Finite Group Databasebeta

← #639828

Subgroups — #639828

Order 512.

5258 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2625C212yesmaximal, complemented
3C2×#2280C211yesmaximal, characteristic
4#16366C212yesmaximal, complemented
5#16365C212yesmaximal, complemented
6#16364C214yesmaximal, complemented
7#15179C214yesmaximal, complemented
8#14822C212yesmaximal, complemented
9#14821C212yesmaximal, complemented
10#14727C214yesmaximal
11#14680C211yesmaximal, characteristic
12#14258C212yesmaximal, complemented
13#14130C212yesmaximal, complemented
14#13922C212yesmaximal, complemented
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×#117C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C22⋊C4C2212yes—
28#2625—216no—
29C2×#400C2214yescomplemented
30C2×#398C2214yescomplemented
31C2×#397C2218yes—
32C2×#391C2214yescomplemented
33C2×#391C2214yescomplemented
34C2×#389C2214yescomplemented
35C2×#385C2214yes—
36#2280—28no—
37C2×#381C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#379C2212yes—
46C2×#379C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1446C2214yescomplemented
50#1446C2214yescomplemented
51#1444C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2218yescomplemented
54#1441C2218yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1440C2218yes—
58#1440C2218yes—
59#1439C2218yes—
60C4⋊#95C2214yescomplemented
61C22⋊#95C2214yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C26C2311yescharacteristic
68C23×Q8—28no—
69C23×Q8—24no—
70C23×Q8—28no—
71C23×Q8—44no—
72C23×Q8C2312yes—
73C23×Q8C2312yes—
74C24×C4—28no—
75C24×C4—28no—
76C24×C4—44no—
77C24×C4D418yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×C22⋊Q8—2128no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8D4116yes—
97C2×#117—28no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—44no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—216no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2318yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—28no—
116C22×C22⋊C4—216no—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130#400—216no—
131#398—216no—
132#397—232no—
133#391—216no—
134#391—48no—
135#389—216no—
136#385—48no—
137#381—28no—
138#380—28no—
139#380—24no—
140#380—24no—
141#380—24no—
142#380—24no—
143#380—28no—
144#380—44no—
145#379—28no—
146#379—44no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C2×#95C2318yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C25—24no—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—22no—
168C25—22no—
169C25—42no—
170C25—42no—
171C25C2×D414yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×Q8—232no—
176C22×Q8—28no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—28no—
182C22×Q8—28no—
183C22×Q8—44no—
184C22×Q8—464no—
185C22×Q8—416no—
186C22×Q8—416no—
187C22×Q8C2×D418yes—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—232no—
197C23×C4—232no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—416no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—464no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D418yes—
223C23×C4C2×D414yes—
224C23×C4#4118yes—
225C23×C4C2412yes—
226C23×C4C2418yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C22⋊Q8—264no—
237C22⋊Q8—264no—
238C22⋊Q8—264no—
239C22⋊Q8—264no—
240#117—416no—
241#117—416no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—264no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D4116yes—
269C2×C22⋊C4—264no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D4116yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—28no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—216no—
308#95—216no—
309#95—216no—
310#95—28no—
311#95—48no—
312#95—44no—
313#95—48no—
314#95—48no—
315#95—48no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—232no—
321C24—28no—
322C24—28no—
323C24—28no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—28no—
339C24—28no—
340C24—44no—
341C24—44no—
342C24—416no—
343C24—48no—
344C24—48no—
345C24—416no—
346C24—48no—
347C24—48no—
348C24—416no—
349C24—48no—
350C24—48no—
351C24C22×D412yes—
352C24C22×D414yes—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C2×#4114yes—
356C24H(5,𝔽2)12yes—
357C242−1+414yes—
358C242−1+414yes—
359C242−1+412yes—
360C24C2511yescharacteristic
361C2×Q8—216no—
362C2×Q8—216no—
363C2×Q8—232no—
364C2×Q8—432no—
365C2×Q8—48no—
366C2×Q8—4128no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—416no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—232no—
380C22×C4—232no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—416no—
434C22×C4—4128no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—432no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4#120116yes—
446C22×C4#123116yes—
447C22×C4C22×D418yes—
448C22×C4C22×D412yes—
449C22×C4C22×D418yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)14yes—
458C4⋊C4—216no—
459C4⋊C4—232no—
460C4⋊C4—216no—
461C4⋊C4—232no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—416no—
467C4⋊C4—816no—
468C22⋊C4—232no—
469C22⋊C4—232no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C42—48no—
480C42—48no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—232no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—24no—
513C23—24no—
514C23—416no—
515C23—416no—
516C23—416no—
517C23—416no—
518C23—464no—
519C23—464no—
520C23—416no—
521C23—416no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23C2×#12014yes—
535C23C2×#12314yes—
536C23#53614yes—
537C23#53614yes—
538C23#53714yes—
539C23#54118yes—
540C23#54218yes—
541C23C23×D411yescharacteristic
542C23C23×D412yes—
543C23C22×#4112yes—
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×2−1+412yes—
547C23C2×2−1+412yes—
548C23C2×2−1+411yescharacteristic
549C23#58512yes—
550Q8—432no—
551Q8—432no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—232no—
559C2×C4—232no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—216no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—416no—
571C2×C4—432no—
572C2×C4—432no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4C2×#12014yes—
608C2×C4C2×#12014yes—
609C2×C4C2×#12314yes—
610C2×C4C2×#12314yes—
611C2×C4#53414yes—
612C2×C4#546116yes—
613C22—216no—
614C22—216no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—464no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—48no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22#262518yescomplemented
641C22C22×#12011yescharacteristic
642C22C22×#12311yescharacteristic
643C22C2×#53411yescharacteristic
644C22C2×#53611yescharacteristic
645C22C2×#53611yescharacteristic
646C22C2×#53711yescharacteristic
647C22#344911yescharacteristic
648C22C2×#54112yes—
649C22C2×#54212yes—
650C22C2×#54614yes—
651C22#348614yes—
652C22#348812yes—
653C22#348912yes—
654C22#349514yes—
655C22#353612yes—
656C22#353812yes—
657C22#354114yes—
658C4—24no—
659C4—24no—
660C4—42no—
661C4—48no—
662C4—48no—
663C4—88no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#240518yes—
669C2—28no—
670C2—44no—
671C2—44no—
672C2C2×#262511yescharacteristic
673C2C2×#240511yescharacteristic
674C2#3524211yescharacteristic
675C2#3525111yescharacteristic
676C2#3531111yescharacteristic
677C2#3743811yescharacteristic
678C2#3744211yescharacteristic
679C2#3766314yes—
680C2#3766514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2625C2semidirect
4. #16366C2semidirect
5. #16365C2semidirect
6. #16364C2semidirect
7. #15179C2semidirect
8. #14822C2semidirect
9. #14821C2semidirect
12. #14258C2semidirect
13. #14130C2semidirect
14. #13922C2semidirect
29. C2×#400C22semidirect
30. C2×#398C22semidirect
32. C2×#391C22semidirect
33. C2×#391C22semidirect
34. C2×#389C22semidirect
49. #1446C22semidirect
50. #1446C22semidirect
51. #1444C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
60. C4⋊#95C22semidirect
640. C22#2625semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#26252
C2×#22802
#163662
#163652
#163642
#151792
#148222
#148212
#147272
#146802
#142582
#141302
#139222