Finite Group Databasebeta

← #6490782

Subgroups — #6490782

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#33115C214yesmaximal, complemented
4#30068C214yesmaximal, complemented
5#29267C214yesmaximal, complemented
6#29208C211yesmaximal, characteristic, complemented
7#29206C211yesmaximal, characteristic, complemented
8#28750C211yesmaximal, characteristic, complemented
9#28750C211yesmaximal, characteristic, complemented
10#27908C211yesmaximal, characteristic, complemented
11#27907C211yesmaximal, characteristic
12#25903C211yesmaximal, characteristic
13#25902C211yesmaximal, characteristic, complemented
14#24075C214yesmaximal, complemented
15#22221C211yesmaximal, characteristic, complemented
16#22220C211yesmaximal, characteristic, complemented
17#22219C211yesmaximal, characteristic, complemented
18#22218C211yesmaximal, characteristic, complemented
19#20782C211yesmaximal, characteristic, complemented
20C2×#542C2212yes—
21C22×C4⋊Q8C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#485C2214yes—
27C2×#482C2214yes—
28#2868C2211yescharacteristic
29#2858C2211yescharacteristic
30#2857C2211yescharacteristic
31#2856C2211yescharacteristic
32#2640C2212yes—
33#2620C2212yes—
34#2512—21no—
35#2512—21no—
36#2491—21no—
37#2491—21no—
38#2452—21no—
39#2452—21no—
40#2422—21no—
41#2422—21no—
42#2369—21no—
43#2369—21no—
44#2365—21no—
45#2365—21no—
46#2345—21no—
47#2345—21no—
48Q8×C22⋊C4—21no—
49Q8×C22⋊C4—21no—
50#2331C2212yes—
51#2331C2212yes—
52#2326C2212yes—
53#2316C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×#127C2211yescharacteristic
60#2285C2211yescharacteristic
61#2274C2212yes—
62C4×#117C2211yescharacteristic
63#2097C2214yes—
64#2092C2214yes—
65#2058C2214yes—
66#2057C2214yes—
67#2042C2214yes—
68#2042C2214yes—
69#2024C2214yes—
70#2023C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1983C2214yes—
76#1983C2214yes—
77#1957C2211yescharacteristic
78#1957C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1948C2214yes—
82#1947C2214yes—
83#1928C2214yes—
84#1927C2214yes—
85#1926C2214yes—
86#1925C2214yes—
87#1874C2214yes—
88#1872C2214yes—
89#1865C2214yes—
90#1864C2214yes—
91#1863C2214yes—
92#1862C2214yes—
93#1845C2211yescharacteristic
94#1845C2211yescharacteristic
95#1844C2211yescharacteristic
96#1844C2211yescharacteristic
97#1840C2212yes—
98#1832C2212yes—
99#1818C2212yes—
100#1798C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1744C2211yescharacteristic
106#1744C2211yescharacteristic
107C23×Q8—21no—
108#542—48no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8—24no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×#127C2311yescharacteristic
121C2×#126—22no—
122C2×#124—22no—
123C2×#124C2312yes—
124C2×#124C2312yes—
125C2×#123—22no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8C2312yes—
128C2×#121—21no—
129C2×#121—21no—
130C2×#121C2312yes—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8C2312yes—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4C2312yes—
139C2×#117C2312yes—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2312yes—
144C22×C42C2311yescharacteristic
145#487—24no—
146#487—24no—
147#486—24no—
148#486—24no—
149#485—48no—
150#482—48no—
151C2×#107C2314yes—
152C2×#107C2314yes—
153C2×#106C2314yes—
154C2×#106C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
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—22no—
184#391—22no—
185#391—22no—
186#391—22no—
187#391C2312yes—
188#391C2312yes—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390C2312yes—
198#390C2312yes—
199#389—22no—
200#389—22no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#388—22no—
214#388—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#385—22no—
220#385—22no—
221#384—21no—
222#384—21no—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2312yes—
228#382C2312yes—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—22no—
234#380—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C22⋊C4—22no—
254C4×C22⋊C4—22no—
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259#376C2311yescharacteristic
260#376C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—28no—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—22no—
279C22×Q8—22no—
280C22×Q8—82no—
