Finite Group Databasebeta

← #6496418

Subgroups — #6496418

Order 512.

1821 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3111C211yesmaximal, characteristic
2C2×#3079C211yesmaximal, characteristic, complemented
3#37632C211yesmaximal, characteristic, complemented
4#34197C211yesmaximal, characteristic, complemented
5#33206C212yesmaximal, complemented
6#33035C212yesmaximal, complemented
7#29874C212yesmaximal, complemented
8#29838C212yesmaximal, complemented
9#24831C212yesmaximal, complemented
10#24828C212yesmaximal, complemented
11#24696C212yesmaximal, complemented
12#24693C212yesmaximal, complemented
13#24595C212yesmaximal, complemented
14#24594C212yesmaximal, complemented
15#24543C212yesmaximal, complemented
16#24542C212yesmaximal
17#21979C211yesmaximal, characteristic, complemented
18#21895C211yesmaximal, characteristic, complemented
19#21841C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C2×#529C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23#3111—28no—
24C2×#485C2211yescharacteristic
25C2×#482C2211yescharacteristic
26#3079—28no—
27C2×#479C2211yescharacteristic
28C2×#477C2212yes—
29C2×#475C2212yes—
30C2×#478C2211yescharacteristic
31C2×#476C2212yes—
32C2×#474C2212yes—
33C2×#464C2211yescharacteristic
34C2×#463C2211yescharacteristic
35C2×#452C2211yescharacteristic
36C2×#449C2211yescharacteristic
37C2×#442C2211yescharacteristic
38C2×#441C2211yescharacteristic
39C2×#440C2211yescharacteristic
40C2×#440C2211yescharacteristic
41C2×#432C2211yescharacteristic
42C2×#428C2212yes—
43C2×#423C2211yescharacteristic
44C2×#417C2212yes—
45#2821—21no—
46#2821—21no—
47#2660C2212yes—
48#2640C2212yes—
49#2628—22no—
50#2614—22no—
51#2567C2212yes—
52#2563C2212yes—
53#2503C2212yes—
54#2501C2212yes—
55#2486—22no—
56#2357—22no—
57#2357C2212yes—
58#2351C2212yes—
59#2346—22no—
60#2346C2212yes—
61#2341C2212yes—
62#2319—22no—
63#2319C2212yes—
64#2283—21no—
65#2283—21no—
66#2283C2211yescharacteristic
67#2283C2211yescharacteristic
68#2276C2212yes—
69#2085C2214yes—
70#2084C2214yes—
71#2073C2214yes—
72#2071C2214yes—
73#2066C2214yes—
74#2065C2214yes—
75#2055C2214yes—
76#2054C2214yes—
77#1993C2212yes—
78#1993C2212yes—
79#1989C2212yes—
80#1988C2212yes—
81#1988C2212yes—
82#1983C2212yes—
83#1949C2214yes—
84#1947C2214yes—
85#1940C2212yes—
86#1940C2212yes—
87#1928C2214yes—
88#1925C2214yes—
89#1895C2212yes—
90#1894C2212yes—
91#1880C2214yes—
92#1878C2214yes—
93#1851C2212yes—
94#1851C2212yes—
95#1843C2212yes—
96#1832C2212yes—
97#1824C2212yes—
98#1824C2212yes—
99#1805C2212yes—
100#1805C2214yes—
101#1804C2212yes—
102#1803C2214yes—
103C22×#41C2311yescharacteristic
104C23×Q8—21no—
105C22×Q16C2311yescharacteristic
106C22×SD16C2311yescharacteristic
107C22×SD16C2311yescharacteristic
108C22×D8C2311yescharacteristic
109C22×M16C2312yes—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116#529—24no—
117#529—24no—
118C2×#126—21no—
119C2×#125C2311yescharacteristic
120C2×#124C2311yescharacteristic
121C2×#123—21no—
122C2×#123—24no—
123C2×C22⋊Q8C2311yescharacteristic
124C2×#121C2311yescharacteristic
125C4×#41—24no—
126C4×#41D418yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2314yes—
129C2×C4×Q8C2311yescharacteristic
130C2×C4×Q8C2311yescharacteristic
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2311yescharacteristic
136C2×C4×D4C2311yescharacteristic
137C2×#117—21no—
138C2×#117—21no—
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2312yes—
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144#485—24no—
145#482—24no—
146#479—24no—
147#478—24no—
148#477—28no—
149#476—28no—
150#475—28no—
151#474—28no—
