Finite Group Databasebeta

← #6686266

Subgroups — #6686266

Order 512.

1772 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3364C211yesmaximal, characteristic, complemented
2C2×#3340C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic, complemented
4#36185C211yesmaximal, characteristic
5#33195C211yesmaximal, characteristic, complemented
6#30638C214yesmaximal, complemented
7#30594C214yesmaximal, complemented
8#30280C211yesmaximal, characteristic, complemented
9#29830C211yesmaximal, characteristic, complemented
10#28677C211yesmaximal, characteristic, complemented
11#28600C211yesmaximal, characteristic, complemented
12#26560C214yesmaximal, complemented
13#26351C214yesmaximal, complemented
14#25836C211yesmaximal, characteristic, complemented
15#25700C211yesmaximal, characteristic
16#25650C211yesmaximal, characteristic, complemented
17#25570C211yesmaximal, characteristic, complemented
18#22208C211yesmaximal, characteristic, complemented
19#21117C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#555C2211yescharacteristic
24#3364—28no—
25#3340—28no—
26C2×#498C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#488C2212yes—
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#482C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#474C2211yescharacteristic
36C2×#474C2211yescharacteristic
37C2×#465C2212yes—
38C2×#464C2212yes—
39C2×#462C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#427C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2850—21no—
48#2721—24no—
49#2658—21no—
50#2567C2211yescharacteristic
51#2547C2211yescharacteristic
52#2546—21no—
53#2546C2211yescharacteristic
54#2545—21no—
55#2542C2211yescharacteristic
56#2521—24no—
57#2500—21no—
58#2499—21no—
59#2441C2214yes—
60#2427C2214yes—
61#2423C2214yes—
62#2362C2214yes—
63#2327—21no—
64C4×#124C2211yescharacteristic
65#2295C2211yescharacteristic
66#2285—21no—
67C42×D4C2211yescharacteristic
68#2096C2214yes—
69#2096C2214yes—
70#2088C2214yes—
71#2068C2214yes—
72#2059C2214yes—
73#2056C2214yes—
74#2042C2214yes—
75#2035C2214yes—
76#2034C2214yes—
77#2001C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1947C2214yes—
85#1941C2214yes—
86#1927C2212yes—
87#1923C2214yes—
88#1920C2212yes—
89#1892C2214yes—
90#1877C2212yes—
91#1873C2214yes—
92#1865C2214yes—
93#1864C2212yes—
94#1863C2214yes—
95#1854C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1802C2214yes—
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2312yes—
104C22×SD16C2311yescharacteristic
105C23×C8C2312yes—
106#557—24no—
107#557—42no—
108#556—24no—
109#556—24no—
110#556—42no—
111#556—42no—
112#555—24no—
113#555—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×#126—22no—
123C2×#126—22no—
124C2×#125—22no—
125C2×#125C2312yes—
126C2×#124—21no—
127C2×#124—21no—
128C2×#124—22no—
129C2×#124D411yescharacteristic
130C2×#124C2312yes—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8C2312yes—
133C2×C22⋊Q8C2314yes—
134C2×#121—21no—
135C2×#121C2312yes—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×D4C2312yes—
143C2×C4×D4C2311yescharacteristic
144C2×C4×D4C2311yescharacteristic
145C2×#117—21no—
146C2×#117—21no—
147C2×#117C2312yes—
148C2×#117C2312yes—
149C22×C42—21no—
150#498—24no—
151#498—42no—
152#488—28no—
153#487—24no—
154#487—42no—
155#485—28no—
156#482—28no—
157#479—28no—
158#477—24no—
159#474—24no—
160#474—24no—
161#465—28no—
162#464—28no—
163#462—24no—
164C8×Q8—24no—
165C4×Q16—28no—
166C4×SD16D418yes—
167C8×D4—24no—
168#427—28no—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2314yes—
176C2×#106C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2311yescharacteristic
180C2×C4⋊C8C2314yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2314yes—
187C2×#103C2314yes—
188C2×#103C2311yescharacteristic
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2311yescharacteristic
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2314yes—
195C2×C4×C8C2311yescharacteristic
196C2×C4×C8C2312yes—
197C2×C4×C8C2311yescharacteristic
198#399—24no—
199#399—24no—
200#399—22no—
201#398—22no—
202#398—24no—
203#397—22no—
204#397—24no—
205#396—24no—
206#395—22no—
207#394—24no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—22no—
212#391D411yescharacteristic
213#391C2312yes—
214#391C2314yes—
215#390—21no—
216#390D411yescharacteristic
217#389—24no—
218#389C2314yes—
219#389C2312yes—
220#389C2314yes—
221#388—24no—
222#388—24no—
223#388—24no—
224#388D414yes—
225#387—22no—
226#387—24no—
227#387—24no—
228#387—24no—
229#386—24no—
230#384—21no—
231#384—21no—
232#384—22no—
233#384C2314yes—
234#382—21no—
235#382—21no—
236#382D411yescharacteristic
237#380—24no—
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—24no—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2311yescharacteristic
