Finite Group Databasebeta

← #557846

Subgroups — #557846

Order 512.

2566 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#14699C212yesmaximal, complemented
3#14639C212yesmaximal, complemented
4#14603C212yesmaximal, complemented
5#14584C214yesmaximal, complemented
6#14401C212yesmaximal
7#14398C214yesmaximal, complemented
8#14130C212yesmaximal, complemented
9#14045C212yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13973C212yesmaximal, complemented
12#13907C212yesmaximal, complemented
13#13872C211yesmaximal, characteristic
14#13832C212yesmaximal
15C4×#390C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2487C4116yescomplemented
20C2×#400C2212yes—
21C2×#399C2212yes—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2214yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C43C2211yescharacteristic
43#1444C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2212yes—
58#1437C2212yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C2×C4×D4C2×C418yes—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4C2×C418yescomplemented
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119#400C2×C4116yescomplemented
120#399C2×C4116yes—
121#390—24no—
122#390Q818yes—
123#389—44no—
124#389—44no—
125#389—44no—
126#389C2×C4116yescomplemented
127#387—48no—
128#386—28no—
129#386C2×C4116yes—
130#385—48no—
131#385C2×C4116yescomplemented
132#380—48no—
133#380—44no—
134#380—42no—
135#380C2×C4116yes—
136#380C2×C418yescomplemented
137C4×C4⋊C4—42no—
138C4×C4⋊C4—44no—
139C4×C22⋊C4—44no—
140C4×C22⋊C4—44no—
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×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C43—42no—
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—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4C22×C414yes—
193C23×C4C22×C414yes—
194C23×C4C22×C418yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C4×D4—416no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
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—28no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C22×C4116yes—
248C2×C4⋊C4C2×Q814yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4C22×C414yes—
274C2×C22⋊C4C22×C414yes—
275C2×C22⋊C4C22×C4116yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C2×Q818yes—
278C2×C22⋊C4C2×Q818yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—41no—
290C2×C42—41no—
291C2×C42—42no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42C22×C414yes—
299C2×C42C2×Q814yes—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95C22×C4116yes—
322#95C22×C418yes—
323#95C22×C4116yes—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
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—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
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—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—48no—
426C22×C4—416no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—88no—
440C22×C4—84no—
441C22×C4C23×C412yes—
442C22×C4C23×C418yes—
443C22×C4C23×C412yes—
444C22×C4C23×C418yes—
445C22×C4C23×C412yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4C22×Q814yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q812yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C42−1+414yes—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—48no—
460C4⋊C4—416no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—84no—
479C22⋊C4—88no—
480C42—28no—
481C42—44no—
482C42—44no—
483C42—48no—
484C42—48no—
485C42—48no—
486C42—88no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C2×C4×Q814yes—
532C23C2×C4×Q814yes—
533C23C24×C411yescharacteristic
534C23C23×Q811yescharacteristic
535C23C22×#4111yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)12yes—
540C23C2×2−1+412yes—
541C23#58511yescharacteristic
542C23#58511yescharacteristic
543C23#58512yes—
544D4—416no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—816no—
625C2×C4—816no—
626C2×C4C2×C4×Q814yes—
627C2×C4#51814yes—
628C2×C4#52018yes—
629C2×C4#56014yes—
630C2×C4#56218yes—
631C2×C4#56314yes—
632C22—28no—
633C22—28no—
634C22—24no—
635C22—24no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22C22×C4×Q811yescharacteristic
655C22C2×#51811yescharacteristic
656C22C2×#52012yes—
657C22#342811yescharacteristic
658C22C4×H(5,𝔽2)12yes—
659C22#350112yes—
660C22#350111yescharacteristic
661C22#350111yescharacteristic
662C22#350312yes—
663C22#350511yescharacteristic
664C22#350511yescharacteristic
665C22#351512yes—
666C22#351912yes—
667C22C2×#56011yescharacteristic
668C22C2×#56212yes—
669C22C2×#56311yescharacteristic
670C22#354212yes—
671C22#354612yes—
672C22#354911yescharacteristic
673C22#355111yescharacteristic
674C22#355112yes—
675C22#355912yes—
676C22#356512yes—
677C4—24no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—48no—
682C4—44no—
683C4—48no—
684C4—44no—
685C4—44no—
686C4—82no—
687C4—84no—
688C4—84no—
689C4#237618yes—
690C2—44no—
691C2—44no—
692C2#3421211yescharacteristic
693C2#3433212yes—
694C2C2×#237611yescharacteristic
695C2#3441511yescharacteristic
696C2#3448312yes—
697C2#4549812yes—
698C2#4568812yes—
699C2#4615211yescharacteristic
700C2#4619011yescharacteristic
701C2#4633011yescharacteristic
702C2#4633811yescharacteristic
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. #14699C2semidirect
3. #14639C2semidirect
4. #14603C2semidirect
5. #14584C2semidirect
7. #14398C2semidirect
8. #14130C2semidirect
9. #14045C2semidirect
10. #13993C2semidirect
11. #13973C2semidirect
12. #13907C2semidirect
19. #2487C4semidirect
88. C22×C4⋊C4C2×C4semidirect
119. #400C2×C4semidirect
126. #389C2×C4semidirect
131. #385C2×C4semidirect
136. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#146992
#146392
#146032
#145842
#144012
#143982
#141302
#140452
#139932
#139732
#139072
#138722
#138322
C4×#3902