Finite Group Databasebeta

← #4309027

Subgroups — #4309027

Order 512.

5705 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#123C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic, complemented
3C2×#2622C211yesmaximal, characteristic, complemented
4C2×#2613C211yesmaximal, characteristic
5#16657C218yesmaximal, complemented
6#16654C218yesmaximal, complemented
7#15073C212yesmaximal, complemented
8#15069C212yesmaximal, complemented
9#14918C212yesmaximal, complemented
10#14911C212yesmaximal, complemented
11#14681C211yesmaximal, characteristic, complemented
12#14341C211yesmaximal, characteristic, complemented
13#14335C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C22×#123—28no—
17C22×#123—28no—
18C22×#123—28no—
19C22×#123C2218yes—
20C22×#123C2212yes—
21C22×#123C2211yescharacteristic
22C22×#120C2211yescharacteristic
23C22×#120C2211yescharacteristic
24C23×C4⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2211yescharacteristic
27#2849—28no—
28#2622—28no—
29#2613—28no—
30C2×#397C2214yescomplemented
31C2×#394C2214yes—
32C2×#394C2214yescomplemented
33C2×#392C2214yes—
34C2×#386C2212yes—
35C2×#386C2214yescomplemented
36C2×#385C2212yes—
37C2×#385C2214yescomplemented
38C2×#381C2212yes—
39C2×#381C2211yescharacteristic
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#379C2211yescharacteristic
44C2×#379C2211yescharacteristic
45C2×#379C2211yescharacteristic
46C2×#379C2211yescharacteristic
47C2×#379C2212yes—
48#1444C2218yescomplemented
49#1442C2218yescomplemented
50#1441C2218yescomplemented
51#1440C2218yescomplemented
52#1440C2218yescomplemented
53#1439C2218yescomplemented
54#1439C2218yescomplemented
55#1439C2218yescomplemented
56#1438C2218yes—
57#1438C2218yescomplemented
58C22⋊#95C2212yes—
59C22⋊#95C2212yes—
60C22⋊#95C2218yescomplemented
61C26C2311yescharacteristic
62C26C2311yescharacteristic
63C23×D4—24no—
64C23×D4—24no—
65C23×D4—44no—
66C23×D4—44no—
67C23×D4D418yes—
68C23×D4C2311yescharacteristic
69C23×D4C2312yes—
70C23×D4C2314yes—
71C23×D4C2312yes—
72C24×C4—24no—
73C24×C4—24no—
74C24×C4—44no—
75C24×C4—44no—
76C24×C4D418yescomplemented
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2314yes—
80C24×C4C2312yes—
81C2×#123—2128no—
82C2×#123—232no—
83C2×#123—232no—
84C2×#123—28no—
85C2×#123—24no—
86C2×#123—232no—
87C2×#123—44no—
88C2×#123—44no—
89C2×#123—44no—
90C2×#120—24no—
91C2×#120—24no—
92C2×#120—24no—
93C2×#120—24no—
94C2×#120—44no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4C2318yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4D418yescomplemented
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2318yes—
125#397—216no—
126#394—216no—
127#394—216no—
128#392—216no—
129#386—216no—
130#386—28no—
131#385—216no—
132#385—28no—
133#381—28no—
134#381—24no—
135#381—44no—
136#381—44no—
137#381—44no—
138#379—24no—
139#379—24no—
140#379—24no—
141#379—24no—
142#379—44no—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C25—24no—
154C25—24no—
155C25—22no—
156C25—22no—
157C25—24no—
158C25—24no—
159C25—48no—
160C25—48no—
161C25C2×D414yes—
162C25C2×D414yes—
163C25C2×D414yes—
164C25C2×D414yes—
165C25C2411yescharacteristic
166C25C2412yes—
167C25C2412yes—
168C25C2411yescharacteristic
169C22×D4—232no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—416no—
182C22×D4—416no—
183C22×D4—48no—
184C22×D4—48no—
185C22×D4—44no—
186C22×D4—48no—
187C22×D4—48no—
188C22×D4—432no—
189C22×D4—432no—
190C22×D4C2×D4116yes—
191C23×C4—232no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—432no—
207C23×C4—48no—
208C23×C4—48no—
209C23×C4—48no—
210C23×C4—416no—
211C23×C4—416no—
212C23×C4—432no—
213C23×C4C2×D418yes—
214C23×C4C2×D414yes—
215C23×C4C2×D418yes—
216C23×C4C2×D4116yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226#123—2256no—
227#123—432no—
228#123—416no—
229#120—416no—
230#120—416no—
231C2×C4⋊C4—264no—
232C2×C4⋊C4—216no—
233C2×C4⋊C4—216no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—48no—
242C2×C22⋊C4—264no—
243C2×C22⋊C4—232no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4C2×D4116yes—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—48no—
292#95—48no—
293#95—48no—
294#95—48no—
295#95—48no—
296C24—28no—
297C24—216no—
298C24—28no—