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—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2412yes—
263C22×Q8C2411yescharacteristic
264C22×D4—22no—
265C22×D4C2×D411yescharacteristic
266C22×D4C2411yescharacteristic
267C23×C4—21no—
268C23×C4—22no—
269C23×C4C2412yes—
270C2×Q16—24no—
271C2×Q16—44no—
272C2×SD16—42no—
273C2×SD16C2×D414yes—
274C22×C8—24no—
275C22×C8—44no—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
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—44no—
290C4⋊Q8—44no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293#126—48no—
294#126—48no—
295#125—24no—
296#125—42no—
297#125—48no—
298#124—24no—
299#124—24no—
300#124—42no—
301#124—44no—
302#124—48no—
303#124—82no—
304C22⋊Q8—24no—
305C22⋊Q8—28no—
306C22⋊Q8—44no—
307C22⋊Q8—42no—
308C22⋊Q8—44no—
309#121—24no—
310#121—42no—
311#121—44no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8C2×D414yes—
322C4×D4—24no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—42no—
327C4×D4C2×D414yes—
328#117—24no—
329#117—24no—
330#117—42no—
331#117—42no—
332#117—44no—
333#117—44no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
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—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D414yes—
373C2×C22⋊C4C2×D414yes—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—44no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D412yes—
385C2×C42C2×D411yescharacteristic
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389#107—24no—
390#107—24no—
391#107—24no—
392#106—24no—
393#106—24no—
394#106—24no—
395#106—44no—
396#106C2×D414yes—
397C4⋊C8—24no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400C4⋊C8—44no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—44no—
407#103—44no—
408#103C2×D414yes—
409#102—24no—
410#102—24no—
411#102—42no—
412#102C2×D414yes—
413C22⋊C8—24no—
414C22⋊C8—44no—
415C4×C8—24no—
416C4×C8—22no—
417C4×C8C2×D414yes—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—21no—
422#95—24no—
423#95—22no—
424#95—21no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95C2×D414yes—
430#95C2×D414yes—
431C24—21no—
432C24C22×D412yes—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—82no—
451C2×Q8C22×D412yes—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—44no—
459C2×D4—84no—
460C2×D4C22×D412yes—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—82no—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488Q16—48no—
489SD16—44no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—48no—
500C2×C8C22×D412yes—
501C2×C8C22×D412yes—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—48no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4C22×D412yes—
534C4⋊C4C22×D412yes—
535C4⋊C42−1+414yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—28no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—84no—
549C42—24no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—44no—
560C42—44no—
561C42—42no—
562C42—42no—
563C42#13514yes—
564C42C22×D412yes—
565C42H(5,𝔽2)12yes—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—22no—
571C23—22no—
572C23—42no—
573C23—42no—
574C23—41no—
575C23—41no—
576C23C23×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—44no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—44no—
584Q8—84no—
585Q8—84no—
586Q8—162no—
587D4—22no—
588D4—22no—
589D4—44no—
590D4—42no—
591D4—164no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—88no—
633C2×C4D4212yes—
634C2×C4D4212yes—
635C2×C4C2×#13514yes—
636C2×C4C2×#13512yes—
637C2×C4#57712yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+412yes—
641C2×C4#58512yes—
642C8—22no—
643C8—22no—
644C8—42no—
645C8—42no—
646C8—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—82no—
656C22—82no—
657C22C2×D4211yescharacteristic
658C22#348912yes—
659C22C22×#13511yescharacteristic
660C22C2×#57711yescharacteristic
661C22#358612yes—
662C4—24no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#328414yes—
678C4#344911yescharacteristic
679C4#349911yescharacteristic
680C4#355412yes—
681C2—82no—
682C2—82no—
683C2#3768011yescharacteristic
684C2C2×#328411yescharacteristic
685C2#6108311yescharacteristic
686C2#6209712yes—
687C2#6218412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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. #30638C2semidirect
7. #30594C2semidirect
8. #30280C2semidirect
9. #29830C2semidirect
10. #28677C2semidirect
11. #28600C2semidirect
12. #26560C2semidirect
13. #26351C2semidirect
14. #25836C2semidirect
16. #25650C2semidirect
17. #25570C2semidirect
18. #22208C2semidirect
19. #21117C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33642
C2×#33402
#397442
#361852
#331952
#306382
#305942
#302802
#298302
#286772
#286002
#265602
#263512
#258362
#257002
#256502
#255702
#222082
#211172