Finite Group Databasebeta

← #7501929

Subgroups — #7501929

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic, complemented
2C2×#3096C211yesmaximal, characteristic
3#35620C211yesmaximal, characteristic, complemented
4#35264C211yesmaximal, characteristic
5#30804C214yesmaximal, complemented
6#30791C214yesmaximal, complemented
7#30781C214yesmaximal, complemented
8#30778C214yesmaximal, complemented
9C2×#2092C211yesmaximal, characteristic, complemented
10#23639C211yesmaximal, characteristic, complemented
11#23618C211yesmaximal, characteristic, complemented
12#23556C211yesmaximal, characteristic, complemented
13#23483C211yesmaximal, characteristic
14#23456C211yesmaximal, characteristic, complemented
15#21172C211yesmaximal, characteristic, complemented
16#21091C211yesmaximal, characteristic, complemented
17#21051C211yesmaximal, characteristic, complemented
18#21032C211yesmaximal, characteristic, complemented
19#20848C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C2×#520C2211yescharacteristic
24#3164—28no—
25C2×#501C2212yescomplemented
26C2×#501C2212yescomplemented
27C2×#500C2212yescomplemented
28C2×#499C2212yescomplemented
29#3096—28no—
30C2×#481C2212yes—
31C2×#482C2212yes—
32C2×#482C2212yes—
33C2×#480C2212yes—
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#428C2211yescharacteristic
37C2×#427C2211yescharacteristic
38C22×#107C2211yescharacteristic
39C22×#106C2211yescharacteristic
40C2×#423C2211yescharacteristic
41C2×#418C2211yescharacteristic
42C2×#418C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C2×#408C2211yescharacteristic
46C2×#408C2211yescharacteristic
47C2×#405C2211yescharacteristic
48#2861—21no—
49#2858—21no—
50#2641—21no—
51#2640—21no—
52#2568—21no—
53#2566—21no—
54#2565—21no—
55#2563—21no—
56#2491C2214yes—
57#2490C2214yes—
58#2472—24no—
59#2450—24no—
60#2442C2214yescomplemented
61#2439C2214yescomplemented
62#2317C2211yescharacteristic
63#2311C2211yescharacteristic
64#2311C2211yescharacteristic
65#2309C2211yescharacteristic
66#2297C2211yescharacteristic
67#2296C2211yescharacteristic
68C2×#390C2211yescharacteristic
69C2×#382C2211yescharacteristic
70C4×#117C2211yescharacteristic
71#2092C2214yes—
72#2092C2214yes—
73#2076C2214yescomplemented
74#2075C2214yescomplemented
75#2067C2214yes—
76#2065C2214yes—
77#2059C2214yescomplemented
78#2058C2214yescomplemented
79#2034C2214yes—
80#2033C2214yes—
81#2025C2214yes—
82#2023C2214yes—
83#1949C2214yescomplemented
84#1949C2214yescomplemented
85#1936C2214yes—
86#1934C2214yes—
87#1924C2214yescomplemented
88#1923C2214yescomplemented
89#1921C2212yescomplemented
90#1920C2212yescomplemented
91#1919C2212yescomplemented
92#1918C2212yescomplemented
93#1854C2214yescomplemented
94#1853C2214yescomplemented
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1735C2211yescharacteristic
99#1735C2211yescharacteristic
100#1735C2211yescharacteristic
101#1729C2211yescharacteristic
102C2×#336C2211yescharacteristic
103C22×#41C2311yescharacteristic
104C23×D4C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#544—24no—
110#544—24no—
111#544—42no—
112#544—42no—
113#535—24no—
114#535—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2312yes—
119C2×#127—21no—
120C2×#125—21no—
121C2×#125—21no—
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×#124—21no—
125C2×#124—24no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—22no—
128C2×#121C2312yes—
129C2×#121C2312yes—
130C2×#120—22no—
131#520—24no—
132#520—42no—
133C2×C4×Q8—21no—
134C2×C4×D4—21no—
135C2×C4×D4—22no—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2312yes—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2312yes—
146C22×C22⋊C4—22no—
147C22×C42C2311yescharacteristic
148#501—28no—
149#501—28no—
150#500—28no—
151#499—28no—
152#482—28no—
153#482—28no—
154#481—28no—
155#480—28no—
156#428—24no—
157#428—24no—
158#428—24no—
159#427D418yes—
160C2×#107—24no—
161C2×#107C2312yes—
162C2×#107C2312yes—
163C2×#107C2312yes—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#107C2311yescharacteristic
