Finite Group Databasebeta

← #6595402

Subgroups — #6595402

Order 512.

1631 conjugacy classes of subgroups.

The subgroup lattice has 773 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#33316C211yesmaximal, characteristic, complemented
4#33174C211yesmaximal, characteristic, complemented
5#33170C211yesmaximal, characteristic, complemented
6#29144C211yesmaximal, characteristic, complemented
7#29139C211yesmaximal, characteristic, complemented
8#28235C214yesmaximal, complemented
9#27892C211yesmaximal, characteristic, complemented
10#27891C211yesmaximal, characteristic, complemented
11#27890C211yesmaximal, characteristic, complemented
12#27889C211yesmaximal, characteristic, complemented
13#27800C214yesmaximal, complemented
14#27795C214yesmaximal, complemented
15#22819C214yesmaximal, complemented
16#21692C211yesmaximal, characteristic, complemented
17#20824C211yesmaximal, characteristic, complemented
18#20817C211yesmaximal, characteristic, complemented
19C4×#435C211yesmaximal, characteristic, complemented
20C2×#501C2214yes—
21C2×#492C2211yescharacteristic
22C2×#468C2214yes—
23C2×#465C2214yes—
24C2×#440C2214yes—
25C2×#435C2212yes—
26C2×C4×M16C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2856C2211yescharacteristic
33#2856C2211yescharacteristic
34#2571—21no—
35#2563—21no—
36#2538C2212yes—
37#2535C2212yes—
38#2513—21no—
39#2513—21no—
40#2497—21no—
41#2497—21no—
42#2492C2212yes—
43#2492C2212yes—
44#2490C2212yes—
45#2490C2212yes—
46#2439—21no—
47#2439—21no—
48#2420—21no—
49#2420—21no—
50#2414—21no—
51#2414—21no—
52#2369—21no—
53#2363—21no—
54#2363—21no—
55#2357C2212yes—
56#2355C2212yes—
57#2354—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×#127C2211yescharacteristic
60#2285C2211yescharacteristic
61#2285C2211yescharacteristic
62#2285C2211yescharacteristic
63#2285C2211yescharacteristic
64C42×D4C2211yescharacteristic
65#2036C2214yes—
66#2035C2214yes—
67#2034C2214yes—
68#2033C2214yes—
69#2026C2214yes—
70#2023C2214yes—
71#1973C2214yes—
72#1973C2214yes—
73#1968C2214yes—
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1928C2214yes—
81#1927C2214yes—
82#1926C2214yes—
83#1925C2214yes—
84#1847C2214yes—
85#1845C2211yescharacteristic
86#1845C2211yescharacteristic
87#1845C2211yescharacteristic
88#1844C2211yescharacteristic
89#1844C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1809C2214yes—
95#1808C2214yes—
96#1805C2214yes—
97#1802C2214yes—
98#1776C2214yes—
99C4×#107C2214yes—
100#1771C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102#1753C2212yes—
103#1752C2212yes—
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×SD16C2314yes—
108C22×M16C2312yes—
109C23×C8C2312yes—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×#127C2311yescharacteristic
115C2×#125C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×#121—21no—
119C2×#121—21no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×#121C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—21no—
133C2×#117—21no—
134C2×#117—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#501—48no—
138#492—24no—
139#468—48no—
140#465—48no—
141#440—216no—
142#435—28no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C4×M16—24no—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161C2×C4×C8—24no—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2312yes—
166C2×C4×C8C2311yescharacteristic
167#400—22no—
168#400—22no—
169#397—22no—
170#397—22no—
171#397—22no—
172#396—22no—
173#396—22no—
174#394—22no—
175#394—22no—
176#393—22no—
177#393—22no—
178#392—22no—
179#391—22no—
180#391—22no—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#390—22no—
185#390—22no—
186#390—22no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390C2312yes—
194#390C2312yes—
195#390C2312yes—
196#389—22no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#387—22no—
214#387—22no—
215#386—22no—
216#386—22no—
217#384—21no—
218#384—21no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2312yes—
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#382—22no—
227#382—21no—
228#382—21no—
229#382—21no—
230#382—21no—
231#382C2312yes—
232#381—22no—
233#380—21no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—21no—
241#380C2312yes—
242#380C2312yes—
243#380C2312yes—
244#380C2312yes—
245C4×C4⋊C4—22no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2311yescharacteristic
259C4×C22⋊C4—22no—
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4D411yescharacteristic
264C4×C22⋊C4C2311yescharacteristic
265C4×C22⋊C4C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274C22×Q8—22no—
275C22×Q8C2412yes—
276C22×D4—22no—
277C22×D4—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2×D411yescharacteristic
