Finite Group Databasebeta

← #4100357

Subgroups — #4100357

Order 512.

4134 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2678C214yesmaximal, complemented
2#17135C211yesmaximal, characteristic, complemented
3#17118C212yesmaximal, complemented
4#15947C211yesmaximal, characteristic, complemented
5#15782C212yesmaximal, complemented
6#15272C214yesmaximal, complemented
7#15268C214yesmaximal, complemented
8#15257C212yesmaximal, complemented
9#14976C214yesmaximal, complemented
10#14926C212yesmaximal, complemented
11#14358C212yesmaximal, complemented
12#14228C212yesmaximal, complemented
13#14152C211yesmaximal, characteristic, complemented
14C22×#123C2214yescomplemented
15C22×#121C2212yes—
16C22×#121C2211yescharacteristic
17C22×#120C2212yes—
18C22×#120C2214yes—
19C22×#120C2212yescomplemented
20#2678—232no—
21C2×#397C2214yescomplemented
22C2×#396C2214yescomplemented
23C2×#396C2214yescomplemented
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#392C2214yes—
27C2×#392C2212yes—
28C2×#390C2214yes—
29C2×#390C2212yes—
30C2×#390C2214yescomplemented
31C2×#388C2214yescomplemented
32C2×#388C2214yescomplemented
33C2×#388C2214yescomplemented
34C2×#386C2214yescomplemented
35C2×#386C2214yescomplemented
36C2×#379C2214yescomplemented
37C2×#379C2212yescomplemented
38C2×C4×C22⋊C4C2211yescharacteristic, complemented
39C22×#95C2211yescharacteristic, complemented
40#1445C2211yescharacteristic, complemented
41#1445C2211yescharacteristic, complemented
42#1444C2212yescomplemented
43#1444C2212yescomplemented
44#1444C2212yescomplemented
45#1441C2214yescomplemented
46#1440C2212yescomplemented
47#1440C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2212yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1438C2211yescharacteristic
54#1438C2212yes—
55#1438C2214yes—
56#1438C2211yescharacteristic
57#1438C2212yes—
58#1438C2214yes—
59#1438C2212yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2212yescomplemented
63#1437C2214yescomplemented
64#1437C2214yescomplemented
65#1436C2214yescomplemented
66C22⋊#95C2212yescomplemented
67C4×#95C2212yescomplemented
68C26C2312yes—
69C23×D4C2312yes—
70C23×D4C2314yes—
71C23×D4C2312yes—
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C2×#123—216no—
78C2×#123—48no—
79C2×#123—48no—
80C2×#121—216no—
81C2×#121—24no—
82C2×#121—42no—
83C2×#121—42no—
84C2×#121—44no—
85C2×#120—216no—
86C2×#120—216no—
87C2×#120—28no—
88C2×#120—48no—
89C2×#120—48no—
90C2×#120—48no—
91C2×#120—44no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C42C2314yes—
118C22×C42C2312yes—
119#397—216no—
120#396—216no—
121#396—48no—
122#394—216no—
123#394—216no—
124#392—216no—
125#392D4116yescomplemented
126#390—216no—
127#390—216no—
128#390—44no—
129#388—216no—
130#388—48no—
131#388—48no—
132#386—216no—
133#386—48no—
134#379—216no—
135#379—44no—
136C4×C22⋊C4—42no—
137C2×#95—42no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C25—28no—
156C25—24no—
157C25—44no—
158C25—44no—
159C25—44no—
160C25—42no—
161C25—44no—
162C25—44no—
163C25—42no—
164C25C2414yes—
165C25C2412yes—
166C25C2411yescharacteristic
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—24no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4C2×D4116yes—
198C22×D4C2×D418yes—
199C22×D4#4114yes—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2412yes—
219#123—464no—
220#121—416no—
221#121—432no—
222#120—432no—
223#120—464no—
224#120—432no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
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—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
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—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
294C2×C22⋊C4C2×D418yes—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—42no—
307#95—41no—
308#95—41no—
309#95—42no—
310#95—44no—
311#95—42no—
312#95—41no—
313#95—41no—
314#95—44no—
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—22no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—48no—
346C24—416no—
347C24—416no—
348C24—48no—
