Finite Group Databasebeta

← #617722

Subgroups — #617722

Order 512.

5905 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#124C211yesmaximal, characteristic, complemented
2C2×#2411C212yesmaximal, complemented
3C2×#2614C211yesmaximal, characteristic, complemented
4C2×#2613C211yesmaximal, characteristic
5C2×#2310C211yesmaximal, characteristic, complemented
6#15321C218yesmaximal, complemented
7#15317C218yesmaximal, complemented
8#14918C212yesmaximal, complemented
9#14911C212yesmaximal, complemented
10#14706C211yesmaximal, characteristic, complemented
11#14705C211yesmaximal, characteristic, complemented
12#14335C211yesmaximal, characteristic, complemented
13#13868C212yesmaximal, complemented
14C24×Q8C2211yescharacteristic
15C24×D4C2211yescharacteristic
16C22×#124—28no—
17C22×#124—216no—
18C22×#124C2218yes—
19C22×#123C2211yescharacteristic
20C22×#123C2211yescharacteristic
21C22×#123C2211yescharacteristic
22C22×#120C2212yes—
23C22×#117C2212yes—
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C42C2211yescharacteristic
27#2614—28no—
28#2613—28no—
29#2411—216no—
30C2×#396C2214yescomplemented
31C2×#395C2214yescomplemented
32C2×#394C2214yes—
33C2×#394C2214yescomplemented
34C2×#393C2214yescomplemented
35C2×#392C2214yes—
36#2310—28no—
37C2×#391C2218yescomplemented
38C2×#390C2214yescomplemented
39C2×#388C2214yescomplemented
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#381C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×#381C2211yescharacteristic
45C2×#381C2212yes—
46C2×#381C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×#379C2212yes—
51#1440C2218yescomplemented
52#1439C2218yescomplemented
53#1439C2218yescomplemented
54#1439C2218yescomplemented
55#1438C2218yes—
56#1438C2218yescomplemented
57#1437C2218yescomplemented
58#1437C2214yescomplemented
59#1436C2218yescomplemented
60#1436C2214yescomplemented
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C26C2312yes—
64C23×Q8—24no—
65C23×Q8—28no—
66C23×Q8—24no—
67C23×Q8—44no—
68C23×Q8D418yescomplemented
69C23×Q8C2314yes—
70C23×D4—28no—
71C23×D4—24no—
72C23×D4—24no—
73C23×D4—24no—
74C23×D4—44no—
75C23×D4C2311yescharacteristic
76C23×D4C2314yes—
77C23×D4C2314yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×#124—2128no—
83C2×#124—232no—
84C2×#124—232no—
85C2×#124—232no—
86C2×#123—24no—
87C2×#123—24no—
88C2×#123—24no—
89C2×#123—24no—
90C2×#123—28no—
91C2×#123—24no—
92C2×#123—44no—
93C2×#120—28no—
94C2×#120—28no—
95C2×#120—44no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4D4116yescomplemented
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2318yes—
127C22×C42—28no—
128C22×C42—24no—
129C22×C42C2314yes—
130C22×C42C2314yes—
131#396—216no—
132#395—216no—
133#394—216no—
134#394—216no—
135#393—216no—
136#392—216no—
137#391—232no—
138#390—216no—
139#388—216no—
140#386—28no—
141#385—28no—
142#381—24no—
143#381—24no—
144#381—24no—
145#381—24no—
146#381—28no—
147#381D418yes—
148#379—28no—
149#379—28no—
150#379—44no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2314yes—
161C25—24no—
162C25—24no—
163C25—24no—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—48no—
168C25—48no—
169C25C2×D418yes—
170C25C2411yescharacteristic
171C25C2414yes—
172C25C2411yescharacteristic
173C22×Q8—232no—
174C22×Q8—232no—
175C22×Q8—28no—
176C22×Q8—28no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—416no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—432no—
183C22×Q8—432no—
184C22×Q8C2×D4116yes—
185C22×D4—232no—
186C22×D4—216no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—28no—
193C22×D4—28no—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—48no—
198C22×D4—432no—
199C22×D4—432no—
200C22×D4—416no—
201C22×D4—44no—
202C22×D4—48no—
203C22×D4—48no—
204C22×D4—416no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—48no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D418yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D418yes—
223C23×C4#4114yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234#124—2256no—
235#123—416no—
236#123—416no—
237#123—416no—
238#120—432no—
239#117—432no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—48no—
253C2×C22⋊C4—264no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—264no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4C2×D4116yes—
298C2×C22⋊C4C2×D4116yes—
299C2×C22⋊C4C2×D418yes—
