Finite Group Databasebeta

← #360570

Subgroups — #360570

Order 512.

2488 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#417C211yesmaximal, characteristic
2C2×#2341C211yesmaximal, characteristic
3C2×#1851C211yesmaximal, characteristic, complemented
4C2×#1805C212yesmaximal, complemented
5C2×#1788C211yesmaximal, characteristic
6#13881C211yesmaximal, characteristic, complemented
7#11076C212yesmaximal, complemented
8#11051C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10722C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C2×#417—28no—
18C2×#417—24no—
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26#2341—24no—
27#2341C418yescomplemented
28C2×#390—22no—
29C2×#390C2212yes—
30C2×#382C2212yes—
31C2×#380—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1851—24no—
37#1805—28no—
38#1788—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1442—21no—
45#1442—21no—
46C4⋊#95C2212yes—
47C22⋊#95—22no—
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—21no—
63C24×C4C2311yescharacteristic
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—22no—
67C22×M16—24no—
68C22×M16—42no—
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2312yes—
72C2×C4×Q8—24no—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—42no—
76C2×C4×Q8—42no—
77C2×C4×Q8C2×C418yescomplemented
78C22×C4⋊C4—22no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D412yes—
93C22×C42—22no—
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—24no—
105C2×#103D418yes—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—42no—
112C2×#102—42no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116#390—216no—
117#390—24no—
118#390C2×C418yescomplemented
119#382—24no—
120#382—42no—
121#380—22no—
122#380—28no—
123#380—22no—
124#380—22no—
125#380C2×C414yes—
126C4×C22⋊C4—28no—
127C4×C22⋊C4C2×C414yes—
128C4×C22⋊C4D414yes—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137#340—24no—
138#340—24no—
139#340—42no—
140#340—42no—
141#336—24no—
142C25—21no—
143C25C2×D411yescharacteristic
144C2×#41—28no—
145C2×#41—28no—
146C2×#41—28no—
147C2×#41—216no—
148C2×#41—28no—
149C2×#41—216no—
150C2×#41—28no—
151C2×#41—48no—
152C2×#41—44no—
153C2×#41—44no—
154C2×#41—48no—
155C2×#41—48no—
156C2×#41—48no—
157C2×#41—48no—
158C2×#41—48no—
159C2×#41—48no—
160C2×#41—48no—
161C2×#41C4⋊C4116yes—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8C22×C412yes—
168C22×Q8C2×D412yes—
169C22×Q8C2×D414yes—
170C22×Q8C2×Q812yes—
171C22×Q8#4114yes—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—21no—
177C22×D4—22no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4C22×C412yes—
185C22×D4C2×D412yes—
186C22×D4C2×Q812yes—
187C23×C4—21no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—81no—
205C23×C4—81no—
206C23×C4C22×C412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2×Q812yes—
213C23×C4#4112yes—
214C23×C4#4112yes—
215C23×C4C2411yescharacteristic
216C2×M16—28no—
217C2×M16—48no—
218C2×M16—48no—
219C2×M16—44no—
220C2×M16—44no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—21no—
228C22×C8—22no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8C2×D414yes—
238C4×Q8—216no—
239C4×Q8—48no—
240C4×Q8—48no—
241C4×Q8—48no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—82no—
280C2×C22⋊C4—82no—
281C2×C22⋊C4—81no—
282C2×C22⋊C4—81no—
283C2×C22⋊C4C22⋊C418yes—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—28no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C22×C414yes—
292C2×C42C2×D414yes—
293C4⋊C8—48no—
294#103—216no—
295#103—44no—
296#103—44no—
297#102—216no—
298#102—44no—
299#102—44no—
300C22⋊C8—44no—
301#95—24no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—82no—
306#95—82no—
307#95C22⋊C414yes—
308#95C22⋊C414yes—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—41no—
325C24—41no—
326C24—41no—
327C24—41no—
328C24C2×C22⋊C412yes—
329C24#12112yes—
330C24C22⋊Q812yes—
331C24C22×D411yescharacteristic
332#41—232no—
333#41—416no—
334#41—432no—
335#41—416no—
336#41—416no—
337#41—432no—
338C2×Q8—22no—
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8C2×C4⋊C414yes—
352C2×Q8C4×D418yes—
353C2×Q8#12014yes—
354C2×Q8C22⋊Q818yes—
