Finite Group Databasebeta

← #6526635

Subgroups — #6526635

Order 512.

1972 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic, complemented
2C2×#2990C211yesmaximal, characteristic, complemented
3#35279C211yesmaximal, characteristic, complemented
4#34976C211yesmaximal, characteristic
5#28103C214yesmaximal, complemented
6#27668C214yesmaximal, complemented
7#25480C212yesmaximal, complemented
8#25477C212yesmaximal, complemented
9#25340C212yesmaximal, complemented
10#25339C212yesmaximal, complemented
11#24838C212yesmaximal, complemented
12#24835C212yesmaximal, complemented
13#24710C212yesmaximal, complemented
14#24709C212yesmaximal, complemented
15#21964C211yesmaximal, characteristic, complemented
16#21833C211yesmaximal, characteristic, complemented
17#20737C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21#3079—28no—
22C2×#477C2212yes—
23C2×#475C2212yes—
24C2×#476C2212yes—
25C2×#474C2212yes—
26#2990—28no—
27C2×#446C2212yes—
28C2×#435C2214yes—
29C2×C8×D4C2212yes—
30C2×#432C2212yes—
31C2×#431C2211yescharacteristic
32C2×#428C2212yes—
33C2×#423C2211yescharacteristic
34C22×C4⋊C8C2212yes—
35C2×#417C2212yes—
36#2857C2212yes—
37#2857C2212yes—
38#2628—22no—
39#2609—22no—
40#2582C2211yescharacteristic
41#2568—21no—
42#2567C2212yes—
43#2563C2212yes—
44#2542—21no—
45#2538C2211yescharacteristic
46#2538C2211yescharacteristic
47#2537C2211yescharacteristic
48#2487C2214yes—
49#2485—21no—
50#2485C2212yes—
51#2409—22no—
52#2407—21no—
53#2407C2212yes—
54#2392—22no—
55#2317—21no—
56#2311—21no—
57C4×C22⋊Q8—21no—
58C4×C22⋊Q8C2212yes—
59C4×#121—21no—
60C4×#121C2212yes—
61#2285C2211yescharacteristic
62#2285C2211yescharacteristic
63#2088C2214yes—
64#2086C2214yes—
65#2077C2214yes—
66#2076C2214yes—
67#2067C2214yes—
68#2066C2214yes—
69#2036C2214yes—
70#2034C2214yes—
71#1990C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1988C2212yes—
75#1948C2212yes—
76#1948C2212yes—
77#1947C2212yes—
78#1947C2212yes—
79#1944C2214yes—
80#1943C2214yes—
81#1936C2214yes—
82#1935C2214yes—
83#1893C2214yes—
84#1892C2214yes—
85#1825C2212yes—
86#1823C2214yes—
87#1809C2214yes—
88#1808C2214yes—
89#1807C2214yes—
90#1803C2211yescharacteristic
91#1803C2211yescharacteristic
92#1802C2211yescharacteristic
93#1802C2211yescharacteristic
94#1788C2212yes—
95C4×#103C2212yes—
96C4×#102C2212yes—
97C22×#41D411yescharacteristic
98C22×#41C2311yescharacteristic
99C23×Q8—21no—
100C22×M16C2312yes—
101C23×C8C2314yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×#121—21no—
113C2×#121C2312yes—
114C4×#41—24no—
115C4×#41—24no—
116C4×#41—24no—
117C4×#41—24no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4D412yes—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117—21no—
133C2×#117—21no—
134C2×#117D411yescharacteristic
135C2×#117D412yes—
136C2×#117C2311yescharacteristic
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2312yes—
139C22×C42—21no—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#477—28no—
143#476—28no—
144#475—28no—
145#474—28no—
146#446—28no—
147#435—216no—
148C8×D4—28no—
149#432—28no—
150#431—24no—
151#428—28no—
152C2×#107C2314yes—
153C2×#107C2314yes—
154C2×#106C2314yes—
155C2×#106C2314yes—
156#423D418yescomplemented
157C2×C4⋊C8—28no—
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2312yes—
163#417—28no—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×C22⋊C8C2314yes—
171C2×C22⋊C8C2312yes—
172C2×#97C2314yes—
173C2×C4×C8C2312yes—
174#400—24no—
175#399—24no—
176#398—22no—
177#397—22no—
178#396—22no—
179#396—22no—
180#395—24no—
181#394—22no—
182#394—22no—
183#394—22no—
184#393—24no—
