Finite Group Databasebeta

← #878105

Subgroups — #878105

Order 512.

2430 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16707C212yesmaximal, complemented
2#16339C212yesmaximal, complemented
3#16120C214yesmaximal, complemented
4#15516C214yesmaximal, complemented
5#15216C212yesmaximal, complemented
6#15216C212yesmaximal, complemented
7#15146C212yesmaximal, complemented
8#14697C212yesmaximal, complemented
9#14513C212yesmaximal
10#14440C212yesmaximal
11#14130C212yesmaximal, complemented
12#14054C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic
14#13872C211yesmaximal, characteristic
15C4×#386C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#400C2214yes—
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#386C2212yes—
25C2×#386C2211yescharacteristic
26C2×#386C2211yescharacteristic
27C2×#385C2214yes—
28C2×#385C2214yes—
29C2×#384C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1445C2214yes—
38#1444C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2212yes—
48#1441C2214yes—
49#1439C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1437C2212yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2214yes—
73C22⋊#95C2211yescharacteristic
74C4×#95C2212yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#125—28no—
82C2×#125—44no—
83C2×#125—44no—
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C2×C4×D4Q818yes—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C42—44no—
112C22×C42—42no—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118#400—48no—
119#390—28no—
120#389—44no—
121#389—44no—
122#386—24no—
123#386—24no—
124#386—28no—
125#385—48no—
126#385—48no—
127#384—42no—
128#380—28no—
129#380—28no—
130#380—44no—
131#380—42no—
132C4×C4⋊C4—42no—
133C4×C22⋊C4—42no—
134C4×C22⋊C4—44no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C22×D4—41no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4C2×Q814yes—
168C22×D4#4114yes—
169C22×D4#4114yes—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—41no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4C2×Q814yes—
194C23×C4C2×Q814yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208#125—432no—
209C4×D4—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×Q814yes—
244C2×C4⋊C4#4118yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C2×Q814yes—
274C2×C22⋊C4C2×Q814yes—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—416no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42C2×Q814yes—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—44no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24C22×Q812yes—
334C24C22×Q812yes—
335C24C2×#4112yes—
336C24C2×#4112yes—
337C24C2×#4112yes—
338C24C2×#4112yes—
339C24C2511yescharacteristic
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4#125116yescomplemented
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—84no—
438C22×C4—88no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4C22×Q812yes—
442C22×C4C22×Q812yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4H(5,𝔽2)14yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+414yes—
455C4⋊C4—216no—
456C4⋊C4—44no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—84no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C4⋊C4—84no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—84no—
476C22⋊C4—88no—
477C42—28no—
478C42—48no—
479C42—48no—
480C42—48no—
481C42—84no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C2×#12514yes—
527C23C2×#12514yes—
528C23C23×Q811yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4112yes—
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)12yes—
536C23C2×2−1+412yes—
537C23#58511yescharacteristic
538C23#58511yescharacteristic
539D4—416no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—816no—
622C2×C4C2×#12514yes—
623C2×C4#53818yes—
624C2×C4#54314yes—
625C2×C4#55418yes—
626C2×C4#55914yes—
627C2×C4H(3,𝔽4)14yes—
628C22—28no—
629C22—28no—
630C22—24no—
631C22—24no—
632C22—44no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22C22×#12511yescharacteristic
651C22C2×#53812yes—
652C22C2×#54311yescharacteristic
653C22#345811yescharacteristic
654C22C2×#55412yes—
655C22C2×#55911yescharacteristic
656C22#348112yes—
657C22#348212yes—
658C22#350312yes—
659C22#350511yescharacteristic
660C22#350511yescharacteristic
661C22#350511yescharacteristic
662C22#350511yescharacteristic
663C22#351512yes—
664C22#351712yes—
665C22#351912yes—
666C22#352012yes—
667C22C2×H(3,𝔽4)11yescharacteristic
668C22#353711yescharacteristic
669C22#353811yescharacteristic
670C22#354112yes—
671C22#354212yes—
672C22#354812yes—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—42no—
679C4—48no—
680C4—82no—
681C4—84no—
682C4—84no—
683C4—82no—
684C4—84no—
685C4#280418yes—
686C2—44no—
687C2—44no—
688C2#3502011yescharacteristic
689C2#3629012yes—
690C2#3730711yescharacteristic
691C2#3810412yes—
692C2#4014812yes—
693C2#4022112yes—
694C2#4564511yescharacteristic
695C2#4566711yescharacteristic
696C2C2×#280411yescharacteristic
697C2#5963111yescharacteristic
698C2#5963211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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
2. #16339C2semidirect
3. #16120C2semidirect
4. #15516C2semidirect
5. #15216C2semidirect
6. #15216C2semidirect
7. #15146C2semidirect
8. #14697C2semidirect
11. #14130C2semidirect
12. #14054C2semidirect
346. C2×D4#125semidirect

Maximal subgroups

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

subgroupindex
#167072
#163392
#161202
#155162
#152162
#151462
#146972
#145132
#144402
#141302
#140542
#139062
#138722
C4×#3862