Finite Group Databasebeta

← #613201

Subgroups — #613201

Order 512.

7249 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic, complemented
2C2×#2853C211yesmaximal, characteristic, complemented
3C2×#2411C212yesmaximal, complemented
4#15092C211yesmaximal, characteristic, complemented
5#15091C211yesmaximal, characteristic, complemented
6#14812C214yesmaximal, complemented
7#14806C214yesmaximal, complemented
8#14724C211yesmaximal, characteristic, complemented
9#14715C211yesmaximal, characteristic
10#14694C212yesmaximal, complemented
11#14693C212yesmaximal, complemented
12#14190C214yesmaximal, complemented
13#14189C214yesmaximal, complemented
14#13868C212yesmaximal, complemented
15#13756C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C25×C4C2212yes—
18C22×#123C2211yescharacteristic
19C22×#120C2212yes—
20C22×C4×D4—28no—
21C22×C4×D4—216no—
22C22×C4×D4C2214yes—
23C22×C4×D4C2214yes—
24C22×#117C2212yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C22⋊C4C2212yes—
27C23×C42C2211yescharacteristic
28#2853—28no—
29#2411—216no—
30C2×#396C2214yescomplemented
31C2×#394C2214yes—
32C2×#394C2214yescomplemented
33C2×#392C2214yes—
34C2×#390C2214yescomplemented
35C2×#388C2212yescomplemented
36C2×#388C2212yescomplemented
37C2×#388C2214yescomplemented
38C2×#387C2212yescomplemented
39C2×#387C2212yescomplemented
40C2×#386C2211yescharacteristic
41C2×#386C2211yescharacteristic
42C2×#386C2212yes—
43C2×#386C2218yescomplemented
44C2×#385C2211yescharacteristic
45C2×#385C2211yescharacteristic
46C2×#385C2212yes—
47C2×#385C2218yescomplemented
48C2×#381C2211yescharacteristic
49C2×#379C2212yes—
50C2×#379C2212yes—
51C2×#379C2212yes—
52#1444C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1439C2218yescomplemented
56#1439C2214yescomplemented
57#1438C2214yes—
58#1438C2214yes—
59#1437C2218yescomplemented
60C4⋊#95C2214yescomplemented
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2218yescomplemented
67C22⋊#95C2218yescomplemented
68C4×#95C2214yescomplemented
69C26C2312yes—
70C23×D4—28no—
71C23×D4—216no—
72C23×D4—24no—
73C23×D4C2311yescharacteristic
74C23×D4C2314yes—
75C23×D4C2314yes—
76C24×C4—28no—
77C24×C4—28no—
78C24×C4—28no—
79C24×C4—416no—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2318yes—
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C2×#123—24no—
86C2×#123—28no—
87C2×#120—28no—
88C2×#120—28no—
89C2×#120—44no—
90C2×C4×D4—2128no—
91C2×C4×D4—232no—
92C2×C4×D4—232no—
93C2×C4×D4—216no—
94C2×C4×D4—216no—
95C2×#117—28no—
96C2×#117—28no—
97C2×#117—44no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4D4116yescomplemented
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2318yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42—28no—
128C22×C42D418yescomplemented
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#396—216no—
133#394—216no—
134#394—216no—
135#392—216no—
136#390—216no—
137#388—216no—
138#388—28no—
139#388—28no—
140#387—28no—
141#387—28no—
142#386—24no—
143#386—24no—
144#386—28no—
145#386—416no—
146#385—24no—
147#385—24no—
148#385—28no—
149#385—416no—
150#381—24no—
151#379—28no—
152#379—28no—
153#379—44no—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2318yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C25—24no—
168C25—24no—
169C25—216no—
170C25—24no—
171C25—24no—
172C25—24no—
173C25—24no—
174C25—48no—
175C25C2×D418yes—
176C25C2411yescharacteristic
177C25C2414yes—
178C25C2411yescharacteristic
179C22×D4—232no—
180C22×D4—2128no—
181C22×D4—264no—
182C22×D4—216no—
183C22×D4—216no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—28no—
190C22×D4—216no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—28no—
194C22×D4—44no—
195C22×D4—48no—
196C22×D4#41116yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—264no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—216no—
208C23×C4—216no—
209C23×C4—216no—
210C23×C4—216no—
211C23×C4—216no—
212C23×C4—48no—
213C23×C4—48no—
214C23×C4—48no—
215C23×C4—464no—
216C23×C4—464no—
217C23×C4—464no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D418yes—
222C23×C4C2×D418yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235#123—416no—
236#120—432no—
237C4×D4—2128no—
238C4×D4—2128no—
239#117—432no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—48no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—264no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4C2×D4116yes—
297C2×C22⋊C4C2×D4116yes—
298C2×C22⋊C4#4118yes—
299C2×C42—232no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—44no—
305C2×C42C2×D418yes—
306C2×C42C2×D418yes—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—216no—
318#95—48no—
319#95—48no—
320C24—24no—
321C24—24no—
322C24—232no—
323C24—232no—
324C24—216no—
325C24—232no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—28no—
331C24—24no—
332C24—28no—
333C24—28no—
334C24—24no—
335C24—216no—
336C24—216no—
