Finite Group Databasebeta

← #618318

Subgroups — #618318

Order 512.

5150 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#126C211yesmaximal, characteristic, complemented
2C2×#2405C211yesmaximal, characteristic
3C2×#2405C211yesmaximal, characteristic
4C2×#2308C211yesmaximal, characteristic
5#16024C214yesmaximal, complemented
6#15986C214yesmaximal, complemented
7#15908C218yesmaximal, complemented
8#15091C212yesmaximal, complemented
9#15069C212yesmaximal, complemented
10#14706C212yesmaximal, complemented
11#14341C211yesmaximal, characteristic, complemented
12#14256C212yesmaximal, complemented
13#13869C211yesmaximal, characteristic, complemented
14#13869C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C22×#126—28no—
17C22×#126—28no—
18C22×#126—28no—
19C22×#126C2214yes—
20C22×#126C2214yescomplemented
21C22×#123C2212yes—
22C22×#123C2212yes—
23C23×C4⋊C4C2211yescharacteristic, complemented
24C23×C4⋊C4C2211yescharacteristic, complemented
25C23×C4⋊C4C2211yescharacteristic, complemented
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C22⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic, complemented
31#2405—28no—
32#2405—28no—
33C2×#394C2214yes—
34C2×#394C2214yes—
35#2308—28no—
36C2×#386C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yescomplemented
45C2×#381C2212yescomplemented
46C2×#379C2211yescharacteristic
47C2×#379C2211yescharacteristic
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×#379C2212yes—
51#1444C2218yescomplemented
52#1442C2218yescomplemented
53#1441C2218yescomplemented
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yescomplemented
58#1439C2218yescomplemented
59#1439C2218yescomplemented
60#1439C2218yescomplemented
61#1437C2218yescomplemented
62#1436C2218yescomplemented
63C22⋊#95C2212yescomplemented
64C22⋊#95C2212yescomplemented
65C22⋊#95C2214yescomplemented
66C22⋊#95C2214yescomplemented
67C22⋊#95C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×D4—24no—
72C23×D4—28no—
73C23×D4—24no—
74C23×D4—24no—
75C23×D4—44no—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C2×#126—2128no—
87C2×#126—216no—
88C2×#126—216no—
89C2×#126—216no—
90C2×#126—216no—
91C2×#126—216no—
92C2×#126—216no—
93C2×#123—28no—
94C2×#123—28no—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—44no—
98C2×#123—44no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—44no—
121C22×C22⋊C4D418yes—
122C22×C22⋊C4D418yes—
123C22×C22⋊C4D418yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C42—24no—
145C22×C42—24no—
146C22×C42—24no—
147C22×C42C2314yes—
148#394—216no—
149#394—216no—
150#386—28no—
151#386—28no—
152#386—28no—
153#386—28no—
154#386—216no—
155#385—28no—
156#385—28no—
157#381—28no—
158#381—44no—
159#381—44no—
160#379—24no—
161#379—24no—
162#379—28no—
163#379—28no—
164#379—28no—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C25—24no—
176C25—24no—
177C25—22no—
178C25—24no—
179C25—22no—
180C25—22no—
181C25—42no—
182C25—44no—
183C25—44no—
184C25—44no—
185C25—44no—
186C25C2×D414yes—
187C25C2×D414yes—
188C25C2×D414yes—
189C25C2412yes—
190C25C2411yescharacteristic
191C25C2411yescharacteristic
192C25C2411yescharacteristic
193C22×D4—232no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—432no—
208C22×D4—432no—
209C22×D4—416no—
210C22×D4—416no—
211C22×D4#4118yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—24no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—42no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4C2414yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257#126—2128no—
258#126—2128no—
259#123—432no—
260#123—432no—
261C2×C4⋊C4—232no—
262C2×C4⋊C4—232no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C22⋊C4—232no—
280C2×C22⋊C4—232no—
281C2×C22⋊C4—232no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—416no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4—48no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4C2×D418yes—
329C2×C22⋊C4C2×D418yes—
330C2×C22⋊C4C2×D418yes—
331C2×C22⋊C4C2×D418yes—
332C2×C22⋊C4C2×D418yes—
333C2×C22⋊C4C2×D418yes—
334C2×C42—232no—
335C2×C42—28no—
336C2×C42—28no—
337C2×C42—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—48no—
343#95—48no—
344#95—48no—
345#95—48no—
346#95—48no—
347#95—48no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—28no—