299C24—28no—
300C24—216no—
301C24—24no—
302C24—24no—
303C24—28no—
304C24—28no—
305C24—28no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—28no—
311C24—28no—
312C24—28no—
313C24—28no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24—28no—
320C24—416no—
321C24—416no—
322C24—416no—
323C24—416no—
324C24—432no—
325C24—432no—
326C24—416no—
327C24—416no—
328C24—48no—
329C24—48no—
330C24—816no—
331C24—816no—
332C24#120116yes—
333C24C22×D414yes—
334C24C22×D414yes—
335C24C22×D414yes—
336C24C22×D414yes—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C22×D412yes—
340C24H(5,𝔽2)14yes—
341C24H(5,𝔽2)14yes—
342C24C2511yescharacteristic
343C2×D4—232no—
344C2×D4—416no—
345C2×D4—416no—
346C2×D4—4128no—
347C2×D4—432no—
348C2×D4—432no—
349C2×D4—48no—
350C2×D4—416no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—44no—
356C2×D4—48no—
357C2×D4—88no—
358C2×D4—88no—
359C2×D4—88no—
360C2×D4—816no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—28no—
364C22×C4—232no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—432no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—4128no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—432no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4C22×D418yes—
423C22×C4C22×D418yes—
424C22×C4C22×D414yes—
425C4⋊C4—264no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—88no—
429C22⋊C4—264no—
430C22⋊C4—232no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C22⋊C4—416no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—48no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—816no—
446C23—28no—
447C23—28no—
448C23—28no—
449C23—216no—
450C23—216no—
451C23—232no—
452C23—28no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—432no—
468C23—464no—
469C23—432no—
470C23—464no—
471C23—432no—
472C23—432no—
473C23—416no—
474C23—416no—
475C23—416no—
476C23—44no—
477C23—44no—
478C23—48no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—48no—
483C23—48no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—48no—
493C23—48no—
494C23—816no—
495C23—816no—
496C23—832no—
497C23—832no—
498C23C2×#12018yes—
499C23C2×#12014yes—
500C23#53418yes—
501C23#53414yes—
502C23#53414yes—
503C23#53518yes—
504C23C23×D411yescharacteristic
505C23C23×D411yescharacteristic
506C23C23×D411yescharacteristic
507C23C2×H(5,𝔽2)12yes—
508C23C2×H(5,𝔽2)12yes—
509C23#58518yes—
510D4—464no—
511D4—816no—
512D4—816no—
513D4—88no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—432no—
524C2×C4—464no—
525C2×C4—432no—
526C2×C4—416no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C22—28no—
561C22—216no—
562C22—28no—
563C22—216no—
564C22—28no—
565C22—28no—
566C22—28no—
567C22—28no—
568C22—464no—
569C22—416no—
570C22—48no—
571C22—416no—
572C22—48no—
573C22—48no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—832no—
579C22—88no—
580C22—88no—
581C22—82no—
582C22—84no—
583C22—82no—
584C22—84no—
585C22—82no—
586C22—81no—
587C22#261318yes—
588C22C22×#12011yescharacteristic
589C22C2×#53412yes—
590C22C2×#53411yescharacteristic
591C22C2×#53411yescharacteristic
592C22C2×#53512yes—
593C22#344914yes—
594C22#348514yes—
595C22#348514yes—
596C22#353318yes—
597C22#353318yes—
598C4—48no—
599C4—48no—
600C4—44no—
601C4—82no—
602C4—84no—
603C4—82no—
604C4—84no—
605C4—82no—
606C4—81no—
607C2—28no—
608C2—48no—
609C2—84no—
610C2—161no—
611C2C2×#261311yescharacteristic
612C2#3508812yes—
613C2#3695512yes—
614C2#3695614yes—
615C2#3696512yes—
616C2#3744314yes—
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×#123C2semidirect
2. C2×#2849C2semidirect
3. C2×#2622C2semidirect
5. #16657C2semidirect
6. #16654C2semidirect
7. #15073C2semidirect
8. #15069C2semidirect
9. #14918C2semidirect
10. #14911C2semidirect
11. #14681C2semidirect
12. #14341C2semidirect
13. #14335C2semidirect
30. C2×#397C22semidirect
32. C2×#394C22semidirect
35. C2×#386C22semidirect
37. C2×#385C22semidirect
48. #1444C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
57. #1438C22semidirect
60. C22⋊#95C22semidirect
76. C24×C4D4semidirect
107. C22×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C23×#1232
C2×#28492
C2×#26222
C2×#26132
#166572
#166542
#150732
#150692
#149182
#149112
#146812
#143412
#143352