281C22×Q8—82no—
282C22×Q8C2412yes—
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C2412yes—
287C23×C4—22no—
288C23×C4—22no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2411yescharacteristic
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C4⋊Q8—28no—
297C4⋊Q8—28no—
298C4⋊Q8—24no—
299C4⋊Q8—24no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—44no—
305C4⋊Q8—82no—
306C4⋊Q8—82no—
307C4⋊Q8#4114yes—
308C4⋊Q8#4114yes—
309#127—22no—
310#127—22no—
311#126—48no—
312#124—28no—
313#124—44no—
314#124—48no—
315#123—48no—
316C22⋊Q8—28no—
317C22⋊Q8—48no—
318#121—28no—
319#121—82no—
320#121—82no—
321C4×Q8—44no—
322C4×Q8—82no—
323C4×Q8—82no—
324C4×Q8—82no—
325C4×Q8—82no—
326C4×D4—44no—
327C4×D4—82no—
328C4×D4—82no—
329#117—24no—
330#117—24no—
331#117—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4C2×D411yescharacteristic
381C2×C22⋊C4C2×D411yescharacteristic
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408#107—44no—
409#107—44no—
410#106—44no—
411#106—44no—
412C4⋊C8—24no—
413C4⋊C8—24no—
414#103—28no—
415#103—28no—
416#103—44no—
417#103—44no—
418#102—28no—
419#102—28no—
420#102—44no—
421#102—44no—
422C22⋊C8—24no—
423C22⋊C8—24no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450C24—22no—
451C24C22×D411yescharacteristic
452C2×Q8—28no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—44no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—82no—
465C2×Q8—82no—
466C2×Q8—82no—
467C2×Q8—82no—
468C2×Q8—816no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—24no—
472C2×D4—24no—
473C2×D4—44no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—82no—
481C2×D4—82no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—28no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—44no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4H(5,𝔽2)14yes—
518C22×C4C2511yescharacteristic
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C4⋊C4—28no—
532C4⋊C4—28no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4C2×#4114yes—
565C4⋊C4C2×#4114yes—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—21no—
593C42—21no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—44no—
598C42—44no—
599C42—82no—
600C42—82no—
601C42—82no—
602C42—82no—
603C42C2×#4112yes—
604C42C2×#4112yes—
605C422−1+414yes—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23#53512yes—
620C23C23×D411yescharacteristic
621Q8—48no—
622Q8—48no—
623Q8—44no—
624Q8—44no—
625Q8—88no—
626Q8—164no—
627D4—44no—
628D4—44no—
629D4—84no—
630D4—84no—
631D4—164no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—88no—
688C2×C4—164no—
689C2×C4#52512yes—
690C2×C4#52512yes—
691C2×C4#53512yes—
692C2×C4#53612yes—
693C2×C4#53612yes—
694C2×C4#57612yes—
695C2×C4#57712yes—
696C2×C4#57712yes—
697C2×C4#57812yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×H(5,𝔽2)12yes—
701C2×C4C2×2−1+412yes—
702C8—42no—
703C8—42no—
704C8—42no—
705C8—42no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C2×#52511yescharacteristic
721C22C2×#53511yescharacteristic
722C22C2×#53611yescharacteristic
723C22C2×#57611yescharacteristic
724C22C2×#57711yescharacteristic
725C22C2×#57711yescharacteristic
726C22C2×#57811yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—41no—
731C4—44no—
732C4—44no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4#311814yes—
746C4#312114yes—
747C4#352212yes—
748C4#352212yes—
749C2—42no—
750C2—162no—
751C2#3524911yescharacteristic
752C2C2×#311811yescharacteristic
753C2C2×#312111yescharacteristic
754C2#6082211yescharacteristic
755C2#6082411yescharacteristic
756C2#6088011yescharacteristic
757C2#6088111yescharacteristic
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
2. #34216C2semidirect
3. #33115C2semidirect
4. #30068C2semidirect
5. #29267C2semidirect
6. #29208C2semidirect
7. #29206C2semidirect
8. #28750C2semidirect
9. #28750C2semidirect
10. #27908C2semidirect
13. #25902C2semidirect
14. #24075C2semidirect
15. #22221C2semidirect
16. #22220C2semidirect
17. #22219C2semidirect
18. #22218C2semidirect
19. #20782C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#342162
#331152
#300682
#292672
#292082
#292062
#287502
#279082
#279072
#259032
#259022
#240752
#222212
#222202
#222192
#222182
#207822