Finite Group Databasebeta

← #4933891

Subgroups — #4933891

Order 512.

3702 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2679C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic, complemented
3#16691C212yesmaximal, complemented
4#16368C214yesmaximal, complemented
5#15941C212yesmaximal, complemented
6#15761C212yesmaximal, complemented
7#15474C214yesmaximal, complemented
8#15259C214yesmaximal, complemented
9#15107C212yesmaximal, complemented
10#14725C214yesmaximal, complemented
11#14357C212yesmaximal, complemented
12#13967C212yesmaximal, complemented
13#13832C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic, complemented
15C22×#124C2212yes—
16C22×#120C2212yes—
17C22×#120C2214yes—
18C22×C4×D4C2211yescharacteristic
19C23×C22⋊C4C2212yescomplemented
20#2679—28no—
21#2563—28no—
22C2×#400C2212yescomplemented
23C2×#394C2212yes—
24C2×#394C2214yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#393C2214yescomplemented
28C2×#393C2214yescomplemented
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#386C2212yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#382C2211yescharacteristic, complemented
36C2×#382C2211yescharacteristic, complemented
37C2×#380C2214yescomplemented
38C2×#379C2214yescomplemented
39C2×#379C2212yescomplemented
40C2×C4×C4⋊C4C2211yescharacteristic, complemented
41#1446C2212yescomplemented
42#1445C2212yescomplemented
43#1443C2214yescomplemented
44#1442C2214yescomplemented
45#1440C2214yescomplemented
46#1440C2214yescomplemented
47#1440C2214yescomplemented
48#1440C2214yescomplemented
49#1440C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1438C2212yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1437C2214yescomplemented
60#1437C2214yescomplemented
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63#1436C2212yescomplemented
64#1436C2212yescomplemented
65#1434C2212yescomplemented
66C22⋊#95C2214yescomplemented
67C22⋊#95C2212yescomplemented
68C22⋊#95C2212yescomplemented
69C26C2312yes—
70C23×Q8C2312yes—
71C23×D4C2314yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—42no—
79C2×C4⋊Q8—42no—
80C2×#124—48no—
81C2×#124—44no—
82C2×#120—28no—
83C2×#120—28no—
84C2×#120—28no—
85C2×#120—216no—
86C2×#120—48no—
87C2×#120—48no—
88C2×C4×D4—24no—
89C2×C4×D4—42no—
90C2×C4×D4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4—48no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#400—44no—
121#394—28no—
122#394—216no—
123#394—44no—
124#394—44no—
125#393—216no—
126#393—48no—
127#390—24no—
128#390—28no—
129#390D418yes—
130#386—28no—
131#386—48no—
132#386—48no—
133#382—24no—
134#382—24no—
135#380—216no—
136#379—44no—
137#379—48no—
138C4×C4⋊C4—24no—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C25—24no—
158C25—28no—
159C25—24no—
160C25—24no—
161C25—42no—
162C25—44no—
163C25—42no—
164C25—44no—
165C25—44no—
166C25—42no—
167C25—42no—
168C25C2414yes—
169C25C2412yes—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—44no—
173C22×Q8—42no—
174C22×Q8—44no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4C2×D414yes—
196C22×D4#4118yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4⋊Q8—416no—
219#124—432no—
220#120—464no—
221#120—432no—
222C4×D4—416no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×D414yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
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—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
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⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—432no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
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⋊C4—44no—
292C2×C22⋊C4C2×D4116yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—44no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42C2×D414yes—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95—44no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—416no—
343C24—48no—
344C24—416no—
345C24—416no—
346C24—48no—
347C24—416no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—84no—
352C24—84no—
353C24—88no—
354C24—84no—
355C24—84no—
356C24—88no—
357C24—88no—
358C24—88no—
359C24—88no—
360C24—84no—
361C24—88no—
362C24—84no—
363C24—88no—
364C24—88no—
365C24C22×D414yes—
366C24C2×#4114yes—
367C24C2×#4114yes—
368C24H(5,𝔽2)18yes—
369C24C2511yescharacteristic
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—88no—
373C2×Q8—84no—
374C2×Q8—88no—
375C2×D4—216no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—416no—
381C2×D4—416no—
382C2×D4—48no—
383C2×D4—416no—
384C2×D4—416no—
385C2×D4—416no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—84no—
396C2×D4#124116yescomplemented
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
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—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—84no—
455C22×C4—88no—
456C22×C4—84no—
457C22×C4—88no—
458C22×C4—84no—
459C22×C4—88no—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C2×#4114yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C4H(5,𝔽2)12yes—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—84no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—88no—
484C22⋊C4—84no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C42—28no—
491C42—44no—
492C42—44no—
493C42—48no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—432no—
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—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—832no—
553C23—816no—
554C23—816no—
555C23—816no—
556C23—832no—
557C23—832no—
558C23—832no—
559C23—816no—
560C23—816no—
561C23—816no—
562C23C2×#12418yes—
563C23C23×D411yescharacteristic
564C23C22×#4112yes—
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×H(5,𝔽2)14yes—
568C23C2×H(5,𝔽2)12yes—
569C23#58512yes—
570C23#58512yes—
571Q8—816no—
572D4—416no—
573D4—816no—
574D4—816no—
575D4—832no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
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—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
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—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—816no—
635C2×C4—816no—
636C2×C4C2×#12414yes—
637C2×C4H(3,𝔽4)14yes—
638C22—28no—
639C22—216no—
640C22—48no—
641C22—48no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—44no—
649C22—48no—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—48no—
654C22—48no—
655C22—48no—
656C22—864no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22C22×#12411yescharacteristic
666C22#348411yescharacteristic
667C22#348412yes—
668C22#348511yescharacteristic
669C22#348812yes—
670C22#350512yes—
671C22#350512yes—
672C22#351314yes—
673C22#351914yes—
674C22C2×H(3,𝔽4)11yescharacteristic
675C22#353212yes—
676C22#353712yes—
677C22#354214yes—
678C22#354514yes—
679C22#354511yescharacteristic
680C22#355812yes—
681C4—24no—
682C4—24no—
683C4—44no—
684C4—42no—
685C4—44no—
686C4—84no—
687C4—82no—
688C4—84no—
689C4—84no—
690C4—82no—
691C4—82no—
692C2—48no—
693C2—88no—
694C2#3537711yescharacteristic
695C2#3590312yes—
696C2#4564512yes—
697C2#4597711yescharacteristic
698C2#4839214yes—
699C2#4877712yes—
700C2#4977112yes—
701C2#5200411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2679C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2563C2semidirect
3. #16691C2semidirect
4. #16368C2semidirect
5. #15941C2semidirect
6. #15761C2semidirect
7. #15474C2semidirect
8. #15259C2semidirect
9. #15107C2semidirect
10. #14725C2semidirect
11. #14357C2semidirect
12. #13967C2semidirect
13. #13832C2semidirect
14. C22×C4⋊Q8C22semidirect
19. C23×C22⋊C4C22semidirect
22. C2×#400C22semidirect
27. C2×#393C22semidirect
28. C2×#393C22semidirect
35. C2×#382C22semidirect
36. C2×#382C22semidirect
37. C2×#380C22semidirect
38. C2×#379C22semidirect
39. C2×#379C22semidirect
40. C2×C4×C4⋊C4C22semidirect
41. #1446C22semidirect
42. #1445C22semidirect
43. #1443C22semidirect
44. #1442C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
59. #1437C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1434C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect
396. C2×D4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#26792
C2×#25632
#166912
#163682
#159412
#157612
#154742
#152592
#151072
#147252
#143572
#139672
#138322