Finite Group Databasebeta

← #556887

Subgroups — #556887

Order 512.

2886 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#14671C214yesmaximal, complemented
3#14352C212yesmaximal, complemented
4#14344C212yesmaximal, complemented
5#14286C212yesmaximal, complemented
6#14278C212yesmaximal, complemented
7#14275C212yesmaximal, complemented
8#14271C212yesmaximal
9#14140C214yesmaximal, complemented
10#13899C212yesmaximal, complemented
11#13894C212yesmaximal, complemented
12#13891C212yesmaximal, complemented
13#13887C212yesmaximal, complemented
14C4×#380C211yesmaximal, characteristic, complemented
15C2×#1434C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C4×#121C4116yescomplemented
21C2×#390C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388C2214yes—
24C2×#388C2212yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2212yes—
28C2×#385C2214yes—
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42#1445C2214yes—
43#1445C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2214yes—
58#1437C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2214yes—
68#1434C2212yes—
69#1434C2212yes—
70#1434C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C2×#121—28no—
85C2×#121C2×C418yes—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4—48no—
89C2×C4×D4C2×C4116yes—
90C2×C4×D4C2×C418yescomplemented
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C42—42no—
110C22×C42—42no—
111C22×C42C2×C418yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119#390C2×C418yescomplemented
120#389C2×C418yescomplemented
121#388—48no—
122#388C2×C4116yescomplemented
123#386—48no—
124#386—48no—
125#386C2×C4116yes—
126#385—48no—
127#381—44no—
128#380—28no—
129#380—44no—
130#380C2×C418yes—
131C4×C4⋊C4—42no—
132C4×C4⋊C4—42no—
133C4×C4⋊C4C2×C418yescomplemented
134C4×C22⋊C4—44no—
135C4×C22⋊C4C2×C4116yescomplemented
136C2×#95—28no—
137C2×#95—42no—
138C2×#95—42no—
139C2×#95—44no—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—42no—
176C22×D4C22×C418yes—
177C22×D4C22×C414yes—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C22×C414yes—
210C23×C4C22×C414yes—
211C23×C4C22×C418yes—
212C23×C4C22×C414yes—
213C23×C4#4118yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#121—232no—
228C4×D4—416no—
229C4×D4—432no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C22×C418yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C414yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C22×C418yes—
274C2×C22⋊C4C22×C418yes—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C418yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—416no—
285C2×C42—44no—
286C2×C42—41no—
287C2×C42—41no—
288C2×C42—41no—
289C2×C42—41no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C22×C414yes—
295C2×C42C22×C414yes—
296C2×C42C22×C418yes—
297C2×C42C22×C414yes—
298C2×C42C22×C414yes—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—44no—
308#95—42no—
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#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C22×C418yes—
323#95C22×C418yes—
324#95C22×C418yes—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24C23×C414yes—
342C24C23×C412yes—
343C24C2×#4114yes—
344C24H(5,𝔽2)14yes—
345C24H(5,𝔽2)12yes—
346C242−1+412yes—
347C24C2511yescharacteristic
348C2×D4—216no—
349C2×D4—216no—
350C2×D4—216no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—416no—
357C2×D4—416no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—28no—
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—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4#117116yes—
456C22×C4C23×C412yes—
457C22×C4C23×C414yes—
458C22×C4C23×C414yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+412yes—
471C4⋊C4—28no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C22⋊C4—216no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C42—28no—
490C42—28no—
491C42—44no—
492C42—44no—
493C42—48no—
494C42—84no—
495C42—84no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23C2×#11714yes—
538C23#51814yes—
539C23#52014yes—
540C23#54018yes—
541C23#54218yes—
542C23C24×C411yescharacteristic
543C23C22×#4112yes—
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×2−1+411yescharacteristic
550C23C2×2−1+411yescharacteristic
551C23#58512yes—
552C23#58512yes—
553D4—416no—
554D4—432no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—416no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
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—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—816no—
632C2×C4—816no—
633C2×C4—816no—
634C2×C4C2×#11714yes—
635C2×C4C2×#11714yes—
636C2×C4#51814yes—
637C2×C4#52014yes—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—28no—
645C22—28no—
646C22—24no—
647C22—24no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22#231918yes—
660C22C22×#11711yescharacteristic
661C22C2×#51811yescharacteristic
662C22C2×#51811yescharacteristic
663C22C2×#52011yescharacteristic
664C22C2×#52011yescharacteristic
665C22#342811yescharacteristic
666C22#342811yescharacteristic
667C22C2×#54012yes—
668C22C2×#54212yes—
669C22#345912yes—
670C22#345912yes—
671C22#346012yes—
672C22#346112yes—
673C22#353412yes—
674C22#353412yes—
675C22#353611yescharacteristic
676C22#353611yescharacteristic
677C22#353711yescharacteristic
678C22#353811yescharacteristic
679C22#355812yes—
680C22#355912yes—
681C22#356012yes—
682C22#356412yes—
683C4—24no—
684C4—24no—
685C4—24no—
686C4—24no—
687C4—44no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—44no—
692C4—84no—
693C4—84no—
694C4—84no—
695C4—84no—
696C2—28no—
697C2—84no—
698C2C2×#231911yescharacteristic
699C2#3405711yescharacteristic
700C2#3406011yescharacteristic
701C2#3435811yescharacteristic
702C2#3436011yescharacteristic
703C2#3439811yescharacteristic
704C2#3440111yescharacteristic
705C2#3745912yes—
706C2#3746112yes—
707C2#3746312yes—
708C2#3746512yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14671C2semidirect
3. #14352C2semidirect
4. #14344C2semidirect
5. #14286C2semidirect
6. #14278C2semidirect
7. #14275C2semidirect
9. #14140C2semidirect
10. #13899C2semidirect
11. #13894C2semidirect
12. #13891C2semidirect
13. #13887C2semidirect
14. C4×#380C2semidirect
15. C2×#1434C2semidirect
20. C4×#121C4semidirect
90. C2×C4×D4C2×C4semidirect
119. #390C2×C4semidirect
120. #389C2×C4semidirect
122. #388C2×C4semidirect
133. C4×C4⋊C4C2×C4semidirect
135. 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×#1212
#146712
#143522
#143442
#142862
#142782
#142752
#142712
#141402
#138992
#138942
#138912
#138872
C4×#3802
C2×#14342