167C2×#107C2311yescharacteristic
168C2×#106—42no—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2314yes—
173C2×#106C2311yescharacteristic
174C2×#106C2311yescharacteristic
175#423—24no—
176C2×C4⋊C8C2314yes—
177#418—24no—
178#418—24no—
179C2×#102—24no—
180C2×#102—42no—
181C2×#102C2312yes—
182C2×#102C2312yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2314yes—
186C2×#102C2314yes—
187#408—24no—
188#408—24no—
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2314yes—
192#405—24no—
193C2×#97C2312yes—
194C2×C4×C8C2312yes—
195#400—24no—
196#398—24no—
197#397—24no—
198#396—24no—
199#394—24no—
200#394—24no—
201#393—24no—
202#393—24no—
203#391—21no—
204#391—21no—
205#391—24no—
206#391D411yescharacteristic
207#391D411yescharacteristic
208#390—21no—
209#390—21no—
210#390—21no—
211#390—21no—
212#390—24no—
213#390—21no—
214#390—21no—
215#390D411yescharacteristic
216#390D411yescharacteristic
217#390C2314yes—
218#390C2314yes—
219#389C2314yes—
220#389C2312yes—
221#389C2312yes—
222#388—24no—
223#388—24no—
224#387—24no—
225#386—22no—
226#386—22no—
227#386D414yes—
228#385—22no—
229#385—24no—
230#385—22no—
231#384—21no—
232#382—21no—
233#382—21no—
234#382—21no—
235#382C2314yes—
236#380—24no—
237#380C2314yes—
238#380C2314yes—
239#379—22no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4—22no—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C2×#95—22no—
248C43—21no—
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2312yes—
253#340C2314yes—
254#340C2312yes—
255#340C2312yes—
256#340C2314yes—
257#340C2312yes—
258#336C2314yes—
259#336C2312yes—
260#336C2312yes—
261C25—21no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41—22no—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8C2×D411yescharacteristic
271C22×D4—22no—
272C22×D4—22no—
273C22×D4—22no—
274C22×D4—21no—
275C22×D4—22no—
276C22×D4—22no—
277C22×D4—22no—
278C22×D4—22no—
279C22×D4—22no—
280C22×D4C2×D412yes—
281C22×D4C2×D411yescharacteristic
282C22×D4C2412yes—
283C23×C4—21no—
284C23×C4—21no—
285C23×C4—22no—
286C23×C4—21no—
287C23×C4C2×D412yes—
288C23×C4C2412yes—
289C23×C4C2411yescharacteristic
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16#4114yes—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8C2×D414yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2411yescharacteristic
305C22×C8C2411yescharacteristic
306C22×C8C2412yes—
307C4⋊Q8—24no—
308C4⋊Q8—24no—
309C4⋊Q8—42no—
310C4⋊Q8—42no—
311C4⋊Q8—44no—
312C4⋊Q8—44no—
313#127—44no—
314#125—24no—
315#125—24no—
316#125—42no—
317#125—42no—
318#125—44no—
319#125—44no—
320#124—44no—
321#124—88no—
322C22⋊Q8—48no—
323C22⋊Q8—48no—
324#121—24no—
325#121—24no—
326#121—42no—
327#121—42no—
328#120—48no—
329C4×Q8—44no—
330C4×D4—44no—
331C4×D4—48no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—24no—
341#117—42no—
342#117C2×D414yes—
343#117C2×D414yes—
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—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2411yescharacteristic
374C2×C4⋊C4C2411yescharacteristic
375C2×C4⋊C4C2411yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—84no—
390C2×C22⋊C4C2×D412yes—
391C2×C22⋊C4C2×D414yes—
392C2×C22⋊C4C2×D414yes—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2412yes—
405C2×C42C2412yes—
406#107—24no—
407#107—24no—
408#107—24no—
409#107—24no—
410#107—24no—
411#107C2×D414yes—
412#107C2×D414yes—
413#106—24no—
414#106—24no—
415#106—24no—
416#106—44no—
417#106C2×D414yes—
418#106C2×D414yes—
419C4⋊C8—28no—
420#102—24no—
421#102—24no—
422#102—28no—
423#102—24no—
424#102—24no—
425#102—44no—
426C22⋊C8—24no—
427C22⋊C8—24no—
428C22⋊C8—28no—
429#97—24no—
430C4×C8—24no—
431#95—24no—
432#95—24no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—24no—
438#95—24no—
439#95—24no—
440#95C2×D414yes—
441#95C2×D414yes—
442C24—22no—
443C24—42no—
444C24—42no—
445C24—42no—
446C24—82no—
447C24—82no—
448C24C22×D412yes—
449C24C22×D411yescharacteristic
450#41—44no—
451#41—44no—
452#41—44no—
453#41—44no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—84no—
