Finite Group Databasebeta

← #875736

Subgroups — #875736

Order 512.

4446 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2697C214yesmaximal, complemented
2C2×#2486C211yesmaximal, characteristic, complemented
3C2×#2408C211yesmaximal, characteristic
4#17015C214yesmaximal, complemented
5#15561C214yesmaximal, complemented
6#15480C214yesmaximal, complemented
7#15150C212yesmaximal, complemented
8#15134C212yesmaximal, complemented
9#15130C212yesmaximal, complemented
10#14700C212yesmaximal, complemented
11#14137C212yesmaximal, complemented
12#13902C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#123C2212yes—
16C22×#121C2212yes—
17C22×#120C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic, complemented
20#2697—232no—
21#2486—28no—
22#2408—28no—
23C2×#398C2212yescomplemented
24C2×#397C2218yes—
25C2×#397C2212yes—
26C2×#396C2218yescomplemented
27C2×#394C2212yes—
28C2×#394C2218yescomplemented
29C2×#392C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2212yescomplemented
37C2×#388C2218yescomplemented
38C2×#386C2218yescomplemented
39C2×#384C2212yescomplemented
40C2×#381C2212yescomplemented
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic, complemented
44C2×#379C2218yes—
45C2×#379C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic, complemented
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yescomplemented
49C2×C43C2211yescharacteristic, complemented
50#1445C2218yescomplemented
51#1439C2218yescomplemented
52#1439C2218yescomplemented
53#1439C2218yescomplemented
54#1438C2218yes—
55#1437C2212yes—
56#1437C2218yescomplemented
57#1437C2212yescomplemented
58C4⋊#95C2212yescomplemented
59C4⋊#95C2212yescomplemented
60C4⋊#95C2212yescomplemented
61C4⋊#95C2212yescomplemented
62C26C2312yes—
63C23×D4—28no—
64C23×D4—24no—
65C23×D4—42no—
66C23×D4—42no—
67C23×D4—44no—
68C23×D4C2312yes—
69C23×D4C2314yes—
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C2×#123—216no—
76C2×#123—28no—
77C2×#121—28no—
78C2×#121—28no—
79C2×#121—44no—
80C2×#120—28no—
81C2×#120—216no—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C2×C4×D4D418yes—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2318yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2318yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2318yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115#398—28no—
116#397—232no—
117#397—28no—
118#396—232no—
119#394—28no—
120#394—232no—
121#392—28no—
122#390—28no—
123#390—28no—
124#390—44no—
125#390—44no—
126#390—44no—
127#390D4116yescomplemented
128#389—28no—
129#388—232no—
130#386—232no—
131#384—44no—
132#381—28no—
133#380—24no—
134#380—28no—
135#380—24no—
136#379—28no—
137#379—232no—
138C4×C4⋊C4—42no—
139C4×C22⋊C4—44no—
140C4×C22⋊C4—44no—
141C2×#95C2312yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2312yes—
145C2×#95C2318yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C43—42no—
152C25—24no—
153C25—28no—
154C25—28no—
155C25—44no—
156C25—42no—
157C25—42no—
158C25—48no—
159C25—44no—
160C25C2414yes—
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—216no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—432no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—44no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—416no—
182C22×D4—44no—
183C22×D4—48no—
184C22×D4—416no—
185C22×D4—432no—
186C22×D4—416no—
187C22×D4C2×D414yes—
188C22×D4C2×D418yes—
189C22×D4#4118yes—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215#123—432no—
216#121—432no—
217#120—432no—
218C4×D4—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—216no—
223C2×C4⋊C4—28no—
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—216no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D418yescomplemented
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D4116yes—
279C2×C22⋊C4C2×D4116yescomplemented
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—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42C2×D414yes—
293C2×C42C2×D418yescomplemented
294#95—28no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—216no—
299#95—216no—
300#95—216no—
301#95—44no—
302#95—44no—
303#95—42no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—28no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—28no—
324C24—28no—
325C24—28no—
326C24—416no—
327C24—48no—
328C24—432no—
329C24—48no—
330C24—416no—
331C24—48no—
332C24—48no—
333C24—48no—
334C24—816no—
335C24—88no—
336C24—88no—
