Finite Group Databasebeta

← #6811117

Subgroups — #6811117

Order 512.

1893 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3255C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic
3#34929C211yesmaximal, characteristic
4#33163C211yesmaximal, characteristic, complemented
5#30929C214yesmaximal, complemented
6#30864C214yesmaximal, complemented
7#29664C212yesmaximal, complemented
8#28489C212yesmaximal
9#28440C211yesmaximal, characteristic, complemented
10#28283C211yesmaximal, characteristic, complemented
11#26462C212yesmaximal, complemented
12#25394C212yesmaximal
13#25370C212yesmaximal, complemented
14#24760C212yesmaximal, complemented
15#24157C212yesmaximal, complemented
16#22650C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#533C2212yes—
20#3255—216no—
21C2×#494C2212yes—
22C2×#475C2212yes—
23C2×#475C2214yes—
24C2×#475C2211yescharacteristic
25C2×#470C2214yes—
26C2×#470C2214yes—
27C2×#470C2214yes—
28C2×#462C2212yes—
29C2×#462C2214yes—
30C2×#462C2212yes—
31C2×#462C2212yes—
32C2×#462C2212yes—
33C2×#462C2212yes—
34C2×#436C2212yes—
35C2×#423C2212yes—
36#2868C2211yescharacteristic
37#2857C2212yes—
38#2556—22no—
39#2552—21no—
40#2551C2211yescharacteristic
41#2546C2211yescharacteristic
42#2542C2212yes—
43#2542C2211yescharacteristic
44#2538—21no—
45#2538C2211yescharacteristic
46#2506—21no—
47#2492C2212yes—
48#2452—21no—
49#2441—22no—
50#2429C2214yes—
51#2429C2212yes—
52#2423—22no—
53#2396—22no—
54#2365C2212yes—
55#2354—21no—
56Q8×C4⋊C4—21no—
57C4×C4⋊Q8—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C22⋊Q8C2212yes—
61#2293—21no—
62C42×Q8C2211yescharacteristic
63C42×D4C2211yescharacteristic
64#2066C2214yes—
65#2066C2214yes—
66#2056C2214yes—
67#2035C2214yes—
68#2024C2214yes—
69#2016C2214yes—
70#2016C2214yes—
71#2001C2214yes—
72#2001C2214yes—
73#2001C2214yes—
74#1988C2211yescharacteristic
75#1981C2212yes—
76#1981C2212yes—
77#1981C2212yes—
78#1981C2212yes—
79#1956C2212yes—
80#1948C2212yes—
81#1948C2211yescharacteristic
82#1939C2212yes—
83#1919C2212yes—
84#1894C2214yes—
85#1878C2214yes—
86#1865C2214yes—
87#1865C2212yes—
88#1851C2214yes—
89#1842C2211yescharacteristic
90#1803C2211yescharacteristic
91#1803C2212yes—
92#1797C2212yes—
93C4×#106C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×M16C2314yes—
100#533—28no—
101#533—28no—
102#533—44no—
103#533—44no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×#125—22no—
113C2×#125—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2314yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2314yes—
121C2×C4×Q8—22no—
122C2×C4×Q8—21no—
123C2×C4×Q8—22no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117—22no—
133C2×#117—22no—
134C22×C42C2312yes—
135#494—44no—
136#475—28no—
137#475—216no—
138#475D418yescomplemented
139#470—216no—
140#470—216no—
141#470—48no—
142#462—28no—
143#462—28no—
144#462—216no—
145#462—44no—
146#462—44no—
147#462—44no—
148#436—44no—
149C2×#106C2314yes—
150C2×#106C2312yes—
151C2×#106C2312yes—
152#423—28no—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2311yescharacteristic
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×C22⋊C8C2314yes—
169C2×C4×C8C2312yes—
170#400—22no—
171#399—22no—
172#399—24no—
173#398—22no—
174#398—22no—
175#398—24no—
176#396—22no—
177#396—22no—
178#396—24no—
179#395—22no—
180#393—22no—
181#391—21no—
182#391—22no—
183#391C2312yes—
184#391C2312yes—
185#391C2314yes—
186#390—22no—
187#390—21no—
188#390—21no—
189#389—21no—
190#389—21no—
191#389—22no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—22no—
198#389—22no—
199#389—22no—
200#389—24no—
201#389C2312yes—
202#389C2312yes—
203#388—24no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—24no—
208#385—22no—
209#384—21no—
210#384—22no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384—21no—
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#380—22no—
218#380—22no—
219#380C2312yes—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4D411yescharacteristic
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2311yescharacteristic
234C4×C4⋊C4C2311yescharacteristic
235C4×C22⋊C4—22no—
236C4×C22⋊C4D412yes—
237C4×C22⋊C4D412yes—
238C4×C22⋊C4C2312yes—
239C43—21no—
240C43C2311yescharacteristic
241#340C2312yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#336C2314yes—
247#336C2312yes—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8C2414yes—
251C22×Q8C2412yes—
252C22×Q8C2411yescharacteristic
253C22×D4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C2×Q16—28no—
258C2×Q16—28no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×Q16—44no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×M16—28no—
267C2×M16—44no—
