Finite Group Databasebeta

← #414293

Subgroups — #414293

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2074C211yesmaximal, characteristic, complemented
2C2×#2073C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic, complemented
6#14273C211yesmaximal, characteristic
7#11373C211yesmaximal, characteristic, complemented
8#11371C211yesmaximal, characteristic, complemented
9#11009C211yesmaximal, characteristic
10#11009C211yesmaximal, characteristic
11#10928C211yesmaximal, characteristic, complemented
12#10927C211yesmaximal, characteristic
13#10926C211yesmaximal, characteristic, complemented
14#10925C211yesmaximal, characteristic
15#10694C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#376C2211yescharacteristic
35#2074—24no—
36#2073—24no—
37#2024—24no—
38#2023—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51#1446—22no—
52#1445—22no—
53#1439—22no—
54#1439—22no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C23×Q8—21no—
60C23×Q8C2311yescharacteristic
61C23×D4C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—41no—
71C2×C4⋊Q8—41no—
72C2×#124—22no—
73C2×#124—22no—
74C2×#124—22no—
75C2×#124—22no—
76C2×#124—41no—
77C2×#124—41no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—41no—
102C2×#103—41no—
103C2×#103—41no—
104C2×#103—41no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—41no—
114C2×#102—41no—
115C2×#102—41no—
116C2×#102—41no—
117C2×#97—22no—
118C2×#97—41no—
119C2×#97—41no—
120C2×C4×C8—22no—
121C2×C4×C8—41no—
122C2×C4×C8—41no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#390—22no—
128#390—22no—
129#390—22no—
130#390—22no—
131#384—22no—
132#384—22no—
133#382—22no—
134#382—22no—
135#376—41no—
136#376—41no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95D412yes—
147C2×#95D412yes—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#336—22no—
157#336—22no—
158#336D414yes—
159#336D414yes—
160C25—21no—
161C22×Q8—24no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—82no—
178C22×Q8—82no—
179C22×Q8#4112yes—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4#4112yes—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2411yescharacteristic
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8C2×D412yes—
231C22×C8C2×D412yes—
232C22×C8C2×D412yes—
233C22×C8C2×D412yes—
234C4⋊Q8—44no—
235C4⋊Q8—44no—
236C4⋊Q8—44no—
237C4⋊Q8—44no—
238#124—44no—
239#124—44no—
240#124—44no—
241#124—44no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4#4112yes—
283C2×C4⋊C4#4112yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—82no—
291C2×C22⋊C4—82no—
292C2×C22⋊C4—82no—
293C2×C22⋊C4—82no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42#4112yes—
309C2×C42#4112yes—
310C2×C42#4112yes—
311C2×C42#4112yes—
312#103—44no—
313#103—44no—
314#103—44no—
315#103—44no—
316#102—44no—
317#102—44no—
318#102—44no—
319#102—44no—
320#97—44no—
321C4×C8—44no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—42no—
329#95—82no—
330#95—82no—
331#95—82no—
332#95—82no—
333C24—22no—
334C24—41no—
335C24—41no—
336C24—41no—
337C24—41no—
338C24—41no—
339C24—41no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24C22×D411yescharacteristic
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—48no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—84no—
375C2×Q8—84no—
376C2×Q8—88no—
377C2×Q8—88no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4#12414yes—
464C22×C4#12414yes—
465C22×C4#12712yes—
466C22×C4#12712yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4111yescharacteristic
475C22×C4C2×#4111yescharacteristic
476C22×C4C2×#4111yescharacteristic
477C22×C4H(5,𝔽2)11yescharacteristic
478C22×C4H(5,𝔽2)11yescharacteristic
479C22×C4H(5,𝔽2)11yescharacteristic
480C22×C42−1+411yescharacteristic
