Finite Group Databasebeta

← #6811809

Subgroups — #6811809

Order 512.

1797 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3233C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic, complemented
3#34932C211yesmaximal, characteristic
4#33197C211yesmaximal, characteristic, complemented
5#30784C214yesmaximal, complemented
6#30640C214yesmaximal, complemented
7#29692C212yesmaximal, complemented
8#28814C212yesmaximal, complemented
9#28511C211yesmaximal, characteristic, complemented
10#28283C211yesmaximal, characteristic, complemented
11#27768C212yesmaximal, complemented
12#26231C212yesmaximal, complemented
13#25011C212yesmaximal
14#24702C212yesmaximal, complemented
15#24303C212yesmaximal, complemented
16#22795C211yesmaximal, characteristic, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#529C2212yes—
20#3233—216no—
21C2×#499C2214yes—
22C2×#488C2214yes—
23C2×#485C2214yes—
24C2×#482C2214yes—
25C2×#477C2212yes—
26C2×#477C2212yes—
27C2×#475C2212yes—
28C2×#476C2212yes—
29C2×#474C2211yescharacteristic
30C2×#431C2212yes—
31#2868C2211yescharacteristic
32#2857C2212yes—
33#2553—22no—
34#2552C2212yes—
35#2552C2211yescharacteristic
36#2547C2211yescharacteristic
37#2542—21no—
38#2542C2211yescharacteristic
39#2538—21no—
40#2537C2211yescharacteristic
41#2496—21no—
42#2489C2212yes—
43#2441C2214yes—
44#2429—22no—
45#2427C2212yes—
46#2423—22no—
47#2422—21no—
48#2395—22no—
49#2365—21no—
50#2354C2212yes—
51Q8×C4⋊C4—21no—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C22⋊Q8C2212yes—
55C4×#121—21no—
56#2285—21no—
57C42×D4C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2095C2214yes—
60#2076C2214yes—
61#2066C2214yes—
62#2059C2214yes—
63#2059C2214yes—
64#2042C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#2033C2214yes—
68#2026C2214yes—
69#1988C2211yescharacteristic
70#1987C2212yes—
71#1981C2212yes—
72#1956C2212yes—
73#1954C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1948C2212yes—
77#1947C2211yescharacteristic
78#1942C2214yes—
79#1933C2214yes—
80#1923C2214yes—
81#1918C2212yes—
82#1873C2214yes—
83#1862C2212yes—
84#1854C2214yes—
85#1845C2212yes—
86#1842C2211yescharacteristic
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C8×C4⋊C4C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#102C2212yes—
96#533—28no—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103#529—28no—
104#529—44no—
105C2×#126—22no—
106C2×#125C2312yes—
107C2×#125C2314yes—
108C2×#124C2312yes—
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C2×#117C2314yes—
130C22×C42—21no—
131C22×C42C2312yes—
132#499—216no—
133#488—216no—
134#485—216no—
135#482—216no—
136#477—28no—
137#477—44no—
138#476—28no—
139#475—28no—
140#474D418yescomplemented
141#431—28no—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2311yescharacteristic
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2314yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2314yes—
164#400—22no—
165#399—22no—
166#399—24no—
167#398—22no—
168#398—24no—
169#397—22no—
170#397—22no—
171#397—22no—
172#396—24no—
173#394—22no—
174#394—22no—
175#391—21no—
176#391—22no—
177#391C2312yes—
178#391C2312yes—
179#390—22no—
180#390—21no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—22no—
187#389—24no—
188#389C2312yes—
189#389C2312yes—
190#389C2314yes—
191#388—24no—
192#388—22no—
193#387—22no—
194#387—24no—
195#386—22no—
196#385—22no—
197#384—22no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2312yes—
206#382—22no—
207#380—22no—
208#380—21no—
209#380—21no—
210#380—21no—
211#380—22no—
212#380C2312yes—
213#380C2314yes—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—22no—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4C2312yes—
233C43C2311yescharacteristic
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2312yes—
240#340C2312yes—
241#336C2314yes—
242#336C2312yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8C2412yes—
246C22×D4—22no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265C4⋊Q8C2×D414yes—
