Finite Group Databasebeta

← #4452986

Subgroups — #4452986

Order 512.

2913 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2622C211yesmaximal, characteristic
2C22×#397C211yesmaximal, characteristic, complemented
3#16858C214yesmaximal, complemented
4#16856C214yesmaximal, complemented
5#16736C214yesmaximal, complemented
6#16722C214yesmaximal, complemented
7#15073C211yesmaximal, characteristic, complemented
8#15069C211yesmaximal, characteristic, complemented
9#14955C211yesmaximal, characteristic, complemented
10#14945C211yesmaximal, characteristic, complemented
11#14918C212yesmaximal, complemented
12#14911C212yesmaximal
13#13996C212yesmaximal, complemented
14#13972C212yesmaximal, complemented
15#13775C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#120C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21#2622—28no—
22C2×#397—28no—
23C2×#397C2212yes—
24C2×#397C2214yes—
25C2×#397C2214yes—
26C2×#397C2212yes—
27C2×#394C2212yes—
28C2×#394C2212yes—
29C2×#386C2211yescharacteristic
30C2×#386C2211yescharacteristic
31C2×#386C2212yes—
32C2×#385C2211yescharacteristic
33C2×#385C2211yescharacteristic
34C2×#385C2212yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#379C2212yes—
39C2×#379C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C22×#95C2211yescharacteristic
42C22×#95C2212yes—
43#1445C2214yescomplemented
44#1444C2214yescomplemented
45#1443C2214yescomplemented
46#1443C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yescomplemented
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1438C2214yes—
62#1438C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yescomplemented
66#1436C2214yescomplemented
67C4⋊#95C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2211yescharacteristic
73C26C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×#123—28no—
82C2×#123—44no—
83C2×#123—44no—
84C2×#120—24no—
85C2×#120—44no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4D418yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C42C2314yes—
118#397—28no—
119#397—28no—
120#397—216no—
121#397—216no—
122#394—28no—
123#394—28no—
124#386—24no—
125#386—24no—
126#386—44no—
127#385—24no—
128#385—24no—
129#385—44no—
130#381—28no—
131#381—44no—
132#381—44no—
133#379—24no—
134#379—28no—
135C4×C22⋊C4—44no—
136C2×#95—28no—
137C2×#95—24no—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25—22no—
161C25—22no—
162C25—24no—
163C25—24no—
164C25—44no—
165C25—44no—
166C25C2×D414yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×D4—22no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—44no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211#123—432no—
212#120—416no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—416no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—432no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C2×D418yes—
276C2×C22⋊C4C2×D418yes—
277C2×C22⋊C4#4118yes—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—44no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
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—
303C24—24no—
304C24—24no—
305C24—28no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—28no—
319C24—28no—
320C24—21no—
321C24—22no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—48no—
328C24—48no—
329C24—416no—
330C24—416no—
331C24—48no—
332C24—48no—
333C24—88no—
334C24—88no—
335C24—88no—
336C24—88no—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C22×D412yes—
340C24C2×#4114yes—
341C24H(5,𝔽2)14yes—
342C24H(5,𝔽2)12yes—
343C24H(5,𝔽2)12yes—
344C24C2511yescharacteristic
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—88no—
355C2×D4—88no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C2×#4118yes—
433C4⋊C4—416no—
434C4⋊C4—416no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—88no—
438C22⋊C4—216no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—48no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C22⋊C4—88no—
457C42—88no—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—432no—
477C23—416no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—88no—
508C23—816no—
509C23—88no—
510C23—832no—
511C23—832no—
512C23#53214yes—
513C23#53414yes—
514C23#53514yes—
515C23#53818yes—
516C23#54018yes—
517C23C23×D411yescharacteristic
518C23C22×#4112yes—
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23#58514yes—
523C23#58514yes—
524D4—88no—
525D4—88no—
526D4—88no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—416no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
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—88no—
588C2×C4—816no—
589C2×C4—816no—
590C22—216no—
591C22—216no—
592C22—28no—
593C22—24no—
594C22—28no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—832no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—81no—
614C22—82no—
615C22—81no—
616C22—82no—
617C22—81no—
618C22#266118yes—
619C22C2×#53211yescharacteristic
620C22C2×#53411yescharacteristic
621C22C2×#53511yescharacteristic
622C22C2×#53812yes—
623C22C2×#54012yes—
624C22#348512yes—
625C22#348512yes—
626C22#352114yes—
627C22#352114yes—
628C22#353314yes—
629C22#353314yes—
630C22#353314yes—
631C22#354514yes—
632C4—48no—
633C4—44no—
634C4—44no—
635C4—84no—
636C4—84no—
637C4—81no—
638C4—82no—
639C4—84no—
640C4—81no—
641C4—82no—
642C4—81no—
643C2—28no—
644C2—82no—
645C2—82no—
646C2—161no—
647C2C2×#266111yescharacteristic
648C2#3546712yes—
649C2#3695612yes—
650C2#3696511yescharacteristic
651C2#3711911yescharacteristic
652C2#4147214yes—
653C2#4147314yes—
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
2. C22×#397C2semidirect
3. #16858C2semidirect
4. #16856C2semidirect
5. #16736C2semidirect
6. #16722C2semidirect
7. #15073C2semidirect
8. #15069C2semidirect
9. #14955C2semidirect
10. #14945C2semidirect
11. #14918C2semidirect
13. #13996C2semidirect
14. #13972C2semidirect
15. #13775C2semidirect
43. #1445C22semidirect
44. #1444C22semidirect
45. #1443C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
54. #1440C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
65. #1437C22semidirect
66. #1436C22semidirect
67. C4⋊#95C22semidirect
68. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26222
C22×#3972
#168582
#168562
#167362
#167222
#150732
#150692
#149552
#149452
#149182
#149112
#139962
#139722
#137752