Finite Group Databasebeta

← #614202

Subgroups — #614202

Order 512.

4690 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2855C212yesmaximal, complemented
3C2×#2848C212yesmaximal
4C22×C4×C22⋊C4C211yesmaximal, characteristic
5#15178C212yesmaximal, complemented
6#15135C212yesmaximal, complemented
7#14969C214yesmaximal, complemented
8#14936C212yesmaximal, complemented
9#14734C212yesmaximal
10#14230C214yesmaximal, complemented
11#13997C214yesmaximal, complemented
12#13965C212yesmaximal, complemented
13#13903C212yesmaximal
14C4×#380C211yesmaximal, characteristic, complemented
15C25×C4C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2214yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2857—28no—
24#2855—216no—
25#2848—216no—
26C2×#399C2214yes—
27C2×#391C2214yescomplemented
28C2×#391C2212yescomplemented
29C2×#389C2214yescomplemented
30C2×#389C2212yescomplemented
31C2×#389C2212yescomplemented
32C2×#387C2218yescomplemented
33C2×#385C2214yes—
34C2×#385C2218yes—
35C2×#385C2214yes—
36C2×#384C2211yescharacteristic, complemented
37C2×#381C2214yes—
38C2×#380C2214yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#379C2214yes—
43C2×C4×C4⋊C4C2212yescomplemented
44C2×C4×C22⋊C4—28no—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1445C2214yescomplemented
53#1445C2218yescomplemented
54#1444C2214yescomplemented
55#1444C2214yescomplemented
56#1444C2214yescomplemented
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2218yes—
61#1437C2218yes—
62C4⋊#95C2212yescomplemented
63C4⋊#95C2218yescomplemented
64#1434C2212yescomplemented
65C22⋊#95C2212yes—
66C4×#95C2212yescomplemented
67C26C2311yescharacteristic
68C23×Q8C2312yes—
69C24×C4—24no—
70C24×C4—28no—
71C24×C4—28no—
72C24×C4—44no—
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—44no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—48no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2318yes—
93C22×C4⋊C4C2318yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4Q818yescomplemented
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42—28no—
112C22×C42—24no—
113C22×C42—44no—
114C22×C42D418yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#399—216no—
123#391—216no—
124#391—28no—
125#389—216no—
126#389—28no—
127#389—28no—
128#387—232no—
129#385—216no—
130#385—216no—
131#385—232no—
132#384—24no—
133#381—216no—
134#380—28no—
135#380—28no—
136#380—216no—
137#380D4116yes—
138#379—216no—
139C4×C4⋊C4—44no—
140C4×C22⋊C4—28no—
141C4×C22⋊C4—28no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—28no—
145C2×#95—24no—
146C2×#95—24no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25—22no—
162C25—24no—
163C25—22no—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—42no—
168C25—42no—
169C25C2×Q814yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—28no—
176C22×Q8—28no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—232no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—464no—
207C23×C4—416no—
208C23×C4—416no—
209C23×C4C2×D414yes—
210C23×C4C2×D418yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×Q814yes—
214C23×C4C2×Q814yes—
215C23×C4#4118yes—
216C23×C4#4114yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2418yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C4⋊Q8—416no—
230C22⋊Q8—464no—
231C22⋊Q8—432no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D4116yes—
255C2×C22⋊C4—232no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—232no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4C2×D4116yes—
278C2×C22⋊C4C2×Q818yes—
279C2×C22⋊C4C2×Q818yes—
280C2×C22⋊C4#4118yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—416no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—416no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—216no—
303#95—28no—
304#95—28no—
305#95—216no—
306#95—216no—
307#95—216no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95C2×D4116yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—28no—
317C24—28no—
318C24—216no—
319C24—28no—
320C24—216no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—216no—
330C24—48no—
331C24—416no—
332C24—416no—
333C24—416no—
334C24—416no—
335C24—44no—
336C24—44no—
337C24—48no—
338C24—44no—
339C24—44no—
340C24—48no—
341C24C22×D412yes—
342C24C22×D414yes—
