Finite Group Databasebeta

← #4100375

Subgroups — #4100375

Order 512.

3880 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2673C214yesmaximal, complemented
2#17083C212yesmaximal, complemented
3#17082C211yesmaximal, characteristic, complemented
4#15980C211yesmaximal, characteristic, complemented
5#15658C214yesmaximal, complemented
6#15310C212yesmaximal, complemented
7#15278C212yesmaximal, complemented
8#15270C214yesmaximal, complemented
9#14926C212yesmaximal, complemented
10#14722C214yesmaximal, complemented
11#14571C212yesmaximal, complemented
12#14175C212yesmaximal, complemented
13#14152C211yesmaximal, characteristic, complemented
14C22×#127C2212yes—
15C22×#123C2214yes—
16C22×#121C2212yes—
17C22×#121C2211yescharacteristic
18C22×#121C2214yes—
19C22×C4×D4C2212yes—
20#2673—232no—
21C2×#394C2214yes—
22C2×#394C2214yes—
23C2×#394C2212yes—
24C2×#394C2214yes—
25C2×#392C2214yes—
26C2×#392C2214yes—
27C2×#392C2214yes—
28C2×#392C2212yes—
29C2×#392C2214yes—
30C2×#392C2212yes—
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1445C2211yescharacteristic
41#1445C2211yescharacteristic
42#1444C2212yes—
43#1444C2212yes—
44#1442C2214yes—
45#1442C2212yes—
46#1441C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2212yes—
53#1438C2214yes—
54#1438C2211yescharacteristic
55#1438C2211yescharacteristic
56#1438C2212yes—
57#1438C2212yes—
58#1438C2212yes—
59#1438C2212yes—
60#1437C2212yes—
61#1437C2214yes—
62#1437C2214yes—
63#1436C2212yes—
64#1434C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C23×D4C2312yes—
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2314yes—
72C23×D4C2314yes—
73C23×D4C2314yes—
74C23×D4C2311yescharacteristic
75C23×D4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2314yes—
79C2×#127—216no—
80C2×#127—44no—
81C2×#123—216no—
82C2×#123—48no—
83C2×#123—48no—
84C2×#121—216no—
85C2×#121—24no—
86C2×#121—216no—
87C2×#121—42no—
88C2×#121—48no—
89C2×#121—48no—
90C2×#121—42no—
91C2×#121—44no—
92C2×C4×D4—44no—
93C2×C4×D4—48no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
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⋊C4C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116#394—28no—
117#394—216no—
118#394—216no—
119#394—48no—
120#392—216no—
121#392—216no—
122#392—216no—
123#392—48no—
124#392—44no—
125#392D4116yescomplemented
126#388—216no—
127#387—216no—
128#386—216no—
129#386—216no—
130#386—216no—
131#386—48no—
132#386—48no—
133C4×C22⋊C4—42no—
134C2×#95—42no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2311yescharacteristic
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C25C2412yes—
156C25C2414yes—
157C25C2411yescharacteristic
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—24no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—22no—
170C22×D4—24no—
171C22×D4—28no—
172C22×D4—42no—
173C22×D4—44no—
174C22×D4—42no—
175C22×D4—44no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4C2×D4116yes—
202C22×D4C2×D418yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2414yes—
229#127—432no—
230#123—464no—
231#121—464no—
232#121—416no—
233#121—432no—
234C4×D4—432no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4#4114yes—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
292C2×C22⋊C4C2×D418yes—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42—42no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—42no—
306#95—41no—
307#95—41no—
308#95—42no—
309#95—42no—
310#95—41no—
311#95—41no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—44no—
316#95C2×D418yes—
317C24—21no—
318C24—24no—
319C24—22no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24C22×D418yes—
347C24C22×D414yes—
348C24H(5,𝔽2)12yes—
349C24C2511yescharacteristic
350C2×D4—232no—
351C2×D4—48no—
352C2×D4—416no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—44no—
363C2×D4—416no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—416no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—84no—
371C2×D4—88no—
372C2×D4—84no—
373C2×D4—88no—
374C2×D4—84no—
375C2×D4—88no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4—88no—
393C2×D4—84no—
394C2×D4—88no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—48no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D418yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C4H(5,𝔽2)18yes—
464C22×C4H(5,𝔽2)12yes—
465C4⋊C4—232no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—48no—
486C42—48no—
487C23—24no—
488C23—28no—
489C23—22no—
490C23—24no—
491C23—28no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—41no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
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—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23D4214yes—
559C23C23×D412yes—
560C23C22×#4111yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)12yes—
563C23C2×H(5,𝔽2)14yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23#58512yes—
566C23#58512yes—
567D4—432no—
568D4—832no—
569D4—816no—
570D4—816no—
571D4—816no—
572D4—832no—
573D4—832no—
574D4—88no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—88no—
630C2×C4—832no—
631C2×C4—816no—
632C22—24no—
633C22—28no—
634C22—44no—
635C22—48no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—48no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—48no—
645C22—48no—
646C22—48no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—82no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22C2×D4211yescharacteristic
665C22D4×#4112yes—
666C22#348414yes—
667C22#349011yescharacteristic
668C22#349614yes—
669C22#349612yes—
670C22#350812yes—
671C22#351712yes—
672C22#352911yescharacteristic
673C22#353212yes—
674C22#353412yes—
675C22#353914yes—
676C22#354214yes—
677C22#354312yes—
678C22#354511yescharacteristic
679C22#354911yescharacteristic
680C4—44no—
681C4—44no—
682C4—44no—
683C4—48no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—48no—
688C4—84no—
689C4—82no—
690C2—44no—
691C2—88no—
692C2—162no—
693C2#4299612yes—
694C2#4390014yes—
695C2#4547612yes—
696C2#4631811yescharacteristic
697C2#4703111yescharacteristic
698C2#4779412yes—
699C2#5485112yes—
700C2#5981411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2673C2semidirect
2. #17083C2semidirect
3. #17082C2semidirect
4. #15980C2semidirect
5. #15658C2semidirect
6. #15310C2semidirect
7. #15278C2semidirect
8. #15270C2semidirect
9. #14926C2semidirect
10. #14722C2semidirect
11. #14571C2semidirect
12. #14175C2semidirect
13. #14152C2semidirect
125. #392D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26732
#170832
#170822
#159802
#156582
#153102
#152782
#152702
#149262
#147222
#145712
#141752
#141522