Finite Group Databasebeta

← #6131859

Subgroups — #6131859

Order 512.

2990 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16569C213yesmaximal, complemented
2#16568C213yesmaximal, complemented
3#16521C213yesmaximal, complemented
4#16520C213yesmaximal, complemented
5#16305C213yesmaximal, complemented
6#16304C213yesmaximal, complemented
7#16304C213yesmaximal, complemented
8#16303C213yesmaximal, complemented
9#16298C213yesmaximal
10#16297C211yesmaximal, characteristic
11#14680C213yesmaximal
12C25×C4C2211yescharacteristic
13C2×#400C2213yescomplemented
14C2×#399C2213yes—
15C2×#398C2213yescomplemented
16C2×#398C2213yescomplemented
17C2×#396C2213yes—
18C2×#396C2213yes—
19C2×#395C2213yescomplemented
20C2×#393C2213yes—
21C2×#380C2213yes—
22C2×#380C2213yes—
23C2×#380C2213yes—
24C2×#380C2213yes—
25C2×#380C2213yes—
26C2×#380C2213yes—
27C2×#379C2213yes—
28C2×#379C2213yes—
29#1446C2211yescharacteristic, complemented
30#1445C2213yescomplemented
31#1444C2213yescomplemented
32#1444C2211yescharacteristic, complemented
33#1444C2213yescomplemented
34#1444C2213yescomplemented
35#1444C2213yescomplemented
36#1443C2211yescharacteristic, complemented
37#1443C2211yescharacteristic, complemented
38#1442C2213yescomplemented
39#1442C2213yescomplemented
40#1442C2213yescomplemented
41#1442C2213yescomplemented
42#1442C2213yescomplemented
43#1441C2213yescomplemented
44#1441C2213yescomplemented
45#1441C2213yescomplemented
46#1441C2213yescomplemented
47#1441C2213yescomplemented
48#1441C2213yescomplemented
49#1439C2213yes—
50#1439C2213yes—
51#1439C2213yes—
52#1439C2213yes—
53#1439C2213yes—
54#1439C2213yes—
55#1439C2213yes—
56#1439C2213yes—
57#1439C2213yes—
58#1439C2213yes—
59#1439C2213yes—
60#1439C2213yes—
61C22⋊#95C2213yes—
62C22⋊#95C2213yes—
63C22⋊#95C2213yes—
64C22⋊#95C2213yes—
65C22⋊#95C2213yes—
66C22⋊#95C2213yes—
67C26C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C24×C4—46no—
70C24×C4—46no—
71C24×C4Q818yescomplemented
72C24×C4C2313yes—
73C24×C4C2313yes—
74C22×C4⋊C4C2313yes—
75C22×C4⋊C4C2313yes—
76C22×C4⋊C4C2313yes—
77C22×C4⋊C4C2313yes—
78C22×C4⋊C4C2313yes—
79C22×C4⋊C4C2313yes—
80C22×C4⋊C4C2313yes—
81C22×C4⋊C4C2313yes—
82C22×C4⋊C4C2313yes—
83C22×C4⋊C4C2313yes—
84C22×C4⋊C4C2313yes—
85C22×C22⋊C4C2313yes—
86C22×C22⋊C4C2313yes—
87C22×C22⋊C4C2313yes—
88C22×C22⋊C4C2313yes—
89C22×C22⋊C4C2313yes—
90C22×C22⋊C4C2313yes—
91C22×C22⋊C4C2313yes—
92C22×C22⋊C4C2313yes—
93C22×C22⋊C4C2313yes—
94C22×C22⋊C4C2313yes—
95C22×C22⋊C4C2313yes—
96C22×C22⋊C4C2313yes—
97#400—46no—
98#399—46no—
99#398—46no—
100#398—46no—
101#396—46no—
102#396—46no—
103#395—46no—
104#393—46no—
105#380—46no—
106#380—46no—
107#380—46no—
108#380—46no—
109#380—46no—
110#380—46no—
111#379—46no—
112#379—46no—
113C2×#95C2313yes—
114C2×#95C2313yes—
115C2×#95C2313yes—
