Finite Group Databasebeta

← #6686327

Subgroups — #6686327

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3363C211yesmaximal, characteristic, complemented
2C2×#3356C211yesmaximal, characteristic
3#39172C211yesmaximal, characteristic
4#36562C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#30636C214yesmaximal, complemented
7#30616C214yesmaximal, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29830C211yesmaximal, characteristic, complemented
10#28677C211yesmaximal, characteristic, complemented
11#28608C211yesmaximal, characteristic
12#26558C214yesmaximal, complemented
13#26358C214yesmaximal, complemented
14#25841C211yesmaximal, characteristic, complemented
15#25706C211yesmaximal, characteristic
16#25647C211yesmaximal, characteristic, complemented
17#25570C211yesmaximal, characteristic
18#22210C211yesmaximal, characteristic
19#21116C211yesmaximal, characteristic, complemented
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3363—28no—
25#3356—28no—
26C2×#500C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#487C2212yes—
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#483C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2211yescharacteristic
35C2×#477C2212yes—
36C2×#475C2211yescharacteristic
37C2×#470C2212yes—
38C2×#462C2211yescharacteristic
39C2×#462C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#428C2212yes—
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2859—21no—
48#2716—24no—
49#2640—21no—
50#2568—21no—
51#2567C2211yescharacteristic
52#2566—21no—
53#2565—21no—
54#2563C2211yescharacteristic
55#2556—24no—
56#2546C2211yescharacteristic
57#2545—21no—
58#2542C2211yescharacteristic
59#2439C2214yes—
60#2429C2214yes—
61#2421C2214yes—
62#2360C2214yes—
63#2327—21no—
64C4×#124—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2097C2214yes—
69#2097C2214yes—
70#2088C2214yes—
71#2068C2214yes—
72#2057C2214yes—
73#2056C2214yes—
74#2043C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2001C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1949C2214yes—
85#1941C2214yes—
86#1927C2212yes—
87#1923C2214yes—
88#1918C2212yes—
89#1894C2214yes—
90#1878C2212yes—
91#1873C2214yes—
92#1865C2214yes—
93#1865C2212yes—
94#1863C2214yes—
95#1852C2214yes—
96#1851C2214yes—
97#1842C2211yescharacteristic
98#1805C2214yes—
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2312yes—
104C22×Q16C2311yescharacteristic
105C22×M16C2312yes—
106Q82—24no—
107Q82—42no—
108#557—24no—
109#557—42no—
110#556—24no—
111#556—42no—
112#554—24no—
113#554—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×#126—22no—
124C2×#125—21no—
125C2×#125—21no—
126C2×#125—22no—
127C2×#124—21no—
128C2×#124D411yescharacteristic
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8C2312yes—
131C2×C22⋊Q8C2312yes—
132C2×C22⋊Q8C2314yes—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×D4—21no—
143C2×C4×D4C2311yescharacteristic
144C2×#117C2312yes—
145C22×C4⋊C4C2312yes—
146#500—42no—
147#498—24no—
148#487—28no—
149#487—24no—
150#487—42no—
151#485—28no—
152#483—28no—
153#477—24no—
154#477—24no—
155#477—28no—
156#475—24no—
157#470—28no—
158#462—28no—
159#462—24no—
160C8×Q8—24no—
161C4×Q16—28no—
162C4×Q16D418yes—
163#436—24no—
164#428—28no—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2314yes—
168C2×#107C2311yescharacteristic
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2314yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2312yes—
187C2×#103C2311yescharacteristic
188C2×#103C2311yescharacteristic
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2314yes—
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2312yes—
193C2×C4×C8C2311yescharacteristic
194#400—24no—
195#399—22no—
196#398—22no—
197#398—24no—
198#397—24no—
199#396—24no—
200#396—24no—
201#395—22no—
202#393—22no—
203#393—24no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391—22no—
210#391—24no—
211#391D411yescharacteristic
212#391C2312yes—
213#391C2314yes—
214#390—21no—
215#390—21no—
216#390D411yescharacteristic
217#389—24no—
218#389—24no—
219#389C2314yes—
220#389C2312yes—
221#389C2314yes—
222#388—24no—
223#388—24no—
224#388D414yes—
225#387—22no—
226#387—24no—
227#387—24no—
228#387—24no—
