Finite Group Databasebeta

← #6454171

Subgroups — #6454171

Order 512.

2160 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C22×C4×SD16C211yesmaximal, characteristic
3#35248C211yesmaximal, characteristic, complemented
4#34194C211yesmaximal, characteristic
5#24816C212yesmaximal, complemented
6#24730C212yesmaximal, complemented
7#24681C212yesmaximal, complemented
8#24630C212yesmaximal, complemented
9#24579C212yesmaximal, complemented
10#24527C212yesmaximal
11#24482C212yesmaximal, complemented
12#24358C212yesmaximal
13#23689C214yesmaximal, complemented
14#21514C214yesmaximal, complemented
15#21400C211yesmaximal, characteristic, complemented
16#21351C211yesmaximal, characteristic, complemented
17#20740C211yesmaximal, characteristic, complemented
18C23×SD16C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#3110—28no—
23C2×#484C2212yes—
24C2×#474C2211yescharacteristic
25C2×#474C2211yescharacteristic
26C2×#461C2214yes—
27C2×#461C2211yescharacteristic
28C2×#461C2211yescharacteristic
29C2×#450C2212yes—
30C2×#440C2212yes—
31C2×#440C2212yes—
32C2×C4×SD16—28no—
33C2×C4×SD16C2212yes—
34C2×C4×SD16C2212yes—
35C2×C4×SD16C2212yes—
36C2×C4×SD16C2212yes—
37C2×#432C2211yescharacteristic
38C22×#106C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42#2855—21no—
43#2855—21no—
44#2634—22no—
45#2581—22no—
46#2568C2212yescomplemented
47#2546C2212yescomplemented
48#2503C2212yes—
49#2484—22no—
50#2434C2212yes—
51#2354C2212yes—
52#2352—22no—
53D4×C4⋊C4C2212yes—
54#2345C2212yes—
55#2342—22no—
56D4×C22⋊C4C2212yes—
57#2327C2214yescomplemented
58#2319C2212yes—
59#2318—22no—
60#2285C2211yescharacteristic
61#2283C2211yescharacteristic
62#2280—21no—
63C2×#380—21no—
64#2275C2212yes—
65#2085C2214yescomplemented
66#2073C2214yes—
67#2068C2214yescomplemented
68#2066C2214yescomplemented
69#2055C2214yes—
70#2055C2214yes—
71#2043C2214yescomplemented
72#2013C2214yes—
73#1990C2212yescomplemented
74#1981C2214yescomplemented
75#1956C2212yescomplemented
76#1949C2214yescomplemented
77#1939C2212yescomplemented
78#1939C2212yescomplemented
79#1928C2214yescomplemented
80#1927C2214yescomplemented
81#1892C2212yes—
82#1892C2212yes—
83#1877C2214yes—
84#1877C2214yes—
85#1850C2212yes—
86#1850C2212yes—
87#1823C2212yes—
88#1804C2214yes—
89#1803C2212yescomplemented
90#1803C2212yescomplemented
91#1787C2212yes—
92#1768C2214yescomplemented
93#1763C2214yescomplemented
94C4×#103C2214yescomplemented
95C23×Q8C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16—24no—
99C22×SD16—28no—
100C22×SD16C2312yes—
101C22×SD16C2312yes—
102C22×SD16C2312yes—
103C22×SD16C2312yes—
104C23×C8C2311yescharacteristic
105C23×C8C2311yescharacteristic
106#533—24no—
107#533—24no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#125—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—24no—
125C2×C4×D4D418yes—
126C2×C4×D4C2314yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—21no—
133C2×#117—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—21no—
139C22×C22⋊C4—22no—
140C22×C42—21no—
141C22×C42C2311yescharacteristic
142#484—28no—
143#474—24no—
144#474—24no—
145#461—216no—
146#461—24no—
147#461—24no—
148#450—28no—
149#440—28no—
150#440—28no—
151C4×SD16—28no—
152C4×SD16—28no—
153C4×SD16—28no—
154C4×SD16—28no—
155#432—24no—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106—24no—
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×C4⋊C8C2314yes—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2314yes—
165C2×#103—24no—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#102—24no—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C2×#102C2312yes—
179C2×C22⋊C8C2312yes—
180C2×#97C2312yes—
181C2×C4×C8—24no—
182C2×C4×C8C2312yes—
183C2×C4×C8C2312yes—
184#400—24no—
185#400—24no—
186#398—24no—
187#395—24no—
188#391—22no—
189#391—22no—
190#391C2314yes—
191#391C2314yes—
192#390C2314yes—
193#389—22no—
194#389—22no—
195#389C2314yes—
196#388—24no—
197#388—24no—
198#387—24no—
199#387—24no—
200#387—24no—
201#387—24no—
202#387—24no—
203#386—24no—
204#385—22no—
205#385—22no—
206#385—24no—
207#384C2312yes—
208#383D412yes—
209#382—21no—
210#382D412yes—
211#382C2314yes—
212#381—22no—
213#381—22no—
214#381—22no—
215#381D412yes—
216#380—21no—
217#380—22no—
218#380—22no—
219#380—22no—
220#380—22no—
221#380D411yescharacteristic
222#380D411yescharacteristic
223#380C2312yes—
224#379—22no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2314yes—
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—24no—
232C4×C22⋊C4C2312yes—
233#376C2312yes—
234C2×#95—22no—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C25—21no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—42no—
248C22×Q8C2412yes—
249C22×Q8C2412yes—
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—42no—
254C22×D4C2×D414yes—
255C22×D4C2411yescharacteristic
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—28no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—24no—
266C23×C4—84no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×SD16—232no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—28no—
