Finite Group Databasebeta

← #945709

Subgroups — #945709

Order 512.

4580 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2435C212yesmaximal, complemented
3C2×#2408C211yesmaximal, characteristic
4C2×#2620C212yesmaximal, complemented
5#17088C214yesmaximal, complemented
6#16514C214yesmaximal, complemented
7#16298C214yesmaximal, complemented
8#15281C214yesmaximal, complemented
9#15243C212yesmaximal, complemented
10#15215C212yesmaximal, complemented
11#15182C212yesmaximal, complemented
12#14001C212yesmaximal, complemented
13#13839C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic, complemented
16C22×#124C2212yes—
17C22×#124C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yescomplemented
20C22×#121C2212yes—
21C22×#120C2212yes—
22C22×C4×D4C2211yescharacteristic
23C22×#117C2211yescharacteristic, complemented
24#2620—216no—
25#2542—28no—
26#2435—216no—
27#2408—28no—
28C2×#399C2212yes—
29C2×#399C2212yescomplemented
30C2×#396C2218yescomplemented
31C2×#394C2212yes—
32C2×#394C2218yes—
33C2×#394C2218yescomplemented
34C2×#392C2212yes—
35C2×#391C2212yescomplemented
36C2×#391C2212yescomplemented
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#388C2218yescomplemented
42C2×#384C2212yescomplemented
43C2×#384C2212yescomplemented
44C2×#380C2211yescharacteristic
45C2×#379C2212yes—
46C2×#379C2218yes—
47C2×C4×C4⋊C4C2211yescharacteristic, complemented
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic, complemented
50#1441C2212yescomplemented
51#1441C2218yescomplemented
52#1440C2212yescomplemented
53#1439C2212yes—
54#1439C2218yes—
55#1439C2218yescomplemented
56#1439C2218yescomplemented
57#1439C2218yescomplemented
58#1439C2218yescomplemented
59#1438C2212yes—
60#1438C2218yescomplemented
61C4⋊#95C2212yescomplemented
62C4×#95C2212yescomplemented
63C26C2312yes—
64C23×Q8C2312yes—
65C23×D4—28no—
66C23×D4—24no—
67C23×D4—42no—
68C23×D4—42no—
69C23×D4—44no—
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2314yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—42no—
79C2×#124—28no—
80C2×#124—28no—
81C2×#124—28no—
82C2×#124—44no—
83C2×#124—44no—
84C2×#124—44no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—44no—
90C2×C22⋊Q8—44no—
91C2×#121—28no—
92C2×#121—28no—
93C2×#121—44no—
94C2×#120—28no—
95C2×#120—48no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4D418yes—
99C2×#117—28no—
100C2×#117—42no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2318yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2318yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#399—28no—
130#399—28no—
131#396—232no—
132#394—28no—
133#394—232no—
134#394—232no—
135#392—28no—
136#391—28no—
137#391—44no—
138#390—28no—
139#390—28no—
140#390—28no—
141#390D4116yescomplemented
142#388—416no—
143#384—28no—
144#384—28no—
145#380—24no—
146#379—28no—
147#379—232no—
148C4×C4⋊C4—24no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—42no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2312yes—
160C25—24no—
161C25—28no—
162C25—28no—
163C25—44no—
164C25—42no—
165C25—42no—
166C25—44no—
167C25—48no—
168C25C2414yes—
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×D4—24no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—432no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—416no—
197C22×D4—48no—
198C22×D4—44no—
199C22×D4—48no—
200C22×D4—416no—
201C22×D4—432no—
202C22×D4—416no—
203C22×D4C2×D418yes—
204C22×D4C2×D414yes—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C4⋊Q8—416no—
226#124—432no—
227#124—432no—
228C22⋊Q8—432no—
229C22⋊Q8—432no—
230#121—432no—
231#120—432no—
232C4×D4—416no—
233#117—416no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×D414yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D4132yes—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42C2×D418yes—
310C2×C42C2×D414yes—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—216no—
315#95—28no—
316#95—44no—
317#95—42no—
318#95—48no—
319#95—48no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—28no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—28no—
340C24—28no—
341C24—24no—
342C24—28no—
343C24—416no—
344C24—48no—
345C24—432no—
346C24—48no—
347C24—416no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—88no—
352C24—88no—
353C24—88no—
354C24—816no—
355C24—816no—
356C24—88no—
357C24—816no—
358C24—88no—
359C24C22×D418yes—
360C24C22×D412yes—