152#464—24no—
153#463—24no—
154#452—24no—
155#449—24no—
156#442—24no—
157#441—24no—
158#440—24no—
159#440—24no—
160#432—24no—
161#428—28no—
162C2×#107C2311yescharacteristic
163C2×#107C2314yes—
164C2×#107C2311yescharacteristic
165C2×#106C2311yescharacteristic
166C2×#106C2314yes—
167C2×#106C2311yescharacteristic
168#423—24no—
169C2×C4⋊C8C2311yescharacteristic
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2314yes—
174C2×C4⋊C8C2314yes—
175#417—28no—
176C2×#103C2314yes—
177C2×#103C2311yescharacteristic
178C2×#103C2311yescharacteristic
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#103C2314yes—
182C2×#102C2311yescharacteristic
183C2×#102C2311yescharacteristic
184C2×#102C2314yes—
185C2×#102C2311yescharacteristic
186C2×#102C2311yescharacteristic
187C2×#102C2314yes—
188C2×C22⋊C8C2311yescharacteristic
189C2×C22⋊C8C2311yescharacteristic
190C2×#97C2311yescharacteristic
191C2×#97C2311yescharacteristic
192#400—22no—
193#400—22no—
194#399—22no—
195#399—22no—
196#398—22no—
197#398—22no—
198#397—22no—
199#397—22no—
200#395—22no—
201#395—22no—
202#395—24no—
203#393—22no—
204#393—22no—
205#393—24no—
206#391C2314yes—
207#390—24no—
208#390C2314yes—
209#390C2314yes—
210#389—24no—
211#389—24no—
212#389C2314yes—
213#388—24no—
214#388—24no—
215#388—24no—
216#384C2314yes—
217#383—21no—
218#383D411yescharacteristic
219#383D411yescharacteristic
220#382—22no—
221#382—22no—
222#382D412yes—
223#382C2314yes—
224#382C2312yes—
225#381—21no—
226#381—24no—
227#381—22no—
228#381—22no—
229#381D412yes—
230#380—24no—
231#380—22no—
232#380—22no—
233#380D412yes—
234#380C2312yes—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—24no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4C2312yes—
239C4×C4⋊C4C2314yes—
240C4×C4⋊C4C2312yes—
241C4×C22⋊C4—21no—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—22no—
244C4×C22⋊C4C2312yes—
245#376—21no—
246#376—22no—
247#376—22no—
248#376C2312yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#336C2314yes—
256#336C2314yes—
257C2×#41—22no—
258C2×#41—24no—
259C2×#41—42no—
260C2×#41C2×D414yes—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—44no—
264C22×Q8—84no—
265C22×Q8C2411yescharacteristic
266C22×Q8C2411yescharacteristic
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C2411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—21no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2411yescharacteristic
279C2×Q16—22no—
280C2×Q16—24no—
281C2×SD16—22no—
282C2×SD16—24no—
283C2×SD16—22no—
284C2×SD16—42no—
285C2×D8—22no—
286C2×D8—42no—
287C2×M16—24no—
288C2×M16—44no—
289C22×C8C2414yes—
290C22×C8C2411yescharacteristic
291C22×C8C2411yescharacteristic
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C4⋊Q8—24no—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300#126—44no—
301#125—22no—
302#125—22no—
303#124—22no—
304#124—22no—
305#123—44no—
306#123—88no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309#121—22no—
310#121—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—28no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—44no—
317C4×Q8C2×D414yes—
318C4×Q8C2×D414yes—
319C4×D4—28no—
320C4×D4—28no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—44no—
326C4×D4C2×D414yes—
327C4×D4C2×D414yes—
328#117—22no—
329#117—28no—
330#117—44no—
331#117C2×D414yes—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4C2×D414yes—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—28no—
373C2×C42—21no—
374C2×C42—24no—
375C2×C42—24no—
376C2×C42—22no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D414yes—
380C2×C42#4114yes—
