Finite Group Databasebeta

← #760516

Subgroups — #760516

Order 512.

3654 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2608C211yesmaximal, characteristic, complemented
2C22×#381C211yesmaximal, characteristic
3#16667C214yesmaximal, complemented
4#14773C214yesmaximal, complemented
5#14769C214yesmaximal, complemented
6#14726C214yesmaximal
7#14165C211yesmaximal, characteristic, complemented
8#14028C214yesmaximal, complemented
9#14004C214yesmaximal, complemented
10#13953C212yesmaximal, complemented
11C4×#379C212yesmaximal
12C25×C4C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14#2608—28no—
15C2×#396C2214yes—
16C2×#396C2214yes—
17C2×#389C2214yescomplemented
18C2×#387C2214yescomplemented
19C2×#385C2214yes—
20C2×#385C2214yes—
21C2×#381—28no—
22C2×#381C2212yes—
23C2×#381C2211yescharacteristic
24C2×#381C2214yes—
25C2×#381C2211yescharacteristic
26C2×#380C2214yes—
27C2×#379C2214yes—
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2212yes—
34C22×#95C2214yes—
35#1445C2214yescomplemented
36#1444C2214yescomplemented
37#1443C2214yescomplemented
38#1442C2214yescomplemented
39#1442C2214yescomplemented
40#1441C2214yescomplemented
41#1441C2214yescomplemented
42#1441C2214yescomplemented
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1436C2214yescomplemented
50#1436C2214yescomplemented
51#1436C2214yescomplemented
52#1436C2214yescomplemented
53C4⋊#95C2214yescomplemented
54#1434C2214yescomplemented
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C26C2311yescharacteristic
60C24×C4—24no—
61C24×C4—24no—
62C24×C4—44no—
63C24×C4—44no—
64C24×C4Q818yes—
65C24×C4C2314yes—
66C24×C4C2312yes—
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C22⋊C4—28no—
77C22×C22⋊C4—44no—
78C22×C22⋊C4—44no—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C42C2312yes—
89C22×C42C2312yes—
90C22×C42C2312yes—
91C22×C42C2312yes—
92#396—216no—
93#396—48no—
94#389—216no—
95#387—216no—
96#385—216no—
97#385—48no—
98#381—24no—
99#381—28no—
100#381—216no—
101#381D418yes—
102#380—216no—
103#379—216no—
104C4×C22⋊C4—28no—
105C4×C22⋊C4—28no—
106C4×C22⋊C4—44no—
107C4×C22⋊C4—44no—
108C4×C22⋊C4—44no—
109C4×C22⋊C4—44no—
110C2×#95—216no—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C25—22no—
132C25—22no—
133C25—24no—
134C25—24no—
135C25—28no—
136C25—44no—
137C25C2×Q814yes—
138C25C2411yescharacteristic
139C25C2411yescharacteristic
140C25C2411yescharacteristic
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—232no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—28no—
154C23×C4—28no—
155C23×C4—28no—
156C23×C4—416no—
157C23×C4—416no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—432no—
170C23×C4—432no—
171C23×C4C2×D414yes—
172C23×C4C2×Q818yes—
173C23×C4C2×Q818yes—
174C23×C4C2×Q814yes—
175C23×C4C2×Q814yes—
176C23×C4#4114yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
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×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—432no—
229C2×C22⋊C4C2×D418yes—
230C2×C22⋊C4#41116yes—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95C2×D4116yes—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—24no—
267C24—216no—
268C24—216no—
269C24—28no—
270C24—28no—
271C24—28no—
272C24—21no—
273C24—24no—
274C24—24no—
275C24—21no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—24no—
280C24—24no—
281C24—216no—
282C24—48no—
283C24—416no—
284C24—416no—
285C24—416no—
286C24—416no—
287C24—416no—
288C24—416no—
289C24C22×D412yes—
290C24C22×Q812yes—
291C24C22×Q812yes—
292C24C22×Q812yes—
293C24C2×#4112yes—
294C24C2×#4118yes—
295C24H(5,𝔽2)14yes—
296C24H(5,𝔽2)14yes—
297C24C2511yescharacteristic
