Finite Group Databasebeta

← #612622

Subgroups — #612622

Order 512.

6866 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic, complemented
3C2×#2482C211yesmaximal, characteristic, complemented
4C2×#2280C211yesmaximal, characteristic, complemented
5#14918C212yesmaximal, complemented
6#14825C214yesmaximal, complemented
7#14824C214yesmaximal, complemented
8#14724C212yesmaximal
9#14705C212yesmaximal, complemented
10#14681C212yesmaximal, complemented
11#14681C211yesmaximal, characteristic, complemented
12#14173C218yesmaximal, complemented
13#13870C211yesmaximal, characteristic, complemented
14#13868C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×C4×D4—28no—
19C22×C4×D4—28no—
20C22×C4×D4—28no—
21C22×C4×D4C2214yes—
22C22×C4×D4C2214yes—
23C22×#117C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C42C2211yescharacteristic
29#2484—28no—
30#2482—28no—
31C2×#400C2212yescomplemented
32C2×#399C2212yescomplemented
33C2×#398C2212yescomplemented
34C2×#397C2212yescomplemented
35C2×#394C2218yes—
36C2×#388C2214yescomplemented
37C2×#387C2214yescomplemented
38C2×#385C2212yes—
39C2×#385C2212yes—
40#2280—28no—
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#380C2212yescomplemented
48C2×#380C2212yescomplemented
49C2×#379C2212yes—
50C2×#379C2212yes—
51C2×#379C2212yes—
52C2×#379C2212yes—
53C2×C4×C22⋊C4C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2218yescomplemented
56#1439C2214yescomplemented
57#1439C2218yescomplemented
58#1438C2218yes—
59#1437C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2218yescomplemented
62#1436C2218yescomplemented
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yescomplemented
66C22⋊#95C2218yescomplemented
67C22⋊#95C2218yescomplemented
68C22⋊#95C2214yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×D4—24no—
72C23×D4—24no—
73C23×D4—216no—
74C23×D4—24no—
75C23×D4C2314yes—
76C24×C4—24no—
77C24×C4—24no—
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—28no—
82C24×C4—28no—
83C24×C4—44no—
84C24×C4—44no—
85C24×C4D418yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2314yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×C4×D4—2128no—
95C2×C4×D4—216no—
96C2×C4×D4—216no—
97C2×C4×D4—216no—
98C2×C4×D4—216no—
99C2×C4×D4—216no—
100C2×C4×D4—216no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117—44no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4D418yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
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⋊C4D418yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2318yes—
132C22×C22⋊C4C2318yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C42—24no—
136C22×C42—24no—
137C22×C42—24no—
138C22×C42C2314yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141#400—28no—
142#399—28no—
143#398—28no—
144#397—28no—
145#394—232no—
146#388—216no—
147#387—216no—
148#385—28no—
149#385—28no—
150#381—24no—
151#381—24no—
152#381—28no—
153#381—28no—
154#381—28no—
155#381—44no—
156#380—28no—
157#380—28no—
158#379—28no—
159#379—28no—
160#379—44no—
161#379—44no—
162C4×C22⋊C4—216no—
163C2×#95C2318yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2318yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2318yes—
176C2×#95C2318yes—
177C25—22no—
178C25—22no—
179C25—22no—
180C25—22no—
181C25—28no—
182C25—28no—
183C25—24no—
184C25—24no—
185C25—24no—
186C25—24no—
187C25—44no—
188C25—44no—
189C25C2×D414yes—
190C25C2×D414yes—
191C25C2412yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C22×D4—232no—
195C22×D4—264no—
196C22×D4—264no—
197C22×D4—264no—
198C22×D4—28no—
199C22×D4—28no—
200C22×D4—28no—
201C22×D4—28no—
202C22×D4—28no—
203C22×D4—28no—
204C22×D4#41116yes—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—232no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—232no—
222C23×C4—232no—
223C23×C4—28no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—28no—
229C23×C4—28no—
230C23×C4—28no—
231C23×C4—216no—
232C23×C4—416no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—48no—
236C23×C4—44no—
237C23×C4—432no—
238C23×C4—432no—
239C23×C4—432no—
240C23×C4—432no—
241C23×C4—416no—
242C23×C4—416no—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D418yes—
246C23×C4C2×D418yes—
247C23×C4C2×D414yes—
248C23×C4C2×D414yes—
249C23×C4C2×D414yes—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4#4118yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2414yes—
256C23×C4C2414yes—
257C23×C4C2414yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2412yes—
263C23×C4C2412yes—
264C4×D4—2128no—
265C4×D4—2128no—
266#117—416no—
267#117—416no—
268C2×C4⋊C4—232no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—216no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C22⋊C4—232no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—232no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
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—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—216no—
308C2×C22⋊C4—216no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—216no—
315C2×C22⋊C4—216no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4C2×D418yes—
