Finite Group Databasebeta

← #363717

Subgroups — #363717

Order 512.

1772 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×#2352C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13907C212yesmaximal, complemented
5#11043C214yesmaximal
6#10854C212yesmaximal, complemented
7#10846C212yesmaximal, complemented
8#10801C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#117C411yescharacteristic, complemented
11C23×C4⋊C4C2211yescharacteristic
12C22×#99C2212yes—
13C22×#99C2212yes—
14#2352—22no—
15#2352—22no—
16#2352—24no—
17C2×#389—22no—
18C2×#388C412yescomplemented
19C2×#387C412yescomplemented
20C2×#386C2212yes—
21C2×#382C411yescharacteristic, complemented
22C2×#380C411yescharacteristic, complemented
23C2×#380C2212yes—
24C2×#380C2214yes—
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C411yescharacteristic, complemented
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2212yes—
29#1799—24no—
30C4×#99—24no—
31C2×#342C2214yes—
32C2×#342C2211yescharacteristic
33C2×#342C2212yes—
34C2×#342C2211yescharacteristic
35C2×#342C2214yes—
36#1441—24no—
37C4⋊#95—22no—
38C22⋊#95C2214yes—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2312yes—
42C24×C4C2311yescharacteristic
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4D414yes—
47C2×C4×D4Q814yes—
48C2×#117—21no—
49C2×#117—22no—
50C2×#117—21no—
51C2×#117—44no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—44no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D414yes—
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2312yes—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2311yescharacteristic
72C22×C42—22no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C412yes—
75C22×C42C2×C412yes—
76C2×#99—24no—
77C2×#99—24no—
78C2×#99—24no—
79C2×#99—24no—
80C2×#99—42no—
81C2×#99—42no—
82#389—48no—
83#388—24no—
84#388—24no—
85#387—24no—
86#387—24no—
87#386—24no—
88#386—24no—
89#382—22no—
90#382—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—24no—
96#380—24no—
97#380—28no—
98#380—44no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—21no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—24no—
106C2×#95—24no—
107C2×#95—24no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—24no—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116#342—24no—
117#342—22no—
118#342—22no—
119#342—28no—
120#342—44no—
121C25C22×C411yescharacteristic
122C25C2411yescharacteristic
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4C2×D412yes—
132C22×D4C2×Q812yes—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—82no—
148C23×C4C4⋊C414yescomplemented
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D414yes—
159C23×C4C2×D412yes—
160C23×C4C2×D414yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×Q812yes—
164C23×C4#4114yes—
165C4×D4—28no—
166C4×D4—28no—
167C4×D4—28no—
168C4×D4—28no—
169#117—44no—
170#117—48no—
171#117—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—84no—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C414yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C4⋊C414yescomplemented
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4C4⋊C414yescomplemented
230C2×C22⋊C4C2×D412yes—
231C2×C22⋊C4C2×Q812yes—
232C2×C22⋊C4#4114yes—
233C2×C22⋊C4#4114yes—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—82no—
246C2×C42C4⋊C414yescomplemented
247#99—216no—
248#99—48no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—24no—
254#95—24no—
255#95—48no—
256#95—48no—
257#95—44no—
258#95—44no—
259#95—84no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24C2×C4⋊C412yes—
275C24C23×C411yescharacteristic
276C24C22×D412yes—
277C24C22×D411yescharacteristic
278C24C22×Q811yescharacteristic
279C24C2×#4112yes—
280C24C2×#4112yes—
281C24C2×#4112yes—
282C24H(5,𝔽2)11yescharacteristic
283C24H(5,𝔽2)11yescharacteristic
284C242−1+411yescharacteristic
285C242−1+411yescharacteristic
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—48no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—84no—
383C22×C4—88no—
384C22×C4—84no—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C4⋊C412yes—
392C22×C4C2×C4⋊C412yes—
393C22×C4C2×C4⋊C412yes—
394C22×C4C2×C4⋊C412yes—
395C22×C4C2×C4⋊C412yes—
396C22×C4#11712yes—
397C22×C4#11712yes—
398C22×C4#12114yes—
399C22×C4#12114yes—
400C22×C4C22⋊Q814yes—
401C22×C4C22⋊Q814yes—
402C4⋊C4—28no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—88no—
414C4⋊C4—816no—
415C4⋊C4#9914yescomplemented
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—28no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—44no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—84no—
436C42—24no—
437C42—24no—
438C42—44no—
439C42—44no—
440C42—82no—
441C42—82no—
442C42—88no—
443C23—22no—
444C23—21no—
445C23—22no—
446C23—21no—
447C23—22no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23C22×C22⋊C411yescharacteristic
493C23C22×C4⋊C411yescharacteristic
494C23C2×#11711yescharacteristic
495C23C4×#4111yescharacteristic
496C23C4×#4111yescharacteristic
497C23#52011yescharacteristic
498C23#52011yescharacteristic
499C23C2×#12112yes—
500C23C2×C22⋊Q812yes—
501C23#52512yes—
502C23#53312yes—
503C23#53511yescharacteristic
504C23#53611yescharacteristic
505C23#54411yescharacteristic
506C23#54411yescharacteristic
507C23#54612yes—
508C23#54712yes—
509C23#54812yes—
510C23#55012yes—
511C23#55212yes—
512C23#55312yes—
513C23#55512yes—
514C23#55612yes—
515D4—44no—
516D4—44no—
517D4—44no—
518D4—44no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—82no—
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—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4#38012yes—
598C2×C4#38014yes—
599C2×C4#38012yes—
600C2×C4C2×#9914yes—
601C2×C4C2×#9912yes—
602C2×C4#41012yes—
603C22—22no—
604C22—22no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—21no—
616C22—21no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—21no—
622C22—21no—
623C22—22no—
624C22—22no—
625C22—21no—
626C22—21no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22C2×#38011yescharacteristic
649C22#229611yescharacteristic
650C22#231711yescharacteristic
651C22#233911yescharacteristic
652C22#234111yescharacteristic
653C22#234611yescharacteristic
654C22#234711yescharacteristic
655C22#241412yes—
656C22#241512yes—
657C22#243912yes—
658C22#244112yes—
659C22C22×#9911yescharacteristic
660C22C2×#41011yescharacteristic
661C22#288311yescharacteristic
662C22#288411yescharacteristic
663C4—22no—
664C4—24no—
665C4—24no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—84no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4—164no—
676C4#179914yes—
677C2—22no—
678C2—22no—
679C2—21no—
680C2—21no—
681C2—21no—
682C2—21no—
683C2—22no—
684C2—22no—
685C2—82no—
686C2#1395111yescharacteristic
687C2C2×#179911yescharacteristic
688C2#2063411yescharacteristic
689C2#2247611yescharacteristic
690C2#2247811yescharacteristic
691C2#2253411yescharacteristic
692C2#2253511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13907C2semidirect
6. #10854C2semidirect
7. #10846C2semidirect
8. #10801C2semidirect
10. C22×#117C4semidirect
18. C2×#388C4semidirect
19. C2×#387C4semidirect
21. C2×#382C4semidirect
22. C2×#380C4semidirect
26. C2×C4×C4⋊C4C4semidirect
148. C23×C4C4⋊C4semidirect
199. C2×C4⋊C4C4⋊C4semidirect
229. C2×C22⋊C4C4⋊C4semidirect
246. C2×C42C4⋊C4semidirect
415. C4⋊C4#99semidirect

Maximal subgroups

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

subgroupindex
C2×#23522
C2×#17992
C2×C4×#992
#139072
#110432
#108542
#108462
#108012