Finite Group Databasebeta

← #2746514

Subgroups — #2746514

Order 512.

2798 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2826C212yesmaximal, complemented
2#17196C212yesmaximal, complemented
3#17141C214yesmaximal, complemented
4#17109C212yesmaximal, complemented
5#16288C212yesmaximal, complemented
6#16283C214yesmaximal, complemented
7#15152C212yesmaximal, complemented
8#15138C212yesmaximal, complemented
9#15134C212yesmaximal, complemented
10#15130C212yesmaximal, complemented
11#15127C212yesmaximal, complemented
12#15124C211yesmaximal, characteristic
13#14137C212yesmaximal, complemented
14#14130C211yesmaximal, characteristic, complemented
15C4×#399C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19#2826—216no—
20C2×#400C2214yes—
21C2×#399C2211yescharacteristic
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#396C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#392C2212yes—
30C2×#392C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385C2212yes—
40C2×#384C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46#1445C2211yescharacteristic
47#1445C2212yes—
48#1445C2214yes—
49#1444C2211yescharacteristic
50#1444C2212yes—
51#1442C2214yes—
52#1442C2212yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2211yescharacteristic
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66#1438C2212yes—
67#1438C2214yes—
68#1438C2214yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×D4C2312yes—
77C23×D4C2314yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×#127—42no—
83C2×#127—44no—
84C2×#121—28no—
85C2×#121—44no—
86C2×#121—44no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120#400—216no—
121#399D418yescomplemented
122#397—216no—
123#396—216no—
124#396—48no—
125#394—216no—
126#394—216no—
127#394—48no—
128#394—48no—
129#392—28no—
130#392—44no—
131#390—44no—
132#390—44no—
133#390—44no—
134#390—44no—
135#390—44no—
136#389—44no—
137#386—216no—
138#386—48no—
139#385—28no—
140#384—44no—
141#380—24no—
142C4×C4⋊C4—44no—
143C4×C4⋊C4—42no—
144C4×C22⋊C4—42no—
145C4×C22⋊C4—44no—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C25C2414yes—
169C25C2411yescharacteristic
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—44no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4#4114yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217#127—416no—
218#121—432no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—416no—
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⋊C4C2×D418yes—
250C2×C4⋊C4C2×D414yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
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—28no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—44no—
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⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D418yes—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—41no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—41no—
321#95—41no—
322#95—42no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95C2×D414yes—
331C24—22no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
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—24no—
354C24C22×D412yes—
355C24C2×#4112yes—
356C24H(5,𝔽2)12yes—
357C24C2511yescharacteristic
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—416no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—84no—
371C2×D4—88no—
372C2×D4—84no—
373C2×D4—88no—
374C2×D4—88no—
375C2×D4—88no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—84no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—84no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—216no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—84no—
456C22×C4—88no—
457C22×C4—84no—
458C22×C4—88no—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)14yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C42−1+412yes—
470C22×C42−1+412yes—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C42—48no—
499C42—44no—
500C42—44no—
501C42—48no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
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#55014yes—
560C23C23×D411yescharacteristic
561C23C22×#4111yescharacteristic
562C23C2×H(5,𝔽2)12yes—
563C23C2×H(5,𝔽2)12yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×H(5,𝔽2)12yes—
567C23C2×2−1+411yescharacteristic
568C23C2×2−1+411yescharacteristic
569C23#58512yes—
570C23#58511yescharacteristic
571D4—816no—
572D4—816no—
573D4—832no—
574D4—816no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—416no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—816no—
651C2×C4#53818yes—
652C2×C4#53914yes—
653C2×C4#56014yes—
654C2×C4#56014yes—
655C2×C4#56218yes—
656C22—28no—
657C22—24no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22C2×#53812yes—
683C22C2×#53911yescharacteristic
684C22C2×#55011yescharacteristic
685C22#349612yes—
686C22#349611yescharacteristic
687C22#349812yes—
688C22#349911yescharacteristic
689C22#350612yes—
690C22#350711yescharacteristic
691C22C2×#56011yescharacteristic
692C22C2×#56011yescharacteristic
693C22C2×#56212yes—
694C22#353612yes—
695C22#353611yescharacteristic
696C22#353711yescharacteristic
697C22#353712yes—
698C22#354612yes—
699C22#354712yes—
700C22#355112yes—
701C22#355111yescharacteristic
702C22#355412yes—
703C22#355812yes—
704C22#355911yescharacteristic
705C4—42no—
706C4—44no—
707C4—42no—
708C4—44no—
709C4—42no—
710C4—44no—
711C4—42no—
712C4—84no—
713C4—84no—
714C4—84no—
715C4—81no—
716C4—82no—
717C4#282618yescomplemented
718C2—44no—
719C2—164no—
720C2C2×#282611yescharacteristic
721C2#3938312yes—
722C2#3938411yescharacteristic
723C2#4252012yes—
724C2#4252211yescharacteristic
725C2#4548612yes—
726C2#4548711yescharacteristic
727C2#4632811yescharacteristic
728C2#5296011yescharacteristic
729C2#5297012yes—
730C2#5298811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2826C2semidirect
2. #17196C2semidirect
3. #17141C2semidirect
4. #17109C2semidirect
5. #16288C2semidirect
6. #16283C2semidirect
7. #15152C2semidirect
8. #15138C2semidirect
9. #15134C2semidirect
10. #15130C2semidirect
11. #15127C2semidirect
13. #14137C2semidirect
14. #14130C2semidirect
15. C4×#399C2semidirect
121. #399D4semidirect
717. C4#2826semidirect

Maximal subgroups

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

subgroupindex
C2×#28262
#171962
#171412
#171092
#162882
#162832
#151522
#151382
#151342
#151302
#151272
#151242
#141372
#141302
C4×#3992