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—28no—
303C22×C4—216no—
304C22×C4—216no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—216no—
330C22×C4—216no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
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—24no—
351C22×C4—24no—
352C22×C4—48no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—416no—
369C22×C4—416no—
370C22×C4—416no—
371C22×C4—4128no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—416no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C22⋊Q8116yes—
379C22×C4C22×D418yes—
380C22×C4C22×D414yes—
381C22×C4C22×Q814yes—
382C22×C4C22×Q814yes—
383C22×C4C2×#4118yes—
384C22×C4C2×#4118yes—
385C22×C4C2×#4114yes—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—88no—
389C4⋊C4—88no—
390C4⋊C4—88no—
391C4⋊C4—88no—
392C22⋊C4—232no—
393C22⋊C4—48no—
394C22⋊C4—416no—
395C22⋊C4—416no—
396C22⋊C4—416no—
397C22⋊C4—416no—
398C22⋊C4—416no—
399C22⋊C4—48no—
400C22⋊C4—416no—
401C42—84no—
402C42—84no—
403C42—84no—
404C42—84no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—28no—
409C23—28no—
410C23—24no—
411C23—24no—
412C23—28no—
413C23—28no—
414C23—28no—
415C23—232no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—28no—
425C23—28no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—28no—
431C23—28no—
432C23—432no—
433C23—432no—
434C23—464no—
435C23—416no—
436C23—464no—
437C23—416no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23C2×C22⋊Q814yes—
447C23#53318yes—
448C23#54314yes—
449C23#54314yes—
450C23#553116yes—
451C23C23×D411yescharacteristic
452C23C23×Q811yescharacteristic
453C23C22×#4111yescharacteristic
454C23C22×#4114yes—
455C23C2×H(5,𝔽2)12yes—
456C23C2×H(5,𝔽2)12yes—
457C23#58514yes—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—416no—
477C2×C4—432no—
478C2×C4—432no—
479C2×C4—416no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
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—832no—
518C2×C4C2×C22⋊Q818yes—
519C22—216no—
520C22—24no—
521C22—24no—
522C22—28no—
523C22—216no—
524C22—28no—
525C22—24no—
526C22—24no—
527C22—28no—
528C22—28no—
529C22—464no—
530C22—48no—
531C22—48no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—44no—
537C22—48no—
538C22—48no—
539C22—44no—
540C22#260818yescomplemented
541C22C22×C22⋊Q811yescharacteristic
542C22C2×#53312yes—
543C22C2×#54311yescharacteristic
544C22C2×#54311yescharacteristic
545C22#345812yes—
546C22D4×#4114yes—
547C22C2×#55314yes—
548C22#347514yes—
549C22#348412yes—
550C22#349012yes—
551C22#351314yes—
552C22#351314yes—
553C22#353314yes—
554C4—28no—
555C4—48no—
556C4—48no—
557C4—44no—
558C4—48no—
559C4—84no—
560C4—84no—
561C4—84no—
562C2—28no—
563C2—44no—
564C2—44no—
565C2#3473912yes—
566C2C2×#260811yescharacteristic
567C2#3498211yescharacteristic
568C2#3506512yes—
569C2#3747911yescharacteristic
570C2#5919014yes—
571C2#5936814yes—
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. C2×#2608C2semidirect
3. #16667C2semidirect
4. #14773C2semidirect
5. #14769C2semidirect
7. #14165C2semidirect
8. #14028C2semidirect
9. #14004C2semidirect
10. #13953C2semidirect
17. C2×#389C22semidirect
18. C2×#387C22semidirect
35. #1445C22semidirect
36. #1444C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1441C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
49. #1436C22semidirect
50. #1436C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect
53. C4⋊#95C22semidirect
54. #1434C22semidirect
540. C22#2608semidirect

Maximal subgroups

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

subgroupindex
C2×#26082
C22×#3812
#166672
#147732
#147692
#147262
#141652
#140282
#140042
#139532
C4×#3792