116C2×#95C2313yes—
117C2×#95C2313yes—
118C2×#95C2313yes—
119C2×#95C2313yes—
120C2×#95C2313yes—
121C2×#95C2313yes—
122C2×#95C2313yes—
123C2×#95C2313yes—
124C2×#95C2313yes—
125C2×#95C2313yes—
126C2×#95C2313yes—
127C2×#95C2313yes—
128C2×#95C2313yes—
129C2×#95C2313yes—
130C2×#95C2313yes—
131C2×#95C2313yes—
132C2×#95C2313yes—
133C2×#95C2313yes—
134C2×#95C2313yes—
135C2×#95C2313yes—
136C2×#95C2313yes—
137C2×#95C2311yescharacteristic
138C2×#95C2313yes—
139C2×#95C2311yescharacteristic
140C2×#95C2311yescharacteristic
141C25—26no—
142C25—26no—
143C25—43no—
144C25—43no—
145C25—41no—
146C25—41no—
147C25C2×Q814yes—
148C25C2413yes—
149C22×Q8—41no—
150C22×Q8—43no—
151C22×Q8—43no—
152C23×C4—26no—
153C23×C4—26no—
154C23×C4—26no—
155C23×C4—26no—
156C23×C4—26no—
157C23×C4—26no—
158C23×C4—448no—
159C23×C4—43no—
160C23×C4—43no—
161C23×C4—43no—
162C23×C4—43no—
163C23×C4—43no—
164C23×C4—43no—
165C23×C4—448no—
166C23×C4—416no—
167C23×C4C2×Q8112yes—
168C23×C4C2×Q8112yes—
169C23×C4C2413yes—
170C23×C4C2413yes—
171C23×C4C2413yes—
172C23×C4C2413yes—
173C23×C4C2413yes—
174C23×C4C2413yes—
175C23×C4C2413yes—
176C23×C4C2413yes—
177C23×C4C2411yescharacteristic
178C23×C4C2413yes—
179C2×C4⋊C4—26no—
180C2×C4⋊C4—26no—
181C2×C4⋊C4—26no—
182C2×C4⋊C4—26no—
183C2×C4⋊C4—43no—
184C2×C4⋊C4—43no—
185C2×C4⋊C4—43no—
186C2×C4⋊C4—43no—
187C2×C4⋊C4—43no—
188C2×C4⋊C4—43no—
189C2×C4⋊C4—43no—
190C2×C4⋊C4—43no—
191C2×C4⋊C4—43no—
192C2×C4⋊C4—43no—
193C2×C4⋊C4—43no—
194C2×C4⋊C4—43no—
195C2×C4⋊C4—43no—
196C2×C4⋊C4—43no—
197C2×C4⋊C4—43no—
198C2×C4⋊C4—43no—
199C2×C4⋊C4—43no—
200C2×C4⋊C4—43no—
201C2×C4⋊C4—43no—
202C2×C4⋊C4—43no—
203C2×C4⋊C4—43no—
204C2×C4⋊C4—43no—
205C2×C4⋊C4—43no—
206C2×C4⋊C4—43no—
207C2×C4⋊C4—43no—
208C2×C4⋊C4—43no—
209C2×C4⋊C4—43no—
210C2×C4⋊C4—43no—
211C2×C4⋊C4—43no—
212C2×C22⋊C4—26no—
213C2×C22⋊C4—26no—
214C2×C22⋊C4—26no—
215C2×C22⋊C4—26no—
216C2×C22⋊C4—26no—
217C2×C22⋊C4—26no—
218C2×C22⋊C4—26no—
219C2×C22⋊C4—26no—
220C2×C22⋊C4—26no—
221C2×C22⋊C4—26no—
222C2×C22⋊C4—26no—
223C2×C22⋊C4—26no—
224C2×C22⋊C4—43no—
225C2×C22⋊C4—43no—
226C2×C22⋊C4—43no—
227C2×C22⋊C4—43no—
228C2×C22⋊C4—43no—
229C2×C22⋊C4—43no—
230C2×C22⋊C4—43no—
231C2×C22⋊C4—43no—
232C2×C22⋊C4—43no—
233C2×C22⋊C4—43no—
234C2×C22⋊C4—43no—
235C2×C22⋊C4—43no—
236C2×C22⋊C4—43no—
237C2×C22⋊C4—43no—
238C2×C22⋊C4—43no—
239C2×C22⋊C4—43no—
240C2×C22⋊C4—43no—
241C2×C22⋊C4—43no—
242C2×C22⋊C4—43no—
243C2×C22⋊C4—43no—
244C2×C22⋊C4—43no—
245C2×C22⋊C4—43no—
246C2×C22⋊C4—43no—
247C2×C22⋊C4—43no—
248#95—26no—
249#95—26no—
250#95—26no—
251#95—26no—
252#95—26no—
253#95—26no—
254#95—26no—
255#95—26no—
256#95—41no—
257#95—43no—
258#95—41no—
259#95—41no—
260#95—43no—
261#95—43no—
262#95—43no—
263#95—43no—
264#95—43no—
265#95—43no—
266#95—43no—
267#95—43no—
268#95—43no—
269#95—43no—
