Finite Group Databasebeta

← #613199

Subgroups — #613199

Order 512.

5789 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic
3C2×#2412C212yesmaximal
4#15092C211yesmaximal, characteristic, complemented
5#15091C211yesmaximal, characteristic, complemented
6#15073C211yesmaximal, characteristic, complemented
7#15069C211yesmaximal, characteristic, complemented
8#14892C214yesmaximal, complemented
9#14888C214yesmaximal, complemented
10#14706C212yesmaximal, complemented
11#14705C212yesmaximal, complemented
12#14135C214yesmaximal, complemented
13#13878C214yesmaximal, complemented
14#13869C212yesmaximal, complemented
15C2×C22⋊#95C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×#120C2211yescharacteristic
20C22×#117—28no—
21C22×#117—216no—
22C22×#117C2214yes—
23C22×#117C2214yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C22⋊C4C2212yes—
27C23×C22⋊C4C2212yes—
28C23×C42C2212yes—
29#2849—28no—
30#2412—216no—
31C2×#397C2214yes—
32C2×#394C2214yes—
33C2×#386C2212yes—
34C2×#386C2211yescharacteristic
35C2×#386C2211yescharacteristic
36C2×#386C2214yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#385C2211yescharacteristic
40C2×#385C2211yescharacteristic
41C2×#385C2212yes—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#380C2214yes—
47C2×#379C2211yescharacteristic
48C22×#95C2214yes—
49#1444C2214yescomplemented
50#1444C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2218yes—
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1437C2218yescomplemented
59#1436C2218yescomplemented
60#1436C2218yescomplemented
61#1436C2218yescomplemented
62C22⋊#95C2214yes—
63C22⋊#95C2218yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2211yescharacteristic
69C26C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C24×C4—28no—
74C24×C4—216no—
75C24×C4D418yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2314yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×#123—28no—
84C2×#123—28no—
85C2×#123—216no—
86C2×#123—28no—
87C2×#123—44no—
88C2×#120—24no—
89C2×#120—28no—
90C2×#117—2128no—
91C2×#117—216no—
92C2×#117—216no—
93C2×#117—232no—
94C2×#117—232no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4D4116yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C42—28no—
128C22×C42—28no—
129C22×C42—28no—
130C22×C42C2318yes—
131#397—216no—
132#394—216no—
133#386—24no—
134#386—28no—
135#386—24no—
136#386—28no—
137#386—216no—
138#385—24no—
139#385—28no—
140#385—24no—
141#385—28no—
142#381—28no—
143#381—28no—
144#381—28no—
145#381—44no—
146#380—216no—
147#379—24no—
148C2×#95—216no—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C25—24no—
165C25—28no—
166C25—22no—
167C25—22no—
168C25—44no—
169C25C2×D418yes—
170C25C2×D414yes—
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—22no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—24no—
185C22×D4—44no—
186C23×C4—2128no—
187C23×C4—232no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—264no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—216no—
208C23×C4—28no—
209C23×C4—48no—
210C23×C4—44no—
211C23×C4—48no—
212C23×C4C2×D418yes—
213C23×C4C2×D414yes—
214C23×C4C2×D418yes—
215C23×C4C2×D414yes—
216C23×C4C2×D418yes—
217C23×C4C2×D418yes—
218C23×C4#41116yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230#123—432no—
231#123—432no—
232#120—416no—
233#117—2128no—
234#117—2128no—
235C2×C4⋊C4—232no—
236C2×C4⋊C4—232no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—48no—
250C2×C22⋊C4—264no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—432no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4C2×D4116yes—
289C2×C22⋊C4C2×D4116yes—
290C2×C22⋊C4#4118yes—
291C2×C42—264no—
292C2×C42—216no—
293C2×C42—216no—
294C2×C42—216no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—216no—
305#95—28no—
306#95—216no—
307#95—48no—
308#95—48no—
309#95—48no—
310C24—24no—
311C24—22no—
312C24—21no—
313C24—21no—
314C24—24no—
315C24—24no—
316C24—216no—
317C24—232no—
318C24—28no—
319C24—28no—
320C24—216no—
321C24—216no—
322C24—22no—
323C24—24no—
324C24—21no—
325C24—21no—
326C24—24no—
327C24—21no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—24no—
334C24—48no—
335C24—416no—
