Finite Group Databasebeta

← #6858515

Subgroups — #6858515

Order 512.

1570 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#36608C211yesmaximal, characteristic, complemented
2#36594C211yesmaximal, characteristic
3#33227C211yesmaximal, characteristic
4#32174C214yesmaximal, complemented
5#32171C214yesmaximal, complemented
6#30262C211yesmaximal, characteristic, complemented
7#30261C211yesmaximal, characteristic
8#28750C211yesmaximal, characteristic
9#28748C211yesmaximal, characteristic, complemented
10#28719C211yesmaximal, characteristic, complemented
11#28616C211yesmaximal, characteristic, complemented
12#28613C211yesmaximal, characteristic
13#28444C211yesmaximal, characteristic
14#24637C214yesmaximal, complemented
15#24195C214yesmaximal, complemented
16#22221C211yesmaximal, characteristic, complemented
17#22220C211yesmaximal, characteristic
18C4×#494C211yesmaximal, characteristic, complemented
19#21339C211yesmaximal, characteristic, complemented
20C2×#556C2212yes—
21C2×#494C2211yescharacteristic
22C2×#494C2211yescharacteristic
23C2×#492C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#487C2211yescharacteristic
28C2×#486C2211yescharacteristic
29C2×#486C2211yescharacteristic
30C2×#464C2214yes—
31C2×C4×Q16C2214yes—
32#2868C2211yescharacteristic
33#2862C2211yescharacteristic
34#2862C2211yescharacteristic
35#2862C2211yescharacteristic
36#2858C2211yescharacteristic
37#2858C2211yescharacteristic
38#2581—21no—
39#2581—21no—
40#2581C2212yes—
41#2547C2212yes—
42#2545C2212yes—
43#2545C2212yes—
44#2538—21no—
45#2538—21no—
46#2514—21no—
47#2514—21no—
48#2514—21no—
49#2514—21no—
50#2514C2212yes—
51#2514C2212yes—
52#2512—21no—
53#2512—21no—
54#2512—21no—
55#2512—21no—
56#2356—21no—
57#2356—21no—
58#2356C2212yes—
59#2354—21no—
60#2354—21no—
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×C4⋊Q8C2211yescharacteristic
64C4×C4⋊Q8C2211yescharacteristic
65C4×C4⋊Q8C2211yescharacteristic
66C4×#127C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2074C2214yes—
69#2056C2214yes—
70#2055C2214yes—
71#2043C2214yes—
72#2042C2214yes—
73#2016C2214yes—
74#2016C2214yes—
75#2015C2214yes—
76#2015C2214yes—
77#2015C2214yes—
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1984C2212yes—
81#1983C2214yes—
82#1981C2214yes—
83#1969C2214yes—
84#1958C2214yes—
85#1958C2214yes—
86#1957C2211yescharacteristic
87#1956C2211yescharacteristic
88#1956C2211yescharacteristic
89#1956C2211yescharacteristic
90#1939C2212yes—
91#1928C2214yes—
92#1925C2214yes—
93#1916C2212yes—
94#1916C2212yes—
95#1878C2214yes—
96#1877C2214yes—
97#1863C2214yes—
98#1862C2214yes—
99#1845C2211yescharacteristic
100#1805C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×Q16C2314yes—
108C22×Q16C2314yes—
109C22×SD16C2314yes—
110#556—48no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×#127C2311yescharacteristic
119C2×#126—22no—
120C2×#126—22no—
121C2×#124C2312yes—
122C2×#124C2312yes—
123C2×#124C2312yes—
124C2×#124C2312yes—
125C2×C22⋊Q8—22no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8D411yescharacteristic
129C2×C4×Q8D411yescharacteristic
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2311yescharacteristic
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4C2312yes—
137C2×#117—22no—
138#494—24no—
139#494—42no—
140#492—42no—
141#487—24no—
142#487—42no—
143#487—42no—
144#487—42no—
145#486—24no—
146#486—42no—
147#464—48no—
148C4×Q16—48no—
149C2×#107C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2311yescharacteristic
169#401—22no—
170#401—22no—
171#400—22no—
172#400—22no—
173#400—22no—
174#397—22no—
175#397—22no—
176#396—22no—
177#396—22no—
178#396—22no—
179#396—22no—
180#395—22no—
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#391—21no—
190#391—21no—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#391C2312yes—
195#391C2312yes—
196#391C2312yes—
197#391C2312yes—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390—21no—
203#390—21no—
204#390—21no—
205#390—21no—
206#390—21no—
207#390—21no—
208#390C2312yes—
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#389C2312yes—
218#389C2312yes—
219#388—22no—
220#388—22no—
221#388—22no—
222#388—22no—
223#387—22no—
224#387—22no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#384—21no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#383—22no—
236#383—22no—
237#383—22no—
238#382—21no—
239#382—21no—
240#382—21no—
241#382—21no—
242#382C2312yes—
243#382C2312yes—