270#95—43no—
271#95—43no—
272#95—43no—
273#95—43no—
274#95—43no—
275#95—43no—
276C24—23no—
277C24—23no—
278C24—23no—
279C24—23no—
280C24—23no—
281C24—23no—
282C24—23no—
283C24—23no—
284C24—23no—
285C24—23no—
286C24—23no—
287C24—23no—
288C24—224no—
289C24—412no—
290C24—44no—
291C24—412no—
292C24—412no—
293C24—412no—
294C24—412no—
295C24—412no—
296C24—412no—
297C24—412no—
298C24—412no—
299C24—412no—
300C24—44no—
301C24C22×Q816yes—
302C24H(5,𝔽2)16yes—
303C242−1+416yes—
304C24C2511yescharacteristic
305C2×Q8—82no—
306C2×Q8—86no—
307C2×Q8—86no—
308C22×C4—23no—
309C22×C4—23no—
310C22×C4—23no—
311C22×C4—23no—
312C22×C4—23no—
313C22×C4—23no—
314C22×C4—224no—
315C22×C4—224no—
316C22×C4—224no—
317C22×C4—23no—
318C22×C4—23no—
319C22×C4—23no—
320C22×C4—23no—
321C22×C4—23no—
322C22×C4—23no—
323C22×C4—23no—
324C22×C4—23no—
325C22×C4—23no—
326C22×C4—23no—
327C22×C4—23no—
328C22×C4—23no—
329C22×C4—23no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—23no—
339C22×C4—23no—
340C22×C4—23no—
341C22×C4—23no—
342C22×C4—23no—
343C22×C4—23no—
344C22×C4—23no—
345C22×C4—23no—
346C22×C4—23no—
347C22×C4—23no—
348C22×C4—23no—
349C22×C4—23no—
350C22×C4—23no—
351C22×C4—23no—
352C22×C4—23no—
353C22×C4—23no—
354C22×C4—23no—
355C22×C4—23no—
356C22×C4—23no—
357C22×C4—23no—
358C22×C4—23no—
359C22×C4—23no—
360C22×C4—23no—
361C22×C4—23no—
362C22×C4—23no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—23no—
366C22×C4—23no—
367C22×C4—23no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4—23no—
371C22×C4—23no—
372C22×C4—23no—
373C22×C4—412no—
374C22×C4—412no—
375C22×C4—412no—
376C22×C4—412no—
377C22×C4—412no—
378C22×C4—412no—
379C22×C4—412no—
380C22×C4—412no—
381C22×C4—412no—
382C22×C4—412no—
383C22×C4—412no—
384C22×C4—4128no—
385C22×C4C22×Q812yes—
386C22×C4C22×Q816yes—
387C22×C4H(5,𝔽2)16yes—
388C22×C4H(5,𝔽2)16yes—
389C4⋊C4—86no—
390C4⋊C4—86no—
391C4⋊C4—86no—
392C4⋊C4—86no—
393C4⋊C4—86no—
394C4⋊C4—86no—
395C4⋊C4—86no—
396C4⋊C4—86no—
397C4⋊C4—86no—
398C4⋊C4—86no—
399C4⋊C4—86no—
400C22⋊C4—86no—
401C22⋊C4—86no—
402C22⋊C4—86no—
403C22⋊C4—86no—
404C22⋊C4—86no—
405C22⋊C4—86no—
406C22⋊C4—86no—
407C22⋊C4—86no—
408C22⋊C4—86no—
409C22⋊C4—86no—
410C22⋊C4—86no—
411C22⋊C4—86no—
412C23—26no—
413C23—26no—
414C23—26no—
415C23—26no—
416C23—26no—
417C23—26no—
418C23—26no—
419C23—26no—
420C23—26no—
421C23—26no—
422C23—26no—
423C23—26no—
424C23—26no—
425C23—26no—
426C23—26no—
427C23—26no—
428C23—26no—
429C23—26no—
430C23—448no—
431C23—448no—
432C23—416no—
433C23—448no—
434C23—416no—
435C23—448no—
436C23—416no—
437C23—43no—
438C23—43no—
439C23—43no—
440C23—43no—
441C23—43no—
442C23—43no—
443C23—43no—
444C23—43no—
445C23—43no—
446C23—43no—
447C23—43no—
448C23—43no—
449C23—43no—
450C23—43no—
451C23—43no—
452C23—43no—
453C23#544112yes—
