Finite Group Databasebeta

← #6594908

Subgroups — #6594908

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C211yesmaximal, characteristic
2#36181C211yesmaximal, characteristic
3#33256C211yesmaximal, characteristic, complemented
4#33043C211yesmaximal, characteristic, complemented
5#33043C211yesmaximal, characteristic, complemented
6#29191C211yesmaximal, characteristic, complemented
7#29176C211yesmaximal, characteristic, complemented
8#28847C214yesmaximal, complemented
9#28030C214yesmaximal, complemented
10#27948C211yesmaximal, characteristic, complemented
11#27905C211yesmaximal, characteristic, complemented
12#27889C211yesmaximal, characteristic, complemented
13#27876C211yesmaximal, characteristic, complemented
14#25236C214yesmaximal
15#22920C214yesmaximal, complemented
16C4×#487C211yesmaximal, characteristic, complemented
17#20816C211yesmaximal, characteristic, complemented
18C2×#1834C211yesmaximal, characteristic, complemented
19C4×C8×D4C211yesmaximal, characteristic
20C2×#552—21no—
21C2×#487C2211yescharacteristic
22C2×#483C2214yes—
23C2×C8×D4C2212yes—
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2570—21no—
33#2565—21no—
34#2511C2212yes—
35#2507—21no—
36#2506—21no—
37#2506C2212yes—
38#2503—21no—
39#2499—21no—
40#2497—21no—
41#2496—21no—
42#2496C2212yes—
43#2487—21no—
44#2486—21no—
45#2486C2212yes—
46#2442—21no—
47#2441—21no—
48#2423C2212yes—
49#2417C2212yes—
50#2360C2212yes—
51#2352—21no—
52#2345—21no—
53#2344—21no—
54#2344C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59C2×#384C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C42×D4C2211yescharacteristic
62#2096C2214yes—
63#2088C2214yes—
64#2084C2214yes—
65#2077C2214yes—
66#2066C2214yes—
67#2035C2214yes—
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1956C2211yescharacteristic
72#1956C2211yescharacteristic
73#1956C2211yescharacteristic
74#1941C2214yes—
75#1941C2214yes—
76#1933C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1927C2214yes—
80#1920C2214yes—
81#1919C2214yes—
82#1892C2214yes—
83#1892C2214yes—
84#1865C2214yes—
85#1853C2214yes—
86#1844C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1844C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2214yes—
95#1834C2211yescharacteristic
96#1807C2214yes—
97C4×#106C2214yes—
98C8×C4⋊C4C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×#103C2214yes—
101C8×C22⋊C4C2212yes—
102C8×C22⋊C4C2212yes—
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105C2×#340C2214yes—
106C42×C8C2211yescharacteristic
107C23×C8C2312yes—
108C23×C8C2312yes—
109#552—48no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#126—22no—
113C2×#125—22no—
114C2×#125—21no—
115C2×#125—21no—
116C2×#123—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—21no—
127C2×#117—21no—
128C2×#117—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#487—24no—
138#483—216no—
139C8×D4—28no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C4×C8—24no—
152C2×C4×C8—24no—
153C2×C4×C8C2312yes—
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2312yes—
158C2×C4×C8C2312yes—
159C2×C4×C8C2312yes—
160C2×C4×C8C2311yescharacteristic
161#400—22no—
162#400—22no—
163#400—22no—
164#400—22no—
165#399—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#397—22no—
171#397—22no—
172#396—22no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
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#389—21no—
190#389—21no—
191#389—21no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#388—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#386—22no—
210#385—22no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384—21no—
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2312yes—
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2312yes—
223#382—21no—
224#382—21no—
225#382—21no—
226#382—21no—
227#381—22no—
228#381—22no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—22no—
238#380C2312yes—
239#380C2312yes—
240#380C2312yes—
241#380C2312yes—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2311yescharacteristic
253C4×C22⋊C4—22no—
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4C2311yescharacteristic
259C4×C22⋊C4C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8C2412yes—
