Finite Group Databasebeta

← #851465

Subgroups — #851465

Order 512.

3154 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16707C212yesmaximal, complemented
2#16297C212yesmaximal
3#16118C214yesmaximal, complemented
4#15586C214yesmaximal, complemented
5#15195C211yesmaximal, characteristic, complemented
6#15183C212yesmaximal, complemented
7#15125C211yesmaximal, characteristic, complemented
8#15125C212yesmaximal, complemented
9#14680C212yesmaximal
10#14596C212yesmaximal, complemented
11#14366C212yesmaximal
12#13993C212yesmaximal, complemented
13#13973C212yesmaximal, complemented
14#13876C212yesmaximal, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic, complemented
18C2×#393C2212yes—
19C2×#393C2214yes—
20C2×#389C2212yescomplemented
21C2×#388C2214yes—
22C2×#384C2211yescharacteristic, complemented
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#379C2211yescharacteristic
30C2×#379C2214yes—
31C2×#379C2211yescharacteristic
32C2×C4×C4⋊C4C2212yescomplemented
33C2×C4×C22⋊C4C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1446C2214yescomplemented
38#1446C2212yescomplemented
39#1445C2214yescomplemented
40#1443C2214yescomplemented
41#1443C2214yescomplemented
42#1442C2214yescomplemented
43#1442C2212yescomplemented
44#1442C2214yescomplemented
45#1442C2214yescomplemented
46#1442C2212yescomplemented
47#1442C2212yescomplemented
48#1440C2214yes—
49#1439C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61C4⋊#95C2212yescomplemented
62C4⋊#95C2212yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65C4⋊#95C2211yescharacteristic, complemented
66C4⋊#95C2211yescharacteristic, complemented
67#1434C2214yescomplemented
68C22⋊#95C2211yescharacteristic
69C22⋊#95C2214yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C26C2311yescharacteristic
76C23×Q8C2312yes—
77C24×C4—28no—
78C24×C4—24no—
79C24×C4—42no—
80C24×C4—44no—
81C24×C4—42no—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2314yes—
114#393—28no—
115#393—48no—
116#389—44no—
117#388—48no—
118#384—42no—
119#380—28no—
120#380—28no—
121#380—44no—
122#380—44no—
123#380—44no—
124#380—44no—
125#379—24no—
126#379—48no—
127#379Q818yescomplemented
128C4×C4⋊C4—44no—
129C4×C22⋊C4—28no—
130C4×C22⋊C4—42no—
131C4×C22⋊C4—44no—
132C4×C22⋊C4—42no—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25—22no—
161C25—24no—
162C25—22no—
163C25—24no—
164C25—41no—
165C25—42no—
166C25—41no—
167C25—41no—
168C25—42no—
169C25—41no—
170C25C2×Q814yes—
171C25C2411yescharacteristic
172C25C2412yes—
173C22×Q8—24no—
174C22×Q8—42no—
175C22×Q8—44no—
176C22×Q8—42no—
177C22×Q8—44no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—41no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—41no—
200C23×C4—416no—
201C23×C4—432no—
202C23×C4—432no—
203C23×C4—416no—
204C23×C4—416no—
205C23×C4#4118yes—
206C23×C4#4114yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C4⋊Q8—416no—
221C2×C4⋊C4—24no—
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—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×Q818yes—
280C2×C22⋊C4C2×Q814yes—
281C2×C22⋊C4C2×Q814yes—
282C2×C22⋊C4C2×Q818yes—
283C2×C22⋊C4#4114yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—24no—
299#95—28no—
300#95—42no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—41no—
308#95—41no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—28no—
342C24—216no—
343C24—48no—
344C24—44no—
345C24—44no—
346C24—44no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—48no—
352C24—44no—
353C24—48no—
354C24—44no—
355C24—44no—
356C24—48no—
357C24—44no—
358C24—44no—
359C24—48no—
360C24—48no—
361C24—44no—
362C24—44no—
363C24—48no—
364C24C22×Q812yes—
365C24C22×Q814yes—
366C24C2×#4112yes—
367C24C2×#4112yes—
368C24C2×#4114yes—
369C24H(5,𝔽2)14yes—
370C24C2511yescharacteristic
371C2×Q8—48no—
372C2×Q8—84no—
