Finite Group Databasebeta

← #6792580

Subgroups — #6792580

Order 512.

1749 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36116C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#32906C211yesmaximal, characteristic, complemented
4#31650C214yesmaximal, complemented
5#31536C214yesmaximal, complemented
6#29908C212yesmaximal, complemented
7#28419C212yesmaximal, complemented
8#27824C212yesmaximal, complemented
9#25902C211yesmaximal, characteristic, complemented
10#25324C212yesmaximal, complemented
11#24815C212yesmaximal, complemented
12#24508C212yesmaximal, complemented
13#24299C211yesmaximal, characteristic
14#23937C212yesmaximal
15#22603C211yesmaximal, characteristic, complemented
16#21867C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C2×#487C2212yes—
22C2×#476C2212yes—
23C2×#462C2212yes—
24C2×#451C2212yes—
25C2×#448C2211yescharacteristic
26C2×C4×SD16C2212yes—
27C2×#432C2212yes—
28#2857C2211yescharacteristic
29#2855C2212yes—
30#2567—21no—
31#2567—22no—
32#2567—21no—
33#2567C2212yes—
34#2567C2214yes—
35#2567C2212yes—
36#2552C2211yescharacteristic
37#2547C2211yescharacteristic
38#2496—21no—
39#2486—21no—
40#2478—22no—
41#2453C2212yes—
42#2447C2212yes—
43#2419—22no—
44C2×#395—22no—
45#2365C2211yescharacteristic
46D4×C4⋊C4—21no—
47D4×C4⋊C4C2211yescharacteristic
48#2344—21no—
49C4×#121C2211yescharacteristic
50C2×#391C2212yes—
51C2×#389—21no—
52C2×#389—21no—
53C2×#384C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2099C2214yes—
56#2096C2214yes—
57#2084C2214yes—
58#2072C2214yes—
59#2066C2214yes—
60#2058C2212yes—
61#2056C2214yes—
62#2026C2214yes—
63#2024C2214yes—
64#2023C2212yes—
65#2015C2214yes—
66#1992C2214yes—
67#1990C2212yes—
68#1987C2212yes—
69#1971C2214yes—
70#1959C2214yes—
71#1939C2212yes—
72#1935C2214yes—
73#1917C2214yes—
74#1880C2212yes—
75#1878C2214yes—
76#1878C2214yes—
77#1877C2214yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1872C2212yes—
83#1865C2212yes—
84#1853C2212yes—
85#1853C2211yescharacteristic
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×Q16C2314yes—
98C22×SD16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105#529—28no—
106#529—44no—
107C2×#126—22no—
108C2×#125C2312yes—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8C2312yes—
112C2×#121C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—44no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×#117—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#487—44no—
137#476—44no—
138#462—44no—
139#451—44no—
140#448—24no—
141C4×SD16—44no—
142#432—44no—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161#400—22no—
162#398—22no—
163#398—22no—
164#397—22no—
165#397—22no—
166#397—22no—
167#395—24no—
168#395—24no—
169#395—24no—
170#395—24no—
171#391—22no—
172#391—22no—
173#391C2314yes—
174#391C2314yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#390C2311yescharacteristic
179#389—22no—
180#389—22no—
181#389—22no—
182#389—24no—
183#389—24no—
184#389—22no—
185#389—21no—
186#389—22no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389C2311yescharacteristic
191#389C2312yes—
192#389C2312yes—
193#388—21no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#386—21no—
199#385—22no—
200#385—22no—
201#385—22no—
202#384—21no—
203#384—22no—
204#384—22no—
205#384C2314yes—
206#384C2314yes—
207#384C2311yescharacteristic
208#381—22no—
209#381—22no—
210#380—21no—
211#380—21no—
212#380—21no—
213#380—21no—
214#380—21no—
215#380C2311yescharacteristic
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#340C2314yes—
231#340C2314yes—
232#340C2312yes—
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×Q8C2412yes—
250C22×Q8C2412yes—
251C22×D4—21no—
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×Q16—28no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×SD16—24no—
267C2×SD16—24no—
268C2×SD16—42no—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#126—48no—
279#125—24no—
280#125—42no—
281#124—24no—
282#124—42no—
283#123—48no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286#121—22no—
287#121#4114yes—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—42no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—42no—
299C4×D4—48no—
300C4×D4—44no—
301#117—48no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—84no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369#106—42no—
370#106—44no—
371C4⋊C8—24no—
372C4⋊C8—44no—
373#103—24no—
374#103—24no—
375#103—42no—
376#103—44no—
377#103—44no—
378#103—44no—
379#103—44no—
380#102—42no—
381#102—44no—
382C22⋊C8—22no—
383C22⋊C8—22no—
384C22⋊C8—24no—
385#97—44no—
386C4×C8—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
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—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—48no—
419C2×Q8—48no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8—816no—
429C2×Q82−1+414yes—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4C2×#4112yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—82no—
473C22×C4#13714yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4C2511yescharacteristic
486Q16—88no—
487SD16—48no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4#13714yes—
533C4⋊C4C2×#4112yes—
534C22⋊C4—24no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C42—22no—
544C42—22no—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—44no—
553C42—44no—
554C42—44no—
555C42—42no—
556C42H(5,𝔽2)14yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23#54612yes—
569C23C2×#13712yes—
570C23#57712yes—
571C23C23×D411yescharacteristic
572Q8—24no—
573Q8—24no—
574Q8—48no—
575Q8—42no—
576Q8—42no—
577Q8—44no—
578Q8—44no—
579Q8—88no—
580Q8—88no—
581D4—24no—
582D4—48no—
583D4—42no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—88no—
636C2×C4—84no—
637C2×C4—88no—
638C2×C4—84no—
639C2×C4#53512yes—
640C2×C4#53512yes—
641C2×C4#54612yes—
642C2×C4C2×#13712yes—
643C2×C4C2×#13714yes—
644C2×C4C2×#13712yes—
645C2×C4C2×#13712yes—
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×2−1+412yes—
650C8—42no—
651C8—42no—
652C8—44no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22#333114yes—
667C22C2×#53511yescharacteristic
668C22C2×#54611yescharacteristic
669C22#348611yescharacteristic
670C22C22×#13711yescharacteristic
671C22C22×#13711yescharacteristic
672C22C2×#57711yescharacteristic
673C22#358511yescharacteristic
674C4—22no—
675C4—22no—
676C4—24no—
677C4—22no—
678C4—41no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—81no—
685C4—81no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4—82no—
690C4#321814yes—
691C4#352012yes—
692C4#353612yes—
693C2—24no—
694C2—82no—
695C2#3890511yescharacteristic
696C2C2×#321811yescharacteristic
697C2#6081211yescharacteristic
698C2C2×#333111yescharacteristic
699C2#6116211yescharacteristic
700C2#6183011yescharacteristic
701C2#6183811yescharacteristic
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×#2567C2semidirect
3. #32906C2semidirect
4. #31650C2semidirect
5. #31536C2semidirect
6. #29908C2semidirect
7. #28419C2semidirect
8. #27824C2semidirect
9. #25902C2semidirect
10. #25324C2semidirect
11. #24815C2semidirect
12. #24508C2semidirect
15. #22603C2semidirect

Maximal subgroups

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

subgroupindex
#361162
C2×#25672
#329062
#316502
#315362
#299082
#284192
#278242
#259022
#253242
#248152
#245082
#242992
#239372
#226032
#218672
C4×#4482