337C24—88no—
338C24—816no—
339C24—88no—
340C24—88no—
341C24—816no—
342C24C22×D414yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C22×D414yes—
346C24C2×#4118yes—
347C24C2511yescharacteristic
348C2×D4—216no—
349C2×D4—216no—
350C2×D4—216no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—416no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—416no—
359C2×D4—48no—
360C2×D4—4128no—
361C2×D4—416no—
362C2×D4—432no—
363C2×D4—416no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—816no—
369C2×D4—88no—
370C2×D4#127116yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D418yes—
430C22×C4C22×D418yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D414yes—
434C22×C4C22×D412yes—
435C22×C4C2×#4114yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C4H(5,𝔽2)14yes—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—416no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—816no—
447C22⋊C4—416no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—416no—
451C22⋊C4—432no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—816no—
458C22⋊C4—816no—
459C22⋊C4—816no—
460C22⋊C4—88no—
461C42—28no—
462C42—48no—
463C42—48no—
464C42—48no—
465C42—48no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—216no—
485C23—28no—
486C23—216no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—44no—
492C23—44no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—432no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—44no—
513C23—44no—
514C23—48no—
515C23—44no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—832no—
521C23—816no—
522C23—832no—
523C23—832no—
524C23—864no—
525C23—816no—
526C23—832no—
527C23C2×#12714yes—
528C23C2×#12714yes—
529C23C23×D411yescharacteristic
530C23C23×D412yes—
531C23C22×#4112yes—
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23#58512yes—
536C23#58514yes—
537D4—416no—
538D4—432no—
539D4—432no—
540D4—832no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—816no—
591C2×C4C2×#12714yes—
592C2×C4#53414yes—
593C2×C4#53818yes—
594C2×C4#54618yes—
595C2×C4#55114yes—
596C2×C4#56014yes—
597C22—28no—
598C22—28no—
599C22—24no—
600C22—24no—
601C22—44no—
602C22—44no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—48no—
610C22—48no—
611C22—44no—
612C22—44no—
613C22—48no—
614C22—48no—
615C22—48no—
616C22—416no—
617C22—416no—
618C22—864no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—88no—
624C22—88no—
625C22C22×#12711yescharacteristic
626C22C2×#53411yescharacteristic
627C22#344911yescharacteristic
628C22C2×#53812yes—
629C22C2×#54612yes—
630C22D4×#4114yes—
631C22C2×#55111yescharacteristic
632C22#349614yes—
633C22#349612yes—
634C22#349612yes—
635C22#350012yes—
636C22#352114yes—
637C22C2×#56011yescharacteristic
638C22#354012yes—
639C22#355114yes—
640C22#355112yes—
641C4—24no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—48no—
646C4—48no—
647C4—84no—
648C4—84no—
649C4—84no—
650C4#269718yescomplemented
651C2—44no—
652C2—44no—
653C2—88no—
654C2#3500111yescharacteristic
655C2#3626012yes—
656C2#3741911yescharacteristic
657C2#3894712yes—
658C2#3940814yes—
659C2#4632812yes—
660C2C2×#269711yescharacteristic
661C2#5984312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2408C2
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×#2697C2semidirect
2. C2×#2486C2semidirect
4. #17015C2semidirect
5. #15561C2semidirect
6. #15480C2semidirect
7. #15150C2semidirect
8. #15134C2semidirect
9. #15130C2semidirect
10. #14700C2semidirect
11. #14137C2semidirect
12. #13902C2semidirect
13. C4×#390C2semidirect
19. C23×C4⋊C4C22semidirect
23. C2×#398C22semidirect
26. C2×#396C22semidirect
28. C2×#394C22semidirect
36. C2×#389C22semidirect
37. C2×#388C22semidirect
38. C2×#386C22semidirect
39. C2×#384C22semidirect
40. C2×#381C22semidirect
43. C2×#380C22semidirect
46. C2×C4×C4⋊C4C22semidirect
48. C2×C4×C22⋊C4C22semidirect
49. C2×C43C22semidirect
50. #1445C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
56. #1437C22semidirect
57. #1437C22semidirect
58. C4⋊#95C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
61. C4⋊#95C22semidirect
127. #390D4semidirect
242. C2×C4⋊C4C2×D4semidirect
279. C2×C22⋊C4C2×D4semidirect
293. C2×C42C2×D4semidirect
650. C4#2697semidirect

Maximal subgroups

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

subgroupindex
C2×#26972
C2×#24862
C2×#24082
#170152
#155612
#154802
#151502
#151342
#151302
#147002
#141372
#139022
C4×#3902