Finite Group Databasebeta

← #360668

Subgroups — #360668

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2283C211yesmaximal, characteristic
2C2×#1847C211yesmaximal, characteristic, complemented
3C2×#1836C211yesmaximal, characteristic, complemented
4C2×#1749C211yesmaximal, characteristic
5#14686C211yesmaximal, characteristic
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10843C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2283—24no—
21#2283—24no—
22C2×#383—21no—
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#381—21no—
26C2×#380—21no—
27C2×#380—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34C2×#376—21no—
35#1847—24no—
36#1836—24no—
37#1749C418yescomplemented
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2×C414yes—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42Q814yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—42no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8C2×C418yes—
87C2×#97—22no—
88C2×#97—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97C2×C414yes—
92C2×#97C2×C414yes—
93#384—22no—
94#384—22no—
95#384—22no—
96#384—22no—
97#383—28no—
98#382—22no—
99#382—22no—
100#382D414yes—
101#382D414yes—
102#381—28no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—42no—
108#380—42no—
109#380—44no—
110#380—44no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C22⋊C4—28no—
116#376—28no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#340—42no—
134#336C2×C418yes—
135C25C2×D411yescharacteristic
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4C4⋊C414yes—
163C23×C4C4⋊C414yes—
164C23×C4C22×C412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×Q812yes—
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C22×C414yes—
183C22×C8C22×C414yes—
184C22×C8#4114yes—
185C22×C8#4114yes—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—28no—
244C2×C42—28no—
245C2×C42—28no—
246C2×C42—28no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C4⋊C414yes—
254C2×C42C4⋊C414yes—
255C2×C42C22×C412yes—
256C2×C42C22×C412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×Q812yes—
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263#107—48no—
264#106—48no—
265C22⋊C8—216no—
266#97—28no—
267#97—28no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—44no—
277#95—44no—
278#95—44no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24C2×C4⋊C412yes—
292C24AGL(1,ℤ/8)12yes—
293C24#13712yes—
294C24C22×D411yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—28no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4C2×C4⋊C412yes—
392C22×C4C2×C4⋊C412yes—
393C22×C4C2×C4⋊C412yes—
394C22×C4C2×C4⋊C412yes—
395C22×C4C2×C4⋊C412yes—
396C22×C4#12012yes—
397C22×C4#12012yes—
398C22×C4#12312yes—
399C22×C4#12312yes—
400C22×C4AGL(1,ℤ/8)12yes—
401C22×C4#13712yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4H(5,𝔽2)12yes—
411C22×C42−1+412yes—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8#11718yes—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—88no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—28no—
454C42—28no—
455C42—48no—
456C42—48no—
457C42C4×D414yes—
458C42C4×D414yes—
459C42C22⋊Q814yes—
460C42C22⋊Q814yes—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23#42814yes—
496C23C22×C4⋊C411yescharacteristic
497C23C2×#12011yescharacteristic
498C23C2×#12311yescharacteristic
499C23#53211yescharacteristic
500C23#53211yescharacteristic
501C23#53511yescharacteristic
502C23#53611yescharacteristic
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23C2×AGL(1,ℤ/8)11yescharacteristic
505C23C2×#13711yescharacteristic
506C23C2×#13711yescharacteristic
507C23#57511yescharacteristic
508C23#57511yescharacteristic
509C23#57511yescharacteristic
510C23#57511yescharacteristic
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4#38514yes—
592C2×C4#38514yes—
593C2×C4#42814yes—
594C2×C4C2×#11712yes—
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4#52012yes—
598C2×C4C2×C22⋊Q811yescharacteristic
599C2×C4C2×C22⋊Q811yescharacteristic
600C2×C4#53312yes—
601C2×C4#54412yes—
602C2×C4#54612yes—
603C2×C4#54712yes—
604C2×C4#54812yes—
605C2×C4#55012yes—
606C2×C4#55212yes—
607C2×C4#55312yes—
608C2×C4#55512yes—
609C2×C4#55612yes—
610C8—28no—
611C8—44no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22#228511yescharacteristic
628C22C2×#38511yescharacteristic
629C22#231711yescharacteristic
630C22#243111yescharacteristic
631C22#243211yescharacteristic
632C22#249911yescharacteristic
633C22#250011yescharacteristic
634C22C2×#42811yescharacteristic
635C22C2×#42811yescharacteristic
636C22#291111yescharacteristic
637C22#291111yescharacteristic
638C22#300611yescharacteristic
639C22#300611yescharacteristic
640C22#300711yescharacteristic
641C22#300711yescharacteristic
642C22#309411yescharacteristic
643C22#309411yescharacteristic
644C22#309511yescharacteristic
645C22#309511yescharacteristic
646C22#317111yescharacteristic
647C22#317111yescharacteristic
648C22#317211yescharacteristic
649C22#317211yescharacteristic
650C22#318011yescharacteristic
651C22#318011yescharacteristic
652C22#318111yescharacteristic
653C22#318111yescharacteristic
654C22#321711yescharacteristic
655C22#321811yescharacteristic
656C22#321911yescharacteristic
657C22#321911yescharacteristic
658C22#323811yescharacteristic
659C22#323911yescharacteristic
660C22#324011yescharacteristic
661C22#324011yescharacteristic
662C4—24no—
663C4—24no—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—162no—
672C4—162no—
673C4#235212yes—
674C4#235412yes—
675C4#243912yes—
676C4#244112yes—
677C2—44no—
678C2#1387911yescharacteristic
679C2#2081911yescharacteristic
680C2#2082311yescharacteristic
681C2#2084611yescharacteristic
682C2#2084611yescharacteristic
683C2#2198711yescharacteristic
684C2#2198911yescharacteristic
685C2#2560411yescharacteristic
686C2#2560511yescharacteristic
687C2#2567811yescharacteristic
688C2#2567911yescharacteristic
689C2#2788211yescharacteristic
690C2#2788311yescharacteristic
691C2#2789611yescharacteristic
692C2#2789711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1847C2semidirect
3. C2×#1836C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
10. #10843C2semidirect
11. #10843C2semidirect
37. #1749C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22832
C2×#18472
C2×#18362
C2×#17492
#146862
#110962
#110952
#109082
#109072
#108432