Finite Group Databasebeta

← #356087

Subgroups — #356087

Order 512.

2680 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#416C211yesmaximal, characteristic
2C2×#2339C211yesmaximal, characteristic
3C2×#1788C211yesmaximal, characteristic
4C2×#1762C212yesmaximal, complemented
5C2×#1752C211yesmaximal, characteristic, complemented
6#13801C211yesmaximal, characteristic, complemented
7#10932C212yesmaximal, complemented
8#10928C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10721C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2212yes—
15C22×#117C2211yescharacteristic
16C2×#416—28no—
17C2×#416C418yescomplemented
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2212yes—
26#2339—24no—
27#2339C418yescomplemented
28C2×#391C2212yes—
29C2×#386—22no—
30C2×#381—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35C22×#95—21no—
36#1788C418yescomplemented
37#1762—28no—
38#1752—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1437—22no—
45C4⋊#95C2212yes—
46#1434—21no—
47C4×#95—21no—
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×#41C2×C414yes—
57C22×#41C2×C414yes—
58C23×Q8C2311yescharacteristic
59C23×D4—21no—
60C23×D4C2311yescharacteristic
61C24×C4—21no—
62C24×C4—22no—
63C24×C4C2311yescharacteristic
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—24no—
67C23×C8—42no—
68C23×C8C2×C414yescomplemented
69C23×C8C2312yes—
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4—24no—
75C2×C4×D4—42no—
76C2×C4×D4—42no—
77C2×C4×D4C2×C418yes—
78C2×#117—22no—
79C2×#117—24no—
80C2×#117—42no—
81C2×#117C2×C414yescomplemented
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D412yes—
91C22×C42—21no—
92C22×C42—22no—
93C22×C42C2312yes—
94#416—232no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103C2×C414yescomplemented
100C2×#103C2×C414yescomplemented
101C2×#103C2×C418yescomplemented
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102C2×C414yes—
106C2×#102C2×C418yes—
107C2×#102C2×C414yes—
108C2×#102D418yes—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8C2×C414yescomplemented
112C2×#97—24no—
113C2×#97—24no—
114C2×#97—42no—
115#391—24no—
116#391C2×C418yes—
117#386—216no—
118#381—28no—
119#380—22no—
120#380—22no—
121#380C2×C414yes—
122#380C2×C414yescomplemented
123C4×C4⋊C4—24no—
124C4×C4⋊C4—42no—
125C4×C22⋊C4C2×C414yes—
126C4×C22⋊C4D414yes—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—28no—
132C2×#95—22no—
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—28no—
148C2×#41—28no—
149C2×#41—216no—
150C2×#41—216no—
151C2×#41—44no—
152C2×#41—44no—
153C2×#41—48no—
154C2×#41—48no—
155C2×#41—48no—
156C2×#41—48no—
157C2×#41—48no—
158C2×#41—48no—
159C2×#41—48no—
160C2×#41—48no—
161C2×#41C42116yescomplemented
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8C22×C412yes—
170C22×Q8C22×C412yes—
171C22×Q8C22×C412yes—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4C22×C412yes—
183C22×D4C22×C412yes—
184C22×D4C22×C412yes—
185C22×D4C2×D414yes—
186C22×D4#4114yes—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—28no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—82no—
207C23×C4—82no—
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C22×C412yes—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D412yes—
215C23×C4#4112yes—
216C23×C4#4112yes—
217C23×C4C2411yescharacteristic
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—22no—
221C22×C8—28no—
222C22×C8—21no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—21no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—28no—
233C22×C8—44no—
234C22×C8—48no—
235C22×C8—48no—
236C22×C8C22×C412yes—
237C22×C8C22×C414yes—
238C22×C8C22×C412yes—
239C22×C8C2×D414yes—
240C4×D4—216no—
241C4×D4—48no—
242C4×D4—48no—
243C4×D4—48no—
244#117—28no—
245#117—44no—
246#117—48no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
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—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C2×D414yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—81no—
277C2×C22⋊C4—81no—
278C2×C22⋊C4—81no—
279C2×C22⋊C4—81no—
280C2×C22⋊C4C22⋊C418yes—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—82no—
290C2×C42C22×C414yes—
291C2×C42C2×D414yes—
292#103—28no—
293#103—28no—
294#103—48no—
295#102—28no—
296#102—216no—
297#102C22⋊C4116yescomplemented
298C22⋊C8—28no—
299#97—48no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—44no—
305#95—44no—
306#95—82no—
307#95C22⋊C414yes—
308#95C22⋊C414yes—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—24no—
314C24—24no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—42no—
323C24—42no—
324C24—41no—
325C24—41no—
326C24—41no—
327C24—41no—
328C24C2×C22⋊C412yes—
329C24C4×D412yes—
330C24C4×D412yes—
331C24C22×D411yescharacteristic
332#41—232no—
333#41—416no—
334#41—416no—
335#41—432no—
336#41—416no—
