Finite Group Databasebeta

← #6489255

Subgroups — #6489255

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33174C211yesmaximal, characteristic
4#33172C211yesmaximal, characteristic, complemented
5#33049C211yesmaximal, characteristic, complemented
6#33042C211yesmaximal, characteristic
7#30350C214yesmaximal, complemented
8#29208C211yesmaximal, characteristic, complemented
9#29193C211yesmaximal, characteristic, complemented
10#29192C211yesmaximal, characteristic, complemented
11#29141C211yesmaximal, characteristic, complemented
12#28750C211yesmaximal, characteristic, complemented
13#28616C211yesmaximal, characteristic, complemented
14#28613C211yesmaximal, characteristic, complemented
15#28444C211yesmaximal, characteristic, complemented
16#25514C214yesmaximal, complemented
17#24765C214yesmaximal, complemented
18#22163C214yesmaximal, complemented
19C4×#424C211yesmaximal, characteristic, complemented
20C2×#542C2212yes—
21C2×#494C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#468C2214yes—
26C2×#467C2214yes—
27C2×#464C2214yes—
28C2×#424C2212yes—
29C2×C4×M16C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2868C2211yescharacteristic
32#2862C2211yescharacteristic
33#2859C2211yescharacteristic
34#2858C2211yescharacteristic
35#2857C2211yescharacteristic
36#2857C2211yescharacteristic
37#2857C2211yescharacteristic
38#2856C2211yescharacteristic
39#2645C2212yes—
40#2640C2212yes—
41#2625C2212yes—
42#2620C2212yes—
43#2514—21no—
44#2512—21no—
45#2512—21no—
46#2509—21no—
47#2508—21no—
48#2495—21no—
49#2494—21no—
50#2491—21no—
51#2491—21no—
52#2489—21no—
53#2453—21no—
54#2452—21no—
55#2452—21no—
56#2422—21no—
57#2422—21no—
58#2421—21no—
59#2326C2212yes—
60#2323C2212yes—
61#2320C2212yes—
62C4×C4⋊Q8C2211yescharacteristic
63C4×C4⋊Q8C2211yescharacteristic
64C4×C4⋊Q8C2211yescharacteristic
65C4×#127C2211yescharacteristic
66C4×#117C2211yescharacteristic
67#2068C2214yes—
68#2067C2214yes—
69#2057C2214yes—
70#2056C2214yes—
71#2055C2214yes—
72#2042C2214yes—
73#2024C2214yes—
74#2015C2214yes—
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1983C2214yes—
78#1973C2214yes—
79#1969C2214yes—
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1947C2214yes—
87#1935C2214yes—
88#1934C2214yes—
89#1894C2214yes—
90#1893C2214yes—
91#1878C2214yes—
92#1877C2214yes—
93#1863C2214yes—
94#1862C2214yes—
95#1845C2211yescharacteristic
96#1845C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1803C2214yes—
100#1768C2212yes—
101#1768C2212yes—
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106#1729C2212yes—
107C22×Q16C2314yes—
108C22×SD16C2314yes—
109C22×M16C2312yes—
110C23×C8C2312yes—
111#542—48no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×#127C2311yescharacteristic
119C2×#126—22no—
120C2×#125—21no—
121C2×#124—22no—
122C2×#124C2312yes—
123C2×#124C2312yes—
124C2×#123—22no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8C2312yes—
128C2×C22⋊Q8C2312yes—
129C2×#121—21no—
130C2×#121—21no—
131C2×#121C2312yes—
132C2×C4×Q8—21no—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4C2312yes—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#494—42no—
142#492—42no—
143#487—42no—
144#486—42no—
145#468—48no—
146#467—48no—
147#464—216no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150#424—28no—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C4×M16—42no—
164C2×#97C2312yes—
165C2×C4×C8—42no—
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2312yes—
171#401—22no—
172#400—22no—
173#400—22no—
174#400—22no—
175#400—22no—
176#398—22no—
177#397—22no—
178#397—22no—
179#397—22no—
180#397—22no—
181#396—22no—
182#396—22no—
183#394—22no—
184#394—22no—
185#393—22no—
186#393—22no—
187#391—21no—
188#391—21no—
189#391—21no—
190#391—21no—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#391C2312yes—
195#391C2312yes—
196#391C2312yes—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390—21no—
203#390—21no—
204#390—21no—
205#390C2312yes—
206#390C2312yes—
207#389—22no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#388—22no—
225#388—22no—
226#388—22no—
227#388—22no—
228#387—22no—
229#387—22no—
230#387—22no—
231#385—22no—
232#385—22no—
233#384—21no—
234#384—21no—
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2312yes—
240#383—22no—
241#382—21no—
242#382—21no—
243#382—21no—
244#382—22no—
245#380—21no—
246#380—21no—
247#380—21no—
248#380—21no—
249#380—21no—
250#380C2312yes—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4—21no—
255C4×C4⋊C4—21no—
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4D411yescharacteristic
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4C2312yes—
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4D411yescharacteristic
264C4×C22⋊C4D411yescharacteristic
265#376C2311yescharacteristic
266#376C2311yescharacteristic
267C43C2311yescharacteristic
268C43C2311yescharacteristic
