Finite Group Databasebeta

← #6792589

Subgroups — #6792589

Order 512.

1781 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36117C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#32903C211yesmaximal, characteristic
4#31652C214yesmaximal, complemented
5#31534C214yesmaximal, complemented
6#29906C212yesmaximal, complemented
7#28418C212yesmaximal, complemented
8#27824C212yesmaximal, complemented
9#25903C211yesmaximal, characteristic
10#25322C212yesmaximal, complemented
11#24817C212yesmaximal, complemented
12#24509C212yesmaximal, complemented
13#24302C211yesmaximal, characteristic, complemented
14#23937C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16#21867C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C2×#486C2212yes—
22C2×#477C2212yes—
23C2×#460C2212yes—
24C2×#451C2211yescharacteristic
25C2×#448C2212yes—
26C2×C4×Q16C2212yes—
27C2×#432C2212yes—
28#2856C2211yescharacteristic
29#2854C2212yes—
30#2567—21no—
31#2567—22no—
32#2567—21no—
33#2567C2212yes—
34#2567C2214yes—
35#2567C2212yes—
36#2552C2211yescharacteristic
37#2548C2211yescharacteristic
38#2512—21no—
39#2501—21no—
40#2479—22no—
41#2454C2212yes—
42#2448C2212yes—
43#2425—22no—
44C2×#395—22no—
45#2366C2211yescharacteristic
46#2361—21no—
47#2354—21no—
48Q8×C4⋊C4C2211yescharacteristic
49C4×C22⋊Q8C2211yescharacteristic
50C2×#391C2212yes—
51C2×#389—21no—
52C2×#389—21no—
53C2×#384C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2099C2214yes—
56#2093C2214yes—
57#2083C2214yes—
58#2071C2214yes—
59#2068C2214yes—
60#2057C2212yes—
61#2055C2214yes—
62#2026C2214yes—
63#2024C2212yes—
64#2023C2214yes—
65#2015C2214yes—
66#1992C2214yes—
67#1990C2212yes—
68#1987C2212yes—
69#1973C2214yes—
70#1959C2214yes—
71#1939C2212yes—
72#1935C2214yes—
73#1917C2214yes—
74#1880C2214yes—
75#1878C2214yes—
76#1878C2212yes—
77#1877C2214yes—
78#1874C2212yes—
79#1874C2212yes—
80#1874C2212yes—
81#1872C2211yescharacteristic
82#1872C2212yes—
83#1863C2212yes—
84#1852C2212yes—
85#1852C2211yescharacteristic
86#1817C2212yes—
87#1803C2214yes—
88#1803C2212yes—
89#1802C2214yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92#1770C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105#529—28no—
106#529—44no—
107C2×#126—22no—
108C2×#125C2312yes—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—44no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#486—44no—
139#477—44no—
140#460—44no—
141#451—24no—
142#448—44no—
143C4×Q16—44no—
144#432—44no—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2311yescharacteristic
161C2×#97C2314yes—
162C2×C4×C8C2312yes—
163#399—22no—
164#399—22no—
165#398—22no—
166#397—22no—
167#396—22no—
168#395—24no—
169#395—24no—
170#395—24no—
171#395—24no—
172#394—22no—
173#391—22no—
174#391—22no—
175#391C2314yes—
176#391C2314yes—
177#391C2311yescharacteristic
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#389—22no—
187#389—22no—
188#389—24no—
189#389—22no—
190#389—24no—
191#389—22no—
192#389—21no—
193#389—22no—
194#389C2311yescharacteristic
195#389C2312yes—
196#389C2312yes—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#387—21no—
202#386—22no—
203#386—22no—
204#385—21no—
205#385—22no—
206#384—21no—
207#384—21no—
208#384—22no—
209#384—22no—
210#384C2314yes—
211#384C2314yes—
212#384C2311yescharacteristic
213#380—22no—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C2×#95—22no—
226C2×#95—22no—
227C2×#95D412yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×Q8—44no—
247C22×Q8—82no—
248C22×Q8—82no—
249C22×Q8C2411yescharacteristic
250C22×Q8C2412yes—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C2412yes—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C2×Q16—24no—
265C2×Q16—24no—
266C2×Q16—42no—
267C2×SD16—28no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#127—24no—
279#126—48no—
