Finite Group Databasebeta

← #554960

Subgroups — #554960

Order 512.

4018 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2679C211yesmaximal, characteristic
2C2×#2321C212yesmaximal, complemented
3#14700C212yesmaximal, complemented
4#14589C212yesmaximal, complemented
5#14518C214yesmaximal, complemented
6#14445C212yesmaximal, complemented
7#14440C211yesmaximal, characteristic, complemented
8#14358C214yesmaximal, complemented
9#14357C212yesmaximal, complemented
10#14312C212yesmaximal, complemented
11#14199C214yesmaximal, complemented
12#13995C214yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C22×#120C2212yes—
15C22×#120C2214yes—
16C22×C4×D4C2211yescharacteristic
17C23×C22⋊C4C2212yescomplemented
18#2679C4116yescomplemented
19C2×#394C2214yes—
20C2×#394C2212yes—
21#2321—216no—
22C2×#390C2214yes—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#389C2212yescomplemented
30C2×#388C2214yescomplemented
31C2×#388C2214yescomplemented
32C2×#388C2214yescomplemented
33C2×#388C2214yescomplemented
34C2×#386C2212yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#385C2212yescomplemented
38C2×#385C2214yescomplemented
39C2×#379C2212yescomplemented
40C2×#379C2214yescomplemented
41C2×C4×C4⋊C4C2211yescharacteristic, complemented
42C2×C4×C22⋊C4C2212yescomplemented
43C2×C4×C22⋊C4C2214yescomplemented
44C2×C43C2211yescharacteristic, complemented
45#1444C2212yescomplemented
46#1441C2212yescomplemented
47#1440C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1438C2214yes—
54#1438C2212yes—
55#1438C2214yes—
56#1438C2214yes—
57#1437C2214yescomplemented
58#1437C2214yescomplemented
59#1437C2214yescomplemented
60#1437C2214yescomplemented
61#1437C2214yescomplemented
62#1436C2214yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65C4⋊#95C2211yescharacteristic, complemented
66C4⋊#95C2211yescharacteristic, complemented
67#1434C2212yescomplemented
68#1434C2214yescomplemented
69C26C2312yes—
70C23×D4C2314yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C2×#120—28no—
77C2×#120—28no—
78C2×#120—48no—
79C2×#120—48no—
80C2×#120C2×C4116yes—
81C2×#120C2×C4132yescomplemented
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C2×C4×D4D418yescomplemented
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—28no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#394C2×C4116yescomplemented
121#394C2×C4132yescomplemented
122#390—24no—
123#390—216no—
124#390—44no—
125#390—44no—
126#390—44no—
127#390C2×C418yes—
128#390C2×C4116yescomplemented
129#389—44no—
130#388—216no—
131#388—216no—
132#388—48no—
133#388—48no—
134#386—28no—
135#386—216no—
136#386—48no—
137#385—28no—
138#385—48no—
139#379—28no—
140#379—48no—
141C4×C4⋊C4—42no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—48no—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C43—42no—
159C25—24no—
160C25—24no—
161C25—42no—
162C25—42no—
163C25—42no—
164C25—42no—
165C25—44no—
166C25—44no—
167C25—44no—
168C25—44no—
169C25C22×C418yes—
170C25C2414yes—
171C25C2412yes—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4C22×C4116yes—
193C22×D4C22×C418yes—
194C22×D4C22×C418yes—
195C22×D4C22×C414yes—
196C22×D4C2×D414yes—
197C22×D4#4114yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2×D418yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219#120—264no—
220#120—464no—
221C4×D4—416no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C2×D414yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—432no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C22×C4116yes—
292C2×C22⋊C4C22×C418yes—
293C2×C22⋊C4C22×C4116yes—
294C2×C22⋊C4C22×C4116yes—
295C2×C22⋊C4C22×C4116yes—
296C2×C22⋊C4C2×D418yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—41no—
307C2×C42—42no—
308C2×C42—41no—
309C2×C42—41no—
310C2×C42—42no—
311C2×C42—41no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42C22×C414yes—
316C2×C42C2×D414yes—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—42no—
324#95—44no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95C22×C418yes—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—216no—
354C24—48no—
355C24—48no—
356C24—416no—
357C24—416no—
358C24—48no—
359C24—48no—
360C24—48no—
361C24—48no—
362C24—88no—
363C24—88no—
364C24—88no—
365C24—88no—
366C24—88no—
367C24—88no—
368C24—84no—
369C24—84no—
370C24—88no—
371C24—88no—
372C24—84no—
373C24—84no—
374C24—88no—
375C24—88no—
376C24C23×C418yes—
377C24C23×C414yes—