229#385—24no—
230#384—21no—
231#384—21no—
232#382—21no—
233#382—21no—
234#382—21no—
235#382D411yescharacteristic
236#382C2314yes—
237#380C2314yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4—24no—
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×C4—22no—
268C23×C4C2412yes—
269C2×Q16—24no—
270C2×Q16—42no—
271C2×Q16—44no—
272C2×Q16C2×D414yes—
273C2×M16—24no—
274C2×M16—44no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—42no—
291C4⋊Q8—44no—
292C4⋊Q8—42no—
293C4⋊Q8—48no—
294C4⋊Q8—44no—
295#126—48no—
296#125—44no—
297#125—44no—
298#125—48no—
299#124—24no—
300#124—44no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—28no—
304C22⋊Q8—48no—
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×Q8C2×D414yes—
322C4×Q8C2×D414yes—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—44no—
326#117—24no—
327#117—42no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
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—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
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—21no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D412yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383#107—24no—
384#107—24no—
385#107—44no—
386#107C2×D414yes—
387#106—24no—
388#106—24no—
389#106—24no—
390#106—24no—
391C4⋊C8—24no—
392C4⋊C8—22no—
393C4⋊C8—22no—
394C4⋊C8—44no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—42no—
403#103—44no—
404#103—44no—
405#103C2×D414yes—
406#103C2×D414yes—
407C22⋊C8—24no—
408C22⋊C8—44no—
409C4×C8—24no—
410C4×C8—22no—
411C4×C8C2×D414yes—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—22no—
417#95—21no—
418#95—22no—
419#95—21no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95C2×D414yes—
426#95C2×D414yes—
427C24C22×D412yes—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—44no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—84no—
449C2×Q8#13714yes—
450C2×Q8C22×D412yes—
451C2×Q8C22×D412yes—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—84no—
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—
481M16—48no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—24no—
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—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—48no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4C22×D412yes—
528C4⋊C4C22×D412yes—
529C4⋊C42−1+414yes—
530C22⋊C4—24no—
531C22⋊C4—28no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—48no—
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—42no—
552C42—42no—
553C42—82no—
554C42C22×D412yes—
555C42H(5,𝔽2)12yes—
556C23—24no—
557C23—22no—
558C23—24no—
559C23—22no—
560C23—22no—
561C23C23×D411yescharacteristic
562Q8—28no—
563Q8—22no—
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—44no—
572Q8—84no—
573Q8—162no—
574Q8—162no—
575Q8—164no—
576D4—44no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
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—164no—
617C2×C4D4212yes—
618C2×C4D4212yes—
619C2×C4C2×#13714yes—
620C2×C4C2×#13712yes—
621C2×C4#57812yes—
622C2×C4C23×D411yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×2−1+412yes—
625C2×C4#58512yes—
626C8—22no—
627C8—22no—
628C8—42no—
629C8—42no—
630C8—42no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22C2×D4211yescharacteristic
637C22#348912yes—
638C22C22×#13711yescharacteristic
639C22C2×#57811yescharacteristic
640C22#358412yes—
641C4—24no—
642C4—22no—
643C4—42no—
644C4—44no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#327014yes—
658C4#344911yescharacteristic
659C4#349911yescharacteristic
660C4#355412yes—
661C2—82no—
662C2#3768011yescharacteristic
663C2C2×#327011yescharacteristic
664C2#6108811yescharacteristic
665C2#6215312yes—
666C2#6221712yes—
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. #30636C2semidirect
7. #30616C2semidirect
8. #30281C2semidirect
9. #29830C2semidirect
10. #28677C2semidirect
12. #26558C2semidirect
13. #26358C2semidirect
14. #25841C2semidirect
16. #25647C2semidirect
19. #21116C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33632
C2×#33562
#391722
#365622
#332022
#306362
#306162
#302812
#298302
#286772
#286082
#265582
#263582
#258412
#257062
#256472
#255702
#222102
#211162