454C23C23×Q811yescharacteristic
455C23C2×H(5,𝔽2)13yes—
456C23C2×H(5,𝔽2)13yes—
457C23C2×H(5,𝔽2)13yes—
458C23C2×2−1+413yes—
459C23#58511yescharacteristic
460C23#58511yescharacteristic
461Q8—88no—
462C2×C4—26no—
463C2×C4—26no—
464C2×C4—26no—
465C2×C4—26no—
466C2×C4—26no—
467C2×C4—26no—
468C2×C4—26no—
469C2×C4—448no—
470C2×C4—448no—
471C2×C4—43no—
472C2×C4—41no—
473C2×C4—43no—
474C2×C4—43no—
475C2×C4—43no—
476C2×C4—43no—
477C2×C4—43no—
478C2×C4—43no—
479C2×C4—43no—
480C2×C4—43no—
481C2×C4—43no—
482C2×C4—43no—
483C2×C4—43no—
484C2×C4—43no—
485C2×C4—43no—
486C2×C4—43no—
487C2×C4—43no—
488C2×C4—43no—
489C2×C4—43no—
490C2×C4—43no—
491C2×C4—43no—
492C2×C4—43no—
493C2×C4—43no—
494C2×C4—43no—
495C2×C4—43no—
496C2×C4—43no—
497C2×C4—43no—
498C2×C4—43no—
499C2×C4—43no—
500C2×C4—43no—
501C2×C4—43no—
502C2×C4—43no—
503C2×C4—43no—
504C2×C4—43no—
505C2×C4—43no—
506C2×C4—43no—
507C2×C4—43no—
508C2×C4—43no—
509C2×C4—43no—
510C2×C4—43no—
511C2×C4—43no—
512C2×C4—43no—
513C2×C4—43no—
514C2×C4—43no—
515C2×C4—43no—
516C2×C4—43no—
517C2×C4—43no—
518C2×C4—43no—
519C2×C4—43no—
520C2×C4—43no—
521C2×C4—43no—
522C2×C4—43no—
523C2×C4—43no—
524C2×C4—43no—
525C2×C4—43no—
526C2×C4—43no—
527C2×C4—43no—
528C2×C4—43no—
529C2×C4#543112yes—
530C22—212no—
531C22—212no—
532C22—212no—
533C22—464no—
534C22—46no—
535C22—46no—
536C22—46no—
537C22—46no—
538C22—46no—
539C22—46no—
540C22—46no—
541C22—46no—
542C22—46no—
543C22—46no—
544C22—46no—
545C22—46no—
546C22C2×#54313yes—
547C22C2×#54413yes—
548C22#345811yescharacteristic
549C22#353213yes—
550C22#353211yescharacteristic
551C22#353613yes—
552C22#353613yes—
553C22#353713yes—
554C22#353813yes—
555C22#354113yes—
556C22#354113yes—
557C22#354213yes—
558C22#354213yes—
559C4—212no—
560C4—42no—
561C4—83no—
562C4—83no—
563C4—83no—
564C4—83no—
565C4—83no—
566C4—83no—
567C4—83no—
568C4—83no—
569C2—412no—
570C2#3747511yescharacteristic
571C2#3747613yes—
572C2#3747713yes—
573C2#5316713yes—
574C2#5316811yescharacteristic
575C2#5317813yes—
576C2#5317911yescharacteristic
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. #16569C2semidirect
2. #16568C2semidirect
3. #16521C2semidirect
4. #16520C2semidirect
5. #16305C2semidirect
6. #16304C2semidirect
7. #16304C2semidirect
8. #16303C2semidirect
13. C2×#400C22semidirect
15. C2×#398C22semidirect
16. C2×#398C22semidirect
19. C2×#395C22semidirect
29. #1446C22semidirect
30. #1445C22semidirect
31. #1444C22semidirect
32. #1444C22semidirect
33. #1444C22semidirect
34. #1444C22semidirect
35. #1444C22semidirect
36. #1443C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
71. C24×C4Q8semidirect

Maximal subgroups

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

subgroupindex
#165692
#165682
#165212
#165202
#163052
#163042
#163032
#162982
#162972
#146802