Finite Group Databasebeta

← #6686305

Subgroups — #6686305

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3375C211yesmaximal, characteristic, complemented
2C2×#3338C211yesmaximal, characteristic, complemented
3#39617C211yesmaximal, characteristic, complemented
4#36496C211yesmaximal, characteristic
5#33198C211yesmaximal, characteristic, complemented
6#30659C214yesmaximal, complemented
7#30602C214yesmaximal, complemented
8#30276C211yesmaximal, characteristic, complemented
9#29827C211yesmaximal, characteristic, complemented
10#28682C211yesmaximal, characteristic, complemented
11#28599C211yesmaximal, characteristic
12#26571C214yesmaximal, complemented
13#26336C214yesmaximal, complemented
14#25841C211yesmaximal, characteristic, complemented
15#25706C211yesmaximal, characteristic, complemented
16#25648C211yesmaximal, characteristic, complemented
17#25569C211yesmaximal, characteristic
18#22212C211yesmaximal, characteristic, complemented
19#21116C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×Q82C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3375—28no—
25#3338—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#489C2212yes—
29C2×#486C2211yescharacteristic
30C2×#486C2211yescharacteristic
31C2×#484C2212yes—
32C2×#484C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2211yescharacteristic
35C2×#477C2211yescharacteristic
36C2×#474C2212yes—
37C2×#467C2212yes—
38C2×#464C2212yes—
39C2×#460C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×#440C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C22×#107C2212yes—
45#2868C2211yescharacteristic
46#2861—21no—
47#2858C2211yescharacteristic
48#2677—24no—
49#2660—21no—
50#2567C2211yescharacteristic
51#2567C2211yescharacteristic
52#2566—21no—
53#2565—21no—
54#2563C2211yescharacteristic
55#2544—21no—
56#2542—21no—
57#2541C2211yescharacteristic
58#2461—24no—
59#2440C2214yes—
60#2426C2214yes—
61#2423C2214yes—
62#2361C2214yes—
63#2326—21no—
64C4×#127—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2093C2214yes—
69#2093C2214yes—
70#2086C2214yes—
71#2067C2214yes—
72#2057C2214yes—
73#2055C2214yes—
74#2042C2214yes—
75#2036C2214yes—
76#2034C2214yes—
77#2000C2214yes—
78#1987C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1982C2211yescharacteristic
83#1954C2211yescharacteristic
84#1947C2214yes—
85#1944C2214yes—
86#1927C2212yes—
87#1924C2214yes—
88#1918C2212yes—
89#1893C2214yes—
90#1878C2212yes—
91#1874C2214yes—
92#1865C2212yes—
93#1863C2214yes—
94#1862C2214yes—
95#1852C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1803C2214yes—
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2311yescharacteristic
104C22×SD16C2312yes—
105C23×C8C2312yes—
106#559—24no—
107#559—42no—
108Q82—24no—
109Q82—42no—
110#554—24no—
111#554—24no—
112#554—42no—
113#554—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×#127—21no—
122C2×#127C2311yescharacteristic
123C2×#125—21no—
124C2×#125—21no—
125C2×#124—21no—
126C2×#124—22no—
127C2×#124D411yescharacteristic
128C2×#124C2312yes—
129C2×#123—22no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8—24no—
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2312yes—
135C2×#121C2314yes—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×D4—21no—
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4—24no—
146C2×C4×D4C2312yes—
147C2×C4×D4C2311yescharacteristic
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2312yes—
150#500—24no—
151#500—42no—
152#489—28no—
153#486—24no—
154#486—42no—
155#484—28no—
156#484—28no—
157#477—24no—
158#477—24no—
159#477—24no—
160#474—28no—
161#467—28no—
162#464—28no—
163#460—24no—
164#446—24no—
165#440—28no—
166C4×Q16D418yescomplemented
167C8×D4—24no—
168C2×#107—28no—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#107C2314yes—
175C2×#107C2311yescharacteristic
176C2×#106C2312yes—
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2314yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2311yescharacteristic
187C2×#103C2311yescharacteristic
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2314yes—
192C2×#102C2314yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2314yes—
195C2×#97C2312yes—
196C2×C4×C8C2311yescharacteristic
197C2×C4×C8C2311yescharacteristic
198#399—24no—
199#398—24no—
200#398—24no—
201#396—24no—
202#395—22no—
203#395—22no—
204#395—24no—
205#394—24no—
206#393—22no—
207#392—22no—
208#391—21no—
209#391—21no—
210#391D411yescharacteristic
211#391C2312yes—
212#390—21no—
213#390—21no—
214#390—24no—
215#390—21no—
216#390—21no—
217#390D411yescharacteristic
218#390C2314yes—
219#389C2314yes—
220#389C2314yes—
221#389C2312yes—
222#388—24no—
223#388—24no—
224#388—24no—
225#388D414yes—
226#387—22no—
227#387—24no—
228#386—24no—
229#385—24no—
230#385—24no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384—21no—
235#384C2314yes—
236#382—21no—
237#382D411yescharacteristic
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2312yes—
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—22no—
260C22×Q8—21no—
261C22×Q8—21no—
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×D4—24no—
266C22×D4—21no—
267C22×D4—21no—
268C22×D4C2×D411yescharacteristic