281C22×D4C2412yes—
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288C23×C4C2411yescharacteristic
289C2×SD16—28no—
290C2×SD16—28no—
291C2×SD16—28no—
292C2×M16—24no—
293C2×M16—24no—
294C2×M16—24no—
295C22×C8—24no—
296C22×C8—24no—
297C22×C8—24no—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306C4⋊Q8—82no—
307#127—22no—
308#127—22no—
309#125—44no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312#121—44no—
313#121—44no—
314#121—44no—
315#121—44no—
316#121—82no—
317#121—82no—
318C4×Q8—28no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—28no—
322C4×D4—42no—
323C4×D4—82no—
324C4×D4—82no—
325#117—82no—
326#117—82no—
327#117—82no—
328#117—82no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C22⋊C4C2×D411yescharacteristic
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#107—28no—
404#107—28no—
405#107—44no—
406#106—28no—
407#106—44no—
408C4⋊C8—24no—
409#103—28no—
410#103—44no—
411#103—44no—
412#102—28no—
413#102—44no—
414#102—44no—
415C22⋊C8—24no—
416C22⋊C8—24no—
417#97—24no—
418#97—24no—
419#97#4118yes—
420C4×C8—24no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8#4114yes—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447C24—22no—
448C24C22×D411yescharacteristic
449C24C22×D411yescharacteristic
450C2×Q8—24no—
451C2×Q8—22no—
452C2×Q8—24no—
453C2×Q8—22no—
454C2×Q8—44no—
455C2×Q8—82no—
456C2×Q8—82no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—41no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—82no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4—82no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—44no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4#13714yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C2511yescharacteristic
519SD16—416no—
520M16—48no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—48no—
530C2×C8C2×#4114yes—
531C2×C8C2×#4114yes—
532C2×C8C2×#4114yes—
533C2×C8H(5,𝔽2)14yes—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—41no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—41no—
570C22⋊C4—41no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—21no—
595C42—21no—
596C42—22no—
597C42—21no—
598C42—21no—
599C42—44no—
600C42—82no—
601C42—82no—
602C42—82no—
603C42—82no—
604C42C2×#4114yes—
605C42C2×#4112yes—
606C422−1+412yes—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23—21no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23C2×#13712yes—
628C23C23×D411yescharacteristic
629Q8—44no—
630Q8—44no—
631Q8—164no—
632D4—44no—
633D4—44no—
634D4—44no—
635D4—84no—
636D4—84no—
637D4—164no—
638D4—164no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—28no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—164no—
696C2×C4#53212yes—
697C2×C4#53212yes—
698C2×C4#54712yes—
699C2×C4#54712yes—
700C2×C4C2×#13712yes—
701C2×C4C2×#13712yes—
702C2×C4#57512yes—
703C2×C4#57612yes—
704C2×C4#57712yes—
705C2×C4C22×#4111yescharacteristic
706C2×C4C22×#4112yes—
707C2×C4C2×H(5,𝔽2)12yes—
708C2×C4C2×2−1+411yescharacteristic
709C8—24no—
710C8—42no—
711C8—42no—
712C8#55518yes—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—41no—
725C22—41no—
726C22—41no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22C2×#53211yescharacteristic
735C22C2×#54711yescharacteristic
736C22#349811yescharacteristic
737C22C22×#13711yescharacteristic
738C22C2×#57511yescharacteristic
739C22C2×#57611yescharacteristic
740C22C2×#57711yescharacteristic
741C4—22no—
742C4—22no—
743C4—22no—
744C4—41no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—41no—
749C4—41no—
750C4—41no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4—82no—
757C4—82no—
758C4#334614yes—
759C4#335014yes—
760C4C2×#55512yes—
761C4#351112yes—
762C2—42no—
763C2—81no—
764C2—162no—
765C2#3618711yescharacteristic
766C2#6070111yescharacteristic
767C2#6072211yescharacteristic
768C2C2×#334611yescharacteristic
769C2C2×#335011yescharacteristic
770C2#6242211yescharacteristic
771C2#6242611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36211C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36194C2semidirect
3. #33316C2semidirect
4. #33174C2semidirect
5. #33170C2semidirect
6. #29144C2semidirect
7. #29139C2semidirect
8. #28235C2semidirect
9. #27892C2semidirect
10. #27891C2semidirect
11. #27890C2semidirect
12. #27889C2semidirect
13. #27800C2semidirect
14. #27795C2semidirect
15. #22819C2semidirect
16. #21692C2semidirect
17. #20824C2semidirect
18. #20817C2semidirect
19. C4×#435C2semidirect

Maximal subgroups

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

subgroupindex
#362112
#361942
#333162
#331742
#331702
#291442
#291392
#282352
#278922
#278912
#278902
#278892
#278002
#277952
#228192
#216922
#208242
#208172
C4×#4352