Finite Group Databasebeta

← #4320004

Subgroups — #4320004

Order 512.

2682 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C212yesmaximal
2#17059C211yesmaximal, characteristic, complemented
3#16831C212yesmaximal, complemented
4#16823C211yesmaximal, characteristic, complemented
5#16364C214yesmaximal, complemented
6#15586C214yesmaximal, complemented
7#15485C214yesmaximal, complemented
8#15043C214yesmaximal, complemented
9#14820C212yesmaximal
10#14820C212yesmaximal
11#14581C212yesmaximal, complemented
12#14422C212yesmaximal, complemented
13#13794C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18#2561—216no—
19C2×#397C2212yes—
20C2×#397C2212yes—
21C2×#397C2212yes—
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#393C2212yes—
26C2×#393C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#381C2214yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#379C2214yes—
33C2×C4×C22⋊C4C2212yes—
34#1446C2214yescomplemented
35#1446C2214yescomplemented
36#1445C2214yescomplemented
37#1444C2214yescomplemented
38#1443C2214yescomplemented
39#1443C2214yescomplemented
40#1442C2214yescomplemented
41#1442C2214yescomplemented
42#1442C2214yescomplemented
43#1442C2214yescomplemented
44#1441C2214yescomplemented
45#1441C2214yescomplemented
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2212yes—
51#1439C2212yes—
52#1439C2212yes—
53#1439C2212yes—
54#1436C2214yescomplemented
55#1436C2214yescomplemented
56#1434C2214yescomplemented
57#1434C2214yescomplemented
58C22⋊#95C2211yescharacteristic
59C22⋊#95C2212yes—
60C22⋊#95C2214yes—
61C22⋊#95C2214yes—
62C22⋊#95C2211yescharacteristic
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2212yes—
68C26C2311yescharacteristic
69C23×Q8C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—44no—
76C2×C22⋊Q8—44no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—42no—
86C22×C22⋊C4—42no—
87C22×C22⋊C4—42no—
88C22×C22⋊C4—42no—
89C22×C22⋊C4—42no—
90C22×C22⋊C4—42no—
91C22×C22⋊C4D418yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C42C2314yes—
107#397—28no—
108#397—28no—
109#397—28no—
110#397—44no—
111#397—44no—
112#397—44no—
113#393—28no—
114#393—28no—
115#388—216no—
116#388—216no—
117#381—216no—
118#380—28no—
119#380—44no—
120#379—216no—
121C4×C22⋊C4—28no—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
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—
148C25—22no—
149C25—22no—
150C25—22no—
151C25—22no—
152C25—22no—
153C25—22no—
154C25—44no—
155C25—44no—
156C25C2×D414yes—
157C25C2411yescharacteristic
158C25C2411yescharacteristic
159C25C2411yescharacteristic
160C22×Q8—24no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—44no—
164C22×Q8—44no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—42no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4C2×D418yes—
182C23×C4#4114yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C22⋊Q8—432no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—416no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—416no—
246C2×C22⋊C4—416no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×D414yes—
252C2×C22⋊C4C2×D418yes—
253C2×C22⋊C4C2×D414yes—
254C2×C22⋊C4#4114yes—
255C2×C42—28no—
256C2×C42—44no—
257C2×C42—44no—
258#95—24no—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—28no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—42no—
281#95—44no—
282#95—44no—
283#95—44no—
284C24—28no—
285C24—28no—
286C24—28no—
287C24—28no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—28no—
307C24—28no—
308C24—28no—
309C24—416no—
310C24—84no—
311C24—84no—
312C24—88no—
313C24—84no—
314C24—84no—
315C24—84no—
316C24—84no—
317C24—88no—
318C24—88no—
319C24C22×D412yes—
320C24C22×D412yes—
321C24C22×D412yes—
322C24C2×#4112yes—
323C24C2×#4112yes—
324C24H(5,𝔽2)12yes—
325C24H(5,𝔽2)12yes—
326C24H(5,𝔽2)12yes—
327C242−1+412yes—
328C24C2511yescharacteristic
329C2×Q8—48no—
330C2×Q8—88no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8—88no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—216no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
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—44no—
390C22×C4—416no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—88no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—88no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—88no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4C22×D414yes—
407C22×C4C22×D414yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4114yes—
410C4⋊C4—416no—
411C4⋊C4—48no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C22⋊C4—44no—
417C22⋊C4—48no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—416no—
424C22⋊C4—44no—
425C22⋊C4—416no—
426C22⋊C4—416no—
427C22⋊C4—416no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C42—88no—
431C23—232no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—28no—
439C23—28no—
440C23—28no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—28no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—816no—
481C23—816no—
482C23—816no—
483C23—816no—
484C23—816no—
485C23—816no—
486C23—88no—
487C23—88no—
488C23#53214yes—
489C23#53414yes—
490C23#53714yes—
491C23C23×D411yescharacteristic
492C23C22×#4111yescharacteristic
493C23C22×#4111yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×H(5,𝔽2)11yescharacteristic
497C23C2×2−1+411yescharacteristic
498C23#58514yes—
499C23#58514yes—
500Q8—816no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
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—816no—
560C2×C4—88no—
561C2×C4—816no—
562C2×C4—88no—
563C22—28no—
564C22—28no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—28no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—832no—
591C22C2×#53211yescharacteristic
592C22C2×#53411yescharacteristic
593C22C2×#53711yescharacteristic
594C22#348514yes—
595C22#349011yescharacteristic
596C22#349011yescharacteristic
597C22#349112yes—
598C22#349211yescharacteristic
599C22#349211yescharacteristic
600C22#351012yes—
601C22#352114yes—
602C22#353312yes—
603C22#354512yes—
604C22#354512yes—
605C22#354514yes—
606C22#354512yes—
607C22#356414yes—
608C4—44no—
609C4—44no—
610C4—44no—
611C4—44no—
612C4—82no—
613C4—84no—
614C4—82no—
615C4—84no—
616C4—84no—
617C4—84no—
618C2—44no—
619C2—44no—
620C2—44no—
621C2#3505111yescharacteristic
622C2#3601212yes—
623C2#3711912yes—
624C2#4140512yes—
625C2#4683711yescharacteristic
626C2#4688212yes—
627C2#4703511yescharacteristic
628C2#4879712yes—
629C2#4899712yes—
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
2. #17059C2semidirect
3. #16831C2semidirect
4. #16823C2semidirect
5. #16364C2semidirect
6. #15586C2semidirect
7. #15485C2semidirect
8. #15043C2semidirect
11. #14581C2semidirect
12. #14422C2semidirect
34. #1446C22semidirect
35. #1446C22semidirect
36. #1445C22semidirect
37. #1444C22semidirect
38. #1443C22semidirect
39. #1443C22semidirect
40. #1442C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
54. #1436C22semidirect
55. #1436C22semidirect
56. #1434C22semidirect
57. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
#170592
#168312
#168232
#163642
#155862
#154852
#150432
#148202
#145812
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