269C22×D4C2412yes—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4C2412yes—
274C2×Q16—42no—
275C2×Q16C2×D414yes—
276C2×SD16—24no—
277C2×SD16—44no—
278C22×C8—24no—
279C22×C8—44no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2411yescharacteristic
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—42no—
293C4⋊Q8—44no—
294C4⋊Q8—48no—
295C4⋊Q8—84no—
296#127—22no—
297#127—22no—
298#127—44no—
299#125—44no—
300#125—44no—
301#124—24no—
302#124—24no—
303#124—42no—
304#124—44no—
305#124—84no—
306#123—48no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—48no—
310C22⋊Q8—48no—
311C22⋊Q8—42no—
312C22⋊Q8—42no—
313C22⋊Q8—88no—
314#121—28no—
315#121—44no—
316C4×Q8—22no—
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×Q8C2×D414yes—
326C4×Q8C2×D414yes—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—42no—
331C4×D4—44no—
332C4×D4—44no—
333C4×D4—44no—
334C4×D4—88no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—42no—
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—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4C2×D414yes—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D414yes—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42C2×D412yes—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2412yes—
389C2×C42C2412yes—
390C2×C42C2411yescharacteristic
391#107—24no—
392#107—24no—
393#107—24no—
394#107—24no—
395#107—24no—
396#107—44no—
397#107C2×D414yes—
398#106—24no—
399C4⋊C8—24no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8—44no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—24no—
407#103—42no—
408#103C2×D414yes—
409#103C2×D414yes—
410#102—24no—
411#102—24no—
412#102—24no—
413#102—44no—
414#102—44no—
415C22⋊C8—24no—
416C22⋊C8—44no—
417#97—24no—
418C4×C8—22no—
419C4×C8C2×D414yes—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—21no—
424#95—22no—
425#95—21no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95C2×D414yes—
430#95C2×D414yes—
431C24—22no—
432C24C22×D412yes—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—84no—
447C2×Q8—82no—
448C2×Q8—82no—
449C2×Q8—82no—
450C2×Q8C22×D412yes—
451C2×Q8C22×D412yes—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—48no—
459C2×D4—48no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—82no—
465C2×D4—84no—
466C2×D4#13714yes—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
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—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2511yescharacteristic
494Q16—44no—
495SD16—48no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—48no—
506C2×C8C22×D412yes—
507C2×C8C22×D412yes—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4C22×D412yes—
540C4⋊C4C22×D412yes—
541C4⋊C42−1+414yes—
542C22⋊C4—28no—
543C22⋊C4—24no—
544C22⋊C4—28no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C42—24no—
555C42—24no—
556C42—22no—
557C42—24no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—44no—
564C42—44no—
565C42—42no—
566C42—44no—
567C42—42no—
568C42C22×D412yes—
569C42H(5,𝔽2)12yes—
570C23—22no—
571C23—24no—
572C23—24no—
573C23—22no—
574C23—42no—
575C23—44no—
576C23—42no—
577C23—44no—
578C23—42no—
579C23C2×#13714yes—
580C23C23×D411yescharacteristic
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8—22no—
585Q8—44no—
586Q8—42no—
587Q8—42no—
588Q8—162no—
589Q8—162no—
590Q8—162no—
591Q8—164no—
592D4—28no—
593D4—42no—
594D4—42no—
595D4—44no—
596D4—816no—
597D4—84no—
598D4—162no—
599C2×C4—22no—
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—24no—
616C2×C4—22no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
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—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—164no—
636C2×C4—164no—
637C2×C4—164no—
638C2×C4D4212yes—
639C2×C4D4212yes—
640C2×C4C2×#13712yes—
641C2×C4#57614yes—
642C2×C4#57812yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+412yes—
646C2×C4#58512yes—
647C8—22no—
648C8—22no—
649C8—42no—
650C8—42no—
651C8—42no—
652C22—24no—
653C22—28no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—84no—
661C22—84no—
662C22C2×D4211yescharacteristic
663C22#348912yes—
664C22C22×#13711yescharacteristic
665C22C2×#57612yes—
666C22C2×#57811yescharacteristic
667C4—22no—
668C4—24no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#327014yes—
683C4#344911yescharacteristic
684C4#349911yescharacteristic
685C4#355412yes—
686C2—44no—
687C2—162no—
688C2#3768011yescharacteristic
689C2C2×#327011yescharacteristic
690C2#6108911yescharacteristic
691C2#6212712yes—
692C2#6220612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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×#3375C2semidirect
2. C2×#3338C2semidirect
3. #39617C2semidirect
5. #33198C2semidirect
6. #30659C2semidirect
7. #30602C2semidirect
8. #30276C2semidirect
9. #29827C2semidirect
10. #28682C2semidirect
12. #26571C2semidirect
13. #26336C2semidirect
14. #25841C2semidirect
15. #25706C2semidirect
16. #25648C2semidirect
18. #22212C2semidirect
19. #21116C2semidirect
166. C4×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33752
C2×#33382
#396172
#364962
#331982
#306592
#306022
#302762
#298272
#286822
#285992
#265712
#263362
#258412
#257062
#256482
#255692
#222122
#211162