Finite Group Databasebeta

← #6800075

Subgroups — #6800075

Order 512.

1785 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#32957C211yesmaximal, characteristic, complemented
4#32098C214yesmaximal, complemented
5#30894C214yesmaximal, complemented
6#29957C212yesmaximal, complemented
7#28416C212yesmaximal, complemented
8#27777C212yesmaximal, complemented
9#25861C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic, complemented
11#25278C212yesmaximal, complemented
12#24730C212yesmaximal, complemented
13#24209C212yesmaximal
14#24209C211yesmaximal, characteristic
15#22629C211yesmaximal, characteristic, complemented
16#22047C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C2×#491C2214yes—
22C2×#487C2212yes—
23C2×#477C2212yes—
24C2×#461C2212yes—
25C2×#461C2214yes—
26C2×#448C2212yes—
27C2×#448C2211yescharacteristic
28C2×C4×SD16C2212yes—
29#2857C2211yescharacteristic
30#2851C2212yes—
31#2589—22no—
32#2563C2211yescharacteristic
33#2552C2211yescharacteristic
34#2546—22no—
35#2546—21no—
36#2546—21no—
37#2546C2212yes—
38#2546C2214yes—
39#2546C2212yes—
40#2509C2212yes—
41#2497—21no—
42#2487—21no—
43#2454C2212yes—
44#2450—22no—
45C2×#398—22no—
46#2369C2211yescharacteristic
47D4×C4⋊C4—21no—
48D4×C4⋊C4C2211yescharacteristic
49#2345—21no—
50C4×#121C2212yes—
51C4×#121C2211yescharacteristic
52C2×#391C2212yes—
53C2×#389—21no—
54C2×#389—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2097C2214yes—
57#2068C2214yes—
58#2066C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2043C2214yes—
62#2043C2212yes—
63#2043C2214yes—
64#2026C2214yes—
65#2013C2214yes—
66#2013C2212yes—
67#1981C2214yes—
68#1981C2212yes—
69#1973C2214yes—
70#1960C2214yes—
71#1956C2212yes—
72#1939C2212yes—
73#1935C2214yes—
74#1928C2214yes—
75#1917C2214yes—
76#1880C2212yes—
77#1877C2214yes—
78#1874C2211yescharacteristic
79#1865C2212yes—
80#1863C2214yes—
81#1853C2211yescharacteristic
82#1852C2212yes—
83#1803C2212yes—
84#1797C2212yes—
85#1797C2211yescharacteristic
86#1787C2212yes—
87#1787C2212yes—
88#1787C2211yescharacteristic
89#1787C2212yes—
90#1787C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2312yes—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8—48no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125C2312yes—
111C2×C22⋊Q8—21no—
112C2×#121C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—44no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#491—48no—
135#487—44no—
136#477—44no—
137#461—48no—
138#461—44no—
139#448—24no—
140#448—28no—
141C4×SD16—44no—
142C2×#107C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#102C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2311yescharacteristic
161C2×C22⋊C8C2311yescharacteristic
162C2×#97C2314yes—
163C2×C4×C8C2312yes—
164#400—22no—
165#400—22no—
166#399—22no—
167#398—22no—
168#398—24no—
169#398—24no—
170#398—24no—
171#398—24no—
172#396—22no—
173#393—22no—
174#391—21no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391C2314yes—
179#391C2314yes—
180#390C2312yes—
181#390C2311yescharacteristic
182#389—22no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—24no—
189#389—24no—
190#389—22no—
191#389—21no—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2311yescharacteristic
196#389C2312yes—
197#388—21no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#386—21no—
203#386—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#385—22no—
208#384—21no—
209#384C2311yescharacteristic
210#382C2312yes—
211#380—21no—
212#380—21no—
213#380—21no—
214#380—21no—
215#380—22no—
216#380C2311yescharacteristic
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D412yes—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C2×#95—22no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—42no—
248C22×Q8—44no—
249C22×Q8—44no—
250C22×Q8C2414yes—
251C22×Q8C2412yes—
252C22×D4—21no—
253C22×D4C2411yescharacteristic
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—28no—