337C24—21no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—21no—
343C24—21no—
344C24—416no—
345C24—432no—
346C24—432no—
347C24—432no—
348C24—432no—
349C24—416no—
350C24—416no—
351C24—48no—
352C24—48no—
353C24—48no—
354C24—48no—
355C24C22×D418yes—
356C24C22×D412yes—
357C24C22×D414yes—
358C24C2×#41116yes—
359C24C2×#4114yes—
360C24H(5,𝔽2)12yes—
361C242−1+412yes—
362C24C2511yescharacteristic
363C2×D4—232no—
364C2×D4—2256no—
365C2×D4—232no—
366C2×D4—264no—
367C2×D4—264no—
368C2×D4—232no—
369C2×D4—416no—
370C2×D4—432no—
371C2×D4—44no—
372C2×D4—416no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—88no—
378C2×D4—816no—
379C22×C4—232no—
380C22×C4—232no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—264no—
400C22×C4—216no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—416no—
435C22×C4—432no—
436C22×C4—432no—
437C22×C4—432no—
438C22×C4—432no—
439C22×C4—4256no—
440C22×C4—432no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—432no—
444C22×C4#120116yes—
445C22×C4C22×D418yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4118yes—
454C22×C4H(5,𝔽2)18yes—
455C4⋊C4—232no—
456C4⋊C4—416no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—88no—
462C22⋊C4—216no—
463C22⋊C4—232no—
464C22⋊C4—232no—
465C22⋊C4—432no—
466C22⋊C4—416no—
467C22⋊C4—44no—
468C22⋊C4—416no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—816no—
479C42—216no—
480C42—216no—
481C42—416no—
482C23—24no—
483C23—28no—
484C23—24no—
485C23—216no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—264no—
490C23—28no—
491C23—28no—
492C23—216no—
493C23—216no—
494C23—28no—
495C23—28no—
496C23—216no—
497C23—216no—
498C23—28no—
499C23—28no—
500C23—216no—
501C23—216no—
502C23—464no—
503C23—464no—
504C23—432no—
505C23—464no—
506C23—464no—
507C23—464no—
508C23—464no—
509C23—44no—
510C23—48no—
511C23—44no—
512C23—432no—
513C23—44no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—41no—
521C23—42no—
522C23—44no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—48no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23C2×#12014yes—
533C23#525116yes—
534C23#53214yes—
535C23#53714yes—
536C23#54618yes—
537C23#54718yes—
538C23C23×D411yescharacteristic
539C23C23×D412yes—
540C23C22×#4114yes—
541C23C22×#4112yes—
542C23C2×H(5,𝔽2)14yes—
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×2−1+411yescharacteristic
545D4—2128no—
546D4—832no—
547D4—88no—
548C2×C4—232no—
549C2×C4—232no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—216no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—232no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—216no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—216no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—4128no—
571C2×C4—432no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4C2×#12014yes—
603C2×C4C2×#12014yes—
604C2×C4#52518yes—
605C2×C4#53518yes—
606C22—24no—
607C22—28no—
608C22—24no—
609C22—28no—
610C22—216no—
611C22—216no—
612C22—216no—
613C22—216no—
614C22—216no—
615C22—4128no—
616C22—416no—
617C22—48no—
618C22—416no—
619C22—48no—
620C22—48no—
621C22—416no—
622C22—48no—
623C22—416no—
624C22—81no—
625C22—82no—
626C22—81no—
627C22—82no—
628C22—81no—
629C22—84no—
630C22—84no—
631C22#243018yes—
632C22C22×#12011yescharacteristic
633C22C2×#52512yes—
634C22C2×#52514yes—
635C22C2×#53211yescharacteristic
636C22C2×#53512yes—
637C22C2×#53711yescharacteristic
638C22C2×#54612yes—
639C22C2×#54712yes—
640C22#348118yes—
641C22#348414yes—
642C22#351914yes—
643C22#352014yes—
644C4—28no—
645C4—28no—
646C4—24no—
647C4—24no—
648C4—48no—
649C4—48no—
650C4—81no—
651C4—82no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—84no—
656C2—28no—
657C2—416no—
658C2—161no—
659C2C2×#243011yescharacteristic
660C2#3496512yes—
661C2#3522211yescharacteristic
662C2#3523612yes—
663C2#3631614yes—
664C2#3631714yes—
665C2#3661411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14715C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4×D4C2semidirect
2. C2×#2853C2semidirect
3. C2×#2411C2semidirect
4. #15092C2semidirect
5. #15091C2semidirect
6. #14812C2semidirect
7. #14806C2semidirect
8. #14724C2semidirect
10. #14694C2semidirect
11. #14693C2semidirect
12. #14190C2semidirect
13. #14189C2semidirect
14. #13868C2semidirect
15. #13756C2semidirect
30. C2×#396C22semidirect
32. C2×#394C22semidirect
34. C2×#390C22semidirect
35. C2×#388C22semidirect
36. C2×#388C22semidirect
37. C2×#388C22semidirect
38. C2×#387C22semidirect
39. C2×#387C22semidirect
43. C2×#386C22semidirect
47. C2×#385C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
59. #1437C22semidirect
60. C4⋊#95C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C4×#95C22semidirect
109. C22×C22⋊C4D4semidirect
128. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D42
C2×#28532
C2×#24112
#150922
#150912
#148122
#148062
#147242
#147152
#146942
#146932
#141902
#141892
#138682
#137562