Finite Group Databasebeta

← #6686267

Subgroups — #6686267

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3363C211yesmaximal, characteristic, complemented
2C2×#3342C211yesmaximal, characteristic
3#39172C211yesmaximal, characteristic
4#36508C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#30634C214yesmaximal, complemented
7#30604C214yesmaximal, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29831C211yesmaximal, characteristic, complemented
10#28675C211yesmaximal, characteristic, complemented
11#28598C211yesmaximal, characteristic
12#26561C214yesmaximal, complemented
13#26339C214yesmaximal, complemented
14#25837C211yesmaximal, characteristic, complemented
15#25701C211yesmaximal, characteristic
16#25647C211yesmaximal, characteristic, complemented
17#25570C211yesmaximal, characteristic
18#22208C211yesmaximal, characteristic
19#21119C211yesmaximal, characteristic, complemented
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#556C2211yescharacteristic
24#3363—28no—
25#3342—28no—
26C2×#500C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#487C2212yes—
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#485C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2212yes—
35C2×#475C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#467C2212yes—
38C2×#462C2211yescharacteristic
39C2×#462C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#427C2212yes—
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2861—21no—
48#2716—24no—
49#2640—21no—
50#2568—21no—
51#2565—21no—
52#2563C2211yescharacteristic
53#2546C2211yescharacteristic
54#2546C2211yescharacteristic
55#2545—21no—
56#2545—21no—
57#2542C2211yescharacteristic
58#2519—24no—
59#2439C2214yes—
60#2429C2214yes—
61#2421C2214yes—
62#2362C2214yes—
63#2327—21no—
64C4×C4⋊Q8—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2097C2214yes—
69#2097C2214yes—
70#2086C2214yes—
71#2066C2214yes—
72#2058C2214yes—
73#2056C2214yes—
74#2043C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2001C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1949C2214yes—
85#1942C2214yes—
86#1928C2212yes—
87#1923C2214yes—
88#1921C2212yes—
89#1892C2214yes—
90#1878C2212yes—
91#1873C2214yes—
92#1865C2214yes—
93#1865C2214yes—
94#1863C2212yes—
95#1854C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1805C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2312yes—
104C22×Q16C2311yescharacteristic
105C23×C8C2312yes—
106Q82—24no—
107Q82—42no—
108#557—24no—
109#557—42no—
110#556—24no—
111#556—24no—
112#556—42no—
113#556—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2312yes—
124C2×#126—22no—
125C2×#126—22no—
126C2×#125—21no—
127C2×#125—22no—
128C2×#124—21no—
129C2×#124—21no—
130C2×#124D411yescharacteristic
131C2×C22⋊Q8C2312yes—
132C2×C22⋊Q8C2312yes—
133C2×C22⋊Q8C2314yes—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×D4C2311yescharacteristic
145C2×#117C2312yes—
146C2×#117C2312yes—
147#500—24no—
148#498—42no—
149#487—28no—
150#487—24no—
151#487—42no—
152#485—28no—
153#485—28no—
154#477—24no—
155#477—28no—
156#475—24no—
157#475—24no—
158#467—28no—
159#462—28no—
160#462—24no—
161C8×Q8—24no—
162C4×Q16—28no—
163C4×Q16D418yes—
164C8×D4—24no—
165#427—28no—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2314yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2314yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2312yes—
188C2×#103C2311yescharacteristic
189C2×#103C2311yescharacteristic
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2314yes—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2312yes—
194C2×C4×C8C2311yescharacteristic
195#400—24no—
196#399—22no—
197#398—22no—
198#398—22no—
199#398—24no—
200#397—24no—
201#396—24no—
202#396—24no—
203#393—22no—
204#393—24no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—22no—
212#391—24no—
213#391D411yescharacteristic
214#391C2312yes—
215#391C2314yes—
216#390—21no—
217#390—21no—
218#390D411yescharacteristic
219#389—24no—
220#389C2314yes—
221#389C2312yes—
222#389C2314yes—
223#388—24no—
224#388—24no—
225#388—24no—
226#388—24no—
227#388D414yes—
228#387—22no—
229#387—24no—
230#387—24no—
231#387—24no—
232#384—21no—
233#382—21no—
234#382—22no—
235#382D411yescharacteristic
236#382C2314yes—
