Finite Group Databasebeta

← #6499234

Subgroups — #6499234

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34226C211yesmaximal, characteristic
3#33074C214yesmaximal, complemented
4#30002C214yesmaximal, complemented
5#29247C214yesmaximal, complemented
6#29246C214yesmaximal, complemented
7#29192C211yesmaximal, characteristic, complemented
8#29192C211yesmaximal, characteristic, complemented
9#29192C211yesmaximal, characteristic, complemented
10#29190C211yesmaximal, characteristic, complemented
11#29179C211yesmaximal, characteristic, complemented
12#29179C211yesmaximal, characteristic, complemented
13#29178C211yesmaximal, characteristic, complemented
14#29176C211yesmaximal, characteristic, complemented
15#21984C211yesmaximal, characteristic, complemented
16#21982C211yesmaximal, characteristic, complemented
17#21863C211yesmaximal, characteristic
18#21859C211yesmaximal, characteristic
19#20809C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2647C2212yes—
28#2647C2212yes—
29#2571—21no—
30#2571—21no—
31#2552—21no—
32#2552—21no—
33#2442—21no—
34#2442—21no—
35#2442—21no—
36#2442—21no—
37#2367—21no—
38#2367—21no—
39#2363—21no—
40#2363—21no—
41#2357—21no—
42#2349—21no—
43#2341—21no—
44#2339—21no—
45#2324C2212yes—
46#2324C2212yes—
47#2319C2212yes—
48#2317C2212yes—
49#2316C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#127C2211yescharacteristic
53C4×#127C2211yescharacteristic
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2275C2212yes—
58C4×#117C2211yescharacteristic
59#2088C2214yes—
60#2087C2214yes—
61#2087C2214yes—
62#2086C2214yes—
63#2067C2214yes—
64#2067C2214yes—
65#2067C2214yes—
66#2065C2214yes—
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1957C2211yescharacteristic
72#1957C2211yescharacteristic
73#1957C2211yescharacteristic
74#1957C2211yescharacteristic
75#1949C2214yes—
76#1947C2214yes—
77#1921C2214yes—
78#1921C2214yes—
79#1920C2214yes—
80#1918C2214yes—
81#1875C2214yes—
82#1874C2214yes—
83#1864C2214yes—
84#1862C2214yes—
85#1862C2214yes—
86#1862C2214yes—
87#1854C2214yes—
88#1852C2214yes—
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1843C2212yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1832C2212yes—
99#1804C2214yes—
100#1802C2214yes—
101#1798C2212yes—
102#1798C2212yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106C4×C4⋊C8C2211yescharacteristic
107C22×#41—21no—
108#532—24no—
109#532Q818yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#127C2311yescharacteristic
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—21no—
116C2×#124—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137#431—24no—
138#431—24no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#400—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#391—21no—
168#391—21no—
169#390—22no—
170#390—22no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2312yes—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#386—22no—
204#386—22no—
205#385—22no—
206#385—22no—
207#385—22no—
208#385—22no—
209#384—22no—
210#384—21no—
211#384—21no—
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2312yes—
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#382—22no—
220#382—21no—
221#382—21no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382C2312yes—
226#381—22no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—22no—
231#380—22no—
232#380—22no—
233#380C2312yes—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248C4×C22⋊C4—22no—
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253#376C2311yescharacteristic
254#376C2311yescharacteristic
255C2×#95—22no—
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C2×#41—28no—
271C2×#41—82no—
272C2×#41—82no—
273C22×Q8—21no—
274C22×D4—22no—
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C2412yes—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4—22no—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—22no—
290C4⋊Q8C2×Q814yes—
291#127—22no—
292#127C2×Q814yes—
293#125—82no—
294#125—82no—
295#125—82no—
296#125—82no—
297#124—28no—
298#121—28no—
299#121—24no—
300#121C2×Q818yes—
301C4×Q8—82no—
302C4×Q8—82no—
303C4×D4—28no—
304C4×D4—24no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—82no—
308C4×D4—82no—
309C4×D4—82no—
310C4×D4—82no—
311C4×D4C2×Q818yes—
312#117—42no—
313#117—42no—
314#117—44no—
315#117—82no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
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—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—82no—
344C2×C4⋊C4—82no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
367C2×C22⋊C4C2×D411yescharacteristic
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×Q814yes—
385C2×C42#4114yes—
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—44no—
393#107—44no—
394#107—44no—
395#106—44no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—42no—
399C4⋊C8—42no—
400C4⋊C8—42no—
401C4⋊C8—42no—
402#102—44no—
403#102—44no—
404#102—44no—
405#102—44no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408C4×C8—22no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436C24—22no—
437C24C22×D411yescharacteristic
438#41—816no—
439C2×Q8—24no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—82no—
453C2×D4—82no—
454C2×D4C22×Q814yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—28no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4C22⋊Q814yes—
481C22×C4C22⋊Q814yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×Q812yes—
496C22×C4C2×#4112yes—
497C22×C4C2511yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4C22×Q814yes—
548C22⋊C4—28no—
549C22⋊C4—28no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—44no—
575C42—82no—
576C42—82no—
577C42—82no—
578C42C22⋊Q814yes—
579C42C22⋊Q814yes—
580C42C22×Q812yes—
581C42C22×Q812yes—
582C42C2×#4112yes—
583C42C2×#4112yes—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—24no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23C2×C22⋊Q812yes—
598C23C23×D411yescharacteristic
599Q8—48no—
600D4—28no—
601D4—48no—
602D4—48no—
603D4—44no—
604D4—88no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—88no—
661C2×C4—164no—
662C2×C4—164no—
663C2×C4—164no—
664C2×C4C2×C22⋊Q812yes—
665C2×C4C2×C22⋊Q812yes—
666C2×C4C2×C22⋊Q812yes—
667C2×C4C2×C22⋊Q812yes—
668C2×C4C2×C22⋊Q812yes—
669C2×C4#57612yes—
670C2×C4#57612yes—
671C2×C4#57712yes—
672C2×C4#57812yes—
673C2×C4C23×Q811yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4#58512yes—
676C2×C4#58512yes—
677C8—42no—
678C8—42no—
679C8—42no—
680C8—42no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22C22×C22⋊Q811yescharacteristic
696C22#344911yescharacteristic
697C22#344911yescharacteristic
698C22C2×#57611yescharacteristic
699C22C2×#57611yescharacteristic
700C22C2×#57711yescharacteristic
701C22C2×#57811yescharacteristic
702C4—22no—
703C4—22no—
704C4—22no—
705C4—44no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4#308114yes—
721C4#308314yes—
722C4#350412yes—
723C4#351212yes—
724C2—42no—
725C2—84no—
726C2#3533311yescharacteristic
727C2C2×#308111yescharacteristic
728C2C2×#308311yescharacteristic
729C2#6100611yescharacteristic
730C2#6100811yescharacteristic
731C2#6101111yescharacteristic
732C2#6101411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #33074C2semidirect
4. #30002C2semidirect
5. #29247C2semidirect
6. #29246C2semidirect
7. #29192C2semidirect
8. #29192C2semidirect
9. #29192C2semidirect
10. #29190C2semidirect
11. #29179C2semidirect
12. #29179C2semidirect
13. #29178C2semidirect
14. #29176C2semidirect
15. #21984C2semidirect
16. #21982C2semidirect
19. #20809C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#342262
#330742
#300022
#292472
#292462
#291922
#291902
#291792
#291782
#291762
#219842
#219822
#218632
#218592
#208092