Finite Group Databasebeta

← #6795513

Subgroups — #6795513

Order 512.

1689 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32280C214yesmaximal, complemented
5#31934C214yesmaximal, complemented
6#29922C212yesmaximal, complemented
7#28687C211yesmaximal, characteristic, complemented
8#28530C212yesmaximal, complemented
9#28283C211yesmaximal, characteristic, complemented
10#27729C212yesmaximal, complemented
11#27678C212yesmaximal, complemented
12#25702C212yesmaximal, complemented
13#24651C212yesmaximal, complemented
14#24118C212yesmaximal
15#22795C211yesmaximal, characteristic, complemented
16#21981C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#529—21no—
20C2×#498C2212yes—
21C2×#475C2212yes—
22C2×#476C2212yes—
23C2×#461C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#431C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2567C2212yes—
29#2553—22no—
30#2552—21no—
31#2552—21no—
32#2552C2211yescharacteristic
33#2547C2211yescharacteristic
34#2542C2211yescharacteristic
35#2537C2211yescharacteristic
36#2502C2214yes—
37#2496C2212yes—
38#2489—21no—
39#2440—22no—
40#2440—22no—
41#2427—21no—
42#2422C2212yes—
43#2395—22no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8—21no—
49C4×#121C2212yes—
50#2285C2212yes—
51C2×#384C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2100C2214yes—
55#2096C2214yes—
56#2094C2214yes—
57#2084C2214yes—
58#2072C2214yes—
59#2066C2214yes—
60#2035C2214yes—
61#2035C2214yes—
62#2035C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1981C2212yes—
67#1981C2211yescharacteristic
68#1968C2214yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1948C2211yescharacteristic
76#1933C2214yes—
77#1927C2212yes—
78#1923C2214yes—
79#1877C2212yes—
80#1874C2214yes—
81#1853C2214yes—
82#1853C2214yes—
83#1853C2214yes—
84#1844C2214yes—
85#1842C2211yescharacteristic
86#1834C2214yes—
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96#554—44no—
97#554—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103#529—48no—
104C2×#126—22no—
105C2×#125—21no—
106C2×#125—22no—
107C2×#124—21no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×#121—21no—
114C2×#121C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2314yes—
129C22×C42—21no—
130C22×C42C2312yes—
131#498—44no—
132#476—44no—
133#475—44no—
134#461—24no—
135C4×SD16—44no—
136#431—28no—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2314yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#399—22no—
160#398—22no—
161#398—24no—
162#398—24no—
163#397—22no—
164#397—22no—
165#397—22no—
166#395—24no—
167#394—22no—
168#394—22no—
169#391—21no—
170#391—21no—
171#391C2312yes—
172#391C2314yes—
173#390C2312yes—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#387—22no—
186#386—22no—
187#385—24no—
188#385—24no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384—22no—
193#384—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2314yes—
197#384C2312yes—
198#384C2312yes—
199#384C2312yes—
200#384C2312yes—
201#382—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—24no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C2×SD16—24no—
251C2×SD16—24no—
252C2×SD16—42no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—82no—
263C4⋊Q8—82no—
264#126—48no—
265#125—48no—
266#125—44no—
267#124—44no—
268#123—48no—
269C22⋊Q8—416no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274#121—24no—
275#121—42no—
276#121—44no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—44no—
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—41no—
289C4×D4—48no—
290C4×D4—48no—
291C4×D4SD1618yescomplemented
292#117—48no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—84no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#106—42no—
362#106—44no—
363#106—44no—
364#106—44no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—42no—
370C4⋊C8#4114yes—
371#103—24no—
372#103—42no—
373#103—44no—
374#103—42no—
375#103—44no—
376#102—42no—
377#102—44no—
378C22⋊C8—28no—
379C22⋊C8—44no—
380C4×C8—28no—
381C4×C8—24no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24C22×D412yes—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—82no—
422C2×D4C2×SD1614yes—
423C2×D42−1+412yes—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—81no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4C2×SD1618yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2511yescharacteristic
466SD16—48no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8C2×#4114yes—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4C2×SD1614yes—
515C4⋊C4#13714yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4C2×SD1618yes—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—41no—
538C42—44no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42C2×SD1614yes—
544C42C2×#4112yes—
545C42H(5,𝔽2)12yes—
546C42H(5,𝔽2)12yes—
547C422−1+412yes—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C22×SD1614yes—
562C23C23×D411yescharacteristic
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—84no—
568Q8—164no—
569D4—28no—
570D4—22no—
571D4—22no—
572D4—41no—
573D4—41no—
574D4—88no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—216no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—168no—
625C2×C4#53512yes—
626C2×C4#53512yes—
627C2×C4#54712yes—
628C2×C4#54712yes—
629C2×C4C22×SD1612yes—
630C2×C4C22×SD1614yes—
631C2×C4C22×SD1612yes—
632C2×C4C22×SD1612yes—
633C2×C4C2×#13714yes—
634C2×C4C2×#13712yes—
635C2×C4#57712yes—
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C2×C4#58511yescharacteristic
642C8—24no—
643C8—44no—
644C8—42no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22C2×#53511yescharacteristic
660C22C2×#54711yescharacteristic
661C22#348911yescharacteristic
662C22C23×SD1611yescharacteristic
663C22C22×#13711yescharacteristic
664C22C2×#57711yescharacteristic
665C22#358511yescharacteristic
666C4—24no—
667C4—22no—
668C4—24no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—41no—
673C4—82no—
674C4—81no—
675C4—81no—
676C4—82no—
677C4—81no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4#320614yes—
682C4#333514yes—
683C4#351711yescharacteristic
684C4#352011yescharacteristic
685C4#354711yescharacteristic
686C4#355211yescharacteristic
687C2—44no—
688C2—82no—
689C2#3892011yescharacteristic
690C2C2×#320611yescharacteristic
691C2#6083311yescharacteristic
692C2C2×#333511yescharacteristic
693C2#6125411yescharacteristic
694C2#6209611yescharacteristic
695C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36180C2semidirect
3. #33197C2semidirect
4. #32280C2semidirect
5. #31934C2semidirect
6. #29922C2semidirect
7. #28687C2semidirect
8. #28530C2semidirect
9. #28283C2semidirect
10. #27729C2semidirect
11. #27678C2semidirect
12. #25702C2semidirect
13. #24651C2semidirect
15. #22795C2semidirect
16. #21981C2semidirect
17. C4×#461C2semidirect
291. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361802
#349322
#331972
#322802
#319342
#299222
#286872
#285302
#282832
#277292
#276782
#257022
#246512
#241182
#227952
#219812
C4×#4612