300C2×C42—232no—
301C2×C42—216no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—44no—
306#95—216no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—48no—
314#95—48no—
315#95C2×D4116yes—
316C24—28no—
317C24—216no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—28no—
325C24—28no—
326C24—28no—
327C24—28no—
328C24—216no—
329C24—216no—
330C24—21no—
331C24—22no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—28no—
336C24—416no—
337C24—416no—
338C24—416no—
339C24—416no—
340C24—416no—
341C24—416no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—816no—
347C24—816no—
348C24—816no—
349C24—816no—
350C24C22×D418yes—
351C24C22×D412yes—
352C24C22×D414yes—
353C24C22×D412yes—
354C24C2×#4112yes—
355C24H(5,𝔽2)14yes—
356C24C2511yescharacteristic
357C2×Q8—232no—
358C2×Q8—232no—
359C2×Q8—416no—
360C2×Q8—4128no—
361C2×Q8—416no—
362C2×Q8—416no—
363C2×Q8—416no—
364C2×Q8—416no—
365C2×D4—232no—
366C2×D4—432no—
367C2×D4—432no—
368C2×D4—4128no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—816no—
379C2×D4—88no—
380C2×D4—816no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4#120116yes—
438C22×C4#121116yes—
439C22×C4C22×D418yes—
440C22×C4C22×D418yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C2×#4114yes—
448C4⋊C4—416no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—88no—
453C22⋊C4—264no—
454C22⋊C4—216no—
455C22⋊C4—232no—
456C22⋊C4—232no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—416no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—88no—
467C22⋊C4—816no—
468C22⋊C4—88no—
469C22⋊C4—816no—
470C42—232no—
471C42—416no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—216no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—432no—
486C23—432no—
487C23—44no—
488C23—432no—
489C23—44no—
490C23—48no—
491C23—44no—
492C23—44no—
493C23—48no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—48no—
515C23—832no—
516C23—832no—
517C23—832no—
518C23—832no—
519C23—832no—
520C23—832no—
521C23C2×#12014yes—
522C23C2×#12114yes—
523C23#53414yes—
524C23D4218yes—
525C23#54618yes—
526C23C23×D411yescharacteristic
527C23C23×D412yes—
528C23C23×D411yescharacteristic
529C23C22×#4111yescharacteristic
530C23C2×H(5,𝔽2)12yes—
531C23#58518yes—
532Q8—464no—
533D4—464no—
534D4—832no—
535D4—88no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—232no—
539C2×C4—232no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—216no—
556C2×C4—432no—
557C2×C4—416no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4C2×#12018yes—
587C2×C4C2×#12118yes—
588C22—216no—
589C22—28no—
590C22—28no—
591C22—28no—
592C22—216no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—48no—
597C22—48no—
598C22—48no—
599C22—48no—
600C22—864no—
601C22—88no—
602C22—88no—
603C22—88no—
604C22—88no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—81no—
609C22—82no—
610C22—84no—
611C22#241018yes—
612C22C22×#12011yescharacteristic
613C22C22×#12111yescharacteristic
614C22C2×#53411yescharacteristic
615C22#344914yes—
616C22C2×D4212yes—
617C22C2×#54612yes—
618C22#348514yes—
619C22#348714yes—
620C22#350014yes—
621C22#351014yes—
622C22#354518yes—
623C4—28no—
624C4—28no—
625C4—48no—
626C4—44no—
627C4—48no—
628C4—82no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—84no—
633C2—28no—
634C2—88no—
635C2—161no—
636C2C2×#241011yescharacteristic
637C2#3514912yes—
638C2#3567712yes—
639C2#3718712yes—
640C2#3933314yes—
641C2#3946114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2613C2
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×#124C2semidirect
2. C2×#2411C2semidirect
3. C2×#2614C2semidirect
5. C2×#2310C2semidirect
6. #15321C2semidirect
7. #15317C2semidirect
8. #14918C2semidirect
9. #14911C2semidirect
10. #14706C2semidirect
11. #14705C2semidirect
12. #14335C2semidirect
13. #13868C2semidirect
30. C2×#396C22semidirect
31. C2×#395C22semidirect
33. C2×#394C22semidirect
34. C2×#393C22semidirect
37. C2×#391C22semidirect
38. C2×#390C22semidirect
39. C2×#388C22semidirect
51. #1440C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
56. #1438C22semidirect
57. #1437C22semidirect
58. #1437C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
68. C23×Q8D4semidirect
109. C22×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C23×#1242
C2×#24112
C2×#26142
C2×#26132
C2×#23102
#153212
#153172
#149182
#149112
#147062
#147052
#143352
#138682