Finite Group Databasebeta

← #619802

Subgroups — #619802

Order 512.

3398 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic, complemented
2C2×#2280C211yesmaximal, characteristic
3#16071C214yesmaximal, complemented
4#16021C214yesmaximal, complemented
5#15386C214yesmaximal, complemented
6#15282C214yesmaximal
7#15223C211yesmaximal, characteristic, complemented
8#15223C211yesmaximal, characteristic, complemented
9#15195C211yesmaximal, characteristic, complemented
10#15150C211yesmaximal, characteristic, complemented
11#14680C212yesmaximal
12#14343C211yesmaximal, characteristic, complemented
13#14264C212yesmaximal, complemented
14#13994C212yesmaximal, complemented
15#13993C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#397C2214yes—
20#2314—28no—
21C2×#389C2212yescomplemented
22C2×#389C2212yescomplemented
23C2×#388C2214yes—
24C2×#388C2214yes—
25C2×#387C2214yescomplemented
26C2×#384C2211yescharacteristic, complemented
27C2×#382C2211yescharacteristic, complemented
28#2280—28no—
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#379C2212yes—
39C2×#379C2214yes—
40C2×C4×C4⋊C4C2212yescomplemented
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#1445C2214yescomplemented
46#1445C2214yescomplemented
47#1445C2214yescomplemented
48#1445C2214yescomplemented
49#1444C2214yescomplemented
50#1443C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2214yes—
63#1436C2214yescomplemented
64C4⋊#95C2211yescharacteristic, complemented
65C4⋊#95C2211yescharacteristic, complemented
66C4⋊#95C2212yescomplemented
67C4⋊#95C2212yescomplemented
68C4⋊#95C2212yescomplemented
69#1434C2214yescomplemented
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C26C2311yescharacteristic
74C24×C4—24no—
75C24×C4—24no—
76C24×C4—24no—
77C24×C4—28no—
78C24×C4—44no—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#117—28no—
86C2×#117—24no—
87C2×#117—44no—
88C2×#117D418yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115#397—48no—
116#389—44no—
117#389—44no—
118#388—216no—
119#388—48no—
120#387—216no—
121#384—24no—
122#382—24no—
123#381—28no—
124#380—28no—
125#380—28no—
126#380—28no—
127#380—28no—
128#380—28no—
129#380—28no—
130#380—44no—
131#380—44no—
132#379—28no—
133#379—48no—
134C4×C4⋊C4—28no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—24no—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25—22no—
162C25—22no—
163C25—22no—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—42no—
168C25—42no—
169C25—42no—
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—232no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—432no—
201C23×C4—432no—
202C23×C4—416no—
203C23×C4—416no—
204C23×C4C2×D414yes—
205C23×C4#4114yes—
206C23×C4#4114yes—
207C23×C4#4118yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#117—232no—
221#117—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D418yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4C2×D418yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
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—22no—
337C24—22no—
338C24—22no—
339C24—28no—
340C24—216no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—44no—
351C24—44no—
352C24—44no—
353C24—48no—
354C24—48no—
355C24—48no—
356C24—48no—
357C24—44no—
358C24—44no—
359C24—44no—
360C24C22×D412yes—
361C24C2×#4114yes—
362C24C2×#4112yes—
363C24C2×#4112yes—
364C24H(5,𝔽2)14yes—
365C242−1+414yes—
366C24C2511yescharacteristic
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
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—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—4128no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4#123116yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)14yes—
474C4⋊C4—216no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4—88no—
483C22⋊C4—216no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C42—28no—
496C42—28no—
497C42—44no—
498C42—44no—
499C42—88no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—416no—
531C23—432no—
532C23—416no—
533C23—432no—
534C23—432no—
535C23—416no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23C2×#12314yes—
570C23#54014yes—
571C23#54114yes—
572C23C23×D411yescharacteristic
573C23C22×#4112yes—
574C23C22×#4111yescharacteristic
575C23C22×#4111yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)12yes—
578C23C2×2−1+412yes—
579C23#58512yes—
580C23#58512yes—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—232no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—416no—
614C2×C4—416no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4C2×#12314yes—
658C2×C4C2×#12314yes—
659C2×C4#52514yes—
660C2×C4#53514yes—
661C2×C4#55118yes—
662C2×C4#55118yes—
663C22—28no—
664C22—24no—
665C22—28no—
666C22—28no—
667C22—24no—
668C22—24no—
669C22—464no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22C22×#12311yescharacteristic
692C22C2×#52511yescharacteristic
693C22C2×#53511yescharacteristic
694C22C2×#54011yescharacteristic
695C22C2×#54111yescharacteristic
696C22#345511yescharacteristic
697C22#345511yescharacteristic
698C22C2×#55112yes—
699C22C2×#55112yes—
700C22#348812yes—
701C22#348912yes—
702C22#349212yes—
703C22#349312yes—
704C22#349412yes—
705C22#349412yes—
706C22#349512yes—
707C22#349512yes—
708C22#353612yes—
709C22#354012yes—
710C22#354112yes—
711C22#354212yes—
712C4—24no—
713C4—44no—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—82no—
718C4—82no—
719C4—84no—
720C4—84no—
721C4—84no—
722C4—84no—
723C4#248218yes—
724C2—44no—
725C2—44no—
726C2—44no—
727C2C2×#248211yescharacteristic
728C2#3526111yescharacteristic
729C2#3531011yescharacteristic
730C2#3551612yes—
731C2#3727112yes—
732C2#3785012yes—
733C2#3785112yes—
734C2#3785812yes—
735C2#3785912yes—
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
1. C2×#2314C2semidirect
3. #16071C2semidirect
4. #16021C2semidirect
5. #15386C2semidirect
7. #15223C2semidirect
8. #15223C2semidirect
9. #15195C2semidirect
10. #15150C2semidirect
12. #14343C2semidirect
13. #14264C2semidirect
14. #13994C2semidirect
15. #13993C2semidirect
21. C2×#389C22semidirect
22. C2×#389C22semidirect
25. C2×#387C22semidirect
26. C2×#384C22semidirect
27. C2×#382C22semidirect
40. C2×C4×C4⋊C4C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
63. #1436C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
69. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#22802
#160712
#160212
#153862
#152822
#152232
#151952
#151502
#146802
#143432
#142642
#139942
#139932