Finite Group Databasebeta

← #535396

Subgroups — #535396

Order 512.

3534 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic, complemented
2C2×#2341C211yesmaximal, characteristic
3C2×#2311C211yesmaximal, characteristic
4#14376C214yesmaximal, complemented
5#14369C214yesmaximal, complemented
6#14141C212yesmaximal, complemented
7#14107C214yesmaximal, complemented
8#14071C214yesmaximal, complemented
9#14022C212yesmaximal, complemented
10#14017C212yesmaximal, complemented
11#13937C211yesmaximal, characteristic
12#13924C211yesmaximal, characteristic, complemented
13#13852C211yesmaximal, characteristic, complemented
14#13817C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×C4×Q8C2212yes—
18C22×#117C2212yes—
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2344—28no—
22#2341—28no—
23#2311C4116yescomplemented
24C2×#391C2212yes—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#390C2214yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#386C2214yes—
37C2×#381C2212yes—
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2214yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48#1445C2212yes—
49#1445C2212yes—
50#1444C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1438C2212yes—
61#1438C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2214yes—
77C4×#95C2212yes—
78C22×#41—28no—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—44no—
82C22×#41C2×C418yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—44no—
92C2×#117—28no—
93C2×#117—44no—
94C2×#117—48no—
95C2×#117C2×C418yes—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4C2×C418yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128#391—28no—
129#391C2×C418yes—
130#391C2×C418yes—
131#390—28no—
132#390—216no—
133#390—44no—
134#390C2×C418yes—
135#390C2×C418yes—
136#389—28no—
137#388—48no—
138#388—48no—
139#387—216no—
140#386C2×C4132yes—
141#381—28no—
142#380—24no—
143#380—28no—
144#380C2×C4132yes—
145#380D418yes—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—44no—
148C4×C4⋊C4—44no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—28no—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C2×#41—232no—
179C2×#41—232no—
180C2×#41—416no—
181C2×#41—432no—
182C2×#41—432no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8C22×C414yes—
187C22×Q8#4118yes—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—28no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—44no—
196C22×D4C22×C414yes—
197C22×D4C22×C418yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C22×C418yes—
209C23×C4C22×C414yes—
210C23×C4C22×C418yes—
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C4×Q8—432no—
227#117—416no—
228#117—432no—
229C2×C4⋊C4—232no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C22×C414yes—
258C2×C4⋊C4C22×C418yes—
259C2×C4⋊C4C2×D418yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
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—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C22×C418yes—
289C2×C22⋊C4C22×C4116yes—
290C2×C22⋊C4C22×C4116yes—
291C2×C22⋊C4C2×D418yes—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—28no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42C22×C414yes—
310C2×C42C22×C414yes—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C22×C4116yes—
333#95C22×C4116yes—
334#95C2×D418yes—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24C23×C414yes—
350C24C23×C412yes—
351C24C22×D412yes—
352C24C2×#4112yes—
353C24C2511yescharacteristic
354#41—4128no—
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—216no—
358C2×Q8—216no—
359C2×Q8#117116yescomplemented
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—48no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—48no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—416no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4C4×D4116yes—
449C22×C4C23×C414yes—
450C22×C4C23×C414yes—
451C22×C4C23×C414yes—
452C22×C4C23×C414yes—
453C22×C4C23×C414yes—
454C22×C4C23×C412yes—
455C22×C4C23×C412yes—
456C22×C4C22×D418yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C42−1+414yes—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—48no—
471C4⋊C4—416no—
472C4⋊C4—416no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C22⋊C4—28no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—84no—
487C22⋊C4—88no—
488C42—28no—
489C42—28no—
490C42—416no—
491C42—416no—
492C42—48no—
493C42—48no—
494C42—88no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—28no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23C2×C4×D414yes—
537C23C24×C411yescharacteristic
538C23C23×D411yescharacteristic
539C23C22×#4112yes—
540C23C22×#4111yescharacteristic
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)12yes—
543C23C2×2−1+412yes—
544C23#58512yes—
545C23#58512yes—
546Q8—232no—
547D4—432no—
548D4—416no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—232no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—432no—
578C2×C4—432no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4C2×#11718yes—
627C2×C4C2×#11714yes—
628C2×C4C2×C4×D414yes—
629C2×C4C2×C4×D414yes—
630C2×C4#51814yes—
631C2×C4#54618yes—
632C2×C4#55118yes—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—48no—
644C22—48no—
645C22—44no—
646C22—44no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22C22×#11711yescharacteristic
652C22C22×C4×D411yescharacteristic
653C22C2×#51811yescharacteristic
654C22C4×H(5,𝔽2)12yes—
655C22C4×2−1+412yes—
656C22C2×#54612yes—
657C22C2×#55112yes—
658C22#348411yescharacteristic
659C22#348711yescharacteristic
660C22#348912yes—
661C22#350612yes—
662C22#350911yescharacteristic
663C22#351011yescharacteristic
664C22#351612yes—
665C22#351712yes—
666C22#352212yes—
667C22#352412yes—
668C22#354112yes—
669C22#354212yes—
670C22#354812yes—
671C22#355112yes—
672C4—24no—
673C4—28no—
674C4—44no—
675C4—44no—
676C4—48no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—82no—
682C4—84no—
683C4—82no—
684C4#234218yes—
685C2—44no—
686C2—84no—
687C2C2×#234211yescharacteristic
688C2#3426312yes—
689C2#3426511yescharacteristic
690C2#3426811yescharacteristic
691C2#3441412yes—
692C2#4493012yes—
693C2#4496012yes—
694C2#4496212yes—
695C2#4526512yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2344C2semidirect
4. #14376C2semidirect
5. #14369C2semidirect
6. #14141C2semidirect
7. #14107C2semidirect
8. #14071C2semidirect
9. #14022C2semidirect
10. #14017C2semidirect
12. #13924C2semidirect
13. #13852C2semidirect
14. #13817C2semidirect
15. #13777C2semidirect
23. #2311C4semidirect
359. C2×Q8#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
C2×#23412
C2×#23112
#143762
#143692
#141412
#141072
#140712
#140222
#140172
#139372
#139242
#138522
#138172
#137772