355C2×Q8#12314yes—
356C2×D4—216no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—416no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—416no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4C2×C4⋊C414yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
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—216no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—416no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—416no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
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—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C2×C22⋊C414yes—
447C22×C4C2×C22⋊C414yes—
448C22×C4C2×C22⋊C412yes—
449C22×C4C2×C4⋊C414yes—
450C22×C4#11714yes—
451C22×C4#12114yes—
452C22×C4#12112yes—
453C22×C4C22⋊Q814yes—
454C22×C4C22⋊Q812yes—
455C22×C4C23×C411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×Q811yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465M16—416no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—88no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—24no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—416no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—82no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23#38014yes—
555C23C22×C22⋊C411yescharacteristic
556C23C2×#12111yescharacteristic
557C23C2×C22⋊Q811yescharacteristic
558C23D4212yes—
559C23#54612yes—
560C23#57512yes—
561C23#57512yes—
562C23#57611yescharacteristic
563C23#57711yescharacteristic
564C23#57711yescharacteristic
565C23#57811yescharacteristic
566Q8—24no—
567Q8—28no—
568Q8—24no—
569Q8—24no—
570Q8—28no—
571Q8#38518yes—
572D4—24no—
573D4—24no—
574D4—432no—
575D4—44no—
576D4—42no—
577D4—42no—
578D4—44no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—432no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—816no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4#38014yes—
635C2×C4C22×C4⋊C411yescharacteristic
636C2×C4C2×#11711yescharacteristic
637C2×C4C2×C4×D412yes—
638C2×C4#51812yes—
639C2×C4C2×#12011yescharacteristic
640C2×C4C2×#12111yescharacteristic
641C2×C4C2×C22⋊Q811yescharacteristic
642C2×C4C2×C22⋊Q812yes—
643C2×C4C2×#12311yescharacteristic
644C2×C4#53411yescharacteristic
645C2×C4#53411yescharacteristic
646C2×C4#54312yes—
647C2×C4D4212yes—
648C2×C4#54612yes—
649C2×C4#54612yes—
650C2×C4#54612yes—
651C2×C4#55112yes—
652C2×C4#55112yes—
653C8—44no—
654C8—81no—
655C8—81no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—82no—
671C22—82no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—82no—
677C22—82no—
678C22C2×#38011yescharacteristic
679C22D4×C22⋊C412yes—
680C22#241011yescharacteristic
681C22#241211yescharacteristic
682C22#243712yes—
683C22#289612yes—
684C22#289711yescharacteristic
685C22#289811yescharacteristic
686C22#304011yescharacteristic
687C22#304011yescharacteristic
688C22#304511yescharacteristic
689C22#304611yescharacteristic
690C22#308012yes—
691C22#308311yescharacteristic
692C22#308411yescharacteristic
693C22#326812yes—
694C22#327112yes—
695C22#327512yes—
696C22#327712yes—
697C22#330912yes—
698C22#331712yes—
699C22#332612yes—
700C22#332712yes—
701C4—24no—
702C4—28no—
703C4—44no—
704C4—44no—
705C4—84no—
706C4—84no—
707C4—161no—
708C4—161no—
709C4C2×#38511yescharacteristic
710C4#231511yescharacteristic
711C4#234212yes—
712C4#241011yescharacteristic
713C4#241211yescharacteristic
714C4#243712yes—
715C2—44no—
716C2—84no—
717C2#1387711yescharacteristic
718C2#2064911yescharacteristic
719C2#2065211yescharacteristic
720C2#2242212yes—
721C2#2242912yes—
722C2#2437111yescharacteristic
723C2#2437511yescharacteristic
724C2#2454311yescharacteristic
725C2#2454711yescharacteristic
726C2#2596912yes—
727C2#2597712yes—
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×#1805C2semidirect
6. #13881C2semidirect
7. #11076C2semidirect
8. #11051C2semidirect
9. #10789C2semidirect
10. #10730C2semidirect
11. #10722C2semidirect
27. #2341C4semidirect
77. C2×C4×Q8C2×C4semidirect
118. #390C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4172
C2×#23412
C2×#18512
C2×#18052
C2×#17882
#138812
#110762
#110512
#107892
#107302
#107222