Finite Group Databasebeta

← #5486626

Subgroups — #5486626

Order 512.

2726 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17214C214yesmaximal, complemented
2#17135C211yesmaximal, characteristic, complemented
3#16520C214yesmaximal, complemented
4#16352C214yesmaximal, complemented
5#15948C211yesmaximal, characteristic
6#15440C214yesmaximal, complemented
7#15429C214yesmaximal, complemented
8#15262C211yesmaximal, characteristic, complemented
9#15259C211yesmaximal, characteristic
10#14726C214yesmaximal, complemented
11#14662C211yesmaximal, characteristic, complemented
12#14658C211yesmaximal, characteristic, complemented
13#14134C211yesmaximal, characteristic
14C22×#124C2211yescharacteristic
15C22×#123C2211yescharacteristic
16C22×#120C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C2×#400C2214yescomplemented
19C2×#396C2214yes—
20C2×#396C2214yes—
21C2×#396C2214yescomplemented
22C2×#395C2212yescomplemented
23C2×#393C2212yes—
24C2×#386C2214yes—
25C2×#386C2212yes—
26C2×#385C2214yes—
27C2×#385C2214yes—
28C2×#385C2212yescomplemented
29C2×#382C2212yescomplemented
30C2×#380C2214yes—
31C2×#380C2212yescomplemented
32C2×#379C2212yes—
33C2×#379C2212yes—
34#1446C2212yescomplemented
35#1445C2214yescomplemented
36#1444C2214yescomplemented
37#1444C2214yescomplemented
38#1443C2214yescomplemented
39#1442C2214yescomplemented
40#1442C2214yescomplemented
41#1441C2214yescomplemented
42#1441C2214yescomplemented
43#1441C2214yescomplemented
44#1440C2212yes—
45#1440C2212yes—
46#1440C2212yes—
47#1440C2212yescomplemented
48#1440C2212yescomplemented
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yescomplemented
57#1438C2211yescharacteristic
58#1438C2211yescharacteristic
59#1438C2211yescharacteristic
60#1438C2214yes—
61#1438C2211yescharacteristic
62#1438C2212yes—
63#1437C2212yes—
64#1436C2212yescomplemented
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C26C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2314yes—
73C2×#124—24no—
74C2×#124—44no—
75C2×#123—24no—
76C2×#123—44no—
77C2×#120—24no—
78C2×#120—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C22⋊C4—48no—
87C22×C22⋊C4—44no—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C42C2311yescharacteristic
108#400—48no—
109#396—48no—
110#396—48no—
111#396—48no—
112#395—44no—
113#393—44no—
114#386—48no—
115#386—44no—
116#385—48no—
117#385—44no—
118#385—48no—
119#382—44no—
120#380—44no—
121#380—48no—
122#379—44no—
123#379—44no—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C25—22no—
145C25—24no—
146C25—28no—
147C25—44no—
148C25—42no—
149C25—42no—
150C25C2411yescharacteristic
151C25C2411yescharacteristic
152C25C2412yes—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—28no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4—44no—
165C22×D4#4114yes—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187#124—416no—
188#123—416no—
189#120—416no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—432no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C42—22no—
260C2×C42—24no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—44no—
268#95—42no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281C24—22no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—22no—
287C24—24no—
288C24—28no—
289C24—24no—
290C24—24no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—416no—
300C24—416no—
301C24—48no—
302C24—48no—
303C24—48no—
304C24—48no—
305C24—88no—
306C24—88no—
307C24—88no—
308C24—82no—
309C24—82no—
310C24—84no—
311C24—84no—
312C24C2×#4112yes—
313C24C2×#4112yes—
314C24H(5,𝔽2)14yes—
315C242−1+418yes—
316C24C2511yescharacteristic
317C2×Q8—44no—
318C2×Q8—82no—
319C2×Q8—82no—
320C2×Q8—88no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—48no—
325C2×D4—416no—
326C2×D4—48no—
327C2×D4—84no—
328C2×D4—88no—
329C2×D4—88no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
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—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—416no—
385C22×C4—416no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—88no—
391C22×C4—88no—
392C22×C4—88no—
393C22×C4—88no—
394C22×C4C2×#4112yes—
395C22×C4H(5,𝔽2)14yes—
396C22×C4H(5,𝔽2)18yes—
397C22×C4H(5,𝔽2)14yes—
398C4⋊C4—48no—
399C4⋊C4—88no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C4⋊C4—88no—
403C4⋊C4—88no—
404C4⋊C4—88no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C22⋊C4—416no—
408C22⋊C4—416no—
409C22⋊C4—416no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—48no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C22⋊C4—84no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C42—28no—
425C23—24no—
426C23—28no—
427C23—24no—
428C23—28no—
429C23—28no—
430C23—28no—
431C23—28no—
432C23—28no—
433C23—28no—
434C23—28no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—28no—
442C23—28no—
443C23—432no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—832no—
475C23—88no—
476C23—816no—
477C23—816no—
478C23—832no—
479C23#56318yes—
480C23C22×#4111yescharacteristic
481C23C2×H(5,𝔽2)12yes—
482C23C2×H(5,𝔽2)12yes—
483C23C2×H(5,𝔽2)12yes—
484C23C2×H(5,𝔽2)14yes—
485C23C2×2−1+414yes—
486Q8—88no—
487D4—88no—
488D4—816no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
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—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—88no—
547C2×C4—832no—
548C22—28no—
549C22—216no—
550C22—48no—
551C22—48no—
552C22—44no—
553C22—44no—
554C22—48no—
555C22—48no—
556C22—48no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—832no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22#349011yescharacteristic
571C22#349011yescharacteristic
572C22#349011yescharacteristic
573C22#352014yes—
574C22C2×#56312yes—
575C22#353212yes—
576C22#353614yes—
577C22#353614yes—
578C22#353614yes—
579C22#353712yes—
580C22#353714yes—
581C22#353812yes—
582C22#353812yes—
583C22#354912yes—
584C4—48no—
585C4—44no—
586C4—42no—
587C4—44no—
588C4—84no—
589C4—84no—
590C4—84no—
591C4—82no—
592C4—82no—
593C4—82no—
594C2—48no—
595C2—82no—
596C2—82no—
597C2#4572812yes—
598C2#4638411yescharacteristic
599C2#4966012yes—
600C2#5162912yes—
601C2#5247314yes—
602C2#5504314yes—
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
1. #17214C2semidirect
2. #17135C2semidirect
3. #16520C2semidirect
4. #16352C2semidirect
6. #15440C2semidirect
7. #15429C2semidirect
8. #15262C2semidirect
10. #14726C2semidirect
11. #14662C2semidirect
12. #14658C2semidirect
18. C2×#400C22semidirect
21. C2×#396C22semidirect
22. C2×#395C22semidirect
28. C2×#385C22semidirect
29. C2×#382C22semidirect
31. C2×#380C22semidirect
34. #1446C22semidirect
35. #1445C22semidirect
36. #1444C22semidirect
37. #1444C22semidirect
38. #1443C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
56. #1439C22semidirect
64. #1436C22semidirect

Maximal subgroups

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

subgroupindex
#172142
#171352
#165202
#163522
#159482
#154402
#154292
#152622
#152592
#147262
#146622
#146582
#141342