Finite Group Databasebeta

← #4414093

Subgroups — #4414093

Order 512.

5142 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2817C212yesmaximal, complemented
3C2×#2848C211yesmaximal, characteristic
4#16601C214yesmaximal, complemented
5#16366C212yesmaximal, complemented
6#16262C214yesmaximal, complemented
7#16254C212yesmaximal, complemented
8#14874C212yesmaximal, complemented
9#14843C212yesmaximal, complemented
10#14822C212yesmaximal, complemented
11#14727C214yesmaximal
12#14720C212yesmaximal, complemented
13#14681C211yesmaximal, characteristic
14#14660C212yesmaximal, complemented
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C4⋊C4C2211yescharacteristic
28C23×C22⋊C4C2212yes—
29#2848—28no—
30#2817—216no—
31C2×#400C2214yescomplemented
32C2×#400C2214yescomplemented
33C2×#397C2218yes—
34C2×#395C2214yescomplemented
35C2×#395C2214yescomplemented
36C2×#395C2214yescomplemented
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#379C2212yes—
47C2×#379C2212yes—
48#1446C2214yescomplemented
49#1446C2214yescomplemented
50#1445C2214yescomplemented
51#1444C2218yescomplemented
52#1442C2218yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1440C2218yes—
58#1440C2218yes—
59#1439C2218yes—
60C22⋊#95C2212yes—
61C22⋊#95C2214yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C26C2311yescharacteristic
68C23×Q8—28no—
69C23×Q8—24no—
70C23×Q8—44no—
71C23×Q8—44no—
72C23×Q8C2312yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C24×C4—28no—
76C24×C4—44no—
77C24×C4—44no—
78C24×C4D418yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—2128no—
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—28no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—216no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—216no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8—44no—
103C2×C22⋊Q8D4116yes—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—216no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2318yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—28no—
120C22×C22⋊C4—216no—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132#400—216no—
133#400—216no—
134#397—232no—
135#395—216no—
136#395—216no—
137#395—48no—
138#385—28no—
139#385—28no—
140#385—28no—
141#385—48no—
142#381—28no—
143#381—44no—
144#380—24no—
145#380—24no—
146#380—24no—
147#379—28no—
148#379—44no—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2318yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2318yes—
159C2×#95C2314yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C25—24no—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—24no—
168C25—42no—
169C25—42no—
170C25C2×D414yes—
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—232no—
175C22×Q8—28no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—28no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—464no—
188C22×Q8—416no—
189C22×Q8—416no—
190C22×Q8C2×D418yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—232no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—416no—
210C23×C4—416no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—464no—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2418yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C22⋊Q8—264no—
240C22⋊Q8—264no—
241C22⋊Q8—264no—
242C22⋊Q8—264no—
243C22⋊Q8—416no—
244C22⋊Q8—416no—
245C22⋊Q8—416no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—264no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4C2×D4116yes—
273C2×C4⋊C4C2×D418yes—
274C2×C22⋊C4—264no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C2×D4116yes—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—216no—
308#95—216no—
309#95—216no—
310#95—28no—
311#95—48no—
312#95—48no—
313#95—44no—
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—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—28no—
337C24—28no—
338C24—44no—
339C24—44no—
340C24—416no—
341C24—416no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24—416no—
346C24C22×D412yes—
347C24C22×D414yes—
348C24C22×D412yes—
349C24C22×D412yes—
350C24H(5,𝔽2)14yes—
351C242−1+414yes—
352C242−1+414yes—
353C242−1+414yes—
354C24C2511yescharacteristic
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8—416no—
358C2×Q8—432no—
359C2×Q8—4128no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—416no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—48no—
368C2×Q8—88no—
369C22×C4—232no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—432no—
438C22×C4—4128no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4#120116yes—
445C22×C4C22×D418yes—
446C22×C4C22×D418yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4H(5,𝔽2)14yes—
454C4⋊C4—216no—
455C4⋊C4—232no—
456C4⋊C4—216no—
457C4⋊C4—232no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—816no—
465C22⋊C4—232no—
466C22⋊C4—232no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—232no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—416no—
506C23—416no—
507C23—416no—
508C23—416no—
509C23—416no—
510C23—4128no—
511C23—416no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23C2×#12014yes—
521C23#53414yes—
522C23#53614yes—
523C23#53614yes—
524C23#53614yes—
525C23#563116yes—
526C23C23×D411yescharacteristic
527C23C23×D412yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+412yes—
531C23C2×2−1+412yes—
532C23C2×2−1+412yes—
533C23#58512yes—
534Q8—432no—
535Q8—432no—
536Q8—88no—
537C2×C4—232no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—216no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—416no—
556C2×C4—416no—
557C2×C4—416no—
558C2×C4—416no—
559C2×C4—416no—
560C2×C4—432no—
561C2×C4—432no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4C2×#12014yes—
594C2×C4C2×#12014yes—
595C2×C4#53414yes—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—232no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—416no—
617C22—416no—
618C22—42no—
619C22—42no—
620C22#281718yescomplemented
621C22C22×#12011yescharacteristic
622C22C2×#53411yescharacteristic
623C22C2×#53411yescharacteristic
624C22C2×#53611yescharacteristic
625C22C2×#53611yescharacteristic
626C22C2×#53611yescharacteristic
627C22#344911yescharacteristic
628C22#348414yes—
629C22#348614yes—
630C22#348914yes—
631C22C2×#56314yes—
632C22#353514yes—
633C22#353814yes—
634C22#354114yes—
635C4—24no—
636C4—24no—
637C4—24no—
638C4—42no—
639C4—48no—
640C4—48no—
641C4—82no—
642C4—82no—
643C4—88no—
644C4—82no—
645C4—82no—
646C2—28no—
647C2—44no—
648C2—44no—
649C2#3498011yescharacteristic
650C2#3524211yescharacteristic
651C2#3530511yescharacteristic
652C2C2×#281711yescharacteristic
653C2#3743811yescharacteristic
654C2#3744711yescharacteristic
655C2#3745311yescharacteristic
656C2#4544618yes—
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×#2817C2semidirect
4. #16601C2semidirect
5. #16366C2semidirect
6. #16262C2semidirect
7. #16254C2semidirect
8. #14874C2semidirect
9. #14843C2semidirect
10. #14822C2semidirect
12. #14720C2semidirect
14. #14660C2semidirect
31. C2×#400C22semidirect
32. C2×#400C22semidirect
34. C2×#395C22semidirect
35. C2×#395C22semidirect
36. C2×#395C22semidirect
48. #1446C22semidirect
49. #1446C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
620. C22#2817semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#28172
C2×#28482
#166012
#163662
#162622
#162542
#148742
#148432
#148222
#147272
#147202
#146812
#146602