268C2×M16—44no—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—22no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—24no—
281C4⋊Q8—42no—
282C4⋊Q8—42no—
283C4⋊Q8—42no—
284C4⋊Q8—42no—
285C4⋊Q8—44no—
286C4⋊Q8C2×D414yes—
287#125—48no—
288#125—48no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—28no—
292C22⋊Q8—24no—
293C22⋊Q8—42no—
294C22⋊Q8—42no—
295C22⋊Q8—44no—
296C22⋊Q8—42no—
297C22⋊Q8—48no—
298C22⋊Q8—48no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C4×Q8—22no—
302C4×Q8—28no—
303C4×Q8—22no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—48no—
309C4×Q8—41no—
310C4×Q8—44no—
311C4×Q8—48no—
312C4×Q8—44no—
313C4×Q8C2×D414yes—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—22no—
317C4×D4—22no—
318#117—48no—
319#117—48no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C42—21no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—22no—
370C2×C42—24no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#106—28no—
388#106—24no—
389#106C2×D418yes—
390C4⋊C8—28no—
391C4⋊C8—24no—
392C4⋊C8—24no—
393C4⋊C8—22no—
394C4⋊C8—42no—
395C4⋊C8C2×D414yes—
396#103—28no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—42no—
401#103—44no—
402#103—44no—
403#103—44no—
404#103C2×D418yes—
405C22⋊C8—44no—
406C4×C8—24no—
407#95—22no—
408#95—22no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—24no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—24no—
420#95C2×D412yes—
421#95C2×D412yes—
422#95C2×D412yes—
423C24C22×D412yes—
424C2×Q8—24no—
425C2×Q8—21no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—44no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8C22×D412yes—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—42no—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2511yescharacteristic
482Q16—88no—
483Q16—88no—
484Q16—88no—
485M16—88no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8C22×D414yes—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—81no—
531C4⋊C4—84no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4#13714yes—
535C4⋊C4C22×D412yes—
536C4⋊C4C22×D414yes—
537C4⋊C4H(5,𝔽2)12yes—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C22⋊C4—82no—
546C22⋊C4—84no—
547C42—24no—
548C42—24no—
549C42—24no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—21no—
554C42—22no—
555C42—21no—
556C42—41no—
557C42—48no—
558C42—42no—
559C42—42no—
560C42—44no—
561C42—44no—
562C42—42no—
563C42—42no—
564C42—81no—
565C42#13714yes—
566C42C22×D412yes—
567C42H(5,𝔽2)12yes—
568C42H(5,𝔽2)12yes—
569C422−1+412yes—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23C23×D411yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—48no—
581Q8—41no—
582Q8—41no—
583Q8—44no—
584Q8—42no—
585Q8—42no—
586Q8—44no—
587Q8—42no—
588Q8—44no—
589Q8—44no—
590Q8—42no—
591Q8—88no—
592D4—44no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—48no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4#53712yes—
644C2×C4#53712yes—
645C2×C4D4212yes—
646C2×C4D4212yes—
647C2×C4C2×#13712yes—
648C2×C4C2×#13714yes—
649C2×C4C2×#13712yes—
650C2×C4C2×#13712yes—
651C2×C4C2×#13712yes—
652C2×C4#57812yes—
653C2×C4C23×D411yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C2×C4#58511yescharacteristic
659C8—24no—
660C8—44no—
661C8—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22C2×#53711yescharacteristic
670C22C2×D4211yescharacteristic
671C22#348811yescharacteristic
672C22C22×#13711yescharacteristic
673C22C22×#13711yescharacteristic
674C22C2×#57811yescharacteristic
675C22#358611yescharacteristic
676C4—22no—
677C4—22no—
678C4—24no—
679C4—22no—
680C4—41no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—84no—
691C4—81no—
692C4#325514yescomplemented
693C4#327614yes—
694C4#349711yescharacteristic
695C4#349811yescharacteristic
696C4#354611yescharacteristic
697C4#354811yescharacteristic
698C2—82no—
699C2#3934111yescharacteristic
700C2C2×#325511yescharacteristic
701C2#6092511yescharacteristic
702C2C2×#327611yescharacteristic
703C2D4×#13711yescharacteristic
704C2#6196611yescharacteristic
705C2#6200011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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. C2×#3255C2semidirect
4. #33163C2semidirect
5. #30929C2semidirect
6. #30864C2semidirect
7. #29664C2semidirect
9. #28440C2semidirect
10. #28283C2semidirect
11. #26462C2semidirect
13. #25370C2semidirect
14. #24760C2semidirect
15. #24157C2semidirect
17. C4×#475C2semidirect
138. #475D4semidirect
692. C4#3255semidirect

Maximal subgroups

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

subgroupindex
C2×#32552
#361342
#349292
#331632
#309292
#308642
#296642
#284892
#284402
#282832
#264622
#253942
#253702
#247602
#241572
#226502
C4×#4752