381C2×C42C2414yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—28no—
385#107—22no—
386#107—22no—
387#106—28no—
388#106—22no—
389#106—22no—
390C4⋊C8—24no—
391C4⋊C8—24no—
392C4⋊C8—22no—
393C4⋊C8—22no—
394C4⋊C8—44no—
395C4⋊C8—44no—
396#103—28no—
397#103—22no—
398#103—22no—
399#103—22no—
400#103—22no—
401#103—44no—
402#102—28no—
403#102—22no—
404#102—22no—
405#102—22no—
406#102—22no—
407#102—44no—
408C22⋊C8—22no—
409C22⋊C8—22no—
410#97—22no—
411#97—22no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95C2×D412yes—
423#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D414yes—
426#95C2×D412yes—
427#95C2×D412yes—
428#95C2×D412yes—
429C24—21no—
430C24—21no—
431C24C22×D411yescharacteristic
432#41—28no—
433#41—44no—
434#41—48no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—82no—
444C2×Q8—82no—
445C2×Q8—84no—
446C2×Q8—816no—
447C2×Q8C22×D412yes—
448C2×Q82−1+414yes—
449C2×D4—24no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4C22×D412yes—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4#12114yes—
484C22×C4#12114yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C2×#4112yes—
494C22×C4C2511yescharacteristic
495Q16—44no—
496SD16—44no—
497SD16—44no—
498D8—44no—
499M16—48no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—24no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—24no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—24no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4C22×D412yes—
536C4⋊C4C22×D412yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—82no—
551C42—24no—
552C42—24no—
553C42—24no—
554C42—24no—
555C42—24no—
556C42—48no—
557C42—42no—
558C42—82no—
559C42—84no—
560C42#12114yes—
561C42#12114yes—
562C42C22×D412yes—
563C42C22×D412yes—
564C42C2×#4112yes—
565C42C2×#4112yes—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—42no—
577C23C2×#12112yes—
578C23C23×D411yescharacteristic
579Q8—22no—
580Q8—24no—
581Q8—41no—
582Q8—41no—
583Q8—42no—
584Q8—88no—
585Q8—88no—
586Q8#53614yes—
587D4—22no—
588D4—22no—
589D4—48no—
590D4—41no—
591D4—41no—
592D4—42no—
593D4—42no—
594D4—84no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4C2×#12112yes—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#12112yes—
639C2×C4C23×D411yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×2−1+412yes—
642C2×C4#58512yes—
643C8—41no—
644C8—41no—
645C8—44no—
646C8—41no—
647C8—41no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—84no—
656C22—81no—
657C22—81no—
658C22—84no—
659C22—81no—
660C22C22×#12111yescharacteristic
661C22#348912yes—
662C22#358512yes—
663C22#358612yes—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—84no—
675C4—81no—
676C4—81no—
677C4—84no—
678C4—81no—
679C4C2×#53611yescharacteristic
680C4#344911yescharacteristic
681C4#349512yes—
682C2—42no—
683C2—82no—
684C2—161no—
685C2#3530611yescharacteristic
686C2#6042611yescharacteristic
687C2#6042711yescharacteristic
688C2#6223014yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#449C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3079C2semidirect
3. #37632C2semidirect
4. #34197C2semidirect
5. #33206C2semidirect
6. #33035C2semidirect
7. #29874C2semidirect
8. #29838C2semidirect
9. #24831C2semidirect
10. #24828C2semidirect
11. #24696C2semidirect
12. #24693C2semidirect
13. #24595C2semidirect
14. #24594C2semidirect
15. #24543C2semidirect
17. #21979C2semidirect
18. #21895C2semidirect
19. #21841C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31112
C2×#30792
#376322
#341972
#332062
#330352
#298742
#298382
#248312
#248282
#246962
#246932
#245952
#245942
#245432
#245422
#219792
#218952
#218412