Finite Group Databasebeta

← #6858563

Subgroups — #6858563

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#36594C211yesmaximal, characteristic
3#33238C211yesmaximal, characteristic
4#32190C214yesmaximal, complemented
5#32187C214yesmaximal, complemented
6#30262C211yesmaximal, characteristic, complemented
7#30261C211yesmaximal, characteristic
8#28750C211yesmaximal, characteristic
9#28748C211yesmaximal, characteristic, complemented
10#28717C211yesmaximal, characteristic, complemented
11#28616C211yesmaximal, characteristic, complemented
12#28613C211yesmaximal, characteristic
13#28444C211yesmaximal, characteristic
14#24641C214yesmaximal, complemented
15#24207C214yesmaximal, complemented
16#22219C211yesmaximal, characteristic
17#22218C211yesmaximal, characteristic, complemented
18#21691C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, 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×#441C2214yes—
32#2868C2211yescharacteristic
33#2868C2211yescharacteristic
34#2862C2211yescharacteristic
35#2862C2211yescharacteristic
36#2858C2211yescharacteristic
37#2858C2211yescharacteristic
38#2582C2212yes—
39#2571—21no—
40#2570—21no—
41#2570—21no—
42#2568—21no—
43#2547C2212yes—
44#2545C2212yes—
45#2545C2212yes—
46#2538—21no—
47#2538—21no—
48#2512—21no—
49#2512—21no—
50#2512—21no—
51#2512—21no—
52#2511C2212yes—
53#2511C2212yes—
54#2446—21no—
55#2446—21no—
56#2365—21no—
57#2355—21no—
58#2355C2212yes—
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#2072C2214yes—
69#2056C2214yes—
70#2055C2214yes—
71#2043C2214yes—
72#2042C2214yes—
73#2016C2214yes—
74#2016C2214yes—
75#2015C2214yes—
76#2015C2214yes—
77#2015C2214yes—
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1985C2212yes—
81#1983C2214yes—
82#1981C2214yes—
83#1971C2214yes—
84#1959C2214yes—
85#1959C2214yes—
86#1957C2211yescharacteristic
87#1956C2211yescharacteristic
88#1956C2211yescharacteristic
89#1956C2211yescharacteristic
90#1940C2212yes—
91#1927C2214yes—
92#1926C2214yes—
93#1917C2212yes—
94#1917C2212yes—
95#1878C2214yes—
96#1877C2214yes—
97#1865C2214yes—
98#1864C2214yes—
99#1844C2211yescharacteristic
100#1803C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×Q16C2314yes—
108C22×Q16C2314yes—
109C22×SD16C2314yes—
110#556—48no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×#127C2311yescharacteristic
121C2×#126—22no—
122C2×#126—22no—
123C2×#124C2312yes—
124C2×#124C2312yes—
125C2×#124C2312yes—
126C2×#124C2312yes—
127C2×C22⋊Q8—22no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8D411yescharacteristic
131C2×C4×Q8D411yescharacteristic
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4C2312yes—
139C2×#117—22no—
140#494—24no—
141#494—42no—
142#492—42no—
143#487—24no—
144#487—42no—
145#487—42no—
146#487—42no—
147#486—24no—
148#486—42no—
149#464—48no—
150#441—48no—
151C2×#106C2314yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#97C2312yes—
166C2×#97C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171#400—22no—
172#400—22no—
173#400—22no—
174#400—22no—
175#398—22no—
176#398—22no—
177#397—22no—
178#397—22no—
179#396—22no—
180#396—22no—
181#396—22no—
182#396—22no—
183#391—22no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391—21no—
189#391—21no—
190#391C2312yes—
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#390—21no—
206#390—21no—
207#390C2312yes—
208#389—22no—
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#384—21no—
234#384—21no—
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2312yes—
238#384C2312yes—
239#382—22no—
240#382—21no—
241#382—21no—
242#382—21no—
243#382—21no—
244#380—22no—
245#380—22no—
246C4×C4⋊C4—22no—
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⋊C4—21no—
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4D411yescharacteristic
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4D411yescharacteristic
261C4×C4⋊C4C2312yes—
262C4×C4⋊C4C2312yes—
263C4×C4⋊C4C2311yescharacteristic
264C4×C4⋊C4C2311yescharacteristic
265C4×C4⋊C4C2311yescharacteristic
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—
