Finite Group Databasebeta

← #6686326

Subgroups — #6686326

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3364C211yesmaximal, characteristic, complemented
2C2×#3353C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic, complemented
4#36406C211yesmaximal, characteristic
5#33195C211yesmaximal, characteristic, complemented
6#30640C214yesmaximal, complemented
7#30606C214yesmaximal, complemented
8#30279C211yesmaximal, characteristic, complemented
9#29831C211yesmaximal, characteristic, complemented
10#28675C211yesmaximal, characteristic, complemented
11#28606C211yesmaximal, characteristic, complemented
12#26559C214yesmaximal, complemented
13#26362C214yesmaximal, complemented
14#25840C211yesmaximal, characteristic, complemented
15#25703C211yesmaximal, characteristic
16#25650C211yesmaximal, characteristic, complemented
17#25570C211yesmaximal, characteristic, complemented
18#22210C211yesmaximal, characteristic, complemented
19#21114C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#547C2211yescharacteristic
24#3364—28no—
25#3353—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#488C2212yes—
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#484C2212yes—
32C2×#482C2212yes—
33C2×#479C2212yes—
34C2×#475C2211yescharacteristic
35C2×#474C2211yescharacteristic
36C2×#474C2211yescharacteristic
37C2×#468C2212yes—
38C2×#464C2212yes—
39C2×#462C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#428C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2857—21no—
48#2721—24no—
49#2658—21no—
50#2553—24no—
51#2547C2211yescharacteristic
52#2546—21no—
53#2546C2211yescharacteristic
54#2546C2211yescharacteristic
55#2545—21no—
56#2542C2211yescharacteristic
57#2499—21no—
58#2441C2214yes—
59#2432—21no—
60#2427C2214yes—
61#2423C2214yes—
62#2360C2214yes—
63#2327—21no—
64C4×#124C2211yescharacteristic
65#2297—21no—
66#2295C2211yescharacteristic
67C42×D4C2211yescharacteristic
68#2095C2214yes—
69#2095C2214yes—
70#2086C2214yes—
71#2066C2214yes—
72#2059C2214yes—
73#2056C2214yes—
74#2042C2214yes—
75#2036C2214yes—
76#2033C2214yes—
77#2001C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1988C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1947C2214yes—
85#1942C2214yes—
86#1928C2212yes—
87#1923C2214yes—
88#1919C2212yes—
89#1894C2214yes—
90#1877C2212yes—
91#1873C2214yes—
92#1863C2214yes—
93#1863C2214yes—
94#1862C2212yes—
95#1852C2214yes—
96#1851C2214yes—
97#1842C2211yescharacteristic
98#1802C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×#41—21no—
104C22×Q16C2312yes—
105C22×SD16C2311yescharacteristic
106C22×M16C2312yes—
107#557—24no—
108#557—42no—
109#556—24no—
110#556—24no—
111#556—42no—
112#556—42no—
113#547—24no—
114#547—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×#126—22no—
125C2×#126—22no—
126C2×#125—22no—
127C2×#125C2312yes—
128C2×#124—21no—
129C2×#124—21no—
130C2×#124D411yescharacteristic
131C2×#124C2312yes—
132C2×#123—22no—
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8—21no—
135C2×C22⋊Q8C2312yes—
136C2×C22⋊Q8C2314yes—
137C2×#121C2312yes—
138C2×C4×Q8—21no—
139C2×C4×Q8—21no—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×D4C2312yes—
145C2×C4×D4C2311yescharacteristic
146C2×C4×D4C2311yescharacteristic
147C2×#117—21no—
148C2×#117C2312yes—
149C22×C4⋊C4C2312yes—
150C22×C42—21no—
151#500—24no—
152#500—42no—
153#488—28no—
154#487—24no—
155#487—42no—
156#484—28no—
157#482—28no—
158#479—28no—
159#475—24no—
160#474—24no—
161#474—24no—
162#468—28no—
163#464—28no—
164#462—24no—
165C8×Q8—24no—
166C4×Q16—28no—
167C4×SD16D418yes—
168#436—24no—
169#428—28no—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#107C2312yes—
175C2×#107C2314yes—
176C2×#106C2312yes—
177C2×#106C2311yescharacteristic
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2311yescharacteristic
181C2×C4⋊C8C2314yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2314yes—
188C2×#103C2314yes—
189C2×#103C2311yescharacteristic
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2312yes—
193C2×#102C2311yescharacteristic
194C2×C22⋊C8C2312yes—
195C2×C22⋊C8C2314yes—
196C2×C4×C8C2311yescharacteristic
197C2×C4×C8C2312yes—
198C2×C4×C8C2311yescharacteristic
199#399—24no—
200#399—24no—
201#399—22no—
202#398—22no—
203#398—22no—
204#398—24no—
205#397—22no—
206#397—24no—
207#396—24no—
208#394—24no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391D411yescharacteristic
215#391C2312yes—
216#391C2314yes—
217#390—21no—
218#390D411yescharacteristic
219#389—24no—
220#389—24no—
221#389C2314yes—
222#389C2312yes—
223#389C2314yes—
224#388—24no—
225#388D414yes—
226#387—22no—
227#387—24no—
228#387—24no—
229#387—24no—
230#386—24no—
231#385—24no—
232#384—21no—
233#384—22no—
234#384C2314yes—
235#382—22no—
236#382D411yescharacteristic
237#380—24no—
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2311yescharacteristic