274C22×D4C2×D411yescharacteristic
275C23×C4—22no—
276C23×C4—22no—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2411yescharacteristic
282C23×C4C2411yescharacteristic
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8—24no—
287C22×C8—42no—
288C22×C8—42no—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297#126—48no—
298#125—48no—
299#125—44no—
300#125—82no—
301#123—48no—
302C22⋊Q8—28no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4SD1618yes—
311#117—44no—
312#117—48no—
313#117—82no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—82no—
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×C4⋊C4C2412yes—
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⋊C4C2×D411yescharacteristic
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394#106—28no—
395#106—28no—
396#106—44no—
397#106—44no—
398C4⋊C8—24no—
399#103—28no—
400#103—28no—
401#103—28no—
402#103—44no—
403C22⋊C8—24no—
404C22⋊C8#4118yes—
405C4×C8—24no—
406C4×C8—24no—
407C4×C8—24no—
408C4×C8—22no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—42no—
412C4×C8#4114yes—
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#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439C24C22×D411yescharacteristic
440C24C22×D411yescharacteristic
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4C2×SD1614yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4C2×SD1614yes—
482C22×C4C2×SD1614yes—
483C22×C4#13514yes—
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×D411yescharacteristic
496C22×C4C2×#4114yes—
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—48no—
508C2×C8—48no—
509C2×C8C2×#4114yes—
510C2×C8C2×#4114yes—
511C2×C8C2×#4114yes—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4C2×SD1614yes—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4C2×SD1614yes—
565C22⋊C4C2×SD1614yes—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—21no—
574C42—21no—
575C42—22no—
576C42—21no—
577C42—21no—
578C42—22no—
579C42—44no—
580C42—44no—
581C42—82no—
582C42—82no—
583C42C2×SD1614yes—
584C42C2×#4112yes—
585C422−1+412yes—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23#53212yes—
597C23C22×SD1612yes—
598C23C22×SD1612yes—
599C23C2×#13512yes—
600C23C23×D411yescharacteristic
601Q8—44no—
602Q8—44no—
603Q8—84no—
604D4—28no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
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—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
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—42no—
661C2×C4—88no—
662C2×C4—164no—
663C2×C4#53212yes—
664C2×C4#54712yes—
665C2×C4#54712yes—
666C2×C4C22×SD1612yes—
667C2×C4C22×SD1612yes—
668C2×C4C22×SD1612yes—
669C2×C4C22×SD1612yes—
670C2×C4C22×SD1612yes—
671C2×C4C2×#13512yes—
672C2×C4C2×#13512yes—
673C2×C4#57712yes—
674C2×C4#57812yes—
675C2×C4C22×#4111yescharacteristic
676C2×C4C22×#4112yes—
677C2×C4C2×2−1+411yescharacteristic
678C2×C4#58512yes—
679C8—24no—
680C8—24no—
681C8—24no—
682C8—42no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22#316714yes—
697C22C2×#53211yescharacteristic
698C22C2×#54711yescharacteristic
699C22#350011yescharacteristic
700C22C23×SD1611yescharacteristic
701C22C22×#13511yescharacteristic
702C22C2×#57711yescharacteristic
703C22C2×#57811yescharacteristic
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#333514yes—
723C4#333614yes—
724C4#347512yes—
725C4#351512yes—
726C2—24no—
727C2—42no—
728C2#3618411yescharacteristic
729C2C2×#316711yescharacteristic
730C2#6072311yescharacteristic
731C2C2×#333511yescharacteristic
732C2C2×#333611yescharacteristic
733C2#6252611yescharacteristic
734C2#6253111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33256C2semidirect
4. #33043C2semidirect
5. #33043C2semidirect
6. #29191C2semidirect
7. #29176C2semidirect
8. #28847C2semidirect
9. #28030C2semidirect
10. #27948C2semidirect
11. #27905C2semidirect
12. #27889C2semidirect
13. #27876C2semidirect
15. #22920C2semidirect
16. C4×#487C2semidirect
17. #20816C2semidirect
18. C2×#1834C2semidirect

Maximal subgroups

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

subgroupindex
#361892
#361812
#332562
#330432
#291912
#291762
#288472
#280302
#279482
#279052
#278892
#278762
#252362
#229202
C4×#4872
#208162
C2×#18342
C4×C8×D42