266#126—48no—
267#125—24no—
268#125—28no—
269#125—42no—
270#125—44no—
271#124—24no—
272#124—42no—
273#123—48no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—42no—
277C22⋊Q8—42no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—84no—
281#121—24no—
282#121—42no—
283#121—44no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—48no—
287C4×Q8—48no—
288C4×Q8—44no—
289C4×D4—22no—
290C4×D4—24no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—42no—
297C4×D4—41no—
298C4×D4—48no—
299C4×D4C2×D414yes—
300#117—28no—
301#117—48no—
302#117—44no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
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—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—44no—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—28no—
369#107—24no—
370#107—44no—
371#107C2×D418yes—
372#106—28no—
373#106—28no—
374C4⋊C8—24no—
375C4⋊C8—22no—
376C4⋊C8—24no—
377C4⋊C8—42no—
378C4⋊C8C2×D414yes—
379#103—28no—
380#103—28no—
381#103—44no—
382#103—44no—
383#102—28no—
384#102—42no—
385#102C2×D418yes—
386C22⋊C8—28no—
387C22⋊C8—44no—
388C4×C8—28no—
389C4×C8—24no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—21no—
407C24C22×D412yes—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—82no—
431C2×D4C22×D412yes—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2511yescharacteristic
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8C22×D414yes—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—84no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4C22×D414yes—
517C4⋊C4C22×D412yes—
518C4⋊C42−1+412yes—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—84no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42—48no—
543C42—44no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—44no—
548C42#13514yes—
549C42C22×D412yes—
550C42H(5,𝔽2)12yes—
551C422−1+412yes—
552C422−1+412yes—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23C23×D411yescharacteristic
568Q8—48no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—88no—
574D4—22no—
575D4—22no—
576D4—44no—
577D4—41no—
578D4—41no—
579D4—88no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4#53612yes—
632C2×C4#53612yes—
633C2×C4D4212yes—
634C2×C4D4212yes—
635C2×C4C2×#13512yes—
636C2×C4C2×#13512yes—
637C2×C4C2×#13512yes—
638C2×C4#57512yes—
639C2×C4#57812yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C2×C4C2×2−1+411yescharacteristic
645C2×C4#58511yescharacteristic
646C8—24no—
647C8—44no—
648C8—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22C2×#53611yescharacteristic
664C22C2×D4211yescharacteristic
665C22#348911yescharacteristic
666C22C22×#13511yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57811yescharacteristic
669C22#358511yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—41no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4—81no—
684C4—82no—
685C4#323314yescomplemented
686C4#328314yes—
687C4#349711yescharacteristic
688C4#349911yescharacteristic
689C4#354711yescharacteristic
690C4#355011yescharacteristic
691C2—82no—
692C2—82no—
693C2#3934311yescharacteristic
694C2C2×#323311yescharacteristic
695C2#6089311yescharacteristic
696C2C2×#328311yescharacteristic
697C2#6107011yescharacteristic
698C2#6209811yescharacteristic
699C2#6214611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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
1. C2×#3233C2semidirect
2. #36134C2semidirect
4. #33197C2semidirect
5. #30784C2semidirect
6. #30640C2semidirect
7. #29692C2semidirect
8. #28814C2semidirect
9. #28511C2semidirect
10. #28283C2semidirect
11. #27768C2semidirect
12. #26231C2semidirect
14. #24702C2semidirect
15. #24303C2semidirect
16. #22795C2semidirect
17. C4×#474C2semidirect
140. #474D4semidirect
685. C4#3233semidirect

Maximal subgroups

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

subgroupindex
C2×#32332
#361342
#349322
#331972
#307842
#306402
#296922
#288142
#285112
#282832
#277682
#262312
#250112
#247022
#243032
#227952
C4×#4742