Finite Group Databasebeta

← #6484738

Subgroups — #6484738

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3240C212yesmaximal, complemented
2#36631C211yesmaximal, characteristic
3C4×#533C211yesmaximal, characteristic, complemented
4#33112C214yesmaximal, complemented
5#29759C212yesmaximal, complemented
6#29758C212yesmaximal
7#28753C211yesmaximal, characteristic, complemented
8#28753C211yesmaximal, characteristic, complemented
9#28692C211yesmaximal, characteristic, complemented
10#28692C211yesmaximal, characteristic, complemented
11#28659C212yesmaximal, complemented
12#28658C212yesmaximal, complemented
13#28285C214yesmaximal, complemented
14#28209C212yesmaximal, complemented
15#28207C212yesmaximal, complemented
16#21868C212yesmaximal, complemented
17#20810C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2212yes—
20#3240—216no—
21C2×#501C2212yes—
22C2×#501C2212yes—
23C2×#489C2212yes—
24C2×#489C2212yes—
25C2×#485C2214yes—
26C2×#482C2214yes—
27C2×#479C2214yes—
28C2×#478C2214yes—
29C2×#432C2212yes—
30#2861C2212yes—
31#2860—21no—
32#2860—21no—
33#2859C2211yescharacteristic
34#2858C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2854—21no—
38#2854—21no—
39#2850C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125C2212yes—
47C4×#125C2212yes—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50C4×C22⋊Q8—21no—
51C4×C22⋊Q8—21no—
52C4×C22⋊Q8—21no—
53C4×#121—21no—
54C4×#121—21no—
55#2296C2212yes—
56#2295—21no—
57#2295—21no—
58#2293—21no—
59#2293—21no—
60#2285C2211yescharacteristic
61C42×Q8—21no—
62C42×Q8—21no—
63C4×#117C2212yes—
64C2×C43C2211yescharacteristic
65#2084C2212yes—
66#2084C2212yes—
67#2083C2212yes—
68#2083C2212yes—
69#2068C2214yes—
70#2067C2214yes—
71#2066C2214yes—
72#2065C2214yes—
73#2043C2212yes—
74#2043C2212yes—
75#2041C2212yes—
76#2041C2212yes—
77#1992C2212yes—
78#1992C2212yes—
79#1983C2214yes—
80#1983C2214yes—
81#1959C2212yes—
82#1959C2212yes—
83#1949C2214yes—
84#1949C2214yes—
85#1928C2214yes—
86#1927C2214yes—
87#1926C2214yes—
88#1925C2214yes—
89#1875C2214yes—
90#1873C2214yes—
91#1842C2212yes—
92#1831C2212yes—
93#1805C2214yes—
94#1804C2214yes—
95#1797C2212yes—
96#1797C2212yes—
97#1767C2211yescharacteristic
98#1767C2211yescharacteristic
99#1767C2212yes—
100#533—24no—
101#533D418yescomplemented
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106#529—28no—
107#529—28no—
108C2×#126—22no—
109C2×#125C2312yes—
110C2×#125C2312yes—
111C2×#124C2312yes—
112C2×#124C2312yes—
113C2×#123—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121—21no—
119C2×#121C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×#117—22no—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#501—28no—
142#501—28no—
143#489—28no—
144#489—28no—
145#485—216no—
146#482—216no—
147#479—216no—
148#478—216no—
149#432—28no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8C2314yes—
167C2×#97C2312yes—
168C2×#97C2312yes—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391C2312yes—
174#391C2312yes—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390C2312yes—
184#390C2312yes—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#388—22no—
208#388—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#386—22no—
214#386—22no—
215#385—22no—
216#385—22no—
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#382C2312yes—
222#382C2312yes—
223#380—21no—
224#380—21no—
225#380—21no—
226#380—21no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248C4×C22⋊C4—22no—
249#376—22no—
250C43—22no—
251C43D411yescharacteristic
252C43D411yescharacteristic
253C43D411yescharacteristic
254C43D411yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#336C2312yes—
264#336C2312yes—
265#336C2314yes—
266C22×Q8—21no—
267C22×Q8—21no—
268C22×Q8C2412yes—
269C22×D4—22no—
270C22×D4—21no—
271C22×D4—21no—
272C22×D4C2412yes—
273C23×C4—22no—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—42no—
286C4⋊Q8C2×D414yes—
