Finite Group Databasebeta

← #360571

Subgroups — #360571

Order 512.

2668 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#417C211yesmaximal, characteristic
2C2×#2339C211yesmaximal, characteristic
3C2×#1851C211yesmaximal, characteristic, complemented
4C2×#1802C212yesmaximal, complemented
5C2×#1788C211yesmaximal, characteristic
6#13880C211yesmaximal, characteristic, complemented
7#11077C212yesmaximal, complemented
8#11050C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10722C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C4×D4C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C2×#417—28no—
18C2×#417—24no—
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26#2339—24no—
27#2339C418yescomplemented
28C2×#391C2212yes—
29C2×#386—22no—
30C2×#385—22no—
31C2×#384C2212yes—
32C2×#381—21no—
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C22×#95—21no—
37#1851—24no—
38#1802—28no—
39#1788—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1444—21no—
47C4⋊#95C2212yes—
48C22×#41—22no—
49C22×#41—22no—
50C22×#41—22no—
51C22×#41—24no—
52C22×#41—24no—
53C22×#41—42no—
54C22×#41—42no—
55C22×#41C2×C414yes—
56C22×#41D414yes—
57C22×#41Q814yes—
58C23×Q8C2311yescharacteristic
59C23×D4—21no—
60C23×D4C2311yescharacteristic
61C24×C4—21no—
62C24×C4—22no—
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C22×M16—22no—
66C22×M16—22no—
67C22×M16—22no—
68C22×M16—24no—
69C22×M16—42no—
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2312yes—
73C2×C4×D4—24no—
74C2×C4×D4—24no—
75C2×C4×D4—24no—
76C2×C4×D4—42no—
77C2×C4×D4—42no—
78C2×C4×D4C2×C418yescomplemented
79C22×C4⋊C4—22no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4D412yes—
94C22×C42C2312yes—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—42no—
98#417—232no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—42no—
105C2×#103—42no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—24no—
112C2×#102D418yes—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116#391—24no—
117#391C2×C418yescomplemented
118#386—216no—
119#385—48no—
120#384—24no—
121#384—42no—
122#381—28no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380C2×C414yes—
127C4×C22⋊C4C2×C414yes—
128C4×C22⋊C4D414yes—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—28no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95D411yescharacteristic
137C2×#95D411yescharacteristic
138#340—24no—
139#340—24no—
140#340—42no—
141#340—42no—
142#336—24no—
143C25—21no—
144C25C2×D411yescharacteristic
145C2×#41—28no—
146C2×#41—28no—
147C2×#41—28no—
148C2×#41—216no—
149C2×#41—28no—
150C2×#41—216no—
151C2×#41—28no—
152C2×#41—48no—
153C2×#41—44no—
154C2×#41—44no—
155C2×#41—48no—
156C2×#41—48no—
157C2×#41—48no—
158C2×#41—48no—
159C2×#41—48no—
160C2×#41—48no—
161C2×#41—48no—
162C2×#41C4⋊C4116yescomplemented
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8C22×C412yes—
171C22×Q8C2×D412yes—
172C22×Q8C2×Q812yes—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4C22×C412yes—
184C22×D4C2×D412yes—
185C22×D4C2×D414yes—
186C22×D4C2×Q812yes—
187C22×D4#4114yes—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—21no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—28no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—82no—
211C23×C4—82no—
212C23×C4—81no—
213C23×C4—81no—
214C23×C4C22×C412yes—
215C23×C4C2×D412yes—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×Q812yes—
221C23×C4#4112yes—
222C23×C4#4112yes—
223C23×C4C2411yescharacteristic
224C2×M16—28no—
225C2×M16—48no—
226C2×M16—48no—
227C2×M16—44no—
228C2×M16—44no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—21no—
236C22×C8—22no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8C2×D414yes—
246C4×D4—216no—
247C4×D4—48no—
248C4×D4—48no—
249C4×D4—48no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4C22×C414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—81no—
290C2×C22⋊C4—81no—
291C2×C22⋊C4C22⋊C418yes—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42C22×C414yes—
297C2×C42C2×D414yes—
298C4⋊C8—48no—
299#103—216no—
300#103—44no—
301#103—44no—
302#102—216no—
303#102—44no—
304#102—44no—
305C22⋊C8—44no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—44no—
310#95—42no—
311#95—44no—
312#95C22⋊C414yes—
313#95C22⋊C414yes—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—42no—
328C24—42no—
329C24—41no—
330C24—41no—
331C24—41no—
332C24—41no—
333C24C2×C22⋊C412yes—
334C24#12112yes—
335C24C22⋊Q812yes—
336C24C22×D411yescharacteristic
337#41—232no—
338#41—416no—
