Finite Group Databasebeta

← #560059

Subgroups — #560059

Order 512.

2510 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2#14682C212yesmaximal, complemented
3#14669C212yesmaximal, complemented
4#14508C212yesmaximal, complemented
5#14505C212yesmaximal
6#14485C214yesmaximal, complemented
7#14484C214yesmaximal, complemented
8#13909C212yesmaximal, complemented
9#13906C211yesmaximal, characteristic
10#13885C212yesmaximal, complemented
11#13881C212yesmaximal, complemented
12#13875C212yesmaximal, complemented
13#13875C212yesmaximal, complemented
14#13872C211yesmaximal, characteristic
15#13832C212yesmaximal
16C22×#123C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C4×#123C4116yescomplemented
21C2×#390C2212yes—
22C2×#388C2212yes—
23C2×#387C2212yes—
24C2×#386C2212yes—
25C2×#386C2211yescharacteristic
26C2×#385C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2212yes—
29C2×#382C2212yes—
30C2×#382C2212yes—
31C2×#381C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41#1445C2214yes—
42#1443C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1439C2212yes—
55#1437C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C2×#123—24no—
83C2×#123—24no—
84C2×#123C2×C418yes—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C2×C4×D4C2×C418yes—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C42—42no—
111C22×C42—42no—
112C22×C42C2×C418yescomplemented
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2312yes—
119#390—28no—
120#388C2×C4116yes—
121#387C2×C4116yescomplemented
122#386—28no—
123#386C2×C418yes—
124#385C2×C418yescomplemented
125#384—42no—
126#382—44no—
127#382—44no—
128#382—44no—
129#381C2×C418yescomplemented
130#380—44no—
131#380—44no—
132#380—44no—
133#380—42no—
134C4×C4⋊C4—42no—
135C4×C4⋊C4—44no—
136C4×C4⋊C4C2×C4116yescomplemented
137C4×C22⋊C4C2×C4116yes—
138C4×C22⋊C4C2×C418yescomplemented
139C2×#95C2314yes—
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×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4C22×C414yes—
173C22×D4#4114yes—
174C22×D4#4114yes—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4C22×C414yes—
199C23×C4C22×C414yes—
200C23×C4C22×C414yes—
201C23×C4C22×C414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215#123—232no—
216C4×D4—416no—
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—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—24no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C418yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—42no—
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—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4C22×C418yes—
271C2×C22⋊C4C22×C414yes—
272C2×C22⋊C4C22×C414yes—
273C2×C22⋊C4C22×C414yes—
274C2×C22⋊C4C22×C418yes—
275C2×C22⋊C4C22×C414yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—416no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—44no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C22×C414yes—
292C2×C42C22×C418yes—
293C2×C42C22×C418yes—
294C2×C42C22×C418yes—
295#95—28no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95C22×C418yes—
318#95C22×C418yes—
319#95C22×C418yes—
320#95C22×C418yes—
321#95C22×C418yes—
322C24—22no—
323C24—21no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24C23×C412yes—
339C24C23×C412yes—
340C24C2×#4112yes—
341C24C2×#4112yes—
342C24C2×#4112yes—
343C24C2×#4112yes—
344C24C2511yescharacteristic
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4#117116yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
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—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—84no—
444C22×C4—88no—
445C22×C4—84no—
446C22×C4—88no—
447C22×C4C23×C414yes—
448C22×C4C23×C412yes—
449C22×C4C23×C414yes—
450C22×C4C23×C414yes—
451C22×C4C23×C412yes—
452C22×C4C23×C414yes—
453C22×C4C23×C412yes—
454C22×C4C23×C412yes—
455C22×C4C23×C412yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C42−1+414yes—
462C4⋊C4—216no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C22⋊C4—28no—
473C22⋊C4—216no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—84no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—216no—
483C42—28no—
484C42—216no—
485C42—48no—
486C42—84no—
487C42—88no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C2×#11714yes—
533C23C2×#11714yes—
534C23C24×C411yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4111yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)12yes—
540C23C2×2−1+412yes—
541C23#58511yescharacteristic
542C23#58511yescharacteristic
543C23#58512yes—
544C23#58512yes—
545D4—416no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
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—816no—
628C2×C4C2×#11714yes—
629C2×C4#51814yes—
630C2×C4#52018yes—
631C2×C4#56014yes—
632C2×C4#56218yes—
633C2×C4#56314yes—
634C22—24no—
635C22—24no—
636C22—28no—
637C22—28no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22C22×#11711yescharacteristic
657C22C2×#51811yescharacteristic
658C22C2×#52012yes—
659C22#342812yes—
660C22#342811yescharacteristic
661C22#350511yescharacteristic
662C22#350511yescharacteristic
663C22#350511yescharacteristic
664C22#350511yescharacteristic
665C22#351412yes—
666C22#351412yes—
667C22#351512yes—
668C22#351512yes—
669C22C2×#56011yescharacteristic
670C22C2×#56212yes—
671C22C2×#56311yescharacteristic
672C22#353912yes—
673C22#354011yescharacteristic
674C22#354011yescharacteristic
675C22#356212yes—
676C22#356212yes—
677C22#356512yes—
678C22#356512yes—
679C4—24no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—84no—
688C4—84no—
689C4—84no—
690C4—84no—
691C4#237618yes—
692C2—44no—
693C2—44no—
694C2#3418511yescharacteristic
695C2#3419112yes—
696C2C2×#237611yescharacteristic
697C2#3436111yescharacteristic
698C2#3438512yes—
699C2#4568212yes—
700C2#4568311yescharacteristic
701C2#4568411yescharacteristic
702C2#4569512yes—
703C2#4569611yescharacteristic
704C2#4569711yescharacteristic
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. #14682C2semidirect
3. #14669C2semidirect
4. #14508C2semidirect
6. #14485C2semidirect
7. #14484C2semidirect
8. #13909C2semidirect
10. #13885C2semidirect
11. #13881C2semidirect
12. #13875C2semidirect
13. #13875C2semidirect
20. C4×#123C4semidirect
112. C22×C42C2×C4semidirect
121. #387C2×C4semidirect
124. #385C2×C4semidirect
129. #381C2×C4semidirect
136. C4×C4⋊C4C2×C4semidirect
138. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
#146822
#146692
#145082
#145052
#144852
#144842
#139092
#139062
#138852
#138812
#138752
#138722
#138322