287C4⋊Q8C2×D414yes—
288#127—42no—
289#126—48no—
290#125—24no—
291#125—24no—
292#125—24no—
293#125C2×D418yes—
294#124—24no—
295#124—24no—
296#124—28no—
297#123—48no—
298C22⋊Q8—28no—
299C22⋊Q8—28no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302#121—24no—
303#121—24no—
304#121—44no—
305#121—44no—
306C4×Q8—28no—
307C4×Q8—28no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—48no—
319#117—24no—
320#117—44no—
321#117—48no—
322#117C2×D418yes—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
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—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—42no—
373C2×C42—42no—
374C2×C42—42no—
375C2×C42—42no—
376C2×C42—42no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D414yes—
400C2×C42#4114yes—
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407#107—28no—
408#107—28no—
409#106—28no—
410#106—28no—
411C4⋊C8—28no—
412C4⋊C8—24no—
413C4⋊C8—24no—
414C4⋊C8—44no—
415#103—28no—
416#103—28no—
417#103—44no—
418#103—44no—
419#102—28no—
420#102—28no—
421#102—44no—
422#102—44no—
423C22⋊C8—28no—
424#97—24no—
425#97—24no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442C24—22no—
443C24C22×D411yescharacteristic
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—24no—
449C2×Q8—24no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—42no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4#12114yes—
496C22×C4#12114yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D412yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C2×#4112yes—
512C22×C4C2511yescharacteristic
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4C22×D414yes—
550C4⋊C4C22×D414yes—
551C22⋊C4—28no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C42—28no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—44no—
589C42—44no—
590C42—44no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42#12114yes—
595C42#12114yes—
596C42C22×D412yes—
597C42C22×D412yes—
598C42C2×#4112yes—
599C42C2×#4112yes—
600C422−1+414yes—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23C2×#12112yes—
615C23C23×D411yescharacteristic
616Q8—48no—
617Q8—48no—
618Q8—44no—
619Q8—44no—
620D4—44no—
621D4—44no—
622D4—84no—
623D4—84no—
624D4—88no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—88no—
683C2×C4C2×#12112yes—
684C2×C4C2×#12112yes—
685C2×C4C2×#12112yes—
686C2×C4C2×#12112yes—
687C2×C4C2×#12112yes—
688C2×C4#53612yes—
689C2×C4#53612yes—
690C2×C4#57512yes—
691C2×C4#57512yes—
692C2×C4#57512yes—
693C2×C4#57512yes—
694C2×C4C23×D411yescharacteristic
695C2×C4C22×#4111yescharacteristic
696C2×C4C2×2−1+412yes—
697C2×C4#58512yes—
698C8—44no—
699C8—44no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22C22×#12111yescharacteristic
715C22C2×#53611yescharacteristic
716C22#344911yescharacteristic
717C22C2×#57511yescharacteristic
718C22C2×#57511yescharacteristic
719C22C2×#57511yescharacteristic
720C22C2×#57511yescharacteristic
721C4—22no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—41no—
733C4—41no—
734C4—44no—
735C4—44no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4#349712yes—
742C4#350712yes—
743C2—42no—
744C2—162no—
745C2#3517811yescharacteristic
746C2#6041511yescharacteristic
747C2#6041511yescharacteristic
748C2#6085411yescharacteristic
749C2#6086211yescharacteristic
750C2#6095411yescharacteristic
751C2#6095911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3240C2semidirect
3. C4×#533C2semidirect
4. #33112C2semidirect
5. #29759C2semidirect
7. #28753C2semidirect
8. #28753C2semidirect
9. #28692C2semidirect
10. #28692C2semidirect
11. #28659C2semidirect
12. #28658C2semidirect
13. #28285C2semidirect
14. #28209C2semidirect
15. #28207C2semidirect
16. #21868C2semidirect
17. #20810C2semidirect
101. #533D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32402
#366312
C4×#5332
#331122
#297592
#297582
#287532
#286922
#286592
#286582
#282852
#282092
#282072
#218682
#208102