Finite Group Databasebeta

← #6606600

Subgroups — #6606600

Order 512.

2045 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3353C212yesmaximal
2#36188C211yesmaximal, characteristic, complemented
3C4×#547C211yesmaximal, characteristic
4#33195C211yesmaximal, characteristic, complemented
5#33109C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28283C211yesmaximal, characteristic, complemented
8#27646C212yesmaximal, complemented
9#25456C212yesmaximal, complemented
10#25216C214yesmaximal, complemented
11#24759C212yesmaximal, complemented
12#24681C212yesmaximal, complemented
13#24588C214yesmaximal, complemented
14#22650C211yesmaximal, characteristic, complemented
15#22504C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17#21170C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#547C2211yescharacteristic
20C2×C4×#41—21no—
21#3353—216no—
22C2×#500C2212yes—
23C2×#484C2214yes—
24C2×#482C2214yes—
25C2×#475C2212yes—
26C2×#474C2212yes—
27C2×#468C2214yes—
28C2×#461C2211yescharacteristic
29C2×C4×SD16C2212yes—
30C2×#436C2212yes—
31C2×#428C2214yes—
32C2×#423C2212yes—
33#2868C2211yescharacteristic
34#2857C2212yes—
35#2568C2212yes—
36#2547—21no—
37#2547—21no—
38#2499C2214yes—
39#2496—22no—
40#2496—22no—
41#2365—21no—
42#2365—21no—
43#2365C2211yescharacteristic
44#2362—21no—
45#2361C2212yes—
46#2359—22no—
47#2354C2211yescharacteristic
48D4×C4⋊C4C2211yescharacteristic
49#2344C2212yes—
50#2339—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C22⋊Q8—21no—
54C4×#123—22no—
55C4×#121C2212yes—
56#2285C2212yes—
57C42×D4C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C2×C4×C4⋊C4C2212yes—
60#2085C2214yes—
61#2083C2214yes—
62#2073C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2055C2214yes—
66#1990C2212yes—
67#1988C2212yes—
68#1981C2211yescharacteristic
69#1981C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1948C2211yescharacteristic
74#1947C2212yes—
75#1947C2212yes—
76#1947C2212yes—
77#1947C2212yes—
78#1939C2212yes—
79#1928C2214yes—
80#1928C2214yes—
81#1927C2212yes—
82#1894C2214yes—
83#1877C2212yes—
84#1872C2214yes—
85#1851C2214yes—
86#1842C2211yescharacteristic
87#1825C2214yes—
88#1803C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#1736C2214yes—
96C22×#41—21no—
97C22×SD16C2312yes—
98C22×M16C2314yes—
99#556—28no—
100#556—28no—
101#547—24no—
102#547D418yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#126—24no—
109C2×#124—22no—
110C2×#123—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2314yes—
113C2×#121C2312yes—
114C4×#41—48no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—22no—
130C2×#117C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C42—22no—
134C22×C42—21no—
135C22×C42C2312yes—
136#500—28no—
137#484—216no—
138#482—216no—
139#475—28no—
140#474—28no—
141#468—216no—
142#461—24no—
143C4×SD16—28no—
144#436—28no—
145#428—216no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×#106C2312yes—
151#423—28no—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2311yescharacteristic
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×C22⋊C8C2314yes—
166C2×C4×C8C2312yes—
167#400—24no—
168#397—24no—
169#397—24no—
170#391—22no—
171#391—22no—
172#391—21no—
173#391—21no—
174#391C2312yes—
175#391C2314yes—
176#390C2312yes—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389C2312yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—24no—
192#387—24no—
193#386—22no—
194#386—22no—
195#385—22no—
196#385—22no—
197#384—21no—
198#384—21no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203#382C2314yes—
204#381—22no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—21no—
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C22⋊C4—21no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4C2312yes—
234C2×#95—22no—
235C43C2311yescharacteristic
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
243#336C2314yes—
244#336C2312yes—
245C2×#41—28no—
246C2×#41—44no—
247C2×#41—44no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8C2412yes—
251C22×D4—22no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2412yes—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—42no—
265C2×M16—28no—
266C2×M16—28no—
267C2×M16—44no—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—24no—
272C4⋊Q8—22no—
273C4⋊Q8—24no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8C2×D414yes—
279#126—416no—
280#124—48no—
281#123—216no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285#121—24no—