243C4×C22⋊C4—24no—
244C4×C22⋊C4—21no—
245C4×C22⋊C4—22no—
246C4×C22⋊C4C2314yes—
247#376—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C2×#41—44no—
259C2×#41—44no—
260C2×#41—82no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2412yes—
266C22×Q8C2411yescharacteristic
267C22×D4—22no—
268C22×D4C2×D411yescharacteristic
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4C2412yes—
274C2×Q16—24no—
275C2×Q16—44no—
276C2×SD16—42no—
277C2×SD16C2×D414yes—
278C2×M16—24no—
279C2×M16—44no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2411yescharacteristic
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—82no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298#126—48no—
299#126—48no—
300#125—24no—
301#125—42no—
302#125—48no—
303#124—24no—
304#124—42no—
305#124—42no—
306#124—44no—
307#124—82no—
308#123—48no—
309C22⋊Q8—24no—
310C22⋊Q8—28no—
311C22⋊Q8—44no—
312C22⋊Q8—42no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315#121—24no—
316#121—42no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8C2×D414yes—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—42no—
332C4×D4C2×D414yes—
333#117—24no—
334#117—42no—
335#117—44no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2412yes—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D414yes—
377C2×C22⋊C4C2×D414yes—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—82no—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D412yes—
389C2×C42C2412yes—
390C2×C42C2412yes—
391C2×C42C2411yescharacteristic
392#107—24no—
393#107—24no—
394#107—24no—
395#107—24no—
396#107—24no—
397#107—44no—
398#106—24no—
399#106C2×D414yes—
400C4⋊C8—24no—
401C4⋊C8—22no—
402C4⋊C8—22no—
403C4⋊C8—44no—
404#103—24no—
405#103—24no—
406#103—24no—
407#103—24no—
408#103—24no—
409#103—44no—
410#103—44no—
411#103C2×D414yes—
412#102—24no—
413#102—24no—
414#102—42no—
415#102C2×D414yes—
416C22⋊C8—24no—
417C22⋊C8—44no—
418C4×C8—24no—
419C4×C8—22no—
420C4×C8C2×D414yes—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—21no—
425#95—24no—
426#95—24no—
427#95—22no—
428#95—21no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95C2×D414yes—
433#95C2×D414yes—
434C24—21no—
435C24C22×D412yes—
436#41—816no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8C22×D412yes—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—82no—
466C2×D4C22×D412yes—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C2511yescharacteristic
497Q16—48no—
498SD16—44no—
499M16—48no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8C22×D412yes—
510C2×C8C22×D412yes—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4C22×D412yes—
544C4⋊C4C22×D412yes—
545C4⋊C42−1+414yes—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—28no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—48no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—84no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—44no—
567C42—44no—
568C42—44no—
569C42—42no—
570C42—42no—
571C42—82no—
572C42C22×D412yes—
573C42H(5,𝔽2)12yes—
574C23—22no—
575C23—22no—
576C23—24no—
577C23—22no—
578C23—22no—
579C23—24no—
580C23—42no—
581C23—41no—
582C23—41no—
583C23—42no—
584C23C23×D411yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—44no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—44no—
592Q8—84no—
593Q8—84no—
594Q8—84no—
595D4—22no—
596D4—22no—
597D4—44no—
598D4—42no—
599D4—88no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—88no—
642C2×C4—88no—
643C2×C4D4212yes—
644C2×C4D4212yes—
645C2×C4#57512yes—
646C2×C4#57612yes—
647C2×C4#57814yes—
648C2×C4C23×D411yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+412yes—
651C2×C4#58512yes—
652C8—22no—
653C8—22no—
654C8—42no—
655C8—42no—
656C8—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—44no—
664C22—44no—
665C22—82no—
666C22—82no—
667C22C2×D4211yescharacteristic
668C22#348912yes—
669C22C2×#57511yescharacteristic
670C22C2×#57611yescharacteristic
671C22C2×#57812yes—
672C4—22no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4#326814yes—
688C4#344911yescharacteristic
689C4#349911yescharacteristic
690C4#355412yes—
691C2—82no—
692C2—82no—
693C2#3768011yescharacteristic
694C2C2×#326811yescharacteristic
695C2#6109111yescharacteristic
696C2#6214612yes—
697C2#6220112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3364C2semidirect
3. #39744C2semidirect
5. #33195C2semidirect
6. #30640C2semidirect
7. #30606C2semidirect
8. #30279C2semidirect
9. #29831C2semidirect
10. #28675C2semidirect
11. #28606C2semidirect
12. #26559C2semidirect
13. #26362C2semidirect
14. #25840C2semidirect
16. #25650C2semidirect
17. #25570C2semidirect
18. #22210C2semidirect
19. #21114C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33642
C2×#33532
#397442
#364062
#331952
#306402
#306062
#302792
#298312
#286752
#286062
#265592
#263622
#258402
#257032
#256502
#255702
#222102
#211142