237#380C2314yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—24no—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4C2314yes—
246#376—21no—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—22no—
261C22×Q8—21no—
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2412yes—
264C22×Q8C2411yescharacteristic
265C22×Q8C2411yescharacteristic
266C22×D4C2×D411yescharacteristic
267C23×C4C2412yes—
268C2×Q16—24no—
269C2×Q16—42no—
270C2×Q16—44no—
271C2×Q16C2×D414yes—
272C22×C8—24no—
273C22×C8—44no—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—42no—
290C4⋊Q8—44no—
291C4⋊Q8—42no—
292C4⋊Q8—48no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295#126—48no—
296#126—48no—
297#125—44no—
298#125—48no—
299#124—24no—
300#124—44no—
301#124—44no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—28no—
305C22⋊Q8—44no—
306C22⋊Q8—42no—
307C22⋊Q8—42no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—42no—
316C4×Q8—42no—
317C4×Q8—42no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8C2×D414yes—
323C4×Q8C2×D414yes—
324C4×D4—22no—
325C4×D4—22no—
326#117—24no—
327#117—24no—
328#117—42no—
329#117—42no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D414yes—
367C2×C22⋊C4C2×D414yes—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D412yes—
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384#107—24no—
385#107—24no—
386#107—24no—
387#107—24no—
388#107C2×D414yes—
389#106—24no—
390#106—24no—
391#106—44no—
392C4⋊C8—24no—
393C4⋊C8—22no—
394C4⋊C8—22no—
395C4⋊C8—44no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—42no—
404#103—44no—
405#103—44no—
406#103C2×D414yes—
407#103C2×D414yes—
408C22⋊C8—24no—
409C22⋊C8—44no—
410C4×C8—24no—
411C4×C8—22no—
412C4×C8C2×D414yes—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—21no—
417#95—24no—
418#95—22no—
419#95—21no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95C2×D414yes—
427#95C2×D414yes—
428C24C22×D412yes—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—82no—
450C2×Q8—84no—
451C2×Q8#13514yes—
452C2×Q8C22×D412yes—
453C2×Q8C22×D412yes—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C2511yescharacteristic
479Q16—44no—
480Q16—48no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—48no—
491C2×C8C22×D412yes—
492C2×C8C22×D412yes—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—82no—
526C4⋊C4C22×D412yes—
527C4⋊C4C22×D412yes—
528C4⋊C42−1+414yes—
529C22⋊C4—24no—
530C22⋊C4—28no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—84no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—44no—
548C42—44no—
549C42—44no—
550C42—44no—
551C42—44no—
552C42—44no—
553C42—42no—
554C42—42no—
555C42C22×D412yes—
556C42H(5,𝔽2)12yes—
557C23—24no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—22no—
562C23C23×D411yescharacteristic
563Q8—28no—
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—22no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—44no—
573Q8—84no—
574Q8—84no—
575Q8—162no—
576Q8—164no—
577D4—44no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4D4212yes—
619C2×C4D4212yes—
620C2×C4C2×#13514yes—
621C2×C4C2×#13512yes—
622C2×C4#57812yes—
623C2×C4C23×D411yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+412yes—
626C2×C4#58512yes—
627C8—22no—
628C8—22no—
629C8—42no—
630C8—42no—
631C8—42no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—42no—
636C22—44no—
637C22C2×D4211yescharacteristic
638C22#348912yes—
639C22C22×#13511yescharacteristic
640C22C2×#57811yescharacteristic
641C22#358612yes—
642C4—24no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#328514yes—
659C4#344911yescharacteristic
660C4#349911yescharacteristic
661C4#355412yes—
662C2—82no—
663C2#3768011yescharacteristic
664C2C2×#328511yescharacteristic
665C2#6108311yescharacteristic
666C2#6210512yes—
667C2#6219212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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×#3363C2semidirect
5. #33202C2semidirect
6. #30634C2semidirect
7. #30604C2semidirect
8. #30281C2semidirect
9. #29831C2semidirect
10. #28675C2semidirect
12. #26561C2semidirect
13. #26339C2semidirect
14. #25837C2semidirect
16. #25647C2semidirect
19. #21119C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33632
C2×#33422
#391722
#365082
#332022
#306342
#306042
#302812
#298312
#286752
#285982
#265612
#263392
#258372
#257012
#256472
#255702
#222082
#211192