313C4⋊Q8—44no—
314C4⋊Q8—82no—
315#127—22no—
316#127—22no—
317#126—48no—
318#126—48no—
319#124—28no—
320#124—44no—
321#124—44no—
322#124—44no—
323C22⋊Q8—48no—
324C4×Q8—24no—
325C4×Q8—24no—
326C4×Q8—28no—
327C4×Q8—28no—
328C4×Q8—42no—
329C4×Q8—44no—
330C4×Q8—82no—
331C4×Q8—82no—
332C4×D4—44no—
333C4×D4—44no—
334C4×D4—44no—
335#117—48no—
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⋊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—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⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
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⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
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
408C2×C42C2411yescharacteristic
409#106—44no—
410C4⋊C8—24no—
411C4⋊C8—42no—
412C4⋊C8—42no—
413#103—28no—
414#103—44no—
415#103—44no—
416#103—44no—
417#103—44no—
418#103—44no—
419#103—44no—
420#102—28no—
421#102—44no—
422#102—44no—
423#97—24no—
424#97—42no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—22no—
428C4×C8#4114yes—
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#95—22no—
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453C24—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—24no—
462C2×Q8—24no—
463C2×Q8—24no—
464C2×Q8—22no—
465C2×Q8—24no—
466C2×Q8—24no—
467C2×Q8—24no—
468C2×Q8—41no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8—42no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—24no—
478C2×D4—24no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
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×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C2511yescharacteristic
510Q16—416no—
511Q16—88no—
512SD16—88no—
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×C8—22no—
523C2×C8—22no—
524C2×C8C2×#4114yes—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
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—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—82no—
596C42—82no—
597C42#13714yes—
598C42C2×#4112yes—
599C42H(5,𝔽2)14yes—
600C42H(5,𝔽2)12yes—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23C23×D411yescharacteristic
609Q8—44no—
610Q8—44no—
611Q8—44no—
612Q8—44no—
613Q8—44no—
614Q8—44no—
615Q8—44no—
616Q8—84no—
617Q8—84no—
618D4—44no—
619D4—44no—
620D4—84no—
621D4—84no—
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—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
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—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—88no—
676C2×C4#53512yes—
677C2×C4#53512yes—
678C2×C4#55012yes—
679C2×C4#55012yes—
680C2×C4C2×#13712yes—
681C2×C4C2×#13712yes—
682C2×C4C2×#13712yes—
683C2×C4#57512yes—
684C2×C4#57712yes—
685C2×C4#57812yes—
686C2×C4C22×#4111yescharacteristic
687C2×C4C2×H(5,𝔽2)12yes—
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4#58512yes—
690C8—24no—
691C8—42no—
692C8—42no—
693C8—42no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22C2×#53511yescharacteristic
702C22C2×#55011yescharacteristic
703C22#350011yescharacteristic
704C22C22×#13711yescharacteristic
705C22C2×#57511yescharacteristic
706C22C2×#57711yescharacteristic
707C22C2×#57811yescharacteristic
708C4—22no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4#339814yes—
728C4#340114yes—
729C4#350612yes—
730C4#354612yes—
731C2—162no—
732C2#3974711yescharacteristic
733C2#6080411yescharacteristic
734C2#6082411yescharacteristic
735C2C2×#339811yescharacteristic
736C2C2×#340111yescharacteristic
737C2#6254211yescharacteristic
738C2#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. #36598C2semidirect
4. #32190C2semidirect
5. #32187C2semidirect
6. #30262C2semidirect
9. #28748C2semidirect
10. #28717C2semidirect
11. #28616C2semidirect
14. #24641C2semidirect
15. #24207C2semidirect
17. #22218C2semidirect
18. #21691C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#365942
#332382
#321902
#321872
#302622
#302612
#287502
#287482
#287172
#286162
#286132
#284442
#246412
#242072
#222192
#222182
#216912
#213402