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—48no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—88no—
535C22⋊C4—416no—
536C22⋊C4—88no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—81no—
557C23—81no—
558C23—82no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23—82no—
564C23—81no—
565C23—81no—
566C23—81no—
567C23—81no—
568C23—81no—
569C23—82no—
570C23—82no—
571C23—82no—
572C23—82no—
573C23—81no—
574C23#52511yescharacteristic
575C23#52511yescharacteristic
576C23C2×#12411yescharacteristic
577C23C2×#12411yescharacteristic
578C23C2×#12711yescharacteristic
579C23#53411yescharacteristic
580C23#53711yescharacteristic
581C23#57612yes—
582C23#57712yes—
583C23#57712yes—
584C23#57812yes—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—816no—
590Q8—81no—
591Q8—81no—
592Q8—81no—
593Q8—81no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—81no—
599D4—81no—
600D4—81no—
601D4—81no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4—81no—
659C2×C4—82no—
660C2×C4—82no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4#52911yescharacteristic
664C2×C4#52911yescharacteristic
665C2×C4#52911yescharacteristic
666C2×C4#52911yescharacteristic
667C2×C4#53211yescharacteristic
668C2×C4#53211yescharacteristic
669C2×C4#53211yescharacteristic
670C2×C4#53211yescharacteristic
671C2×C4#53811yescharacteristic
672C2×C4#53811yescharacteristic
673C2×C4#53911yescharacteristic
674C2×C4#53911yescharacteristic
675C2×C4#54611yescharacteristic
676C2×C4#54611yescharacteristic
677C2×C4#54611yescharacteristic
678C2×C4#54611yescharacteristic
679C2×C4#54611yescharacteristic
680C2×C4#54611yescharacteristic
681C2×C4#54811yescharacteristic
682C2×C4#54811yescharacteristic
683C2×C4#55111yescharacteristic
684C2×C4#55111yescharacteristic
685C2×C4#55111yescharacteristic
686C2×C4#55111yescharacteristic
687C2×C4#55611yescharacteristic
688C2×C4#55611yescharacteristic
689C2×C4#55911yescharacteristic
690C2×C4#55911yescharacteristic
691C8—42no—
692C8—42no—
693C8—42no—
694C8—42no—
695C22—84no—
696C22—82no—
697C22—84no—
698C22—82no—
699C22—82no—
700C22—161no—
701C22—161no—
702C22—162no—
703C22—161no—
704C22—161no—
705C22—162no—
706C22#239811yescharacteristic
707C22#253911yescharacteristic
708C22#254011yescharacteristic
709C22#256111yescharacteristic
710C22#259211yescharacteristic
711C22#263911yescharacteristic
712C22#285811yescharacteristic
713C22#311812yes—
714C22#312112yes—
715C22#312412yes—
716C22#312512yes—
717C22#313612yes—
718C22#313612yes—
719C22#313712yes—
720C22#313712yes—
721C22#315411yescharacteristic
722C22#315411yescharacteristic
723C22#315511yescharacteristic
724C22#315611yescharacteristic
725C22#319812yes—
726C22#320112yes—
727C22#326212yes—
728C22#326312yes—
729C4—42no—
730C4—42no—
731C4—44no—
732C4—44no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—161no—
737C4—161no—
738C4—161no—
739C4—161no—
740C4#255711yescharacteristic
741C4#255711yescharacteristic
742C4#255711yescharacteristic
743C4#255711yescharacteristic
744C4#256011yescharacteristic
745C4#256011yescharacteristic
746C4#256011yescharacteristic
747C4#256011yescharacteristic
748C2—162no—
749C2#1516711yescharacteristic
750C2#2382712yes—
751C2#2832111yescharacteristic
752C2#2832411yescharacteristic
753C2#2839411yescharacteristic
754C2#2839511yescharacteristic
755C2#2912012yes—
756C2#2936212yes—
757C2#3019512yes—
758C2#3322811yescharacteristic
759C2#3322911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2074C2semidirect
3. C2×#2024C2semidirect
5. #14282C2semidirect
7. #11373C2semidirect
8. #11371C2semidirect
11. #10928C2semidirect
13. #10926C2semidirect
15. #10694C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20742
C2×#20732
C2×#20242
C2×#20232
#142822
#142732
#113732
#113712
#110092
#109282
#109272
#109262
#109252
#106942