280#125—24no—
281#125—42no—
282#124—24no—
283#124—42no—
284#123—48no—
285#123—84no—
286C22⋊Q8—22no—
287C22⋊Q8#4114yes—
288#121—24no—
289#121—24no—
290#121—44no—
291#121—82no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—42no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×D4—24no—
302C4×D4—42no—
303C4×D4—48no—
304C4×D4—44no—
305#117—48no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—84no—
333C2×C4⋊C4—84no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—44no—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D412yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375#107—42no—
376#107—44no—
377C4⋊C8—24no—
378C4⋊C8—44no—
379#103—24no—
380#103—24no—
381#103—42no—
382#103—42no—
383#103—44no—
384#103—44no—
385#102—44no—
386#102—44no—
387#102—44no—
388C22⋊C8—22no—
389C22⋊C8—22no—
390C22⋊C8—24no—
391#97—44no—
392C4×C8—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—22no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—22no—
409C24C22×D411yescharacteristic
410C2×Q8—24no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—48no—
420C2×Q8—48no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8—816no—
430C2×Q8C2×#4112yes—
431C2×Q82−1+414yes—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—81no—
444C2×D4—81no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—82no—
478C22×C4#13714yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4C2511yescharacteristic
491Q16—48no—
492SD16—88no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—81no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—81no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4#13714yes—
537C4⋊C4C2×#4112yes—
538C22⋊C4—24no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C42—22no—
551C42—22no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—44no—
560C42—44no—
561C42—44no—
562C42—42no—
563C42—42no—
564C42H(5,𝔽2)14yes—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23#54612yes—
577C23C2×#13712yes—
578C23#57612yes—
579C23C23×D411yescharacteristic
580Q8—24no—
581Q8—24no—
582Q8—24no—
583Q8—48no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—88no—
588Q8—88no—
589D4—44no—
590D4—44no—
591D4—88no—
592D4—84no—
593D4—162no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—88no—
646C2×C4—84no—
647C2×C4—88no—
648C2×C4#53512yes—
649C2×C4#53512yes—
650C2×C4#54612yes—
651C2×C4C2×#13712yes—
652C2×C4C2×#13714yes—
653C2×C4C2×#13712yes—
654C2×C4C2×#13712yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×2−1+412yes—
659C8—42no—
660C8—42no—
661C8—44no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22#333314yes—
676C22C2×#53511yescharacteristic
677C22C2×#54611yescharacteristic
678C22#348611yescharacteristic
679C22C22×#13711yescharacteristic
680C22C22×#13711yescharacteristic
681C22C2×#57611yescharacteristic
682C22#358511yescharacteristic
683C4—22no—
684C4—22no—
685C4—24no—
686C4—22no—
687C4—41no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—81no—
695C4—82no—
696C4—84no—
697C4—81no—
698C4—82no—
699C4#321814yes—
700C4#352012yes—
701C4#353612yes—
702C2—24no—
703C2—162no—
704C2#3890511yescharacteristic
705C2C2×#321811yescharacteristic
706C2#6081111yescharacteristic
707C2C2×#333311yescharacteristic
708C2#6116211yescharacteristic
709C2#6183011yescharacteristic
710C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
4. #31652C2semidirect
5. #31534C2semidirect
6. #29906C2semidirect
7. #28418C2semidirect
8. #27824C2semidirect
10. #25322C2semidirect
11. #24817C2semidirect
12. #24509C2semidirect
13. #24302C2semidirect
14. #23937C2semidirect
15. #22603C2semidirect
16. #21867C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#361172
C2×#25672
#329032
#316522
#315342
#299062
#284182
#278242
#259032
#253222
#248172
#245092
#243022
#239372
#226032
#218672
C4×#4512