Finite Group Databasebeta

← #368038

Subgroups — #368038

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2324C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3C2×#1875C211yesmaximal, characteristic
4C2×#1874C211yesmaximal, characteristic
5C2×#1862C212yesmaximal, complemented
6C2×#1852C211yesmaximal, characteristic, complemented
7#11328C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#10885C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#107C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2324—24no—
23#2324—24no—
24#2317—28no—
25C2×#390—21no—
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#388—21no—
31C2×#386—21no—
32C2×#385—21no—
33C2×#385—21no—
34C2×#385—21no—
35C2×#384—21no—
36C2×#382—21no—
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1875C418yescomplemented
40#1874—24no—
41#1862—28no—
42#1852—24no—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#336C2212yes—
49C2×#336C2212yes—
50C23×D4—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4—21no—
54C24×C4C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121C2×C414yes—
61C2×#121D414yes—
62C2×#121Q814yes—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4—42no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—24no—
72C2×#117—24no—
73C2×#117—42no—
74C2×#117C2×C414yescomplemented
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42—21no—
99C22×C42—21no—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C2×#107—24no—
103C2×#107—24no—
104C2×#107—42no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—42no—
110C2×#102C2×C418yescomplemented
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8C2×C414yescomplemented
117#390—28no—
118#390—22no—
119#390—22no—
120#389—24no—
121#389—24no—
122#389—22no—
123#389—22no—
124#389—24no—
125#389D418yescomplemented
126#388—28no—
127#386—28no—
128#385—28no—
129#385—28no—
130#385—28no—
131#384—28no—
132#382—28no—
133#380—22no—
134#380D414yes—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—21no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—24no—
146#340C2×C418yescomplemented
147#336—24no—
148#336—42no—
149C25—21no—
150C25C2×D411yescharacteristic
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—82no—
162C22×D4—82no—
163C22×D4C22×C412yes—
164C22×D4C2×D412yes—
165C22×D4C2×Q812yes—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—82no—
182C23×C4—82no—
183C23×C4—82no—
184C23×C4—82no—
185C23×C4C22×C412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×Q812yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8C22×C414yes—
211#121—28no—
212#121—28no—
213#121—28no—
214#121C4⋊C4116yescomplemented
215C4×D4—48no—
216C4×D4—48no—
217#117—216no—
218#117—216no—
219#117—48no—
220#117—48no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4—82no—
256C2×C4⋊C4—82no—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×Q812yes—
265C2×C4⋊C4#4114yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—82no—
285C2×C22⋊C4—82no—
286C2×C22⋊C4—82no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—82no—
299C2×C42C2×D414yes—
300#107—48no—
301#102—216no—
302#102—48no—
303C22⋊C8—28no—
304C22⋊C8—44no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—28no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24—41no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24C4×D412yes—
330C24#12012yes—
331C24C22⋊Q812yes—
332C24C22×D411yescharacteristic
333C2×D4—216no—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—88no—
350C2×D4—88no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4C2×C4⋊C414yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—48no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—48no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—88no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—88no—
427C22×C4C2×C4⋊C414yes—
428C22×C4C4×D414yes—
429C22×C4C4×D412yes—
430C22×C4#12012yes—
431C22×C4#12012yes—
432C22×C4#12012yes—
433C22×C4C22⋊Q814yes—
434C22×C4C22⋊Q812yes—
435C22×C4#12314yes—
436C22×C4C23×C411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×Q811yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4C2×#4111yescharacteristic
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+412yes—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C4⋊C4—84no—
480C4⋊C4C2×C4⋊C414yes—
481C4⋊C4C4×D418yes—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C42—216no—
491C42—24no—
492C42—24no—
493C42—42no—
494C42—42no—
495C42—84no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23#38514yes—
542C23C2×C4×D411yescharacteristic
543C23C2×#12011yescharacteristic
544C23C2×#12011yescharacteristic
545C23C2×C22⋊Q811yescharacteristic
546C23#52511yescharacteristic
547C23#53411yescharacteristic
548C23#53611yescharacteristic
549C23#57612yes—
550C23#57612yes—
551C23#57712yes—
552C23#57812yes—
553D4—28no—
554D4—44no—
555D4—44no—
556D4—44no—
557D4—816no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—816no—
610C2×C4—816no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4#38514yes—
620C2×C4C22×C4⋊C411yescharacteristic
621C2×C4C2×C4×D412yes—
622C2×C4C2×C4×D411yescharacteristic
623C2×C4#52012yes—
624C2×C4C2×C22⋊Q811yescharacteristic
625C2×C4C2×#12311yescharacteristic
626C2×C4#52512yes—
627C2×C4#53912yes—
628C2×C4D4212yes—
629C2×C4D4212yes—
630C2×C4#54712yes—
631C2×C4#54712yes—
632C2×C4#54812yes—
633C2×C4D4×Q812yes—
634C2×C4#55312yes—
635C2×C4#55412yes—
636C8—44no—
637C8—82no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—82no—
653C22—82no—
654C22—84no—
655C22—84no—
656C22—82no—
657C22—82no—
658C22—84no—
659C22—84no—
660C22C2×#38511yescharacteristic
661C22C4×#12011yescharacteristic
662C22#232211yescharacteristic
663C22#240311yescharacteristic
664C22#240611yescharacteristic
665C22#240911yescharacteristic
666C22#257911yescharacteristic
667C22#294612yes—
668C22#295112yes—
669C22#300811yescharacteristic
670C22#300811yescharacteristic
671C22#300812yes—
672C22#300911yescharacteristic
673C22#301111yescharacteristic
674C22#301312yes—
675C22#308112yes—
676C22#308312yes—
677C22#311912yes—
678C22#312412yes—
679C22#319912yes—
680C22#320112yes—
681C22#324112yes—
682C22#324312yes—
683C4—24no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—48no—
688C4—82no—
689C4—81no—
690C4—82no—
691C4—84no—
692C4—81no—
693C4D4×C4⋊C412yes—
694C4#236112yes—
695C4#244312yes—
696C4#244912yes—
697C2—44no—
698C2—84no—
699C2#1401511yescharacteristic
700C2#2087611yescharacteristic
701C2#2088111yescharacteristic
702C2#2146012yes—
703C2#2208812yes—
704C2#2392511yescharacteristic
705C2#2392711yescharacteristic
706C2#2406911yescharacteristic
707C2#2407311yescharacteristic
708C2#2430812yes—
709C2#2924712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2317C2semidirect
5. C2×#1862C2semidirect
6. C2×#1852C2semidirect
7. #11328C2semidirect
8. #11066C2semidirect
9. #10885C2semidirect
10. #10731C2semidirect
11. #10713C2semidirect
39. #1875C4semidirect
74. C2×#117C2×C4semidirect
110. C2×#102C2×C4semidirect
116. C2×C22⋊C8C2×C4semidirect
125. #389D4semidirect
146. #340C2×C4semidirect
214. #121C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23242
C2×#23172
C2×#18752
C2×#18742
C2×#18622
C2×#18522
#113282
#110662
#108852
#107312
#107132