Finite Group Databasebeta

← #618596

Subgroups — #618596

Order 512.

4500 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#126C211yesmaximal, characteristic, complemented
2C2×#2591C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic, complemented
4C2×#2617C211yesmaximal, characteristic
5#15939C218yesmaximal, complemented
6#15928C214yesmaximal, complemented
7#15909C214yesmaximal, complemented
8#14911C212yesmaximal, complemented
9#14911C212yesmaximal, complemented
10#14706C212yesmaximal, complemented
11#14256C212yesmaximal, complemented
12#13923C211yesmaximal, characteristic, complemented
13#13904C211yesmaximal, characteristic, complemented
14#13775C211yesmaximal, characteristic, complemented
15C22×#126—28no—
16C22×#126—28no—
17C22×#126—28no—
18C22×#126C2214yes—
19C22×#126C2214yes—
20C22×#123C2212yes—
21C22×#123C2212yes—
22C22×#121C2211yescharacteristic
23C22×#120C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C22⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic
31#2617—28no—
32#2591—28no—
33#2501—28no—
34C2×#400C2212yescomplemented
35C2×#399C2214yescomplemented
36C2×#397C2212yes—
37C2×#394C2214yes—
38C2×#394C2212yescomplemented
39C2×#392C2212yes—
40C2×#390C2212yescomplemented
41C2×#388C2214yescomplemented
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#384C2212yescomplemented
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#380C2211yescharacteristic
49C2×#379C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2214yescomplemented
53#1441C2218yescomplemented
54#1441C2214yescomplemented
55#1439C2218yes—
56#1439C2218yes—
57#1439C2218yescomplemented
58#1438C2218yes—
59#1438C2218yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2218yescomplemented
63#1436C2214yescomplemented
64#1436C2218yescomplemented
65C4⋊#95C2214yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×#126—2128no—
80C2×#126—216no—
81C2×#126—216no—
82C2×#126—216no—
83C2×#126—216no—
84C2×#126—216no—
85C2×#126—216no—
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—44no—
91C2×#123—44no—
92C2×#121—24no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#120—28no—
96C2×#120—44no—
97C2×#120—44no—
98C22×C4⋊C4—24no—
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⋊C4D418yescomplemented
106C22×C4⋊C4D418yescomplemented
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—24no—
121C22×C22⋊C4—24no—
122C22×C22⋊C4—24no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4D418yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2318yes—
140C22×C42—24no—
141C22×C42—24no—
142C22×C42—24no—
143C22×C42C2314yes—
144C22×C42C2312yes—
145C22×C42C2312yes—
146#400—28no—
147#399—216no—
148#397—28no—
149#394—28no—
150#394—216no—
151#392—28no—
152#390—28no—
153#388—216no—
154#386—28no—
155#386—28no—
156#386—28no—
157#384—28no—
158#381—44no—
159#381—44no—
160#380—24no—
161#379—28no—
162C4×C22⋊C4—24no—
163C4×C22⋊C4—24no—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C25—22no—
176C25—22no—
177C25—24no—
178C25—24no—
179C25—44no—
180C25—44no—
181C25C2×D414yes—
182C25C2412yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—28no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—48no—
200C22×D4—48no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238#126—2128no—
239#126—2128no—
240#123—432no—
241#123—432no—
242#121—416no—
243#120—432no—
244C2×C4⋊C4—232no—
245C2×C4⋊C4—232no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C22⋊C4—232no—
274C2×C22⋊C4—232no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—232no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
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—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—216no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—48no—
313C2×C22⋊C4—48no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D418yes—
317C2×C22⋊C4#4118yes—
318C2×C42—232no—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42—44no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—48no—
333#95—48no—
334#95—48no—
335#95—48no—
336#95—48no—
337C24—24no—
338C24—24no—
339C24—28no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—28no—
345C24—28no—
346C24—28no—
347C24—28no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—28no—
356C24—28no—
357C24—48no—
358C24—48no—
359C24—48no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—816no—
366C24—816no—
367C24C22×D414yes—
368C24C22×D412yes—
369C24C22×D412yes—
370C24C22×D412yes—
371C24C2×#4114yes—
372C24H(5,𝔽2)14yes—
373C24C2511yescharacteristic
374C2×D4—48no—
375C2×D4—416no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—88no—
387C2×D4—88no—
388C2×D4—816no—
389C2×D4—816no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4#120116yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C2×#4118yes—
469C4⋊C4—232no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—44no—
477C4⋊C4—84no—
478C22⋊C4—232no—
479C22⋊C4—232no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—816no—
493C42—232no—
494C42—48no—
495C42—84no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—216no—
503C23—216no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—416no—
510C23—416no—
511C23—44no—
512C23—44no—
513C23—416no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—816no—
540C23—816no—
541C23—832no—
542C23—832no—
543C23C2×#12014yes—
544C23#53214yes—
545C23#53514yes—
546C23#54618yes—
547C23#55018yes—
548C23C23×D411yescharacteristic
549C23C23×D411yescharacteristic
550C23C23×D411yescharacteristic
551C23C22×#4112yes—
552C23C2×H(5,𝔽2)12yes—
553C23#58514yes—
554C23#58514yes—
555D4—88no—
556D4—832no—
557D4—816no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—216no—
577C2×C4—416no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4C2×#12014yes—
616C2×C4C2×#12014yes—
617C22—24no—
618C22—24no—
619C22—28no—
620C22—28no—
621C22—216no—
622C22—216no—
623C22—44no—
624C22—44no—
625C22—48no—
626C22—48no—
627C22—44no—
628C22—832no—
629C22—88no—
630C22—88no—
631C22—82no—
632C22—82no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22#243118yes—
641C22C22×#12011yescharacteristic
642C22C2×#53211yescharacteristic
643C22C2×#53511yescharacteristic
644C22#344912yes—
645C22#344912yes—
646C22C2×#54612yes—
647C22C2×#55012yes—
648C22#348512yes—
649C22#348512yes—
650C22#348512yes—
651C22#348712yes—
652C22#352118yes—
653C22#356018yes—
654C4—24no—
655C4—24no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—48no—
660C4—44no—
661C4—44no—
662C4—84no—
663C4—82no—
664C4—84no—
665C2—28no—
666C2—84no—
667C2—162no—
668C2C2×#243111yescharacteristic
669C2#3508811yescharacteristic
670C2#3514911yescharacteristic
671C2#3546712yes—
672C2#3696912yes—
673C2#3803114yes—
674C2#3912914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2617C2
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
2. C2×#2591C2semidirect
3. C2×#2501C2semidirect
5. #15939C2semidirect
6. #15928C2semidirect
7. #15909C2semidirect
8. #14911C2semidirect
9. #14911C2semidirect
10. #14706C2semidirect
11. #14256C2semidirect
12. #13923C2semidirect
13. #13904C2semidirect
14. #13775C2semidirect
34. C2×#400C22semidirect
35. C2×#399C22semidirect
38. C2×#394C22semidirect
40. C2×#390C22semidirect
41. C2×#388C22semidirect
45. C2×#384C22semidirect
52. #1445C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
57. #1439C22semidirect
62. #1437C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. C4⋊#95C22semidirect
105. C22×C4⋊C4D4semidirect
106. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C23×#1262
C2×#25912
C2×#25012
C2×#26172
#159392
#159282
#159092
#149112
#147062
#142562
#139232
#139042
#137752