Finite Group Databasebeta

← #620107

Subgroups — #620107

Order 512.

3422 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic, complemented
2C2×#2280C211yesmaximal, characteristic
3#15930C212yesmaximal, complemented
4#15916C212yesmaximal, complemented
5#15797C214yesmaximal, complemented
6#15770C214yesmaximal, complemented
7#15467C212yesmaximal
8#15284C212yesmaximal
9#15233C212yesmaximal, complemented
10#15125C212yesmaximal, complemented
11#14680C212yesmaximal
12#14343C212yesmaximal, complemented
13#13974C212yesmaximal, complemented
14#13973C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic, complemented
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2628—28no—
21C2×#400C2212yescomplemented
22C2×#399C2212yes—
23C2×#395C2212yescomplemented
24C2×#393C2212yes—
25C2×#393C2214yes—
26C2×#389C2212yescomplemented
27C2×#388C2214yes—
28C2×#382C2211yescharacteristic, complemented
29#2280—28no—
30C2×#381C2211yescharacteristic
31C2×#381C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#379C2211yescharacteristic
41C2×#379C2211yescharacteristic
42C2×#379C2214yes—
43C2×C4×C4⋊C4C2211yescharacteristic, complemented
44C2×C4×C4⋊C4C2211yescharacteristic, complemented
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47#1446C2214yescomplemented
48#1445C2214yescomplemented
49#1445C2214yescomplemented
50#1443C2214yescomplemented
51#1443C2214yescomplemented
52#1443C2214yescomplemented
53#1442C2214yescomplemented
54#1442C2214yescomplemented
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2212yes—
63#1437C2212yes—
64#1437C2214yes—
65#1436C2214yescomplemented
66#1436C2212yescomplemented
67#1436C2212yescomplemented
68C4⋊#95C2212yescomplemented
69C4⋊#95C2212yescomplemented
70C4⋊#95C2212yescomplemented
71C4⋊#95C2212yescomplemented
72C4⋊#95C2212yescomplemented
73#1434C2214yescomplemented
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×Q8C2312yes—
79C24×C4—24no—
80C24×C4—28no—
81C24×C4—28no—
82C24×C4—42no—
83C24×C4—42no—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8—42no—
91C2×#117—24no—
92C2×#117—28no—
93C2×#117—42no—
94C2×#117—42no—
95C2×#117D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124#400—28no—
125#399—28no—
126#395—28no—
127#393—28no—
128#393—48no—
129#389—28no—
130#388—48no—
131#382—24no—
132#381—24no—
133#381—24no—
134#380—28no—
135#380—28no—
136#380—28no—
137#380—28no—
138#380—28no—
139#380—28no—
140#380—44no—
141#380—44no—
142#379—24no—
143#379—24no—
144#379—48no—
145C4×C4⋊C4—42no—
146C4×C4⋊C4—42no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C25—22no—
175C25—24no—
176C25—24no—
177C25—24no—
178C25—24no—
179C25—41no—
180C25—41no—
181C25—42no—
182C25—42no—
183C25C2411yescharacteristic
184C25C2412yes—
185C22×Q8—24no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—44no—
189C22×Q8—44no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—232no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—432no—
213C23×C4—432no—
214C23×C4—416no—
215C23×C4—416no—
216C23×C4C2×D414yes—
217C23×C4#4118yes—
218C23×C4#4118yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C4⋊Q8—416no—
233#117—416no—
234#117—416no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—21no—
348C24—21no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—28no—
360C24—216no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—48no—
366C24—48no—
367C24—48no—
368C24—48no—
369C24—44no—
370C24—44no—
371C24—44no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—44no—
377C24—44no—
378C24—44no—
379C24—48no—
380C24C22×D412yes—
381C24C2×#4114yes—
382C24C2×#4114yes—
383C24H(5,𝔽2)14yes—
384C242−1+414yes—
385C24C2511yescharacteristic
386C2×Q8—48no—
387C2×Q8—88no—
388C2×Q8—84no—
389C2×Q8—84no—
390C2×Q8—88no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—216no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—4128no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4#124116yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C4H(5,𝔽2)12yes—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—88no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—416no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C42—28no—
519C42—28no—
520C42—48no—
521C42—84no—
522C42—84no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—416no—
552C23—432no—
553C23—432no—
554C23—432no—
555C23—432no—
556C23—416no—
557C23—432no—
558C23—432no—
559C23—416no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—42no—
580C23—42no—
581C23—41no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23C2×#12414yes—
587C23#54218yes—
588C23C23×D411yescharacteristic
589C23C22×#4112yes—
590C23C22×#4112yes—
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×H(5,𝔽2)11yescharacteristic
593C23C2×H(5,𝔽2)12yes—
594C23C2×2−1+412yes—
595C23#58512yes—
596C23#58512yes—
597Q8—816no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—232no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—432no—
615C2×C4—416no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—432no—
633C2×C4—44no—
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—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4#52514yes—
676C2×C4C2×#12414yes—
677C2×C4C2×#12414yes—
678C2×C4#53414yes—
679C2×C4#55118yes—
680C2×C4#55118yes—
681C22—24no—
682C22—28no—
683C22—28no—
684C22—28no—
685C22—28no—
686C22—464no—
687C22—44no—
688C22—42no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—42no—
705C22—44no—
706C22C2×#52511yescharacteristic
707C22C22×#12411yescharacteristic
708C22C2×#53411yescharacteristic
709C22C2×#54212yes—
710C22#345512yes—
711C22C2×#55112yes—
712C22C2×#55112yes—
713C22#348812yes—
714C22#348912yes—
715C22#349212yes—
716C22#349312yes—
717C22#349412yes—
718C22#349412yes—
719C22#349512yes—
720C22#349512yes—
721C22#353711yescharacteristic
722C22#353811yescharacteristic
723C22#354011yescharacteristic
724C22#354011yescharacteristic
725C22#354112yes—
726C22#354212yes—
727C4—24no—
728C4—44no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4—84no—
734C4—84no—
735C4—82no—
736C4—82no—
737C4—84no—
738C4—84no—
739C4#256118yes—
740C2—44no—
741C2—48no—
742C2C2×#256111yescharacteristic
743C2#3526012yes—
744C2#3554212yes—
745C2#3726911yescharacteristic
746C2#3737411yescharacteristic
747C2#3785212yes—
748C2#3785312yes—
749C2#3785812yes—
750C2#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×#2628C2semidirect
3. #15930C2semidirect
4. #15916C2semidirect
5. #15797C2semidirect
6. #15770C2semidirect
9. #15233C2semidirect
10. #15125C2semidirect
12. #14343C2semidirect
13. #13974C2semidirect
14. #13973C2semidirect
15. #13777C2semidirect
17. C22×C4⋊Q8C22semidirect
21. C2×#400C22semidirect
23. C2×#395C22semidirect
26. C2×#389C22semidirect
28. C2×#382C22semidirect
43. C2×C4×C4⋊C4C22semidirect
44. C2×C4×C4⋊C4C22semidirect
47. #1446C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1443C22semidirect
51. #1443C22semidirect
52. #1443C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. C4⋊#95C22semidirect
69. C4⋊#95C22semidirect
70. C4⋊#95C22semidirect
71. C4⋊#95C22semidirect
72. C4⋊#95C22semidirect
73. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#22802
#159302
#159162
#157972
#157702
#154672
#152842
#152332
#151252
#146802
#143432
#139742
#139732
#137772