185#392—22no—
186#391—22no—
187#391—21no—
188#391—22no—
189#391—22no—
190#391C2312yes—
191#390—22no—
192#390—21no—
193#390C2312yes—
194#389—21no—
195#389—21no—
196#389—22no—
197#389—22no—
198#389C2314yes—
199#389C2314yes—
200#389C2312yes—
201#389C2312yes—
202#388—22no—
203#388—22no—
204#387—22no—
205#387—22no—
206#386—24no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#385—24no—
212#384—22no—
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#382—22no—
218#382—21no—
219#382—21no—
220#380—22no—
221#380—22no—
222#380—21no—
223#380—21no—
224#380—21no—
225#380—21no—
226#380C2314yes—
227#380C2314yes—
228#380C2312yes—
229#380C2312yes—
230#380C2312yes—
231#380C2312yes—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4C2312yes—
236C4×C4⋊C4C2312yes—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239#340C2312yes—
240#340C2312yes—
241#340C2312yes—
242#340C2312yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—42no—
256C2×#41—41no—
257C2×#41—41no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—44no—
261C22×Q8—82no—
262C22×Q8—82no—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×D4—21no—
266C22×D4—22no—
267C22×D4C2×D412yes—
268C22×D4C2×D411yescharacteristic
269C22×D4C2411yescharacteristic
270C23×C4—22no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4C2×D412yes—
275C23×C4C2×D412yes—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2412yes—
279C23×C4C2411yescharacteristic
280C2×M16—42no—
281C2×M16—42no—
282C2×M16C2×D418yes—
283C22×C8—28no—
284C22×C8—44no—
285C22×C8—44no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2414yes—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C22⋊Q8—24no—
296C22⋊Q8—42no—
297C22⋊Q8—44no—
298C22⋊Q8—84no—
299C22⋊Q8—84no—
300#121—24no—
301#121—42no—
302#121—44no—
303C4×Q8—24no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×D4—28no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—84no—
323#117—24no—
324#117—28no—
325#117—22no—
326#117—22no—
327#117—44no—
328#117—44no—
329#117—82no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2414yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C42—24no—
379C2×C42—24no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—44no—
386C2×C42C2×D412yes—
387C2×C42C2×D412yes—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D414yes—
391C2×C42#4114yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—28no—
396#107—44no—
397#106—28no—
398#106—44no—
399C4⋊C8—28no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8C2×D418yes—
403C4⋊C8C2×D418yes—
404#103—24no—
405#103—28no—
406#103—42no—
407#102—24no—
408#102—28no—
409#102—42no—
410C22⋊C8—24no—
411C22⋊C8—28no—
412#97—28no—
413C4×C8—24no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428C24—21no—
429C24C22×D411yescharacteristic
430C24C22×D412yes—
431#41—28no—
432#41—44no—
433#41—416no—
434#41—44no—
435#41—44no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—82no—
449C2×Q8—82no—
450C2×Q8—816no—
451C2×Q82−1+412yes—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—82no—
468C2×D4#13714yes—
469C2×D42−1+412yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—24no—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—44no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—82no—
509C22×C4—82no—
510C22×C4#12114yes—
511C22×C4#12114yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C2×#4112yes—
522C22×C4C2511yescharacteristic
523M16—48no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—88no—
533C2×C8C22×D418yes—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—24no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—81no—