336C24—432no—
337C24—416no—
338C24—416no—
339C24#120116yes—
340C24C22×D412yes—
341C24C22×D414yes—
342C24C22×D412yes—
343C24C22×D414yes—
344C24C22×D412yes—
345C24C22×D414yes—
346C24C2×#4118yes—
347C24C2×#4114yes—
348C24H(5,𝔽2)12yes—
349C242−1+412yes—
350C24C2511yescharacteristic
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—416no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—88no—
362C22×C4—2256no—
363C22×C4—232no—
364C22×C4—232no—
365C22×C4—264no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—264no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—432no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—88no—
434C22×C4—816no—
435C22×C4#123132yescomplemented
436C22×C4C22×D418yes—
437C22×C4C22×D418yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C2×#4118yes—
441C22×C4C2×#4118yes—
442C22×C4C2×#4118yes—
443C4⋊C4—232no—
444C4⋊C4—232no—
445C4⋊C4—416no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—88no—
449C22⋊C4—232no—
450C22⋊C4—232no—
451C22⋊C4—216no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—88no—
465C42—264no—
466C23—24no—
467C23—28no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—216no—
475C23—216no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—232no—
483C23—24no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—232no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—416no—
493C23—432no—
494C23—464no—
495C23—464no—
496C23—432no—
497C23—48no—
498C23—44no—
499C23—44no—
500C23—41no—
501C23—42no—
502C23—44no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—48no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23C2×#12014yes—
513C23C2×#12014yes—
514C23C2×#12014yes—
515C23C2×#12318yes—
516C23#525116yes—
517C23#53214yes—
518C23#53714yes—
519C23#54618yes—
520C23#54718yes—
521C23C23×D411yescharacteristic
522C23C23×D412yes—
523C23C22×#4114yes—
524C23C22×#4112yes—
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×2−1+411yescharacteristic
527C23#58514yes—
528D4—88no—
529D4—88no—
530D4—88no—
531C2×C4—232no—
532C2×C4—232no—
533C2×C4—264no—
534C2×C4—264no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—216no—
540C2×C4—216no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—216no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—432no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—432no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—832no—
586C2×C4C2×#12318yes—
587C2×C4C2×#12318yes—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—28no—
593C22—28no—
594C22—216no—
595C22—216no—
596C22—216no—
597C22—216no—
598C22—28no—
599C22—28no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—464no—
604C22—44no—
605C22—48no—
606C22—48no—
607C22—44no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—82no—
612C22—84no—
613C22—84no—
614C22—82no—
615C22#243018yes—
616C22#284918yes—
617C22C22×#12011yescharacteristic
618C22C22×#12312yes—
619C22C2×#52514yes—
620C22C2×#53211yescharacteristic
621C22C2×#53711yescharacteristic
622C22#344912yes—
623C22C2×#54612yes—
624C22C2×#54712yes—
625C22#347518yes—
626C22#348514yes—
627C22#349414yes—
628C22#349514yes—
629C4—28no—
630C4—28no—
631C4—48no—
632C4—48no—
633C4—48no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—82no—
638C4—84no—
639C4—82no—
640C2—28no—
641C2—28no—
642C2—44no—
643C2—161no—
644C2C2×#243011yescharacteristic
645C2#3481012yes—
646C2#3508811yescharacteristic
647C2#3526312yes—
648C2#3634014yes—
649C2#3634114yes—
650C2C2×#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×#117C2semidirect
4. #15092C2semidirect
5. #15091C2semidirect
6. #15073C2semidirect
7. #15069C2semidirect
8. #14892C2semidirect
9. #14888C2semidirect
10. #14706C2semidirect
11. #14705C2semidirect
12. #14135C2semidirect
13. #13878C2semidirect
14. #13869C2semidirect
15. C2×C22⋊#95C2semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1437C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
435. C22×C4#123semidirect

Maximal subgroups

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

subgroupindex
C23×#1172
C2×#28492
C2×#24122
#150922
#150912
#150732
#150692
#148922
#148882
#147062
#147052
#141352
#138782
#138692
C2×C22⋊#952