373C2×Q8—88no—
374C2×Q8—84no—
375C2×Q8—88no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—216no—
409C22×C4—28no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—4128no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4C22×Q812yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C4H(5,𝔽2)12yes—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—88no—
488C22⋊C4—216no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C42—48no—
503C42—44no—
504C42—44no—
505C42—88no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
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—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—416no—
537C23—432no—
538C23—432no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—416no—
543C23—416no—
544C23—416no—
545C23—432no—
546C23—432no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23#53314yes—
574C23#55418yes—
575C23C23×Q811yescharacteristic
576C23C22×#4111yescharacteristic
577C23C22×#4111yescharacteristic
578C23C22×#4112yes—
579C23C2×H(5,𝔽2)11yescharacteristic
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)12yes—
582C23C2×H(5,𝔽2)12yes—
583C23#58511yescharacteristic
584C23#58511yescharacteristic
585C23#58512yes—
586Q8—816no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—432no—
604C2×C4—416no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—432no—
629C2×C4—416no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
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—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4#53818yes—
666C2×C4#53814yes—
667C2×C4#55114yes—
668C2×C4#55118yes—
669C2×C4H(3,𝔽4)14yes—
670C22—28no—
671C22—28no—
672C22—28no—
673C22—28no—
674C22—24no—
675C22—464no—
676C22—42no—
677C22—44no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—44no—
689C22—44no—
690C22—42no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22C2×#53311yescharacteristic
698C22C2×#53811yescharacteristic
699C22C2×#53812yes—
700C22#345511yescharacteristic
701C22C2×#55111yescharacteristic
702C22C2×#55112yes—
703C22C2×#55412yes—
704C22#347512yes—
705C22#349312yes—
706C22#349412yes—
707C22#349412yes—
708C22#350111yescharacteristic
709C22#350112yes—
710C22#350111yescharacteristic
711C22#350511yescharacteristic
712C22#350611yescharacteristic
713C22#351912yes—
714C22C2×H(3,𝔽4)11yescharacteristic
715C22#353711yescharacteristic
716C22#354011yescharacteristic
717C22#354012yes—
718C22#354212yes—
719C22#354212yes—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—44no—
724C4—42no—
725C4—42no—
726C4—44no—
727C4—84no—
728C4—84no—
729C4—84no—
730C4—82no—
731C4—84no—
732C4#273618yes—
733C2—48no—
734C2—44no—
735C2#3501411yescharacteristic
736C2#3619012yes—
737C2#3728411yescharacteristic
738C2#3785812yes—
739C2#3960512yes—
740C2#3993412yes—
741C2#4549411yescharacteristic
742C2#4564511yescharacteristic
743C2C2×#273611yescharacteristic
744C2#5919911yescharacteristic
745C2#5920211yescharacteristic
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. #16707C2semidirect
3. #16118C2semidirect
4. #15586C2semidirect
5. #15195C2semidirect
6. #15183C2semidirect
7. #15125C2semidirect
8. #15125C2semidirect
10. #14596C2semidirect
12. #13993C2semidirect
13. #13973C2semidirect
14. #13876C2semidirect
17. C22×C4⋊Q8C22semidirect
20. C2×#389C22semidirect
22. C2×#384C22semidirect
32. C2×C4×C4⋊C4C22semidirect
37. #1446C22semidirect
38. #1446C22semidirect
39. #1445C22semidirect
40. #1443C22semidirect
41. #1443C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. C4⋊#95C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
127. #379Q8semidirect

Maximal subgroups

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

subgroupindex
#167072
#162972
#161182
#155862
#151952
#151832
#151252
#146802
#145962
#143662
#139932
#139732
#138762
C4×#3792