Finite Group Databasebeta

← #4388147

Subgroups — #4388147

Order 512.

2754 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C211yesmaximal, characteristic
2C2×#2483C211yesmaximal, characteristic
3C2×#2483C211yesmaximal, characteristic
4C2×#2312C211yesmaximal, characteristic
5#16396C214yesmaximal, complemented
6#15929C214yesmaximal, complemented
7#15915C214yesmaximal, complemented
8#15717C214yesmaximal, complemented
9#15560C214yesmaximal, complemented
10#14837C214yesmaximal, complemented
11#14673C211yesmaximal, characteristic
12#14341C211yesmaximal, characteristic
13#14341C211yesmaximal, characteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17#2483—28no—
18#2483—28no—
19#2483—28no—
20C2×#397C2214yes—
21C2×#397C2214yes—
22C2×#397C2214yes—
23#2312—28no—
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×#381C2212yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#380C2212yes—
39C2×#379C2211yescharacteristic
40C2×#379C2211yescharacteristic
41C2×#379C2211yescharacteristic
42C2×#379C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44#1445C2214yescomplemented
45#1443C2214yescomplemented
46#1443C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65C4⋊#95C2214yescomplemented
66#1434C2214yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C26C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—24no—
91C22×C22⋊C4D418yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42C2314yes—
111#397—216no—
112#397—216no—
113#397—216no—
114#388—216no—
115#388—216no—
116#385—28no—
117#385—28no—
118#385—28no—
119#385—28no—
120#385—28no—
121#385—28no—
122#381—28no—
123#381—28no—
124#381—28no—
125#381—28no—
126#381—28no—
127#381—28no—
128#380—28no—
129#379—24no—
130#379—24no—
131#379—24no—
132#379D418yes—
133C4×C22⋊C4—28no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25—42no—
157C25—42no—
158C25—42no—
159C25—42no—
160C25—42no—
161C25—42no—
162C25C2×D414yes—
163C25C2×D414yes—
164C25#4114yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C22⋊C4—232no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—416no—
266C2×C22⋊C4—416no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4C2×D418yes—
276C2×C22⋊C4C2×D418yes—
277C2×C22⋊C4C2×D418yes—
278C2×C22⋊C4C2×D418yes—
279C2×C22⋊C4C2×D418yes—
280C2×C42—28no—
281C2×C42—44no—
282C2×C42—44no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24—44no—
330C24—44no—
331C24—44no—
332C24—44no—
333C24—44no—
334C24—44no—
335C24—88no—
336C24—88no—
337C24—88no—
338C24—88no—
339C24#121116yes—
340C24C22×D412yes—
341C24C22×D412yes—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C22×D412yes—
346C24C2×#4112yes—
347C24C2×#4112yes—
348C24C2×#4112yes—
349C24C2511yescharacteristic
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D418yes—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C22⋊C4—216no—
444C22⋊C4—216no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—88no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—416no—
499C23—416no—
500C23—416no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—88no—
514C23—832no—
515C23—832no—
516C23—88no—
517C23—88no—
518C23—88no—
519C23C2×#12114yes—
520C23C2×#12114yes—
521C23C2×#12114yes—
522C23C23×D411yescharacteristic
523C23C23×D411yescharacteristic
524C23C22×#4111yescharacteristic
525C23#58512yes—
526C23#58512yes—
527C23#58512yes—
528C23#58512yes—
529C23#58512yes—
530C23#58512yes—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—88no—
590C2×C4—88no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—88no—
594C2×C4—88no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—48no—
608C22—48no—
609C22—48no—
610C22—48no—
611C22—48no—
612C22—48no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—832no—
623C22C22×#12111yescharacteristic
624C22#344912yes—
625C22#348512yes—
626C22#348512yes—
627C22#348512yes—
628C22#348512yes—
629C22#348512yes—
630C22#349111yescharacteristic
631C22#349111yescharacteristic
632C22#349111yescharacteristic
633C22#349111yescharacteristic
634C22#349111yescharacteristic
635C22#349111yescharacteristic
636C22#356614yes—
637C22#356614yes—
638C22#356614yes—
639C22#356814yes—
640C4—44no—
641C4—44no—
642C4—48no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—84no—
648C4—84no—
649C4—84no—
650C2—44no—
651C2—44no—
652C2—44no—
653C2#3508911yescharacteristic
654C2#3544711yescharacteristic
655C2#3544711yescharacteristic
656C2#4141414yes—
657C2#4683811yescharacteristic
658C2#4683811yescharacteristic
659C2#4683811yescharacteristic
660C2#4719611yescharacteristic
661C2#4877514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #16396C2semidirect
6. #15929C2semidirect
7. #15915C2semidirect
8. #15717C2semidirect
9. #15560C2semidirect
10. #14837C2semidirect
44. #1445C22semidirect
45. #1443C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. C4⋊#95C22semidirect
66. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C2×#23122
#163962
#159292
#159152
#157172
#155602
#148372
#146732
#143412