Finite Group Databasebeta

← #6497756

Subgroups — #6497756

Order 512.

1493 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#33286C211yesmaximal, characteristic, complemented
5#33212C211yesmaximal, characteristic, complemented
6#33048C211yesmaximal, characteristic
7#30305C211yesmaximal, characteristic, complemented
8#30278C211yesmaximal, characteristic, complemented
9#29877C211yesmaximal, characteristic, complemented
10#29843C211yesmaximal, characteristic
11#28965C214yesmaximal, complemented
12#28861C214yesmaximal, complemented
13#28074C214yesmaximal, complemented
14#28038C214yesmaximal, complemented
15#22247C211yesmaximal, characteristic, complemented
16#22219C211yesmaximal, characteristic, complemented
17C4×#498C211yesmaximal, characteristic, complemented
18C4×#486C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic, complemented
20C2×#542—21no—
21C2×#500C2211yescharacteristic
22C2×#498C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2645—21no—
36#2640—21no—
37#2625—21no—
38#2620—21no—
39#2514—21no—
40#2512—21no—
41#2512C2212yes—
42#2509C2212yes—
43#2508C2212yes—
44#2495C2212yes—
45#2494C2212yes—
46#2491—21no—
47#2491C2212yes—
48#2489—21no—
49#2453—21no—
50#2452—21no—
51#2452C2212yes—
52#2422—21no—
53#2422C2212yes—
54#2421—21no—
55#2326—21no—
56#2323—21no—
57#2320—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2092C2214yes—
64#2092C2214yes—
65#2088C2214yes—
66#2084C2214yes—
67#2077C2214yes—
68#2065C2214yes—
69#2058C2214yes—
70#2034C2214yes—
71#1993C2214yes—
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1944C2214yes—
81#1941C2214yes—
82#1936C2214yes—
83#1936C2214yes—
84#1936C2214yes—
85#1934C2214yes—
86#1926C2214yes—
87#1918C2214yes—
88#1918C2214yes—
89#1895C2214yes—
90#1864C2214yes—
91#1844C2211yescharacteristic
92#1844C2211yescharacteristic
93#1836C2214yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1809C2214yes—
97C4×#106C2214yes—
98#1771C2212yes—
99#1771C2212yes—
100C4×#102C2214yes—
101#1753C2212yes—
102#1753C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×M16C2312yes—
108#542—48no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#126—22no—
117C2×#125—21no—
118C2×#124—22no—
119C2×#124—21no—
120C2×#124—21no—
121C2×#123—22no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—22no—
126C2×#121—21no—
127C2×#121—21no—
128C2×#121C2312yes—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4C2312yes—
135C2×#117D411yescharacteristic
136C2×#117D411yescharacteristic
137C22×C42C2311yescharacteristic
138#500—24no—
139#498—24no—
140#486—24no—
141#486—24no—
142#431—24no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C4×M16—24no—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163#401—22no—
164#400—22no—
165#400—22no—
166#400—22no—
167#400—22no—
168#398—22no—
169#397—22no—
170#397—22no—
171#397—22no—
172#397—22no—
173#396—22no—
174#396—22no—
175#394—22no—
176#394—22no—
177#393—22no—
178#393—22no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390C2312yes—
196#390C2312yes—
197#390C2312yes—
198#390C2312yes—
199#389—22no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#389C2312yes—
216#388—22no—
217#388—22no—
218#388—22no—
219#388—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#385—22no—
224#385—22no—
225#384—21no—
226#384—21no—
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2312yes—
232#383—22no—
233#382—21no—
234#382—22no—
235#382C2312yes—
236#382C2312yes—
237#380—21no—
238#380—21no—
239#380—21no—
240#380C2312yes—
241#380C2312yes—
242#380C2312yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257#376D411yescharacteristic
258#376D411yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×D4—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2412yes—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C2×M16—24no—
286C2×M16—24no—
287C2×M16—42no—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—82no—
301#127—22no—
302#127—22no—
303#126—48no—
304#125—82no—
305#124—44no—
306#124—44no—
307#124—48no—
308#123—48no—
309C22⋊Q8—48no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312C22⋊Q8—82no—
313#121—44no—
314#121—44no—
315#121—82no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×Q8—82no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—82no—
322#117—24no—
323#117—42no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
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⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
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⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
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
394C2×C42C2411yescharacteristic
395#107—28no—
396#106—28no—
397#106—44no—
398#106—44no—
399C4⋊C8—24no—
400#102—28no—
401#102—28no—
402#102—28no—
403#102—44no—
404C22⋊C8—24no—
405#97—24no—
406#97—24no—
407#97—42no—
408C4×C8—24no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—42no—
412C4×C8#4114yes—
413C4×C8#4114yes—
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#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440C24—22no—
441C24C22×D411yescharacteristic
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—82no—
466C2×D4—82no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—82no—
486C22×C4—82no—
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×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C2511yescharacteristic
501M16—48no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8C2×#4114yes—
512C2×C8C2×#4114yes—
513C2×C82−1+414yes—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—21no—
577C42—21no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—44no—
583C42—44no—
584C42—82no—
585C42AGL(1,ℤ/8)14yes—
586C42C2×#4112yes—
587C42C2×#4112yes—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23C23×D411yescharacteristic
603Q8—84no—
604Q8—84no—
605Q8—84no—
606Q8—164no—
607D4—44no—
608D4—44no—
609D4—84no—
610D4—84no—
611D4—164no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
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—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—44no—
644C2×C4—44no—
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—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—164no—
669C2×C4#53212yes—
670C2×C4#53212yes—
671C2×C4C2×AGL(1,ℤ/8)12yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C2×AGL(1,ℤ/8)12yes—
674C2×C4#57512yes—
675C2×C4#57712yes—
676C2×C4#57712yes—
677C2×C4C22×#4111yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×2−1+412yes—
680C2×C4#58512yes—
681C8—24no—
682C8—24no—
683C8—24no—
684C8—42no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22C2×#53211yescharacteristic
700C22#349911yescharacteristic
701C22#350011yescharacteristic
702C22C22×AGL(1,ℤ/8)11yescharacteristic
703C22C2×#57511yescharacteristic
704C22C2×#57711yescharacteristic
705C22C2×#57711yescharacteristic
706C4—22no—
707C4—22no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4#317114yes—
725C4#317314yes—
726C4#346112yes—
727C4#349512yes—
728C2—81no—
729C2—162no—
730C2#3531811yescharacteristic
731C2C2×#317111yescharacteristic
732C2C2×#317311yescharacteristic
733C2#6247711yescharacteristic
734C2#6248311yescharacteristic
735C2#6253911yescharacteristic
736C2#6254311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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. #35291C2semidirect
3. #33293C2semidirect
4. #33286C2semidirect
5. #33212C2semidirect
7. #30305C2semidirect
8. #30278C2semidirect
9. #29877C2semidirect
11. #28965C2semidirect
12. #28861C2semidirect
13. #28074C2semidirect
14. #28038C2semidirect
15. #22247C2semidirect
16. #22219C2semidirect
17. C4×#498C2semidirect
18. C4×#486C2semidirect
19. #20472C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352772
#332932
#332862
#332122
#330482
#303052
#302782
#298772
#298432
#289652
#288612
#280742
#280382
#222472
#222192
C4×#4982
C4×#4862
#204722