361C24C22×D412yes—
362C242−1+414yes—
363C24C2511yescharacteristic
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—216no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—44no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—416no—
383C2×D4—48no—
384C2×D4—4128no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—816no—
392C2×D4—88no—
393C2×D4—816no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—216no—
421C22×C4—216no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4C22×D418yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D418yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)14yes—
459C4⋊C4—216no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—816no—
468C22⋊C4—432no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—432no—
477C22⋊C4—48no—
478C22⋊C4—432no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—816no—
482C22⋊C4—88no—
483C42—28no—
484C42—216no—
485C42—48no—
486C42—48no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—22no—
496C23—28no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—216no—
503C23—216no—
504C23—24no—
505C23—42no—
506C23—41no—
507C23—44no—
508C23—41no—
509C23—42no—
510C23—44no—
511C23—42no—
512C23—432no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—41no—
519C23—44no—
520C23—44no—
521C23—48no—
522C23—48no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—41no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—48no—
531C23—48no—
532C23—42no—
533C23—48no—
534C23—44no—
535C23—48no—
536C23—48no—
537C23—42no—
538C23—48no—
539C23—44no—
540C23—48no—
541C23—48no—
542C23—832no—
543C23—816no—
544C23—832no—
545C23—832no—
546C23—864no—
547C23—832no—
548C23—816no—
549C23#53614yes—
550C23C23×D411yescharacteristic
551C23C23×D412yes—
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×H(5,𝔽2)12yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×2−1+412yes—
557C23#58514yes—
558Q8—816no—
559D4—416no—
560D4—432no—
561D4—432no—
562D4—832no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—216no—
577C2×C4—24no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—416no—
594C2×C4—416no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4#53414yes—
615C2×C4#53514yes—
616C2×C4#53518yes—
617C2×C4D4214yes—
618C2×C4D4218yes—
619C2×C4#56014yes—
620C22—28no—
621C22—24no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—42no—
626C22—44no—
627C22—48no—
628C22—42no—
629C22—44no—
630C22—48no—
631C22—416no—
632C22—42no—
633C22—42no—
634C22—48no—
635C22—44no—
636C22—48no—
637C22—416no—
638C22—864no—
639C22—82no—
640C22—88no—
641C22—84no—
642C22—84no—
643C22—82no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—88no—
648C22C2×#53411yescharacteristic
649C22C2×#53511yescharacteristic
650C22C2×#53512yes—
651C22C2×#53611yescharacteristic
652C22C2×D4212yes—
653C22C2×D4211yescharacteristic
654C22#348914yes—
655C22#349614yes—
656C22#350012yes—
657C22C2×#56011yescharacteristic
658C22#353612yes—
659C22#353612yes—
660C22#354012yes—
661C22#354614yes—
662C22#355112yes—
663C22#356114yes—
664C4—24no—
665C4—42no—
666C4—48no—
667C4—42no—
668C4—42no—
669C4—48no—
670C4—84no—
671C4—84no—
672C4—84no—
673C4#266918yes—
674C2—44no—
675C2—82no—
676C2—88no—
677C2#3741911yescharacteristic
678C2#3745111yescharacteristic
679C2#3767612yes—
680C2#3936212yes—
681C2#4244314yes—
682C2#4510312yes—
683C2C2×#266911yescharacteristic
684C2#5301812yes—
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×#2542C2semidirect
2. C2×#2435C2semidirect
4. C2×#2620C2semidirect
5. #17088C2semidirect
6. #16514C2semidirect
7. #16298C2semidirect
8. #15281C2semidirect
9. #15243C2semidirect
10. #15215C2semidirect
11. #15182C2semidirect
12. #14001C2semidirect
13. #13839C2semidirect
15. C22×C4⋊Q8C22semidirect
19. C22×C22⋊Q8C22semidirect
23. C22×#117C22semidirect
29. C2×#399C22semidirect
30. C2×#396C22semidirect
33. C2×#394C22semidirect
35. C2×#391C22semidirect
36. C2×#391C22semidirect
41. C2×#388C22semidirect
42. C2×#384C22semidirect
43. C2×#384C22semidirect
47. C2×C4×C4⋊C4C22semidirect
49. C2×C4×C22⋊C4C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1440C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
60. #1438C22semidirect
61. C4⋊#95C22semidirect
62. C4×#95C22semidirect
141. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24352
C2×#24082
C2×#26202
#170882
#165142
#162982
#152812
#152432
#152152
#151822
#140012
#138392