244#380—22no—
245#380—22no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4D411yescharacteristic
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4C2312yes—
261C4×C4⋊C4C2312yes—
262C4×C4⋊C4C2311yescharacteristic
263C4×C4⋊C4C2311yescharacteristic
264C4×C4⋊C4C2311yescharacteristic
265#376—22no—
266C43C2311yescharacteristic
267C43C2311yescharacteristic
268C43C2311yescharacteristic
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275#336C2312yes—
276#336C2312yes—
277#336C2312yes—
278C22×Q8—21no—
279C22×Q8—21no—
280C22×Q8C2×D411yescharacteristic
281C22×Q8C2412yes—
282C22×Q8C2412yes—
283C22×Q8C2412yes—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C2412yes—
287C23×C4—22no—
288C2×Q16—28no—
289C2×Q16—44no—
290C2×Q16—44no—
291C2×Q16—44no—
292C2×Q16—44no—
293C2×Q16—44no—
294C2×SD16—44no—
295C2×SD16—44no—
296C2×SD16—44no—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—44no—
312C4⋊Q8—44no—
313#127—22no—
314#127—22no—
315#126—48no—
316#126—48no—
317#124—28no—
318#124—44no—
319#124—44no—
320#124—44no—
321C22⋊Q8—48no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—28no—
325C4×Q8—28no—
326C4×Q8—42no—
327C4×Q8—44no—
328C4×Q8—44no—
329C4×Q8—44no—
330C4×D4—44no—
331C4×D4—44no—
332C4×D4—44no—
333#117—48no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
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×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#107—44no—
404C4⋊C8—24no—
405C4⋊C8—42no—
406C4⋊C8—42no—
407#103—28no—
408#103—44no—
409#103—44no—
410#103—44no—
411#103—44no—
412#103—44no—
413#103—44no—
414#102—28no—
415#102—44no—
416#102—44no—
417C4×C8—24no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—42no—
422C4×C8#4114yes—
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#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#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451C24—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—22no—
463C2×Q8—24no—
464C2×Q8—24no—
465C2×Q8—24no—
466C2×Q8—41no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C2511yescharacteristic
508Q16—416no—
509Q16—88no—
510SD16—88no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8C2×#4114yes—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—44no—
587C42—44no—
588C42—44no—
589C42—44no—
590C42—44no—
591C42#13514yes—
592C42C2×#4112yes—
593C42H(5,𝔽2)14yes—
594C42H(5,𝔽2)12yes—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23C23×D411yescharacteristic
603Q8—44no—
604Q8—44no—
605Q8—44no—
606Q8—44no—
607Q8—44no—
608Q8—44no—
609Q8—44no—
610Q8—84no—
611Q8—84no—
612D4—44no—
613D4—44no—
614D4—84no—
615D4—84no—
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—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
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—42no—
669C2×C4—88no—
670C2×C4#53512yes—
671C2×C4#53512yes—
672C2×C4#55012yes—
673C2×C4#55012yes—
674C2×C4C2×#13512yes—
675C2×C4C2×#13512yes—
676C2×C4C2×#13512yes—
677C2×C4#57512yes—
678C2×C4#57812yes—
679C2×C4#57812yes—
680C2×C4C22×#4111yescharacteristic
681C2×C4C2×H(5,𝔽2)12yes—
682C2×C4C2×H(5,𝔽2)11yescharacteristic
683C2×C4#58512yes—
684C8—24no—
685C8—42no—
686C8—42no—
687C8—42no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22C2×#53511yescharacteristic
696C22C2×#55011yescharacteristic
697C22#350011yescharacteristic
698C22C22×#13511yescharacteristic
699C22C2×#57511yescharacteristic
700C22C2×#57811yescharacteristic
701C22C2×#57811yescharacteristic
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
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—82no—
721C4#339314yes—
722C4#340214yes—
723C4#350612yes—
724C4#354612yes—
725C2—162no—
726C2#3974711yescharacteristic
727C2#6078211yescharacteristic
728C2#6082111yescharacteristic
729C2C2×#339311yescharacteristic
730C2C2×#340211yescharacteristic
731C2#6252811yescharacteristic
732C2#6254511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36608C2semidirect
4. #32174C2semidirect
5. #32171C2semidirect
6. #30262C2semidirect
9. #28748C2semidirect
10. #28719C2semidirect
11. #28616C2semidirect
14. #24637C2semidirect
15. #24195C2semidirect
16. #22221C2semidirect
18. C4×#494C2semidirect
19. #21339C2semidirect

Maximal subgroups

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

subgroupindex
#366082
#365942
#332272
#321742
#321712
#302622
#302612
#287502
#287482
#287192
#286162
#286132
#284442
#246372
#241952
#222212
#222202
C4×#4942
#213392