Finite Group Databasebeta

← #6591601

Subgroups — #6591601

Order 512.

2097 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3302C212yesmaximal
2#36188C211yesmaximal, characteristic, complemented
3C4×#547C211yesmaximal, characteristic
4#33195C211yesmaximal, characteristic, complemented
5#32907C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28255C211yesmaximal, characteristic, complemented
8#27649C212yesmaximal, complemented
9#25323C212yesmaximal, complemented
10#25215C214yesmaximal, complemented
11#24816C212yesmaximal, complemented
12#24752C212yesmaximal, complemented
13#24536C214yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic, complemented
15#22490C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17#20645C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#547C2211yescharacteristic
20C2×C4×#41C2212yes—
21#3302—216no—
22C2×#487C2212yes—
23C2×#485C2214yes—
24C2×#474C2212yes—
25C2×#467C2214yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#461C2211yescharacteristic
29C2×#452C2214yes—
30C2×C4×SD16C2212yes—
31C2×C8×D4C2212yes—
32C22×C4⋊C8C2212yes—
33C2×#416C2214yes—
34#2868C2211yescharacteristic
35#2857C2212yes—
36#2568C2212yes—
37#2547—21no—
38#2547—21no—
39#2499C2214yes—
40#2496—22no—
41#2496—22no—
42#2365—21no—
43#2365—21no—
44#2365C2211yescharacteristic
45#2362C2212yes—
46#2361—21no—
47#2359—22no—
48#2354C2211yescharacteristic
49D4×C4⋊C4C2211yescharacteristic
50#2344—21no—
51#2339C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C22⋊Q8C2212yes—
55C4×#123—22no—
56C4×#121—21no—
57#2285C2212yes—
58C42×D4C2211yescharacteristic
59C42×D4C2211yescharacteristic
60C2×C4×C4⋊C4—21no—
61#2085C2214yes—
62#2074C2214yes—
63#2072C2214yes—
64#2068C2214yes—
65#2056C2214yes—
66#2056C2214yes—
67#1990C2212yes—
68#1988C2212yes—
69#1981C2211yescharacteristic
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1927C2212yes—
75#1892C2214yes—
76#1878C2214yes—
77#1877C2214yes—
78#1877C2212yes—
79#1873C2214yes—
80#1850C2214yes—
81#1842C2211yescharacteristic
82#1825C2214yes—
83#1805C2214yes—
84#1804C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#1735C2214yes—
96C22×#41C2312yes—
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99C23×C8C2314yes—
100#556—28no—
101#556—28no—
102#547—24no—
103#547D418yes—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#126—24no—
110C2×#124—22no—
111C2×#123—22no—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2312yes—
114C2×#121—21no—
115C4×#41—28no—
116C4×#41—44no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C2×#117C2314yes—
132C2×#117C2312yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C42—22no—
136C22×C42—21no—
137C22×C42C2312yes—
138#487—28no—
139#485—216no—
140#474—28no—
141#467—216no—
142#462—28no—
143#462—44no—
144#461—24no—
145#452—216no—
146C4×SD16—28no—
147C8×D4—28no—
148C2×#107C2314yes—
149C2×#106C2312yes—
150C2×C4⋊C8—44no—
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2314yes—
157#416—216no—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×C22⋊C8C2314yes—
168C2×C4×C8C2312yes—
169#400—24no—
170#397—24no—
171#397—24no—
172#391—22no—
173#391—22no—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#391C2314yes—
178#390—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—24no—
194#387—24no—
195#386—22no—
196#386—22no—
197#385—22no—
198#385—22no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2312yes—
205#382C2314yes—
206#381—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—21no—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—22no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4D412yes—
235C4×C22⋊C4C2312yes—
236C2×#95—22no—
237C43C2311yescharacteristic
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2312yes—
245#336C2314yes—
246#336C2312yes—
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41—42no—
251C2×#41—42no—
252C2×#41C2×D418yes—
253C22×Q8—22no—
254C22×Q8C2412yes—
255C22×Q8C2412yes—
256C22×D4—22no—
257C22×D4C2×D411yescharacteristic
258C22×D4C2411yescharacteristic
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2412yes—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—24no—
273C22×C8—28no—
274C22×C8—44no—
275C22×C8—44no—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C22×C8C2412yes—
279C4⋊Q8—24no—
280C4⋊Q8—22no—
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C4⋊Q8C2×D414yes—
287#126—416no—
288#124—48no—
289#123—216no—
290C22⋊Q8—24no—
291C22⋊Q8—28no—
292C22⋊Q8—28no—
293C22⋊Q8C2×D418yes—
