Finite Group Databasebeta

← #612920

Subgroups — #612920

Order 512.

3790 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3C2×C4×#121C211yesmaximal, characteristic
4C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#15234C212yesmaximal, complemented
6#15222C212yesmaximal, complemented
7#14866C214yesmaximal, complemented
8#14759C214yesmaximal, complemented
9#14698C212yesmaximal
10#14682C212yesmaximal, complemented
11#14161C211yesmaximal, characteristic, complemented
12#13973C214yesmaximal, complemented
13#13911C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic, complemented
15#13872C214yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C42C2212yes—
23C23×C42C2211yescharacteristic
24#2857—28no—
25#2563—28no—
26C2×#393C2214yes—
27C4×#121—28no—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#388C2212yes—
36C2×#386C2212yes—
37C2×#386C2218yes—
38C2×#384C2211yescharacteristic
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2214yes—
44C2×#380C2214yes—
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4—28no—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C22×#95C2212yes—
55#1446C2214yes—
56#1442C2214yes—
57#1442C2218yes—
58#1441C2218yes—
59#1439C2218yes—
60#1439C2214yes—
61#1439C2218yes—
62#1437C2218yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2218yes—
70C22⋊#95C2218yes—
71C23×Q8C2312yes—
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2314yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—48no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121Q818yescomplemented
87C2×C4×D4—28no—
88C2×C4×D4—216no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2318yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2312yes—
116C22×C42—24no—
117C22×C42—24no—
118C22×C42—28no—
119C22×C42—28no—
120C22×C42—44no—
121C22×C42D418yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#393—216no—
131#391—28no—
132#390—28no—
133#390—24no—
134#390—44no—
135#389—28no—
136#389—28no—
137#389—24no—
138#388—28no—
139#386—28no—
140#386—416no—
141#384—24no—
142#382—28no—
143#382—44no—
144#380—28no—
145#380—28no—
146#380—24no—
147#380—216no—
148#380—216no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4D418yescomplemented
155C4×C22⋊C4—28no—
156C2×#95—28no—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C2×#95C2318yes—
170C2×#95C2312yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×D4—24no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4C2×Q818yes—
189C22×D4C2×Q814yes—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—22no—
201C23×C4—28no—
202C23×C4—22no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×Q814yes—
217C23×C4#4114yes—
218C23×C4#4118yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C4⋊Q8—416no—
233C22⋊Q8—432no—
234#121—232no—
235C4×D4—432no—
236C4×D4—416no—
237C2×C4⋊C4—264no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×Q814yes—
272C2×C4⋊C4#4118yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—216no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4C2×Q818yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—232no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—432no—
310C2×C42—44no—
311C2×C42C2×D414yes—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42#4114yes—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—216no—
324#95—28no—
325#95—28no—
326#95—216no—
327#95—48no—
328#95—48no—
329#95—48no—
330#95—48no—
331#95C2×D418yes—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24C22×D412yes—
349C24C22×Q814yes—
350C24C22×Q812yes—
351C24C2×#4112yes—
352C24C2×#4114yes—
353C24H(5,𝔽2)14yes—
354C24C2511yescharacteristic
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—88no—
359C2×Q8—88no—
360C2×D4—216no—
361C2×D4—232no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—232no—
392C22×C4—28no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—44no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4C22⋊Q8116yes—
449C22×C4#124116yes—
450C22×C4C22×D418yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×Q814yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q812yes—
463C22×C4C2×#4118yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+412yes—
474C4⋊C4—232no—
475C4⋊C4—28no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—48no—
479C4⋊C4—416no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C22⋊C4—216no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—816no—
494C22⋊C4—816no—
495C42—28no—
496C42—28no—
497C42—28no—
498C42—216no—
499C42—416no—
500C42—48no—
501C42—48no—
502C42#121116yes—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C2×C22⋊Q814yes—
545C23C2×#12414yes—
546C23#53514yes—
547C23#55418yes—
548C23#55518yes—
549C23C23×D411yescharacteristic
550C23C23×D411yescharacteristic
551C23C23×Q811yescharacteristic
552C23C22×#4111yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4112yes—
555C23C22×#4112yes—
556C23C2×H(5,𝔽2)12yes—
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×2−1+411yescharacteristic
559C23#58512yes—
560Q8—816no—
561D4—432no—
562D4—416no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—232no—
578C2×C4—232no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—416no—
591C2×C4—464no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—816no—
626C2×C4C2×#12114yes—
627C2×C4C2×#12114yes—
628C2×C4C2×#12114yes—
629C2×C4C2×C22⋊Q814yes—
630C2×C4C2×C22⋊Q814yes—
631C2×C4#52518yes—
632C2×C4C2×#12414yes—
633C2×C4C2×#12414yes—
634C2×C4#52918yes—
635C2×C4#53214yes—
636C2×C4#53714yes—
637C2×C4#54614yes—
638C2×C4D4×Q814yes—
639C2×C4#55214yes—
640C2×C4#55414yes—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—28no—
650C22—28no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—48no—
655C22—48no—
656C22—82no—
657C22—84no—
658C22—84no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#256318yescomplemented
663C22C22×#12111yescharacteristic
664C22C22×C22⋊Q811yescharacteristic
665C22C2×#52512yes—
666C22C22×#12411yescharacteristic
667C22C2×#52912yes—
668C22C2×#53211yescharacteristic
669C22C2×#53511yescharacteristic
670C22C2×#53711yescharacteristic
671C22#344911yescharacteristic
672C22C2×#54611yescharacteristic
673C22C2×D4×Q811yescharacteristic
674C22C2×#55211yescharacteristic
675C22C2×#55411yescharacteristic
676C22C2×#55412yes—
677C22C2×#55512yes—
678C22#347512yes—
679C22Q8×#4112yes—
680C22#348812yes—
681C22#349412yes—
682C22#350612yes—
683C22#350712yes—
684C22#351912yes—
685C22#352212yes—
686C4—24no—
687C4—24no—
688C4—24no—
689C4—48no—
690C4—48no—
691C4—44no—
692C4—42no—
693C4—82no—
694C4—82no—
695C4—84no—
696C4—84no—
697C4#241718yes—
698C4#285018yes—
699C2—28no—
700C2—84no—
701C2C2×#241711yescharacteristic
702C2C2×#256311yescharacteristic
703C2#3494711yescharacteristic
704C2#3518011yescharacteristic
705C2#3523811yescharacteristic
706C2#3525211yescharacteristic
707C2#3616012yes—
708C2#3617612yes—
709C2#3654412yes—
710C2#3655112yes—
711C2C2×#285011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2857C2semidirect
2. C2×#2563C2semidirect
4. C22×C4×C4⋊C4C2semidirect
5. #15234C2semidirect
6. #15222C2semidirect
7. #14866C2semidirect
8. #14759C2semidirect
10. #14682C2semidirect
11. #14161C2semidirect
12. #13973C2semidirect
13. #13911C2semidirect
14. #13906C2semidirect
15. #13872C2semidirect
86. C2×#121Q8semidirect
154. C4×C4⋊C4D4semidirect
662. C22#2563semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#25632
C2×C4×#1212
C22×C4×C4⋊C42
#152342
#152222
#148662
#147592
#146982
#146822
#141612
#139732
#139112
#139062
#138722