378C24C22×D414yes—
379C24C2×#4114yes—
380C24C2511yescharacteristic
381C2×D4—232no—
382C2×D4—216no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—416no—
390C2×D4—416no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—84no—
404C2×D4—88no—
405C2×D4—84no—
406C2×D4C4×D4116yes—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—216no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4C23×C418yes—
471C22×C4C23×C414yes—
472C22×C4C23×C412yes—
473C22×C4C23×C414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—88no—
491C4⋊C4—84no—
492C22⋊C4—216no—
493C22⋊C4—232no—
494C22⋊C4—232no—
495C22⋊C4—232no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C42—28no—
510C42—44no—
511C42—48no—
512C42—48no—
513C42—44no—
514C42—84no—
515C42—88no—
516C23—28no—
517C23—28no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—432no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—832no—
575C23—832no—
576C23—816no—
577C23—816no—
578C23—816no—
579C23—832no—
580C23—816no—
581C23—832no—
582C23—816no—
583C23—816no—
584C23C2×C4×D418yes—
585C23C24×C411yescharacteristic
586C23C23×D411yescharacteristic
587C23C22×#4111yescharacteristic
588C23C2×H(5,𝔽2)12yes—
589C23C2×H(5,𝔽2)12yes—
590C23C2×H(5,𝔽2)11yescharacteristic
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×H(5,𝔽2)12yes—
593C23#58511yescharacteristic
594C23#58511yescharacteristic
595C23#58512yes—
596D4—416no—
597D4—816no—
598D4—832no—
599D4—816no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
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—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—816no—
665C2×C4—816no—
666C2×C4C2×C4×D414yes—
667C2×C4#51818yes—
668C2×C4#53414yes—
669C22—216no—
670C22—28no—
671C22—44no—
672C22—48no—
673C22—44no—
674C22—48no—
675C22—44no—
676C22—48no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—48no—
681C22—48no—
682C22—48no—
683C22—48no—
684C22—48no—
685C22—48no—
686C22—48no—
687C22—864no—
688C22—84no—
689C22—84no—
690C22—84no—
691C22—84no—
692C22—84no—
693C22—84no—
694C22—84no—
695C22—84no—
696C22C22×C4×D411yescharacteristic
697C22C2×#51812yes—
698C22C4×H(5,𝔽2)12yes—
699C22C4×H(5,𝔽2)11yescharacteristic
700C22C4×H(5,𝔽2)11yescharacteristic
701C22C2×#53411yescharacteristic
702C22#344911yescharacteristic
703C22#345911yescharacteristic
704C22#345911yescharacteristic
705C22#349612yes—
706C22#349812yes—
707C22#351412yes—
708C22#351912yes—
709C22#353412yes—
710C22#353712yes—
711C22#354311yescharacteristic
712C22#354511yescharacteristic
713C22#354512yes—
714C22#354612yes—
715C22#354911yescharacteristic
716C22#354911yescharacteristic
717C22#355812yes—
718C22#356012yes—
719C4—24no—
720C4—24no—
721C4—28no—
722C4—48no—
723C4—44no—
724C4—44no—
725C4—42no—
726C4—44no—
727C4—84no—
728C4—82no—
729C4—84no—
730C2—48no—
731C2—88no—
732C2C4×#53411yescharacteristic
733C2#3431012yes—
734C2#3443712yes—
735C2#3449311yescharacteristic
736C2#3452411yescharacteristic
737C2#3712911yescharacteristic
738C2#3713212yes—
739C2#3715011yescharacteristic
740C2#3715412yes—
741C2#3724911yescharacteristic
742C2#3729711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC2×#2679C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2321C2semidirect
3. #14700C2semidirect
4. #14589C2semidirect
5. #14518C2semidirect
6. #14445C2semidirect
7. #14440C2semidirect
8. #14358C2semidirect
9. #14357C2semidirect
10. #14312C2semidirect
11. #14199C2semidirect
12. #13995C2semidirect
13. C4×#390C2semidirect
17. C23×C22⋊C4C22semidirect
18. #2679C4semidirect
29. C2×#389C22semidirect
30. C2×#388C22semidirect
31. C2×#388C22semidirect
32. C2×#388C22semidirect
33. C2×#388C22semidirect
37. C2×#385C22semidirect
38. C2×#385C22semidirect
39. C2×#379C22semidirect
40. C2×#379C22semidirect
41. C2×C4×C4⋊C4C22semidirect
42. C2×C4×C22⋊C4C22semidirect
43. C2×C4×C22⋊C4C22semidirect
44. C2×C43C22semidirect
45. #1444C22semidirect
46. #1441C22semidirect
47. #1440C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
57. #1437C22semidirect
58. #1437C22semidirect
59. #1437C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. #1436C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
81. C2×#120C2×C4semidirect
84. C2×C4×D4D4semidirect
120. #394C2×C4semidirect
121. #394C2×C4semidirect
128. #390C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26792
C2×#23212
#147002
#145892
#145182
#144452
#144402
#143582
#143572
#143122
#141992
#139952
C4×#3902