294#121—28no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—42no—
301C4×Q8—48no—
302C4×D4—24no—
303C4×D4—22no—
304C4×D4—24no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—42no—
308C4×D4—41no—
309C4×D4—42no—
310C4×D4—48no—
311C4×D4—48no—
312C4×D4C2×D414yes—
313C4×D4C2×D414yes—
314C4×D4#4114yes—
315C4×D4#4114yes—
316#117—28no—
317#117—28no—
318#117—24no—
319#117—48no—
320#117—42no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—48no—
374C2×C42—44no—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#106—24no—
387C4⋊C8—24no—
388C4⋊C8—22no—
389C4⋊C8—22no—
390C4⋊C8—24no—
391C4⋊C8—44no—
392C4⋊C8—42no—
393#103—28no—
394#103—28no—
395#103—28no—
396#103—24no—
397#103—24no—
398#103—42no—
399#103—44no—
400#102—24no—
401#102—28no—
402C22⋊C8—28no—
403C4×C8—24no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95C2×D412yes—
420#95C2×D412yes—
421#95C2×D412yes—
422C24—21no—
423C24C22×D412yes—
424#41—216no—
425#41—48no—
426#41—48no—
427#41—48no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8C22×D414yes—
443C2×D4—28no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4#12114yes—
452C2×D4C22×D412yes—
453C2×D4C2×#4112yes—
454C2×D42−1+412yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—28no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—82no—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D414yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C2×#4114yes—
499C22×C4C2511yescharacteristic
500Q16—88no—
501SD16—48no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—88no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—81no—
543C4⋊C4#12114yes—
544C4⋊C4C22×D412yes—
545C4⋊C4C2×#4112yes—
546C22⋊C4—28no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—42no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—42no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—82no—
557C22⋊C4#12118yes—
558C42—24no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—21no—
565C42—21no—
566C42—41no—
567C42—44no—
568C42—44no—
569C42—48no—
570C42—44no—
571C42—44no—
572C42—42no—
573C42—42no—
574C42—42no—
575C42#13514yes—
576C42C22×D412yes—
577C42C2×#4112yes—
578C42C2×#4112yes—
579C42H(5,𝔽2)12yes—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23C2×#12114yes—
594C23C23×D411yescharacteristic
595Q8—24no—
596Q8—24no—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—88no—
604D4—22no—
605D4—22no—
606D4—22no—
607D4—28no—
608D4—416no—
609D4#54714yes—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—416no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4—48no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—88no—
662C2×C4C2×#12114yes—
663C2×C4C2×#12112yes—
664C2×C4C2×#12112yes—
665C2×C4#54612yes—
666C2×C4#54612yes—
667C2×C4C2×#13512yes—
668C2×C4C2×#13512yes—
669C2×C4C2×#13512yes—
670C2×C4#57512yes—
671C2×C4#57812yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C2×H(5,𝔽2)11yescharacteristic
676C2×C4C2×2−1+411yescharacteristic
677C2×C4#58511yescharacteristic
678C8—42no—
679C8—42no—
680C8—44no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22C22×#12111yescharacteristic
696C22C2×#54611yescharacteristic
697C22#348911yescharacteristic
698C22C22×#13511yescharacteristic
699C22C2×#57511yescharacteristic
700C22C2×#57811yescharacteristic
701C22#358611yescharacteristic
702C4—22no—
703C4—24no—
704C4—22no—
705C4—22no—
706C4—44no—
707C4—41no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—82no—
713C4—82no—
714C4—81no—
715C4—82no—
716C4—84no—
717C4#303514yes—
718C4#330214yescomplemented
719C4C2×#54711yescharacteristic
720C4#347511yescharacteristic
721C4#349611yescharacteristic
722C4#352011yescharacteristic
723C2—44no—
724C2—44no—
725C2#3615311yescharacteristic
726C2C2×#303511yescharacteristic
727C2#6043211yescharacteristic
728C2C2×#330211yescharacteristic
729C2#6116511yescharacteristic
730C2#6209711yescharacteristic
731C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #36188C2semidirect
4. #33195C2semidirect
5. #32907C2semidirect
6. #28717C2semidirect
7. #28255C2semidirect
8. #27649C2semidirect
9. #25323C2semidirect
10. #25215C2semidirect
11. #24816C2semidirect
12. #24752C2semidirect
13. #24536C2semidirect
14. D4×C4⋊C8C2semidirect
15. #22490C2semidirect
16. C4×#461C2semidirect
17. #20645C2semidirect
718. C4#3302semidirect

Maximal subgroups

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

subgroupindex
C2×#33022
#361882
C4×#5472
#331952
#329072
#287172
#282552
#276492
#253232
#252152
#248162
#247522
#245362
D4×C4⋊C82
#224902
C4×#4612
#206452