349C24—48no—
350C24—88no—
351C24—84no—
352C24—84no—
353C24—88no—
354C24—88no—
355C24—88no—
356C24—88no—
357C24—88no—
358C24—88no—
359C24—88no—
360C24—84no—
361C24—88no—
362C24—88no—
363C24—84no—
364C24—88no—
365C24—88no—
366C24C22×D418yes—
367C24C22×D414yes—
368C24C2×#4114yes—
369C24H(5,𝔽2)12yes—
370C24C2511yescharacteristic
371C2×D4—232no—
372C2×D4—416no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—416no—
384C2×D4—416no—
385C2×D4—48no—
386C2×D4—84no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—88no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—84no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—88no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D418yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)18yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C4H(5,𝔽2)12yes—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—44no—
471C4⋊C4—88no—
472C4⋊C4—84no—
473C22⋊C4—216no—
474C22⋊C4—232no—
475C22⋊C4—416no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—84no—
492C42—88no—
493C42—84no—
494C23—24no—
495C23—28no—
496C23—22no—
497C23—24no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—44no—
519C23—41no—
520C23—44no—
521C23—42no—
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—44no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—816no—
566C23—832no—
567C23—832no—
568C23—832no—
569C23—816no—
570C23—832no—
571C23—832no—
572C23—816no—
573C23—832no—
574C23D4214yes—
575C23C23×D412yes—
576C23C22×#4111yescharacteristic
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)12yes—
579C23C2×H(5,𝔽2)14yes—
580C23C2×H(5,𝔽2)11yescharacteristic
581C23#58512yes—
582C23#58512yes—
583D4—832no—
584D4—832no—
585D4—88no—
586D4—816no—
587D4—816no—
588D4—816no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—88no—
644C2×C4—816no—
645C22—24no—
646C22—28no—
647C22—48no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—44no—
653C22—48no—
654C22—48no—
655C22—48no—
656C22—48no—
657C22—44no—
658C22—48no—
659C22—44no—
660C22—48no—
661C22—48no—
662C22—44no—
663C22—48no—
664C22—48no—
665C22—864no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22C2×D4211yescharacteristic
679C22D4×#4112yes—
680C22#348414yes—
681C22#349012yes—
682C22#349011yescharacteristic
683C22#349614yes—
684C22#349612yes—
685C22#351412yes—
686C22#352911yescharacteristic
687C22#353412yes—
688C22#354312yes—
689C22#354312yes—
690C22#354511yescharacteristic
691C22#354614yes—
692C22#354911yescharacteristic
693C22#356014yes—
694C4—48no—
695C4—42no—
696C4—42no—
697C4—44no—
698C4—44no—
699C4—44no—
700C4—48no—
701C4—84no—
702C4—82no—
703C4—82no—
704C2—44no—
705C2—84no—
706C2—88no—
707C2#4299612yes—
708C2#4304314yes—
709C2#4628012yes—
710C2#4628711yescharacteristic
711C2#4638711yescharacteristic
712C2#4641412yes—
713C2#5348912yes—
714C2#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×#2678C2semidirect
2. #17135C2semidirect
3. #17118C2semidirect
4. #15947C2semidirect
5. #15782C2semidirect
6. #15272C2semidirect
7. #15268C2semidirect
8. #15257C2semidirect
9. #14976C2semidirect
10. #14926C2semidirect
11. #14358C2semidirect
12. #14228C2semidirect
13. #14152C2semidirect
14. C22×#123C22semidirect
19. C22×#120C22semidirect
21. C2×#397C22semidirect
22. C2×#396C22semidirect
23. C2×#396C22semidirect
30. C2×#390C22semidirect
31. C2×#388C22semidirect
32. C2×#388C22semidirect
33. C2×#388C22semidirect
34. C2×#386C22semidirect
35. C2×#386C22semidirect
36. C2×#379C22semidirect
37. C2×#379C22semidirect
38. C2×C4×C22⋊C4C22semidirect
39. C22×#95C22semidirect
40. #1445C22semidirect
41. #1445C22semidirect
42. #1444C22semidirect
43. #1444C22semidirect
44. #1444C22semidirect
45. #1441C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
62. #1438C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. #1436C22semidirect
66. C22⋊#95C22semidirect
67. C4×#95C22semidirect
125. #392D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26782
#171352
#171182
#159472
#157822
#152722
#152682
#152572
#149762
#149262
#143582
#142282
#141522