337#41—432no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8C2×C4214yes—
356C2×D4—216no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—48no—
371C2×D4—416no—
372C2×D4—48no—
373C2×D4—416no—
374C2×D4C2×C4214yes—
375C2×D4C2×C22⋊C414yes—
376C2×D4#11714yes—
377C2×D4C4×D418yes—
378C2×D4C4×D418yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
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—22no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
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—42no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4C2×C4214yes—
449C22×C4C2×C22⋊C414yes—
450C22×C4C2×C22⋊C414yes—
451C22×C4C2×C22⋊C412yes—
452C22×C4#11714yes—
453C22×C4C4×D414yes—
454C22×C4C4×D414yes—
455C22×C4C4×D412yes—
456C22×C4C4×D412yes—
457C22×C4C23×C411yescharacteristic
458C22×C4C23×C411yescharacteristic
459C22×C4C23×C411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4111yescharacteristic
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—416no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8C2×C22⋊C414yes—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4C2×C22⋊C414yes—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—24no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—48no—
509C42—88no—
510C23—22no—
511C23—28no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23C4×C22⋊C414yes—
556C23C22×C22⋊C411yescharacteristic
557C23C2×C4×D411yescharacteristic
558C23C2×C4×D411yescharacteristic
559C23D4212yes—
560C23#54612yes—
561C23#57512yes—
562C23#57512yes—
563C23#57611yescharacteristic
564C23#57711yescharacteristic
565C23#57711yescharacteristic
566C23#57811yescharacteristic
567Q8—24no—
568Q8—24no—
569Q8—42no—
570Q8—44no—
571Q8—42no—
572Q8—44no—
573D4—24no—
574D4—24no—
575D4—28no—
576D4—28no—
577D4—24no—
578D4—432no—
579D4C4×C22⋊C418yes—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—432no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—832no—
625C2×C4—82no—
626C2×C4—82no—
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×C4C4×C22⋊C414yes—
636C2×C4C22×C4211yescharacteristic
637C2×C4C22×C22⋊C411yescharacteristic
638C2×C4C2×#11711yescharacteristic
639C2×C4C2×#11711yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4C2×C4×D412yes—
642C2×C4C2×C4×D411yescharacteristic
643C2×C4C2×C4×D412yes—
644C2×C4#51811yescharacteristic
645C2×C4#51812yes—
646C2×C4#51811yescharacteristic
647C2×C4#51812yes—
648C2×C4D4212yes—
649C2×C4#54612yes—
650C2×C4#54612yes—
651C2×C4#54612yes—
652C2×C4#55112yes—
653C2×C4#55112yes—
654C8—24no—
655C8—42no—
656C8—44no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—28no—
661C22—42no—
662C22—44no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—48no—
667C22—44no—
668C22—48no—
669C22—44no—
670C22—48no—
671C22—48no—
672C22—44no—
673C22—42no—
674C22—44no—
675C22—42no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22C2×C4×C22⋊C411yescharacteristic
680C22D4×C22⋊C412yes—
681C22D4×C22⋊C411yescharacteristic
682C22#234211yescharacteristic
683C22#235812yes—
684C22#289612yes—
685C22#289711yescharacteristic
686C22#289811yescharacteristic
687C22#294311yescharacteristic
688C22#294311yescharacteristic
689C22#294312yes—
690C22#294611yescharacteristic
691C22#294711yescharacteristic
692C22#295111yescharacteristic
693C22#295211yescharacteristic
694C22#326712yes—
695C22#326712yes—
696C22#326812yes—
697C22#326912yes—
698C22#330712yes—
699C22#331112yes—
700C22#331512yes—
701C22#331912yes—
702C4—24no—
703C4—44no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—84no—
708C4—84no—
709C4—82no—
710C4C2×C4×C22⋊C411yescharacteristic
711C4#227311yescharacteristic
712C4D4×C22⋊C411yescharacteristic
713C4#234212yes—
714C4#234211yescharacteristic
715C4#235812yes—
716C2—44no—
717C2—48no—
718C2C22⋊C4211yescharacteristic
719C2#2041111yescharacteristic
720C2#2041511yescharacteristic
721C2#2242012yes—
722C2#2242412yes—
723C2#2243811yescharacteristic
724C2#2243911yescharacteristic
725C2#2256411yescharacteristic
726C2#2256811yescharacteristic
727C2#2285612yes—
728C2#2286012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C2×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
4. C2×#1762C2semidirect
5. C2×#1752C2semidirect
6. #13801C2semidirect
7. #10932C2semidirect
8. #10928C2semidirect
9. #10788C2semidirect
10. #10731C2semidirect
11. #10721C2semidirect
17. C2×#416C4semidirect
27. #2339C4semidirect
36. #1788C4semidirect
68. C23×C8C2×C4semidirect
81. C2×#117C2×C4semidirect
99. C2×#103C2×C4semidirect
100. C2×#103C2×C4semidirect
101. C2×#103C2×C4semidirect
111. C2×C22⋊C8C2×C4semidirect
122. #380C2×C4semidirect
161. C2×#41C42semidirect
297. #102C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4162
C2×#23392
C2×#17882
C2×#17622
C2×#17522
#138012
#109322
#109282
#107882
#107312
#107212