286#121C2×D418yes—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—44no—
290C4×Q8—48no—
291C4×Q8—44no—
292C4×D4—22no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—42no—
298C4×D4—41no—
299C4×D4—48no—
300C4×D4—48no—
301C4×D4—44no—
302C4×D4C2×D414yes—
303C4×D4C2×D414yes—
304C4×D4#4114yes—
305#117—28no—
306#117—28no—
307#117—48no—
308#117—44no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D418yes—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—48no—
364C2×C42—44no—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—28no—
376#107—28no—
377#107—28no—
378#106—24no—
379#106—28no—
380C4⋊C8—28no—
381C4⋊C8—24no—
382C4⋊C8—24no—
383C4⋊C8—22no—
384C4⋊C8—42no—
385C4⋊C8#4114yescomplemented
386#103—24no—
387#103—24no—
388#103—28no—
389#103—24no—
390#103—44no—
391#102—24no—
392#102—28no—
393C22⋊C8—28no—
394C4×C8—24no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413C24—21no—
414C24C22×D412yes—
415#41—432no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×D4—28no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4#12114yes—
439C2×D4C22×D412yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—28no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—82no—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D414yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4114yes—
484C22×C4C2511yescharacteristic
485SD16—48no—
486M16—416no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8C2×#4114yes—
496C4⋊C4—28no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4#12114yes—
528C4⋊C4C22×D412yes—
529C4⋊C4C22×D414yes—
530C4⋊C42−1+412yes—
531C22⋊C4—24no—
532C22⋊C4—28no—
533C22⋊C4—24no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—42no—
541C22⋊C4—82no—
542C22⋊C4#12118yes—
543C42—24no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42—21no—
551C42—41no—
552C42—44no—
553C42—44no—
554C42—48no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42—42no—
559C42—44no—
560C42AGL(1,ℤ/8)14yes—
561C42C22×D412yes—
562C42C2×#4112yes—
563C42C2×#4112yes—
564C422−1+412yes—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23C2×#12114yes—
579C23C23×D411yescharacteristic
580Q8—48no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—88no—
586D4—22no—
587D4—22no—
588D4—28no—
589D4—416no—
590D4—41no—
591D4—41no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—416no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—88no—
643C2×C4—84no—
644C2×C4C2×#12114yes—
645C2×C4C2×#12112yes—
646C2×C4C2×#12112yes—
647C2×C4#54712yes—
648C2×C4#54712yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4C2×AGL(1,ℤ/8)12yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4#57512yes—
653C2×C4#57812yes—
654C2×C4C23×D411yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C2×C4C2×2−1+411yescharacteristic
659C2×C4#58511yescharacteristic
660C8—24no—
661C8—44no—
662C8—42no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22C22×#12111yescharacteristic
678C22C2×#54711yescharacteristic
679C22#348911yescharacteristic
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C2×#57511yescharacteristic
682C22C2×#57811yescharacteristic
683C22#358611yescharacteristic
684C4—22no—
685C4—22no—
686C4—22no—
687C4—24no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—41no—
692C4—42no—
693C4—44no—
694C4—82no—
695C4—81no—
696C4—82no—
697C4—82no—
698C4—84no—
699C4#303614yes—
700C4#335314yescomplemented
701C4D4×#4111yescharacteristic
702C4#348211yescharacteristic
703C4#349911yescharacteristic
704C4#351511yescharacteristic
705C2—44no—
706C2—82no—
707C2#3625911yescharacteristic
708C2C2×#303611yescharacteristic
709C2#6043211yescharacteristic
710C2C2×#335311yescharacteristic
711C2#6125711yescharacteristic
712C2#6210911yescharacteristic
713C2#6214611yescharacteristic
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. #33109C2semidirect
6. #28689C2semidirect
7. #28283C2semidirect
8. #27646C2semidirect
9. #25456C2semidirect
10. #25216C2semidirect
11. #24759C2semidirect
12. #24681C2semidirect
13. #24588C2semidirect
14. #22650C2semidirect
15. #22504C2semidirect
16. C4×#461C2semidirect
17. #21170C2semidirect
385. C4⋊C8#41semidirect
700. C4#3353semidirect

Maximal subgroups

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

subgroupindex
C2×#33532
#361882
C4×#5472
#331952
#331092
#286892
#282832
#276462
#254562
#252162
#247592
#246812
#245882
#226502
#225042
C4×#4612
#211702