Finite Group Databasebeta

← #4320411

Subgroups — #4320411

Order 512.

2418 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17057C211yesmaximal, characteristic, complemented
2#16823C211yesmaximal, characteristic, complemented
3#16738C214yesmaximal, complemented
4#16297C212yesmaximal
5#15562C214yesmaximal, complemented
6#15366C214yesmaximal, complemented
7#14969C212yesmaximal, complemented
8#14875C214yesmaximal, complemented
9#14734C212yesmaximal
10#14452C212yesmaximal, complemented
11#14366C212yesmaximal
12#14172C212yesmaximal, complemented
13#13775C211yesmaximal, characteristic
14C22×C4×Q8C2212yescomplemented
15C23×C22⋊C4C2211yescharacteristic
16C2×#399C2212yes—
17C2×#399C2212yes—
18C2×#393C2212yes—
19C2×#393C2214yes—
20C2×#389C2212yescomplemented
21C2×#388C2214yes—
22C2×#387C2214yescomplemented
23C2×#385C2214yes—
24C2×#380C2212yes—
25C2×#379C2211yescharacteristic
26C2×#379C2214yes—
27C2×#379C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1446C2214yescomplemented
31#1446C2212yescomplemented
32#1445C2214yescomplemented
33#1445C2214yescomplemented
34#1444C2212yescomplemented
35#1444C2214yescomplemented
36#1444C2212yescomplemented
37#1444C2212yescomplemented
38#1442C2214yescomplemented
39#1442C2214yescomplemented
40#1441C2214yescomplemented
41#1440C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2212yes—
48#1439C2212yes—
49#1439C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2212yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1434C2214yescomplemented
62#1434C2212yescomplemented
63#1434C2212yescomplemented
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2212yes—
66C22⋊#95C2211yescharacteristic
67C26C2311yescharacteristic
68C23×Q8C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C2×C4×Q8—44no—
72C2×C4×Q8—48no—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—42no—
83C22×C22⋊C4—42no—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2311yescharacteristic
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C42C2312yes—
99C22×C42C2314yes—
100C22×C42C2312yes—
101#399—44no—
102#399—44no—
103#393—28no—
104#393—48no—
105#389—44no—
106#388—48no—
107#387—48no—
108#385—48no—
109#380—28no—
110#379—24no—
111#379—48no—
112#379Q818yescomplemented
113C4×C22⋊C4—42no—
114C4×C22⋊C4—42no—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C25—22no—
140C25—22no—
141C25—41no—
142C25—41no—
143C25—41no—
144C25—41no—
145C25—44no—
146C25—44no—
147C25C2×Q814yes—
148C25C2412yes—
149C25C2411yescharacteristic
150C22×Q8—24no—
151C22×Q8—28no—
152C22×Q8—42no—
153C22×Q8—44no—
154C22×Q8—42no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4#4114yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2412yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C4×Q8—432no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—416no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4C2×Q8116yes—
235C2×C22⋊C4C2×Q814yes—
236C2×C22⋊C4C2×Q814yes—
237C2×C22⋊C4#4114yes—
238C2×C42—24no—
239C2×C42—28no—
240C2×C42—24no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—44no—
245#95—28no—
246#95—28no—
247#95—24no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—44no—
252#95—42no—
253#95—44no—
254#95—42no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—42no—
268#95—42no—
269C24—24no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—24no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—28no—
286C24—44no—
287C24—44no—
288C24—416no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—84no—
294C24—84no—
295C24—88no—
296C24—84no—
297C24—88no—
298C24—84no—
299C24—84no—
300C24—88no—
301C24—84no—
302C24C22×Q814yes—
303C24C22×Q812yes—
304C24C2×#4112yes—
305C24C2×#4112yes—
306C24C2511yescharacteristic
307C2×Q8—48no—
308C2×Q8—416no—
309C2×Q8—48no—
310C2×Q8—416no—
311C2×Q8—48no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
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—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—44no—
368C22×C4—48no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4C22×Q814yes—
375C22×C4C22×Q814yes—
376C22×C4C2×#4114yes—
377C22×C4C2×#4114yes—
378C22×C4H(5,𝔽2)18yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)12yes—
381C4⋊C4—416no—
382C4⋊C4—48no—
383C4⋊C4—416no—
384C4⋊C4—48no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C4⋊C4—84no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—416no—
392C22⋊C4—44no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—416no—
396C22⋊C4—44no—
397C22⋊C4—88no—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C42—416no—
404C42—48no—
405C42—84no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—28no—
414C23—24no—
415C23—28no—
416C23—24no—
417C23—28no—
418C23—24no—
419C23—28no—
420C23—24no—
421C23—416no—
422C23—42no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—816no—
447C23—816no—
448C23—816no—
449C23—816no—
450C23—816no—
451C23—816no—
452C23—88no—
453C23—88no—
454C23#53314yes—
455C23C23×Q811yescharacteristic
456C23C22×#4111yescharacteristic
457C23C22×#4111yescharacteristic
458C23C2×H(5,𝔽2)11yescharacteristic
459C23C2×H(5,𝔽2)11yescharacteristic
460C23C2×H(5,𝔽2)14yes—
461C23#58511yescharacteristic
462C23#58511yescharacteristic
463C23#58514yes—
464Q8—432no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—88no—
527C2×C4—88no—
528C22—24no—
529C22—28no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—42no—
535C22—44no—
536C22—42no—
537C22—48no—
538C22—48no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—832no—
543C22—84no—
544C22—82no—
545C22—82no—
546C22—84no—
547C22—82no—
548C22—82no—
549C22C2×#53311yescharacteristic
550C22#345911yescharacteristic
551C22#346011yescharacteristic
552C22#349011yescharacteristic
553C22#349112yes—
554C22#349311yescharacteristic
555C22#350111yescharacteristic
556C22#350114yes—
557C22#350111yescharacteristic
558C22#350812yes—
559C22#351314yes—
560C22#353212yes—
561C22#353314yes—
562C22#354012yes—
563C22#354014yes—
564C22#354012yes—
565C22#354312yes—
566C4—44no—
567C4—48no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—84no—
574C4—82no—
575C4—84no—
576C4—84no—
577C2—44no—
578C2—84no—
579C2#3501211yescharacteristic
580C2#3583912yes—
581C2#3687112yes—
582C2#4128912yes—
583C2#4699811yescharacteristic
584C2#4701812yes—
585C2#4703111yescharacteristic
586C2#4796212yes—
587C2#5018712yes—
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
1. #17057C2semidirect
2. #16823C2semidirect
3. #16738C2semidirect
5. #15562C2semidirect
6. #15366C2semidirect
7. #14969C2semidirect
8. #14875C2semidirect
10. #14452C2semidirect
12. #14172C2semidirect
14. C22×C4×Q8C22semidirect
20. C2×#389C22semidirect
22. C2×#387C22semidirect
30. #1446C22semidirect
31. #1446C22semidirect
32. #1445C22semidirect
33. #1445C22semidirect
34. #1444C22semidirect
35. #1444C22semidirect
36. #1444C22semidirect
37. #1444C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1441C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1434C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
112. #379Q8semidirect

Maximal subgroups

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

subgroupindex
#170572
#168232
#167382
#162972
#155622
#153662
#149692
#148752
#147342
#144522
#143662
#141722
#137752