semidirect
39. C2×#393C22semidirect
40. C2×#393C22semidirect
57. #1446C22semidirect
58. #1446C22semidirect
59. #1445C22semidirect
60. #1445C22semidirect
61. #1444C22semidirect
62. #1442C22semidirect
63. #1441C22semidirect
67. #1439C22semidirect
78. C23×Q8D4semidirect
116. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#28492
C2×#26142
C2×#26232
#166322
#166062
#166012
#166002
#150732
#150692
#149182
#149112
#146812
#143422
#143362