267C2×SD16—42no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—28no—
274C4⋊Q8—28no—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8—48no—
284C4⋊Q8—48no—
285#125—44no—
286C22⋊Q8—82no—
287#121—22no—
288#121#4114yes—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—42no—
292C4×Q8—42no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×D4—24no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—84no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—48no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C42—24no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365#107—44no—
366#106—42no—
367C4⋊C8—24no—
368C4⋊C8—44no—
369#103—28no—
370#103—24no—
371#103—24no—
372#103—28no—
373#103—42no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103—44no—
379#103—44no—
380#103—44no—
381#102—42no—
382C22⋊C8—24no—
383C22⋊C8—22no—
384C22⋊C8—22no—
385#97—44no—
386C4×C8—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—21no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—21no—
407C24C22×D411yescharacteristic
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8—81no—
429C2×Q8—81no—
430C2×Q8—88no—
431C2×Q8—88no—
432C2×Q8H(5,𝔽2)14yes—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C2×#4112yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4#13714yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4112yes—
487C22×C42−1+412yes—
488C22×C4C2511yescharacteristic
489SD16—48no—
490SD16—88no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—81no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4#13714yes—
534C4⋊C4C2×#4112yes—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—22no—
545C42—22no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42—42no—
550C42—42no—
551C42—44no—
552C42—44no—
553C42—42no—
554C42—82no—
555C42H(5,𝔽2)14yes—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23#54812yes—
568C23C2×#13712yes—
569C23#57712yes—
570C23C23×D411yescharacteristic
571Q8—24no—
572Q8—24no—
573Q8—48no—
574Q8—42no—
575Q8—42no—
576Q8—44no—
577Q8—44no—
578Q8—44no—
579Q8—44no—
580Q8—88no—
581Q8—162no—
582D4—24no—
583D4—48no—
584D4—42no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—84no—
637C2×C4—88no—
638C2×C4—84no—
639C2×C4—164no—
640C2×C4#53512yes—
641C2×C4#53512yes—
642C2×C4#54812yes—
643C2×C4C2×#13712yes—
644C2×C4C2×#13714yes—
645C2×C4C2×#13712yes—
646C2×C4C2×#13712yes—
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×H(5,𝔽2)12yes—
649C2×C4C2×H(5,𝔽2)12yes—
650C2×C4C2×2−1+411yescharacteristic
651C8—42no—
652C8—44no—
653C8—42no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22#337214yes—
668C22C2×#53511yescharacteristic
669C22C2×#54811yescharacteristic
670C22#348411yescharacteristic
671C22C22×#13711yescharacteristic
672C22C22×#13711yescharacteristic
673C22C2×#57711yescharacteristic
674C22#358611yescharacteristic
675C4—22no—
676C4—22no—
677C4—24no—
678C4—22no—
679C4—42no—
680C4—41no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—81no—
686C4—82no—
687C4—82no—
688C4—81no—
689C4—82no—
690C4—84no—
691C4#321814yes—
692C4#351912yes—
693C4#353612yes—
694C2—24no—
695C2—82no—
696C2#3912011yescharacteristic
697C2C2×#321811yescharacteristic
698C2#6081311yescharacteristic
699C2C2×#337211yescharacteristic
700C2#6132211yescharacteristic
701C2#6158411yescharacteristic
702C2#6159311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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×#2546C2semidirect
3. #32957C2semidirect
4. #32098C2semidirect
5. #30894C2semidirect
6. #29957C2semidirect
7. #28416C2semidirect
8. #27777C2semidirect
9. #25861C2semidirect
10. #25856C2semidirect
11. #25278C2semidirect
12. #24730C2semidirect
15. #22629C2semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#25462
#329572
#320982
#308942
#299572
#284162
#277772
#258612
#258562
#252782
#247302
#242092
#226292
#220472
C4×#4482