Finite Group Databasebeta

← #533695

Subgroups — #533695

Order 512.

2758 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#14592C212yesmaximal, complemented
3#14513C212yesmaximal
4#14245C214yesmaximal, complemented
5#14215C212yesmaximal, complemented
6#14075C214yesmaximal, complemented
7#13993C212yesmaximal, complemented
8#13993C212yesmaximal, complemented
9#13981C212yesmaximal, complemented
10#13906C211yesmaximal, characteristic
11#13824C212yesmaximal
12#13818C212yesmaximal, complemented
13C22⋊C4×C4⋊C4C212yesmaximal, complemented
14C4×#380C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2487C4116yescomplemented
20C2×#400C2212yes—
21C2×#399C2212yes—
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#387C2214yes—
26C2×#386C2211yescharacteristic
27C2×#386C2212yes—
28C2×#386C2211yescharacteristic
29C2×#385C2212yes—
30C2×#384C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2214yes—
41C2×C4×C22⋊C4C2214yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1439C2212yes—
48#1439C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1437C2214yes—
52#1437C2214yes—
53#1437C2212yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2214yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2214yes—
73C4×#95C2214yes—
74C4×#95C2212yes—
75C4×#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C2×C4×D4C2×C418yes—
85C22×C4⋊C4—42no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4C2×C418yescomplemented
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C42—28no—
109C22×C42—42no—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2314yes—
117C22×C42C2314yes—
118#400C2×C4116yescomplemented
119#399C2×C4116yes—
120#390—28no—
121#389—28no—
122#389C2×C4116yescomplemented
123#387—48no—
124#386—24no—
125#386C2×C4116yes—
126#386Q818yes—
127#385C2×C4116yescomplemented
128#384—42no—
129#380—28no—
130#380—28no—
131#380C2×C4116yes—
132#380C2×C418yescomplemented
133C4×C4⋊C4—28no—
134C4×C4⋊C4—42no—
135C4×C4⋊C4—44no—
136C4×C4⋊C4—44no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—48no—
139C4×C22⋊C4—48no—
140C4×C22⋊C4—42no—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4C22×C414yes—
173C22×D4C2×Q814yes—
174C22×D4#4114yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—42no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C22×C414yes—
197C23×C4C22×C414yes—
198C23×C4C22×C418yes—
199C23×C4C2×Q814yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C4×D4—416no—
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—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—416no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C22×C418yes—
239C2×C4⋊C4C22×C418yes—
240C2×C4⋊C4C22×C418yes—
241C2×C4⋊C4C22×C414yes—
242C2×C4⋊C4C22×C4116yes—
243C2×C4⋊C4#4118yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4C22×C414yes—
269C2×C22⋊C4C22×C414yes—
270C2×C22⋊C4C22×C4116yes—
271C2×C22⋊C4C22×C418yes—
272C2×C22⋊C4C2×Q818yes—
273C2×C22⋊C4C2×Q814yes—
274C2×C42—24no—
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—28no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—416no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42C22×C414yes—
298C2×C42#4118yes—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—42no—
313#95—44no—
314#95—42no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C22×C4116yes—
321#95C22×C418yes—
322#95C22×C4116yes—
323#95C2×Q818yes—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24C23×C412yes—
341C24C23×C412yes—
342C24C22×Q812yes—
343C24C22×Q812yes—
344C24C2×#4112yes—
345C24C2×#4112yes—
346C24C2511yescharacteristic
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4C4×Q8116yes—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—416no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4C23×C412yes—
444C22×C4C23×C418yes—
445C22×C4C23×C412yes—
446C22×C4C23×C418yes—
447C22×C4C23×C412yes—
448C22×C4C23×C412yes—
449C22×C4C23×C412yes—
450C22×C4C22×Q814yes—
451C22×C4C22×Q812yes—
452C22×C4C22×Q814yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4118yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—48no—
466C4⋊C4—416no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—84no—
470C4⋊C4—88no—
471C4⋊C4—84no—
472C22⋊C4—216no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—44no—
482C42—28no—
483C42—216no—
484C42—48no—
485C42—48no—
486C42—88no—
487C42—84no—
488C42—88no—
489C42—88no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23C2×C4×Q814yes—
535C23C2×C4×Q814yes—
536C23C24×C411yescharacteristic
537C23C23×Q811yescharacteristic
538C23C22×#4111yescharacteristic
539C23C22×#4112yes—
540C23C22×#4112yes—
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×H(5,𝔽2)12yes—
544C23#58511yescharacteristic
545C23#58511yescharacteristic
546C23#58512yes—
547D4—416no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—416no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—816no—
629C2×C4C2×C4×Q814yes—
630C2×C4C4×#4118yes—
631C2×C4#51814yes—
632C2×C4#55114yes—
633C2×C4#55318yes—
634C2×C4#55418yes—
635C2×C4#55914yes—
636C22—28no—
637C22—28no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22C22×C4×Q811yescharacteristic
659C22C2×C4×#4112yes—
660C22C2×#51811yescharacteristic
661C22#342811yescharacteristic
662C22C4×H(5,𝔽2)12yes—
663C22C2×#55111yescharacteristic
664C22C2×#55312yes—
665C22C2×#55412yes—
666C22#347512yes—
667C22Q8×#4112yes—
668C22C2×#55911yescharacteristic
669C22#348112yes—
670C22#349312yes—
671C22#349412yes—
672C22#350111yescharacteristic
673C22#350111yescharacteristic
674C22#350511yescharacteristic
675C22#350511yescharacteristic
676C22#351712yes—
677C22#352112yes—
678C22#353711yescharacteristic
679C22#354011yescharacteristic
680C22#354612yes—
681C4—28no—
682C4—24no—
683C4—44no—
684C4—44no—
685C4—48no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—48no—
690C4—48no—
691C4—82no—
692C4—84no—
693C4#237418yes—
694C2—44no—
695C2—44no—
696C2C2×#237411yescharacteristic
697C2C4×#55412yes—
698C2#3421211yescharacteristic
699C2#3430112yes—
700C2#3441111yescharacteristic
701C2#5919911yescharacteristic
702C2#5920211yescharacteristic
703C2#5921412yes—
704C2#5921512yes—
705C2#5962911yescharacteristic
706C2#5963211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14592C2semidirect
4. #14245C2semidirect
5. #14215C2semidirect
6. #14075C2semidirect
7. #13993C2semidirect
8. #13993C2semidirect
9. #13981C2semidirect
12. #13818C2semidirect
13. C22⋊C4×C4⋊C4C2semidirect
14. C4×#380C2semidirect
19. #2487C4semidirect
87. C22×C4⋊C4C2×C4semidirect
118. #400C2×C4semidirect
122. #389C2×C4semidirect
127. #385C2×C4semidirect
132. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#145922
#145132
#142452
#142152
#140752
#139932
#139812
#139062
#138242
#138182
C22⋊C4×C4⋊C42
C4×#3802
C4×#3862