Finite Group Databasebeta

← #850563

Subgroups — #850563

Order 512.

3822 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C212yesmaximal
2C2×#2489C212yesmaximal, complemented
3#16307C214yesmaximal, complemented
4#15440C214yesmaximal, complemented
5#15282C214yesmaximal
6#15216C211yesmaximal, characteristic, complemented
7#15179C212yesmaximal, complemented
8#15178C211yesmaximal, characteristic, complemented
9#15178C212yesmaximal, complemented
10#14403C214yesmaximal, complemented
11#13997C212yesmaximal, complemented
12#13974C212yesmaximal, complemented
13C4×#379C211yesmaximal, characteristic
14C25×C4C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic, complemented
16C22×#125C2211yescharacteristic, complemented
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20#2848—216no—
21#2489—216no—
22C2×#400C2214yescomplemented
23C2×#397C2214yes—
24C2×#391C2212yescomplemented
25C2×#391C2212yescomplemented
26C2×#391C2212yescomplemented
27C2×#389C2212yescomplemented
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#382C2214yescomplemented
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#379C2214yes—
38C2×#379C2212yes—
39C2×C4×C4⋊C4C2212yescomplemented
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44#1445C2214yescomplemented
45#1444C2214yescomplemented
46#1444C2214yescomplemented
47#1444C2214yescomplemented
48#1443C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1437C2214yes—
61#1436C2214yescomplemented
62C4⋊#95C2214yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2211yescharacteristic, complemented
65C4⋊#95C2211yescharacteristic, complemented
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×Q8C2312yes—
71C24×C4—28no—
72C24×C4—24no—
73C24×C4—44no—
74C24×C4—42no—
75C24×C4—42no—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—44no—
81C2×C4⋊Q8—42no—
82C2×#125—28no—
83C2×#125—42no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—44no—
90C2×C22⋊Q8—44no—
91C2×C22⋊Q8—44no—
92C2×C22⋊Q8—44no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2314yes—
118#400—216no—
119#397—216no—
120#391—28no—
121#391—28no—
122#391—44no—
123#389—44no—
124#388—216no—
125#388—216no—
126#385—216no—
127#385—216no—
128#385—216no—
129#382—216no—
130#380—28no—
131#380—28no—
132#380D4116yes—
133#379—216no—
134#379—28no—
135C4×C4⋊C4—28no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—42no—
138C4×C22⋊C4—44no—
139C4×C22⋊C4—42no—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C25—24no—
163C25—22no—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—42no—
168C25—41no—
169C25—41no—
170C25—42no—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—41no—
200C23×C4—42no—
201C23×C4—41no—
202C23×C4—416no—
203C23×C4—432no—
204C23×C4—432no—
205C23×C4—416no—
206C23×C4—416no—
207C23×C4C2×D418yes—
208C23×C4#4114yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C4⋊Q8—416no—
221#125—416no—
222C22⋊Q8—432no—
223C22⋊Q8—432no—
224C22⋊Q8—432no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
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⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D418yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4C2×D4116yes—
287C2×C22⋊C4#4118yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—44no—
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—28no—
307#95—28no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—41no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—41no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95C2×D4116yes—
320C24—22no—
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—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—28no—
340C24—216no—
341C24—44no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24—44no—
350C24—48no—
351C24—48no—
352C24—44no—
353C24—48no—
354C24—48no—
355C24—48no—
356C24—44no—
357C24—44no—
358C24—44no—
359C24—48no—
360C24C22×D414yes—
361C24C2×#4112yes—
362C24C2×#4114yes—
363C242−1+414yes—
364C242−1+414yes—
365C24C2511yescharacteristic
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
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—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—216no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—44no—
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—44no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—4128no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4C22×D418yes—
461C22×C4C22×D418yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C4H(5,𝔽2)12yes—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—416no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—416no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—88no—
480C22⋊C4—232no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C42—416no—
493C42—48no—
494C42—44no—
495C42—44no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
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—432no—
525C23—416no—
526C23—432no—
527C23—432no—
528C23—432no—
529C23—416no—
530C23—416no—
531C23—432no—
532C23—432no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23#54114yes—
560C23#54718yes—
561C23C23×D412yes—
562C23C22×#4111yescharacteristic
563C23C22×#4112yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)12yes—
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×2−1+412yes—
568C23C2×2−1+412yes—
569C23#58512yes—
570Q8—816no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—416no—
589C2×C4—432no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—432no—
614C2×C4—416no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4#53518yes—
644C2×C4#53814yes—
645C2×C4D4214yes—
646C2×C4#54618yes—
647C2×C4H(3,𝔽4)14yes—
648C22—28no—
649C22—24no—
650C22—28no—
651C22—28no—
652C22—28no—
653C22—464no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22#344411yescharacteristic
674C22C2×#53512yes—
675C22C2×#53811yescharacteristic
676C22C2×#54111yescharacteristic
677C22C2×D4211yescharacteristic
678C22C2×#54612yes—
679C22C2×#54712yes—
680C22D4×#4112yes—
681C22#348612yes—
682C22#348912yes—
683C22#349512yes—
684C22#350512yes—
685C22#352012yes—
686C22#352012yes—
687C22#352012yes—
688C22C2×H(3,𝔽4)11yescharacteristic
689C22#353811yescharacteristic
690C22#353812yes—
691C22#353911yescharacteristic
692C22#354112yes—
693C22#354112yes—
694C4—48no—
695C4—44no—
696C4—44no—
697C4—42no—
698C4—48no—
699C4—44no—
700C4—82no—
701C4—84no—
702C4—82no—
703C4—84no—
704C4#267018yes—
705C2—48no—
706C2—44no—
707C2#3508611yescharacteristic
708C2#3614312yes—
709C2#3738011yescharacteristic
710C2#3766312yes—
711C2#3891712yes—
712C2#3988712yes—
713C2#4566712yes—
714C2C2×#267011yescharacteristic
715C2#5900212yes—
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
2. C2×#2489C2semidirect
3. #16307C2semidirect
4. #15440C2semidirect
6. #15216C2semidirect
7. #15179C2semidirect
8. #15178C2semidirect
9. #15178C2semidirect
10. #14403C2semidirect
11. #13997C2semidirect
12. #13974C2semidirect
15. C22×C4⋊Q8C22semidirect
16. C22×#125C22semidirect
22. C2×#400C22semidirect
24. C2×#391C22semidirect
25. C2×#391C22semidirect
26. C2×#391C22semidirect
27. C2×#389C22semidirect
33. C2×#382C22semidirect
39. C2×C4×C4⋊C4C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
61. #1436C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#24892
#163072
#154402
#152822
#152162
#151792
#151782
#144032
#139972
#139742
C4×#3792