566C4⋊C4—81no—
567C4⋊C4—82no—
568C4⋊C4—84no—
569C4⋊C4—84no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—42no—
578C22⋊C4—44no—
579C22⋊C4—42no—
580C22⋊C4—42no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—84no—
585C22⋊C4—84no—
586C42—24no—
587C42—24no—
588C42—22no—
589C42—22no—
590C42—42no—
591C42—42no—
592C42—42no—
593C42—44no—
594C42—44no—
595C42#12114yes—
596C42#12114yes—
597C42C22×D412yes—
598C42C22×D412yes—
599C42C2×#4112yes—
600C42C2×#4112yes—
601C23—22no—
602C23—22no—
603C23—21no—
604C23—21no—
605C23—22no—
606C23—21no—
607C23—21no—
608C23—22no—
609C23—22no—
610C23—21no—
611C23—22no—
612C23—21no—
613C23—22no—
614C23—41no—
615C23—41no—
616C23—41no—
617C23—41no—
618C23—41no—
619C23—41no—
620C23C2×#12112yes—
621C23C2×#13714yes—
622C23C23×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—41no—
626Q8—41no—
627Q8—44no—
628Q8—88no—
629Q8—88no—
630D4—22no—
631D4—22no—
632D4—28no—
633D4—48no—
634D4—48no—
635D4—41no—
636D4—41no—
637D4—88no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—48no—
661C2×C4—48no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—48no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—41no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—44no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—41no—
690C2×C4—88no—
691C2×C4—84no—
692C2×C4—84no—
693C2×C4C2×#12112yes—
694C2×C4C2×#12112yes—
695C2×C4C2×#12112yes—
696C2×C4C2×#12112yes—
697C2×C4C2×#12112yes—
698C2×C4C2×#13712yes—
699C2×C4#57512yes—
700C2×C4#57612yes—
701C2×C4#57712yes—
702C2×C4C23×D411yescharacteristic
703C2×C4C22×#4111yescharacteristic
704C2×C4C2×2−1+411yescharacteristic
705C2×C4C2×2−1+411yescharacteristic
706C2×C4#58511yescharacteristic
707C2×C4#58511yescharacteristic
708C8—28no—
709C8—42no—
710C8—42no—
711C22—22no—
712C22—24no—
713C22—22no—
714C22—22no—
715C22—24no—
716C22—24no—
717C22—42no—
718C22—42no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—42no—
723C22—42no—
724C22—41no—
725C22—42no—
726C22—42no—
727C22—42no—
728C22—81no—
729C22—81no—
730C22—81no—
731C22—81no—
732C22C22×#12111yescharacteristic
733C22#348911yescharacteristic
734C22#348911yescharacteristic
735C22C22×#13711yescharacteristic
736C22C2×#57511yescharacteristic
737C22C2×#57611yescharacteristic
738C22C2×#57711yescharacteristic
739C4—22no—
740C4—22no—
741C4—22no—
742C4—42no—
743C4—42no—
744C4—42no—
745C4—42no—
746C4—81no—
747C4—81no—
748C4—84no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4#305614yes—
754C4#306214yes—
755C4#348511yescharacteristic
756C4#348611yescharacteristic
757C4#349511yescharacteristic
758C4#349511yescharacteristic
759C2—42no—
760C2—44no—
761C2—82no—
762C2#3563311yescharacteristic
763C2C2×#305611yescharacteristic
764C2C2×#306211yescharacteristic
765C2#6212711yescharacteristic
766C2#6213111yescharacteristic
767C2#6213411yescharacteristic
768C2#6213611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34976C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3079C2semidirect
2. C2×#2990C2semidirect
3. #35279C2semidirect
5. #28103C2semidirect
6. #27668C2semidirect
7. #25480C2semidirect
8. #25477C2semidirect
9. #25340C2semidirect
10. #25339C2semidirect
11. #24838C2semidirect
12. #24835C2semidirect
13. #24710C2semidirect
14. #24709C2semidirect
15. #21964C2semidirect
16. #21833C2semidirect
17. #20737C2semidirect
156. #423D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
C2×#29902
#352792
#349762
#281032
#276682
#254802
#254772
#253402
#253392
#248382
#248352
#247102
#247092
#219642
#218332
#207372