343C24C22×Q814yes—
344C24C22×Q812yes—
345C24C2×#4114yes—
346C24C2×#4114yes—
347C24C2×#4112yes—
348C242−1+414yes—
349C24C2511yescharacteristic
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—416no—
354C2×Q8—416no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—232no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—216no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—232no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—48no—
417C22×C4—4128no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4C22⋊Q8116yescomplemented
430C22×C4#127116yes—
431C22×C4C22×D412yes—
432C22×C4C22×D4116yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q812yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)14yes—
450C4⋊C4—232no—
451C4⋊C4—432no—
452C4⋊C4—432no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C22⋊C4—216no—
460C22⋊C4—232no—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—432no—
465C22⋊C4—432no—
466C22⋊C4—416no—
467C42—28no—
468C42—28no—
469C42—44no—
470C42—44no—
471C42—48no—
472C42—48no—
473C42—88no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—232no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—464no—
500C23—416no—
501C23—416no—
502C23—464no—
503C23—416no—
504C23—416no—
505C23—416no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—416no—
517C23C2×C22⋊Q814yes—
518C23C2×#12714yes—
519C23#53318yes—
520C23#53614yes—
521C23#54718yes—
522C23D4×Q818yes—
523C23C23×D411yescharacteristic
524C23C23×D412yes—
525C23C23×Q811yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4112yes—
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+412yes—
531C23#58512yes—
532Q8—816no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—232no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—216no—
555C2×C4—28no—
556C2×C4—216no—
557C2×C4—28no—
558C2×C4—216no—
559C2×C4—28no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—432no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—416no—
580C2×C4—432no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4C2×C22⋊Q814yes—
592C2×C4C2×C22⋊Q814yes—
593C2×C4#52514yes—
594C2×C4#52514yes—
595C2×C4C2×#12714yes—
596C2×C4C2×#12714yes—
597C2×C4#53414yes—
598C2×C4#546116yes—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—216no—
603C22—216no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—28no—
608C22—216no—
609C22—464no—
610C22—44no—
611C22—44no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22#254218yes—
620C22C22×C22⋊Q811yescharacteristic
621C22C2×#52511yescharacteristic
622C22C2×#52511yescharacteristic
623C22C22×#12711yescharacteristic
624C22C2×#53312yes—
625C22#344412yes—
626C22C2×#53411yescharacteristic
627C22C2×#53611yescharacteristic
628C22#344911yescharacteristic
629C22C2×#54614yes—
630C22C2×#54712yes—
631C22C2×D4×Q812yes—
632C22D4×#4114yes—
633C22#349514yes—
634C22#350112yes—
635C22#350512yes—
636C22#352014yes—
637C4—24no—
638C4—24no—
639C4—24no—
640C4—24no—
641C4—44no—
642C4—42no—
643C4—416no—
644C4—44no—
645C4—44no—
646C4—84no—
647C4—84no—
648C4#253918yes—
649C2—28no—
650C2—48no—
651C2C2×#253911yescharacteristic
652C2C2×#254211yescharacteristic
653C2#3493611yescharacteristic
654C2#3501011yescharacteristic
655C2#3508411yescharacteristic
656C2#3532511yescharacteristic
657C2#3613014yes—
658C2#3614314yes—
659C2#3662511yescharacteristic
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
1. C2×#2857C2semidirect
2. C2×#2855C2semidirect
5. #15178C2semidirect
6. #15135C2semidirect
7. #14969C2semidirect
8. #14936C2semidirect
10. #14230C2semidirect
11. #13997C2semidirect
12. #13965C2semidirect
14. C4×#380C2semidirect
16. C22×C4⋊Q8C22semidirect
27. C2×#391C22semidirect
28. C2×#391C22semidirect
29. C2×#389C22semidirect
30. C2×#389C22semidirect
31. C2×#389C22semidirect
32. C2×#387C22semidirect
36. C2×#384C22semidirect
43. C2×C4×C4⋊C4C22semidirect
52. #1445C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
55. #1444C22semidirect
56. #1444C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. #1434C22semidirect
66. C4×#95C22semidirect
102. C22×C22⋊C4Q8semidirect
429. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#28552
C2×#28482
C22×C4×C22⋊C42
#151782
#151352
#149692
#149362
#147342
#142302
#139972
#139652
#139032
C4×#3802