Finite Group Databasebeta

← #4308086

Subgroups — #4308086

Order 512.

5286 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic
2C2×#2848C211yesmaximal, characteristic
3C2×#2625C212yesmaximal, complemented
4#14906C214yesmaximal, complemented
5#14892C214yesmaximal, complemented
6#14843C212yesmaximal, complemented
7#14786C212yesmaximal, complemented
8#14727C214yesmaximal
9#14702C212yesmaximal, complemented
10#14697C212yesmaximal, complemented
11#14680C211yesmaximal, characteristic
12#14343C212yesmaximal, complemented
13#14264C212yesmaximal, complemented
14#14258C212yesmaximal, complemented
15C25×C4C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117—28no—
20C22×#117—216no—
21C22×#117C2212yes—
22C22×#117C2212yes—
23C22×#117C2212yes—
24C22×#117C2212yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28C23×C42C2212yes—
29#2848—28no—
30#2625—216no—
31C2×#400C2214yescomplemented
32C2×#398C2214yescomplemented
33C2×#397C2218yes—
34C2×#391C2214yescomplemented
35C2×#389C2214yescomplemented
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#379C2212yes—
47C2×#379C2212yes—
48#1445C2214yescomplemented
49#1445C2214yescomplemented
50#1444C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1440C2218yes—
55#1440C2218yes—
56#1439C2218yes—
57#1436C2218yescomplemented
58#1436C2218yescomplemented
59C4⋊#95C2214yescomplemented
60C4⋊#95C2214yescomplemented
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C26C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C24×C4—28no—
70C24×C4—28no—
71C24×C4—28no—
72C24×C4D418yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×#117—2128no—
88C2×#117—28no—
89C2×#117—28no—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—28no—
94C2×#117—28no—
95C2×#117—216no—
96C2×#117—216no—
97C2×#117D4116yes—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2318yes—
114C22×C22⋊C4—28no—
115C22×C22⋊C4—216no—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C42—28no—
128C22×C42—216no—
129C22×C42C2314yes—
130C22×C42C2314yes—
131#400—216no—
132#398—216no—
133#397—232no—
134#391—216no—
135#389—216no—
136#385—28no—
137#385—28no—
138#385—28no—
139#385—48no—
140#380—24no—
141#380—24no—
142#380—24no—
143#380—28no—
144#380—28no—
145#380—28no—
146#379—28no—
147#379—44no—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C25—24no—
162C25—24no—
163C25—24no—
164C25—22no—
165C25—24no—
166C25—22no—
167C25—42no—
168C25—42no—
169C25C2×D414yes—
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×Q8—22no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C23×C4—2128no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—232no—
182C23×C4—232no—
183C23×C4—232no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
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—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D418yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4#4118yes—
214C23×C4#4118yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2418yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C22⋊Q8—416no—
227C22⋊Q8—416no—
228C22⋊Q8—416no—
229#117—264no—
230#117—264no—
231#117—264no—
232#117—264no—
233C2×C4⋊C4—232no—
234C2×C4⋊C4—232no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D418yes—
263C2×C22⋊C4—264no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
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—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C2×D4116yes—
292C2×C42—264no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42C2×D4116yes—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—216no—
304#95—216no—
305#95—216no—
306#95—28no—
307#95—28no—
308#95—48no—
309#95—48no—
310#95—48no—
311#95—48no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—232no—
317C24—28no—
318C24—28no—
319C24—28no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—28no—
335C24—28no—
336C24—44no—
337C24—44no—
338C24—48no—
339C24—48no—
340C24—416no—
341C24—48no—
342C24—416no—
343C24—48no—
344C24—416no—
345C24—48no—
346C24—48no—
347C24C22×D412yes—
348C24C22×D414yes—
349C24C22×D412yes—
350C24C22×D412yes—
351C24C2×#4114yes—
352C24C2×#4114yes—
353C24H(5,𝔽2)12yes—
354C242−1+414yes—
355C242−1+412yes—
356C24C2511yescharacteristic
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—416no—
361C22×C4—232no—
362C22×C4—232no—
363C22×C4—232no—
364C22×C4—2256no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—232no—
387C22×C4—232no—
388C22×C4—232no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4#120116yes—
437C22×C4#123116yes—
438C22×C4#123116yes—
439C22×C4C22×D418yes—
440C22×C4C22×D418yes—
441C22×C4C22×D412yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—816no—
463C22⋊C4—232no—
464C22⋊C4—232no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—232no—
475C42—232no—
476C23—22no—
477C23—22no—
478C23—28no—
479C23—22no—
480C23—22no—
481C23—28no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—24no—
486C23—28no—
487C23—24no—
488C23—232no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—28no—
494C23—28no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—416no—
510C23—416no—
511C23—416no—
512C23—416no—
513C23—416no—
514C23—464no—
515C23—464no—
516C23—416no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23C2×#12014yes—
530C23C2×#12314yes—
531C23C2×#12314yes—
532C23#53614yes—
533C23#53714yes—
534C23#54118yes—
535C23#54218yes—
536C23C23×D411yescharacteristic
537C23C23×D412yes—
538C23C22×#4112yes—
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×2−1+412yes—
542C23C2×2−1+411yescharacteristic
543C23#58512yes—
544C23#58512yes—
545Q8—88no—
546C2×C4—232no—
547C2×C4—232no—
548C2×C4—232no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—232no—
559C2×C4—232no—
560C2×C4—232no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—432no—
568C2×C4—432no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4C2×#12014yes—
603C2×C4C2×#12014yes—
604C2×C4C2×#12314yes—
605C2×C4C2×#12314yes—
606C2×C4C2×#12314yes—
607C2×C4C2×#12314yes—
608C2×C4#525116yes—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—216no—
613C22—24no—
614C22—24no—
615C22—216no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—464no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—48no—
630C22—48no—
631C22—42no—
632C22—42no—
633C22—48no—
634C22—48no—
635C22—42no—
636C22#262518yescomplemented
637C22C22×#12011yescharacteristic
638C22C22×#12311yescharacteristic
639C22C22×#12311yescharacteristic
640C22C2×#52514yes—
641C22C2×#53611yescharacteristic
642C22C2×#53711yescharacteristic
643C22#344911yescharacteristic
644C22#344911yescharacteristic
645C22C2×#54112yes—
646C22C2×#54212yes—
647C22#345514yes—
648C22#348812yes—
649C22#348912yes—
650C22#348914yes—
651C22#349412yes—
652C22#349514yes—
653C22#349512yes—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—48no—
658C4—48no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—88no—
663C4—82no—
664C4#284918yes—
665C2—28no—
666C2—44no—
667C2—44no—
668C2C2×#262511yescharacteristic
669C2#3463511yescharacteristic
670C2#3525111yescharacteristic
671C2#3526014yes—
672C2#3526311yescharacteristic
673C2#3530511yescharacteristic
674C2#3531014yes—
675C2#3531111yescharacteristic
676C2C2×#284911yescharacteristic
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
3. C2×#2625C2semidirect
4. #14906C2semidirect
5. #14892C2semidirect
6. #14843C2semidirect
7. #14786C2semidirect
9. #14702C2semidirect
10. #14697C2semidirect
12. #14343C2semidirect
13. #14264C2semidirect
14. #14258C2semidirect
31. C2×#400C22semidirect
32. C2×#398C22semidirect
34. C2×#391C22semidirect
35. C2×#389C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
636. C22#2625semidirect

Maximal subgroups

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

subgroupindex
C23×#1172
C2×#28482
C2×#26252
#149062
#148922
#148432
#147862
#147272
#147022
#146972
#146802
#143432
#142642
#142582