359C24—28no—
360C24—216no—
361C24—216no—
362C24—28no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—28no—
370C24—28no—
371C24—48no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—48no—
377C24—48no—
378C24—48no—
379C24—44no—
380C24—48no—
381C24—48no—
382C24—44no—
383C24—816no—
384C24—816no—
385C24—84no—
386C24—84no—
387C24—84no—
388C24—84no—
389C24—816no—
390C24—816no—
391C24#120116yes—
392C24C22×D412yes—
393C24C22×D412yes—
394C24C22×D412yes—
395C24C22×D412yes—
396C24C22×D412yes—
397C24C22×D412yes—
398C24C22×D412yes—
399C24C22×D412yes—
400C24C22×D412yes—
401C24C2×#4114yes—
402C24C2×#4114yes—
403C24H(5,𝔽2)14yes—
404C24C2511yescharacteristic
405C2×D4—216no—
406C2×D4—216no—
407C2×D4—216no—
408C2×D4—216no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—4128no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—28no—
469C22×C4—28no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4—816no—
496C22×C4C22×D414yes—
497C22×C4C22×D414yes—
498C22×C4C22×D414yes—
499C22×C4C22×D414yes—
500C22×C4C22×D414yes—
501C22×C4C22×D414yes—
502C22×C4C2×#4114yes—
503C4⋊C4—232no—
504C4⋊C4—232no—
505C4⋊C4—232no—
506C4⋊C4—416no—
507C4⋊C4—416no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—216no—
514C22⋊C4—416no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—232no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—216no—
536C23—216no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—232no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—216no—
562C23—416no—
563C23—416no—
564C23—416no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—48no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—48no—
580C23—48no—
581C23—48no—
582C23—816no—
583C23—816no—
584C23—832no—
585C23—832no—
586C23—88no—
587C23—88no—
588C23—816no—
589C23—832no—
590C23—832no—
591C23—816no—
592C23C2×#12014yes—
593C23C2×#12014yes—
594C23C2×#12014yes—
595C23#52514yes—
596C23#53514yes—
597C23#54618yes—
598C23#54618yes—
599C23C23×D411yescharacteristic
600C23C23×D411yescharacteristic
601C23C23×D411yescharacteristic
602C23C22×#4112yes—
603C23C2×H(5,𝔽2)12yes—
604C23#58514yes—
605C23#58514yes—
606D4—432no—
607D4—432no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—416no—
624C2×C4—416no—
625C2×C4—416no—
626C2×C4—416no—
627C2×C4—416no—
628C2×C4—416no—
629C2×C4—416no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—48no—
658C2×C4—48no—
659C2×C4—48no—
660C2×C4—48no—
661C2×C4—48no—
662C2×C4—48no—
663C2×C4—816no—
664C22—24no—
665C22—24no—
666C22—28no—
667C22—28no—
668C22—216no—
669C22—216no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—48no—
681C22—48no—
682C22—48no—
683C22—48no—
684C22—48no—
685C22—48no—
686C22—44no—
687C22—48no—
688C22—832no—
689C22—832no—
690C22—82no—
691C22—82no—
692C22—84no—
693C22—84no—
694C22—84no—
695C22#240418yes—
696C22C22×#12011yescharacteristic
697C22C2×#52511yescharacteristic
698C22C2×#53511yescharacteristic
699C22#344912yes—
700C22#344912yes—
701C22C2×#54612yes—
702C22C2×#54612yes—
703C22#348512yes—
704C22#348512yes—
705C22#348512yes—
706C22#348512yes—
707C22#352118yes—
708C22#356018yes—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—44no—
714C4—44no—
715C4—44no—
716C4—84no—
717C4—84no—
718C4—84no—
719C2—28no—
720C2—44no—
721C2—44no—
722C2—84no—
723C2C2×#240411yescharacteristic
724C2#3508811yescharacteristic
725C2#3508811yescharacteristic
726C2#3545212yes—
727C2#3696912yes—
728C2#3803014yes—
729C2#3890114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#126C2semidirect
5. #16024C2semidirect
6. #15986C2semidirect
7. #15908C2semidirect
8. #15091C2semidirect
9. #15069C2semidirect
10. #14706C2semidirect
11. #14341C2semidirect
12. #14256C2semidirect
13. #13869C2semidirect
14. #13869C2semidirect
20. C22×#126C22semidirect
23. C23×C4⋊C4C22semidirect
24. C23×C4⋊C4C22semidirect
25. C23×C4⋊C4C22semidirect
30. C23×C42C22semidirect
44. C2×#381C22semidirect
45. C2×#381C22semidirect
51. #1444C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
57. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1437C22semidirect
62. #1436C22semidirect
63. C22⋊#95C22semidirect
64. C22⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C23×#1262
C2×#24052
C2×#23082
#160242
#159862
#159082
#150912
#150692
#147062
#143412
#142562
#138692