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#336C2312yes—
274#336C2312yes—
275#336C2312yes—
276#336C2312yes—
277C22×Q8—21no—
278C22×Q8—22no—
279C22×Q8C2412yes—
280C22×Q8C2412yes—
281C22×D4—22no—
282C22×D4—21no—
283C22×D4—21no—
284C22×D4C2412yes—
285C23×C4—22no—
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2411yescharacteristic
289C2×Q16—28no—
290C2×Q16—44no—
291C2×Q16—44no—
292C2×SD16—28no—
293C2×SD16—44no—
294C2×SD16—44no—
295C2×M16—24no—
296C2×M16—24no—
297C2×M16—42no—
298C22×C8—24no—
299C22×C8—24no—
300C22×C8—42no—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—44no—
312C4⋊Q8—82no—
313C4⋊Q8—82no—
314#127—22no—
315#127—22no—
316#126—48no—
317#125—82no—
318#124—28no—
319#124—44no—
320#124—48no—
321#123—48no—
322C22⋊Q8—28no—
323C22⋊Q8—28no—
324C22⋊Q8—48no—
325C22⋊Q8—82no—
326#121—28no—
327#121—82no—
328#121—82no—
329C4×Q8—28no—
330C4×Q8—44no—
331C4×Q8—82no—
332C4×D4—44no—
333C4×D4—82no—
334C4×D4—82no—
335#117—24no—
336#117—24no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
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—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
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⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C22⋊C4C2×D411yescharacteristic
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—
410C4⋊C8—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413C4⋊C8#4118yes—
414#103—28no—
415#103—44no—
416#103—44no—
417#103—44no—
418#103—44no—
419#102—28no—
420#102—44no—
421#102—44no—
422#97—42no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—42no—
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#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454C24—22no—
455C24C22×D411yescharacteristic
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—22no—
460C2×Q8—24no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—44no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—82no—
469C2×Q8—82no—
470C2×D4—24no—
471C2×D4—22no—
472C2×D4—24no—
473C2×D4—22no—
474C2×D4—44no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—82no—
482C2×D4—82no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
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×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
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×C4H(5,𝔽2)14yes—
516C22×C4C2511yescharacteristic
517Q16—88no—
518SD16—88no—
519M16—48no—
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—48no—
531C2×C8C2×#4114yes—
532C2×C8C2×#4114yes—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
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—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—21no—
596C42—22no—
597C42—22no—
598C42—21no—
599C42—44no—
600C42—44no—
601C42—82no—
602C42C2×#4114yes—
603C42H(5,𝔽2)12yes—
604C422−1+412yes—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23#53512yes—
619C23C23×D411yescharacteristic
620Q8—44no—
621Q8—44no—
622Q8—44no—
623Q8—44no—
624Q8—84no—
625Q8—164no—
626D4—44no—
627D4—44no—
628D4—84no—
629D4—84no—
630D4—164no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
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—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
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—164no—
687C2×C4#53212yes—
688C2×C4#53212yes—
689C2×C4#53512yes—
690C2×C4#53712yes—
691C2×C4#53712yes—
692C2×C4#57612yes—
693C2×C4#57712yes—
694C2×C4#57812yes—
695C2×C4#57812yes—
696C2×C4C22×#4112yes—
697C2×C4C2×H(5,𝔽2)11yescharacteristic
698C2×C4C2×H(5,𝔽2)12yes—
699C2×C4C2×2−1+411yescharacteristic
700C8—24no—
701C8—24no—
702C8—42no—
703C8—42no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22C2×#53211yescharacteristic
719C22C2×#53511yescharacteristic
720C22C2×#53711yescharacteristic
721C22C2×#57611yescharacteristic
722C22C2×#57711yescharacteristic
723C22C2×#57811yescharacteristic
724C22C2×#57811yescharacteristic
725C4—22no—
726C4—22no—
727C4—22no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4#325814yes—
744C4#326214yes—
745C4#351912yes—
746C4#352212yes—
747C2—42no—
748C2—162no—
749C2#3523811yescharacteristic
750C2#6071811yescharacteristic
751C2#6072311yescharacteristic
752C2#6082111yescharacteristic
753C2#6082211yescharacteristic
754C2C2×#325811yescharacteristic
755C2C2×#326211yescharacteristic
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
4. #33172C2semidirect
5. #33049C2semidirect
7. #30350C2semidirect
8. #29208C2semidirect
9. #29193C2semidirect
10. #29192C2semidirect
11. #29141C2semidirect
12. #28750C2semidirect
13. #28616C2semidirect
14. #28613C2semidirect
15. #28444C2semidirect
16. #25514C2semidirect
17. #24765C2semidirect
18. #22163C2semidirect
19. C4×#424C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352772
#331742
#331722
#330492
#330422
#303502
#292082
#291932
#291922
#291412
#287502
#286162
#286132
#284442
#255142
#247652
#221632
C4×#4242