339#41—432no—
340#41—416no—
341#41—416no—
342#41—432no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8C2×C4⋊C414yes—
361C2×D4—216no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—416no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—416no—
379C2×D4C2×C4⋊C414yes—
380C2×D4C4×D418yes—
381C2×D4#12014yes—
382C2×D4C22⋊Q818yes—
383C2×D4#12314yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—216no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—416no—
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—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4C2×C22⋊C414yes—
457C22×C4C2×C22⋊C412yes—
458C22×C4C2×C22⋊C414yes—
459C22×C4C2×C4⋊C414yes—
460C22×C4#11714yes—
461C22×C4#12114yes—
462C22×C4#12112yes—
463C22×C4C22⋊Q814yes—
464C22×C4C22⋊Q812yes—
465C22×C4AGL(1,ℤ/8)14yes—
466C22×C4#13714yes—
467C22×C4C23×C411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×Q811yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477M16—416no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C22⋊C4—216no—
513C22⋊C4—416no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—24no—
517C42—24no—
518C42—24no—
519C42—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23#38014yes—
566C23C22×C22⋊C411yescharacteristic
567C23C2×#12111yescharacteristic
568C23C2×C22⋊Q811yescharacteristic
569C23D4212yes—
570C23#54612yes—
571C23C2×AGL(1,ℤ/8)12yes—
572C23C2×#13712yes—
573C23#57611yescharacteristic
574C23#57711yescharacteristic
575C23#57711yescharacteristic
576C23#57811yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—44no—
580Q8—42no—
581Q8—42no—
582Q8—44no—
583D4—24no—
584D4—28no—
585D4—24no—
586D4—24no—
587D4—28no—
588D4—432no—
589D4#38518yescomplemented
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—432no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—832no—
626C2×C4—816no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—82no—
645C2×C4#38014yes—
646C2×C4#41718yescomplemented
647C2×C4C22×C4⋊C411yescharacteristic
648C2×C4C2×#11711yescharacteristic
649C2×C4C2×C4×D412yes—
650C2×C4#51812yes—
651C2×C4C2×#12011yescharacteristic
652C2×C4C2×#12111yescharacteristic
653C2×C4C2×C22⋊Q811yescharacteristic
654C2×C4C2×C22⋊Q812yes—
655C2×C4C2×#12311yescharacteristic
656C2×C4#53411yescharacteristic
657C2×C4#53411yescharacteristic
658C2×C4#54312yes—
659C2×C4D4212yes—
660C2×C4#54612yes—
661C2×C4#54612yes—
662C2×C4#54612yes—
663C2×C4#55112yes—
664C2×C4#55112yes—
665C8—44no—
666C8—81no—
667C8—81no—
668C22—24no—
669C22—28no—
670C22—24no—
671C22—24no—
672C22—42no—
673C22—44no—
674C22—42no—
675C22—44no—
676C22—44no—
677C22—48no—
678C22—44no—
679C22—48no—
680C22—48no—
681C22—44no—
682C22—44no—
683C22—48no—
684C22—44no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22—42no—
689C22—44no—
690C22C2×#38011yescharacteristic
691C22D4×C22⋊C412yes—
692C22#241011yescharacteristic
693C22#241211yescharacteristic
694C22#243712yes—
695C22C2×#41712yes—
696C22#289711yescharacteristic
697C22#289811yescharacteristic
698C22#303811yescharacteristic
699C22#303911yescharacteristic
700C22#304511yescharacteristic
701C22#304611yescharacteristic
702C22#307912yes—
703C22#308311yescharacteristic
704C22#308411yescharacteristic
705C22#326412yes—
706C22#326612yes—
707C22#327212yes—
708C22#327612yes—
709C22#330612yes—
710C22#331412yes—
711C22#333012yes—
712C22#333112yes—
713C4—24no—
714C4—28no—
715C4—44no—
716C4—44no—
717C4—84no—
718C4—84no—
719C4—161no—
720C4—161no—
721C4C2×#38511yescharacteristic
722C4#231511yescharacteristic
723C4#234212yes—
724C4#241011yescharacteristic
725C4#241211yescharacteristic
726C4#243712yes—
727C2—44no—
728C2—48no—
729C2#1387711yescharacteristic
730C2#2064811yescharacteristic
731C2#2065211yescharacteristic
732C2#2241912yes—
733C2#2243112yes—
734C2#2437211yescharacteristic
735C2#2437611yescharacteristic
736C2#2454411yescharacteristic
737C2#2454811yescharacteristic
738C2#2597012yes—
739C2#2597812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1851C2semidirect
4. C2×#1802C2semidirect
6. #13880C2semidirect
7. #11077C2semidirect
8. #11050C2semidirect
9. #10788C2semidirect
10. #10730C2semidirect
11. #10722C2semidirect
27. #2339C4semidirect
78. C2×C4×D4C2×C4semidirect
117. #391C2×C4semidirect
162. C2×#41C4⋊C4semidirect
589. D4#385semidirect
646. C2×C4#417semidirect

Maximal subgroups

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

subgroupindex
C22×#4172
C2×#23392
C2×#18512
C2×#18022
C2×#17882
#138802
#110772
#110502
#107882
#107302
#107222