278C2×SD16—24no—
279C2×SD16—44no—
280C2×SD16#4118yes—
281C22×C8—22no—
282C22×C8—24no—
283C22×C8—22no—
284C22×C8—42no—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—28no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—44no—
295#125—44no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—44no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—44no—
316C4×D4C2×D414yes—
317C4×D4C2×D414yes—
318C4×D4C2×D414yes—
319C4×D4C2×D414yes—
320#117—44no—
321#117—82no—
322#117—84no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4—84no—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2414yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C22⋊C4—28no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—84no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D414yes—
372C2×C42#4114yes—
373C2×C42C2414yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—24no—
377#107—24no—
378#106—24no—
379#106—24no—
380C4⋊C8—28no—
381C4⋊C8—22no—
382C4⋊C8—22no—
383C4⋊C8—44no—
384#103—28no—
385#103—28no—
386#103—24no—
387#103—24no—
388#103—24no—
389#103—24no—
390#103—44no—
391#103—44no—
392#102—24no—
393#102—24no—
394#102—24no—
395#102—24no—
396C22⋊C8—24no—
397#97—24no—
398C4×C8—24no—
399C4×C8—24no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D414yes—
413#95C2×D414yes—
414#95C2×D414yes—
415C24—21no—
416C24—21no—
417C24—24no—
418C24—44no—
419C24—42no—
420C24—42no—
421C24—82no—
422C24C22×D411yescharacteristic
423C2×Q8—22no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—44no—
434C2×Q8—48no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—44no—
439C2×Q8C2×#4114yes—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—44no—
448C2×D4—48no—
449C2×D4—44no—
450C2×D4C22×D412yes—
451C2×D4C22×D412yes—
452C2×D4C2×#4114yes—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—816no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—82no—
484C22×C4#12114yes—
485C22×C4#12114yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C2×#4112yes—
495C22×C4C2511yescharacteristic
496SD16—216no—
497SD16—216no—
498SD16—48no—
499SD16—48no—
500C2×C8—28no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—44no—
511C2×C8C2×#4114yes—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—84no—
535C4⋊C4—82no—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D412yes—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C42—24no—
550C42—24no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42—82no—
555C42—84no—
556C42#12114yes—
557C42#12114yes—
558C42C22×D412yes—
559C42C22×D412yes—
560C42C2×#4112yes—
561C42C2×#4112yes—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—22no—
567C23—48no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—48no—
573C23—41no—
574C23—42no—
575C23—41no—
576C23—84no—
577C23—88no—
578C23—88no—
579C23C2×#12112yes—
580C23#57712yes—
581C23#57812yes—
582C23C23×D411yescharacteristic
583Q8—24no—
584Q8—24no—
585Q8—48no—
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—44no—
591Q8—84no—
592D4—22no—
593D4—22no—
594D4—22no—
595D4—24no—
596D4—48no—
597D4—42no—
598D4#52514yes—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—168no—
641C2×C4C2×#12112yes—
642C2×C4C2×#12112yes—
643C2×C4C2×#12112yes—
644C2×C4C2×#12112yes—
645C2×C4C2×#12112yes—
646C2×C4#57714yes—
647C2×C4C23×D411yescharacteristic
648C2×C4C22×#4112yes—
649C2×C4C22×#4111yescharacteristic
650C2×C4#58512yes—
651C8—24no—
652C8—42no—
653C8—42no—
654C8—42no—
655C22—22no—
656C22—24no—
657C22—24no—
658C22—22no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—816no—
668C22—82no—
669C22—82no—
670C22#304614yes—
671C22C22×#12111yescharacteristic
672C22D4×#4112yes—
673C22C2×#57712yes—
674C22C2×#57711yescharacteristic
675C22C2×#57811yescharacteristic
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4#304214yes—
690C4C2×#52511yescharacteristic
691C4#344911yescharacteristic
692C4#347512yes—
693C2—24no—
694C2—44no—
695C2—82no—
696C2#3473711yescharacteristic
697C2C2×#304211yescharacteristic
698C2C2×#304611yescharacteristic
699C2#6148112yes—
700C2#6149712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
3. #35248C2semidirect
5. #24816C2semidirect
6. #24730C2semidirect
7. #24681C2semidirect
8. #24630C2semidirect
9. #24579C2semidirect
11. #24482C2semidirect
13. #23689C2semidirect
14. #21514C2semidirect
15. #21400C2semidirect
16. #21351C2semidirect
17. #20740C2semidirect
46. #2568C22semidirect
47. #2546C22semidirect
57. #2327C22semidirect
65. #2085C22semidirect
67. #2068C22semidirect
68. #2066C22semidirect
71. #2043C22semidirect
73. #1990C22semidirect
74. #1981C22semidirect
75. #1956C22semidirect
76. #1949C22semidirect
77. #1939C22semidirect
78. #1939C22semidirect
79. #1928C22semidirect
80. #1927C22semidirect
89. #1803C22semidirect
90. #1803C22semidirect
92. #1768C22semidirect
93. #1763C22semidirect
94. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C22×C4×SD162
#352482
#341942
#248162
#247302
#246812
#246302
#245792
#245272
#244822
#243582
#236892
#215142
#214002
#213512
#207402