460C2×Q8—84no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—44no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—42no—
477C2×D4—44no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—84no—
483C2×D4—84no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—44no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—82no—
507C22×C4—84no—
508C22×C4—82no—
509C22×C4—82no—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C2×#4112yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C42−1+412yes—
520C22×C42−1+412yes—
521C22×C42−1+412yes—
522C22×C4C2511yescharacteristic
523M16—28no—
524M16—44no—
525C2×C8—28no—
526C2×C8—22no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—44no—
539C2×C8C22×D412yes—
540C2×C8C22×D412yes—
541C2×C8C2×#4114yes—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—82no—
569C4⋊C4—84no—
570C4⋊C4—82no—
571C4⋊C4C22×D412yes—
572C4⋊C4C22×D412yes—
573C4⋊C4C22×D412yes—
574C4⋊C4C22×D412yes—
575C22⋊C4—24no—
576C22⋊C4—24no—
577C22⋊C4—24no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C22⋊C4—48no—
584C22⋊C4—42no—
585C22⋊C4—42no—
586C22⋊C4—84no—
587C22⋊C4—84no—
588C22⋊C4—84no—
589C22⋊C4—84no—
590C22⋊C4—84no—
591C42—24no—
592C42—24no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—44no—
600C42—44no—
601C42—42no—
602C42—42no—
603C42—82no—
604C23—22no—
605C23—22no—
606C23—24no—
607C23—24no—
608C23—22no—
609C23—22no—
610C23—22no—
611C23—44no—
612C23—44no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—88no—
617C23—84no—
618C23—164no—
619C23—162no—
620C23—162no—
621C23#53612yes—
622C23D4212yes—
623C23#54712yes—
624C23C23×D411yescharacteristic
625Q8—44no—
626Q8—164no—
627Q8—164no—
628D4—48no—
629D4—44no—
630D4—42no—
631D4—42no—
632D4—42no—
633D4—42no—
634D4—84no—
635D4—164no—
636D4—164no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—84no—
675C2×C4—164no—
676C2×C4—162no—
677C2×C4—162no—
678C2×C4—162no—
679C2×C4#53612yes—
680C2×C4#53614yes—
681C2×C4#54214yes—
682C2×C4D4212yes—
683C2×C4#54712yes—
684C2×C4#54814yes—
685C2×C4#56314yes—
686C2×C4C23×D411yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×2−1+411yescharacteristic
690C2×C4C2×2−1+411yescharacteristic
691C2×C4C2×2−1+411yescharacteristic
692C8—22no—
693C8—22no—
694C8—24no—
695C8—42no—
696C8—42no—
697C22—24no—
698C22—22no—
699C22—22no—
700C22—24no—
701C22—44no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—44no—
706C22—84no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—84no—
711C22—168no—
712C22#267414yes—
713C22C2×#53611yescharacteristic
714C22C2×D4211yescharacteristic
715C22C2×#54711yescharacteristic
716C22#358412yes—
717C22#358512yes—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4#267414yes—
733C4C2×#53611yescharacteristic
734C4C2×#54211yescharacteristic
735C4C2×#54811yescharacteristic
736C4C2×#56311yescharacteristic
737C2—24no—
738C2—82no—
739C2—162no—
740C2C2×#267411yescharacteristic
741C2#6089012yes—
742C2#6108211yescharacteristic
743C2#6109011yescharacteristic
744C2#6127412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3164C2semidirect
3. #35620C2semidirect
5. #30804C2semidirect
6. #30791C2semidirect
7. #30781C2semidirect
8. #30778C2semidirect
9. C2×#2092C2semidirect
10. #23639C2semidirect
11. #23618C2semidirect
12. #23556C2semidirect
14. #23456C2semidirect
15. #21172C2semidirect
16. #21091C2semidirect
17. #21051C2semidirect
18. #21032C2semidirect
19. #20848C2semidirect
25. C2×#501C22semidirect
26. C2×#501C22semidirect
27. C2×#500C22semidirect
28. C2×#499C22semidirect
60. #2442C22semidirect
61. #2439C22semidirect
73. #2076C22semidirect
74. #2075C22semidirect
77. #2059C22semidirect
78. #2058C22semidirect
83. #1949C22semidirect
84. #1949C22semidirect
87. #1924C22semidirect
88. #1923C22semidirect
89. #1921C22semidirect
90. #1920C22semidirect
91. #1919C22semidirect
92. #1918C22semidirect
93. #1854C22semidirect
94. #1853C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C2×#30962
#356202
#352642
#308042
#307912
#307812
#307782
C2×#20922
#236392
#236182
#235562
#234832
#234562
#211722
#210912
#210512
#210322
#208482