Finite Group Databasebeta

← #360667

Subgroups — #360667

Order 512.

1618 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#1847C211yesmaximal, characteristic, complemented
3C2×#1836C211yesmaximal, characteristic, complemented
4C2×#1749C211yesmaximal, characteristic
5#14682C211yesmaximal, characteristic
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10843C211yesmaximal, characteristic, complemented
12C22×#117—21no—
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2285—24no—
20#2285—24no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382—21no—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380—21no—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32#1847—24no—
33#1836—24no—
34#1749C418yescomplemented
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44#1434—21no—
45#1434—21no—
46C22⋊#95—21no—
47C22⋊#95—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#117—28no—
54C2×#117—44no—
55C2×#117—44no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42C2×C414yes—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42Q814yes—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—42no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8C2×C418yes—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97C2×C414yes—
94C2×#97C2×C414yes—
95#384—22no—
96#384—22no—
97#384D414yes—
98#384D414yes—
99#382—28no—
100#382—22no—
101#382—22no—
102#382—22no—
103#382—22no—
104#380—28no—
105#380—24no—
106#380—24no—
107#380—42no—
108#380—42no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130#336C2×C418yes—
131C25C2×D411yescharacteristic
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4C4⋊C414yes—
159C23×C4C4⋊C414yes—
160C23×C4C22×C412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×Q812yes—
167C23×C4C2411yescharacteristic
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C22×C414yes—
179C22×C8C22×C414yes—
180C22×C8#4114yes—
181C22×C8#4114yes—
182#117—432no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
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⋊C4C2×D414yes—
222C2×C4⋊C4C2×D414yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—28no—
241C2×C42—28no—
242C2×C42—28no—
243C2×C42—28no—
244C2×C42—28no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42C4⋊C414yes—
248C2×C42C4⋊C414yes—
249C2×C42C22×C412yes—
250C2×C42C22×C412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×Q812yes—
254C2×C42C2×Q812yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257#107—48no—
258#106—48no—
259C22⋊C8—216no—
260#97—28no—
261#97—28no—
262#95—28no—
263#95—28no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—42no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24C2×C4⋊C412yes—
287C24AGL(1,ℤ/8)12yes—
288C24#13712yes—
289C24C22×D411yescharacteristic
290C22×C4—28no—
291C22×C4—28no—
292C22×C4—28no—
293C22×C4—28no—
294C22×C4—21no—
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—28no—
307C22×C4—28no—
308C22×C4—28no—
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—21no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
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—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
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—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4C2×C4⋊C412yes—
387C22×C4C2×C4⋊C412yes—
388C22×C4C2×C4⋊C412yes—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4#12312yes—
394C22×C4#12312yes—
395C22×C4AGL(1,ℤ/8)12yes—
396C22×C4#13712yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×Q811yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C42−1+412yes—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8#11718yescomplemented
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—42no—
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—
441C22⋊C4—416no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—28no—
449C42—28no—
450C42—416no—
451C42C4×D414yes—
452C42C4×D414yes—
453C42C22⋊Q814yes—
454C42C22⋊Q814yes—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
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—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23#42814yes—
490C23C22×C4⋊C411yescharacteristic
491C23C2×#12011yescharacteristic
492C23C2×#12311yescharacteristic
493C23#53211yescharacteristic
494C23#53211yescharacteristic
495C23#53511yescharacteristic
496C23#53611yescharacteristic
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23C2×#13711yescharacteristic
500C23C2×#13711yescharacteristic
501C23#57511yescharacteristic
502C23#57511yescharacteristic
503C23#57511yescharacteristic
504C23#57511yescharacteristic
505C2×C4—232no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
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—416no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
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—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4#38514yes—
586C2×C4#38514yes—
587C2×C4#42814yes—
588C2×C4C2×#11712yes—
589C2×C4C2×C4×D411yescharacteristic
590C2×C4C2×C4×D411yescharacteristic
591C2×C4#52012yes—
592C2×C4C2×C22⋊Q811yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4#53312yes—
595C2×C4#54412yes—
596C2×C4#54612yes—
597C2×C4#54712yes—
598C2×C4#54812yes—
599C2×C4#55012yes—
600C2×C4#55212yes—
601C2×C4#55312yes—
602C2×C4#55512yes—
603C2×C4#55612yes—
604C8—28no—
605C8—44no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22#228511yescharacteristic
622C22C2×#38511yescharacteristic
623C22#231711yescharacteristic
624C22#243111yescharacteristic
625C22#243211yescharacteristic
626C22#249911yescharacteristic
627C22#250011yescharacteristic
628C22C2×#42811yescharacteristic
629C22C2×#42811yescharacteristic
630C22#291111yescharacteristic
631C22#291111yescharacteristic
632C22#300311yescharacteristic
633C22#300411yescharacteristic
634C22#300511yescharacteristic
635C22#300511yescharacteristic
636C22#309211yescharacteristic
637C22#309211yescharacteristic
638C22#309311yescharacteristic
639C22#309311yescharacteristic
640C22#317111yescharacteristic
641C22#317111yescharacteristic
642C22#317211yescharacteristic
643C22#317211yescharacteristic
644C22#318011yescharacteristic
645C22#318011yescharacteristic
646C22#318111yescharacteristic
647C22#318111yescharacteristic
648C22#321311yescharacteristic
649C22#321411yescharacteristic
650C22#321511yescharacteristic
651C22#321611yescharacteristic
652C22#323411yescharacteristic
653C22#323511yescharacteristic
654C22#323611yescharacteristic
655C22#323711yescharacteristic
656C4—24no—
657C4—24no—
658C4—24no—
659C4—42no—
660C4—42no—
661C4—84no—
662C4—84no—
663C4—161no—
664C4—161no—
665C4—161no—
666C4—161no—
667C4#235212yes—
668C4#235412yes—
669C4#243912yes—
670C4#244112yes—
671C2—44no—
672C2#1387911yescharacteristic
673C2#2081911yescharacteristic
674C2#2082311yescharacteristic
675C2#2084411yescharacteristic
676C2#2084511yescharacteristic
677C2#2198611yescharacteristic
678C2#2198811yescharacteristic
679C2#2560211yescharacteristic
680C2#2560311yescharacteristic
681C2#2567611yescharacteristic
682C2#2567711yescharacteristic
683C2#2788011yescharacteristic
684C2#2788111yescharacteristic
685C2#2789411yescharacteristic
686C2#2789511yescharacteristic
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
34. #1749C4semidirect
420. C2×C8#117semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#18472
C2×#18362
C2×#17492
#146822
#110962
#110952
#109082
#109072
#108432