Finite Group Databasebeta

← #6389773

Subgroups — #6389773

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3166C211yesmaximal, characteristic
2#34394C211yesmaximal, characteristic
3#34378C211yesmaximal, characteristic
4#23169C212yesmaximal, complemented
5#23168C212yesmaximal, complemented
6#23167C212yesmaximal, complemented
7#23166C212yesmaximal, complemented
8#22624C212yesmaximal
9#22623C212yesmaximal
10#22622C212yesmaximal
11#22621C212yesmaximal
12#22252C212yesmaximal, complemented
13#22252C212yesmaximal, complemented
14#21839C211yesmaximal, characteristic
15#21838C211yesmaximal, characteristic
16#21723C212yesmaximal, complemented
17#21722C212yesmaximal, complemented
18#20824C211yesmaximal, characteristic
19#20477C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×#125C2211yescharacteristic
22#3166C4116yescomplemented
23C2×#501C2212yes—
24C2×#501C2212yes—
25C2×#499C2212yes—
26C2×#499C2212yes—
27C2×#432C2211yescharacteristic
28C2×#428C2212yes—
29C2×#428C2212yes—
30C2×C4×M16C2211yescharacteristic
31#2382—21no—
32#2382—21no—
33#2378—22no—
34#2370C2212yes—
35#2370C2212yes—
36#2369C2211yescharacteristic
37#2369C2211yescharacteristic
38#2365C2211yescharacteristic
39#2365C2211yescharacteristic
40#2356—21no—
41#2351—21no—
42#2347C2212yes—
43#2346C2212yes—
44#2345C2212yes—
45#2344—22no—
46#2344C2212yes—
47#2336—22no—
48#2330C2212yes—
49#2330C2212yes—
50#2330C2212yes—
51#2330C2212yes—
52#2323—21no—
53#2323—21no—
54#2320—22no—
55#2316—21no—
56#2313—21no—
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2283C2211yescharacteristic
60#1992C2212yes—
61#1992C2212yes—
62#1992C2212yes—
63#1992C2212yes—
64#1989C2212yes—
65#1989C2212yes—
66#1988C2212yes—
67#1987C2212yes—
68#1944C2212yes—
69#1944C2212yes—
70#1943C2212yes—
71#1943C2212yes—
72#1942C2212yes—
73#1942C2212yes—
74#1941C2212yes—
75#1941C2212yes—
76#1921C2212yes—
77#1921C2212yes—
78#1921C2212yes—
79#1921C2212yes—
80#1920C2212yes—
81#1920C2212yes—
82#1920C2212yes—
83#1920C2212yes—
84#1919C2212yes—
85#1919C2212yes—
86#1919C2212yes—
87#1919C2212yes—
88#1918C2212yes—
89#1918C2212yes—
90#1918C2212yes—
91#1918C2212yes—
92#1845C2211yescharacteristic
93#1844C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1825C2212yes—
97#1825C2212yes—
98#1824C2211yescharacteristic
99#1824C2211yescharacteristic
100#1823C2211yescharacteristic
101#1823C2211yescharacteristic
102#1809C2212yes—
103#1809C2212yes—
104#1776C2212yes—
105#1776C2212yes—
106#1776C2212yes—
107#1776C2212yes—
108#1776C2212yes—
109#1776C2212yes—
110C4×#107C2212yes—
111C4×#107C2212yes—
112C4×#106C2212yes—
113C4×#106C2212yes—
114#1771C2212yes—
115#1771C2212yes—
116#1753C2212yes—
117#1753C2212yes—
118#1746C2212yes—
119#1726C2212yes—
120C22×M16C2312yes—
121#533—24no—
122#533C2×C418yes—
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×#125—21no—
126C2×#125—21no—
127C2×#125—24no—
128C2×#125C2×C418yes—
129C2×#125C2312yes—
130C2×#125C2312yes—
131C2×#125C2312yes—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×#117—21no—
139C2×#117—21no—
140C2×#117—22no—
141C2×#117C2312yes—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#501C2×C4116yes—
148#501C2×C4116yes—
149#499C2×C4116yes—
150#499C2×C4116yes—
151#432—24no—
152#428C2×C4116yes—
153#428C2×C4116yes—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×C4⋊C8C2312yes—
167C2×C22⋊C8C2311yescharacteristic
168C2×C22⋊C8C2311yescharacteristic
169C4×M16C2×C418yes—
170C2×#97C2312yes—
171C2×#97C2312yes—
172C2×#97C2312yes—
173C2×#97C2312yes—
174C2×C4×C8C2312yes—
175C2×C4×C8C2312yes—
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2311yescharacteristic
178#391—21no—
179#391—21no—
180#391—22no—
181#391—22no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—21no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389—22no—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#383—22no—
229#383D412yes—
230#382—21no—
231#382—21no—
232#382—22no—
233#382D411yescharacteristic
234#382D411yescharacteristic
235#381—22no—
236#381D412yes—
237#380—21no—
238#380—21no—
239#380—22no—
240#380D411yescharacteristic
241#380D411yescharacteristic
242#380C2311yescharacteristic
243#380C2311yescharacteristic
244#380C2312yes—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—22no—
250C4×C4⋊C4—22no—
251C4×C4⋊C4—22no—
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2312yes—
259C4×C4⋊C4C2312yes—
260C4×C4⋊C4C2311yescharacteristic
261C4×C4⋊C4C2311yescharacteristic
262C4×C4⋊C4C2312yes—
263C4×C4⋊C4C2311yescharacteristic
264C4×C4⋊C4C2311yescharacteristic
265C4×C22⋊C4—22no—
266C4×C22⋊C4C2312yes—
267#376—22no—
268#376C2312yes—
269#340C2312yes—
270#340C2312yes—
271#340C2312yes—
272#340C2312yes—
273#340C2312yes—
274#340C2312yes—
275#340C2312yes—
276#340C2312yes—
277#340C2312yes—
278#340C2312yes—