323C2×C22⋊C4C2×D418yes—
324C2×C42—232no—
325C2×C42—28no—
326C2×C42—28no—
327C2×C42—28no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—44no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—216no—
344#95—216no—
345#95—48no—
346#95—48no—
347C24—24no—
348C24—24no—
349C24—216no—
350C24—216no—
351C24—216no—
352C24—216no—
353C24—28no—
354C24—28no—
355C24—216no—
356C24—216no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—28no—
370C24—28no—
371C24—28no—
372C24—216no—
373C24—48no—
374C24—48no—
375C24—432no—
376C24—432no—
377C24—432no—
378C24—432no—
379C24—48no—
380C24—48no—
381C24—48no—
382C24—48no—
383C24—44no—
384C24—44no—
385C24—48no—
386C24—48no—
387C24—48no—
388C24C22×D414yes—
389C24C22×D414yes—
390C24C22×D412yes—
391C24C22×D412yes—
392C24C22×D412yes—
393C24C2×#4118yes—
394C24C2×#4118yes—
395C24C2×#4114yes—
396C24H(5,𝔽2)14yes—
397C24C2511yescharacteristic
398C2×D4—232no—
399C2×D4—2256no—
400C2×D4—232no—
401C2×D4—232no—
402C2×D4—232no—
403C2×D4—232no—
404C2×D4—232no—
405C2×D4—232no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—216no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—232no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—216no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—216no—
442C22×C4—216no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—4128no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—432no—
475C22×C4—432no—
476C22×C4—4128no—
477C22×C4—416no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—416no—
484C22×C4#120116yes—
485C22×C4#123116yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D414yes—
495C22×C4C22×D414yes—
496C22×C4C2×#4118yes—
497C22×C4C2×#4118yes—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—416no—
504C4⋊C4—88no—
505C22⋊C4—232no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—88no—
516C22⋊C4—816no—
517C22⋊C4—816no—
518C42—232no—
519C42—48no—
520C42—48no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—216no—
526C23—24no—
527C23—24no—
528C23—216no—
529C23—232no—
530C23—232no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—28no—
544C23—216no—
545C23—216no—
546C23—28no—
547C23—216no—
548C23—216no—
549C23—464no—
550C23—432no—
551C23—432no—
552C23—464no—
553C23—464no—
554C23—464no—
555C23—416no—
556C23—416no—
557C23—432no—
558C23—432no—
559C23—416no—
560C23—416no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—48no—
568C23—48no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—48no—
573C23—48no—
574C23C2×#12014yes—
575C23C2×#12314yes—
576C23#52518yes—
577C23#52518yes—
578C23#53414yes—
579C23#54618yes—
580C23#54618yes—
581C23C23×D411yescharacteristic
582C23C23×D411yescharacteristic
583C23C23×D411yescharacteristic
584C23C22×#4112yes—
585C23C22×#4114yes—
586C23C2×H(5,𝔽2)12yes—
587C23#58514yes—
588D4—2128no—
589C2×C4—232no—
590C2×C4—232no—
591C2×C4—232no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—432no—
613C2×C4—432no—
614C2×C4—464no—
615C2×C4—416no—
616C2×C4—416no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—48no—
636C2×C4—48no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4—48no—
644C2×C4—48no—
645C2×C4C2×#12014yes—
646C2×C4C2×#12014yes—
647C2×C4C2×#12318yes—
648C2×C4#52518yes—
649C22—24no—
650C22—216no—
651C22—216no—
652C22—24no—
653C22—28no—
654C22—28no—
655C22—28no—
656C22—28no—
657C22—216no—
658C22—28no—
659C22—28no—
660C22—464no—
661C22—464no—
662C22—48no—
663C22—416no—
664C22—416no—
665C22—416no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—48no—
672C22—44no—
673C22—48no—
674C22#240518yes—
675C22C22×#12011yescharacteristic
676C22C22×#12311yescharacteristic
677C22C2×#52512yes—
678C22C2×#52512yes—
679C22C2×#52512yes—
680C22C2×#53411yescharacteristic
681C22#344912yes—
682C22C2×#54612yes—
683C22C2×#54612yes—
684C22#347518yes—
685C22#348512yes—
686C22#348712yes—
687C22#351318yes—
688C4—28no—
689C4—28no—
690C4—24no—
691C4—24no—
692C4—44no—
693C4—44no—
694C4—82no—
695C4—84no—
696C4—84no—
697C4—82no—
698C4—84no—
699C2—28no—
700C2—48no—
701C2—48no—
702C2C2×#240511yescharacteristic
703C2#3480912yes—
704C2#3506312yes—
705C2#3514911yescharacteristic
706C2#3615914yes—
707C2#3616214yes—
708C2#3661411yescharacteristic
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
2. C2×#2484C2semidirect
3. C2×#2482C2semidirect
4. C2×#2280C2semidirect
5. #14918C2semidirect
6. #14825C2semidirect
7. #14824C2semidirect
9. #14705C2semidirect
10. #14681C2semidirect
11. #14681C2semidirect
12. #14173C2semidirect
13. #13870C2semidirect
14. #13868C2semidirect
31. C2×#400C22semidirect
32. C2×#399C22semidirect
33. C2×#398C22semidirect
34. C2×#397C22semidirect
36. C2×#388C22semidirect
37. C2×#387C22semidirect
47. C2×#380C22semidirect
48. C2×#380C22semidirect
53. C2×C4×C22⋊C4C22semidirect
54. #1441C22semidirect
55. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
59. #1437C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
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×C4×D42
C2×#24842
C2×#24822
C2×#22802
#149182
#148252
#148242
#147242
#147052
#146812
#141732
#138702
#138682