279#340C2312yes—
280#340C2312yes—
281#340C2312yes—
282#340C2312yes—
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#340C2312yes—
289#336C2312yes—
290#336C2312yes—
291#336C2312yes—
292#336C2312yes—
293C22×Q8—21no—
294C22×Q8—21no—
295C23×C4—21no—
296C23×C4—21no—
297C23×C4C2×D411yescharacteristic
298C23×C4C2×D411yescharacteristic
299C23×C4C2411yescharacteristic
300C2×M16—24no—
301C2×M16—24no—
302C2×M16C22×C418yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C22×C8C2412yes—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8C22×C414yes—
310C4⋊Q8C22×C414yes—
311#125—28no—
312#125—28no—
313#125—24no—
314#125—24no—
315#125—24no—
316#125C22×C418yes—
317#125C22×C418yes—
318#125C22×C418yes—
319C22⋊Q8—28no—
320C22⋊Q8—28no—
321C4×Q8—28no—
322C4×Q8—28no—
323C4×Q8—82no—
324C4×Q8—82no—
325#117—24no—
326#117—82no—
327#117—84no—
328#117—82no—
329#117C22×C418yes—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4C22×C414yes—
361C2×C4⋊C4C22×C414yes—
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42C22×C414yes—
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42#4114yes—
398C2×C42C2412yes—
399C2×C42C2412yes—
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402#107—24no—
403#107—24no—
404#107C22×C418yes—
405#107C22×C418yes—
406#107C22×C418yes—
407#107C22×C418yes—
408#106—24no—
409#106—24no—
410#106C22×C418yes—
411#106C22×C418yes—
412#106C22×C418yes—
413#106C22×C418yes—
414C4⋊C8—24no—
415C22⋊C8—22no—
416C22⋊C8—22no—
417#97—24no—
418#97—24no—
419#97—24no—
420#97C22×C418yes—
421C4×C8—24no—
422C4×C8—24no—
423C4×C8C22×C414yes—
424C4×C8C22×C414yes—
425#95—21no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—21no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—21no—
438#95—21no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95C2×D412yes—
448#95C2×D412yes—
449#95C2×D412yes—
450#95C2×D412yes—
451#95C2×D412yes—
452#95C2×D412yes—
453#95C2×D412yes—
454#95C2×D412yes—
455C24C22×D411yescharacteristic
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—81no—
461C2×Q8—81no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—81no—
484C22×C4C4×D414yes—
485C22×C4C4×D414yes—
486C22×C4C23×C412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4112yes—
497C22×C4C2511yescharacteristic
498M16—216no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8C23×C414yes—
508C2×C8C23×C414yes—
509C2×C82−1+414yes—
510C2×C82−1+414yes—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—28no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—81no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—81no—
546C4⋊C4—81no—
547C4⋊C4—81no—
548C4⋊C4—81no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4C23×C414yes—
554C4⋊C4C23×C414yes—
555C4⋊C4C23×C414yes—
556C4⋊C4C23×C414yes—
557C22⋊C4—28no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—44no—
561C22⋊C4—81no—
562C22⋊C4—81no—
563C42—28no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—42no—
569C42—42no—
570C42—82no—
571C42—82no—
572C42—82no—
573C42—82no—
574C42—82no—
575C42—82no—
576C42—82no—
577C42—82no—
578C42—82no—
579C42C4×D414yes—
580C42C4×D414yes—
581C42C23×C412yes—
582C42C23×C412yes—
583C42C2×#4112yes—
584C42C2×#4112yes—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23C2×C4×D412yes—
592C23C23×D411yescharacteristic
593Q8—48no—
594Q8—84no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—84no—
653C2×C4C2×C4×D412yes—
654C2×C4C2×C4×D412yes—
655C2×C4C2×C4×D412yes—
656C2×C4C2×C4×D412yes—
657C2×C4C2×C4×D412yes—
658C2×C4C24×C411yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×2−1+412yes—
661C2×C4C2×2−1+412yes—
662C8—24no—
663C8—42no—
664C8—42no—
665C8#51918yes—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22C22×C4×D411yescharacteristic
674C22#349911yescharacteristic
675C22#349911yescharacteristic
676C22#358511yescharacteristic
677C22#358511yescharacteristic
678C22#358611yescharacteristic
679C22#358611yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—82no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—82no—
699C4C2×#51912yes—
700C4#350912yes—
701C2—42no—
702C2#3420911yescharacteristic
703C2#6017011yescharacteristic
704C2#6017011yescharacteristic
705C2#6249911yescharacteristic
706C2#6250011yescharacteristic
707C2#6250111yescharacteristic
708C2#6250211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×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
4. #23169C2semidirect
5. #23168C2semidirect
6. #23167C2semidirect
7. #23166C2semidirect
12. #22252C2semidirect
13. #22252C2semidirect
16. #21723C2semidirect
17. #21722C2semidirect
22. #3166C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31662
#343942
#343782
#231692
#231682
#231672
#231662
#226242
#226232
#226222
#226212
#222522
#218392
#218382
#217232
#217222
#208242
#204772