Finite Group Databasebeta

← #4308782

Subgroups — #4308782

Order 512.

3742 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic, complemented
2C22×#380C211yesmaximal, characteristic
3#16304C218yesmaximal, complemented
4#16303C218yesmaximal, complemented
5#14822C212yesmaximal, complemented
6#14734C212yesmaximal
7#14720C212yesmaximal, complemented
8#14717C212yesmaximal
9#14680C212yesmaximal
10#14347C211yesmaximal, characteristic, complemented
11#14275C212yesmaximal, complemented
12C25×C4C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18#2659—28no—
19C2×#400C2214yescomplemented
20C2×#399C2214yes—
21C2×#398C2214yescomplemented
22C2×#395C2214yescomplemented
23C2×#395C2214yescomplemented
24C2×#393C2214yes—
25C2×#385C2214yes—
26C2×#380—28no—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2218yes—
31C2×#380C2214yes—
32C2×#380C2214yes—
33C2×#379C2214yes—
34C22×#95C2211yescharacteristic
35C22×#95C2211yescharacteristic
36#1444C2218yescomplemented
37#1444C2214yescomplemented
38#1442C2218yescomplemented
39#1442C2218yescomplemented
40#1442C2214yescomplemented
41#1441C2214yescomplemented
42#1441C2214yescomplemented
43#1441C2218yescomplemented
44#1439C2218yes—
45#1439C2214yes—
46#1439C2218yes—
47#1439C2214yes—
48C22⋊#95C2212yes—
49C22⋊#95C2212yes—
50C22⋊#95C2214yes—
51C22⋊#95C2218yes—
52C22⋊#95C2214yes—
53C26C2311yescharacteristic
54C23×Q8C2312yes—
55C24×C4—24no—
56C24×C4—24no—
57C24×C4—44no—
58C24×C4—44no—
59C24×C4Q818yescomplemented
60C24×C4C2311yescharacteristic
61C24×C4C2314yes—
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—48no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4—44no—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2318yes—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—44no—
86C22×C22⋊C4—44no—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2318yes—
92C22×C22⋊C4C2318yes—
93#400—48no—
94#399—48no—
95#398—216no—
96#395—216no—
97#395—48no—
98#393—48no—
99#385—48no—
100#380—28no—
101#380—28no—
102#380—28no—
103#380—232no—
104#380—48no—
105#380—48no—
106#379—48no—
107C2×#95—24no—
108C2×#95D418yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2318yes—
112C2×#95C2318yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2318yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2318yes—
120C2×#95C2314yes—
121C2×#95C2318yes—
122C2×#95C2314yes—
123C25—22no—
124C25—22no—
125C25—24no—
126C25—24no—
127C25—24no—
128C25—24no—
129C25—42no—
130C25—42no—
131C25C2×Q814yes—
132C25C2412yes—
133C25C2411yescharacteristic
134C22×Q8—24no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—44no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—232no—
145C23×C4—28no—
146C23×C4—28no—
147C23×C4—28no—
148C23×C4—28no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—432no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—44no—
157C23×C4—44no—
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—416no—
167C23×C4—416no—
168C23×C4—432no—
169C23×C4C2×D414yes—
170C23×C4C2×D414yes—
171C23×C4C2×D414yes—
172C23×C4C2×Q814yes—
173C23×C4C2×Q8116yes—
174C23×C4C2×Q814yes—
175C23×C4#4114yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2418yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C22⋊Q8—432no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—416no—
203C2×C4⋊C4—416no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—44no—
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—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—216no—
224C2×C22⋊C4—216no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—416no—
228C2×C22⋊C4—416no—
229C2×C22⋊C4—48no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—48no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—216no—
238#95—216no—
239#95—48no—
240#95—44no—
241#95—48no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—48no—
246#95C2×D4116yes—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—28no—
252C24—28no—
253C24—216no—
254C24—28no—
255C24—216no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—22no—
263C24—24no—
264C24—216no—
265C24—48no—
266C24—416no—
267C24—416no—
268C24—416no—
269C24—416no—
270C24—44no—
271C24—44no—
272C24—48no—
273C24—44no—
274C24—44no—
275C24—48no—
276C24C22×D412yes—
277C24C22×Q814yes—
278C24C22×Q812yes—
279C24C2×#4112yes—
280C24H(5,𝔽2)14yes—
281C24H(5,𝔽2)14yes—
282C242−1+414yes—
283C242−1+414yes—
284C24C2511yescharacteristic
285C2×Q8—48no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8—816no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—216no—
292C22×C4—28no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—216no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—232no—
309C22×C4—232no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—48no—
338C22×C4—416no—
339C22×C4—48no—
340C22×C4—416no—
341C22×C4—416no—
342C22×C4—416no—
343C22×C4—416no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—4128no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—88no—
356C22×C4—88no—
357C22×C4—88no—
358C22×C4—88no—
359C22×C4C22⋊Q8116yes—
360C22×C4C22×D414yes—
361C22×C4C22×D414yes—
362C22×C4C22×D414yes—
363C22×C4C22×Q814yes—
364C22×C4C22×Q814yes—
365C22×C4C2×#4114yes—
366C22×C4H(5,𝔽2)14yes—
367C22×C4H(5,𝔽2)14yes—
368C4⋊C4—416no—
369C4⋊C4—416no—
370C4⋊C4—48no—
371C4⋊C4—416no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—88no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—816no—
378C22⋊C4—416no—
379C22⋊C4—416no—
380C22⋊C4—416no—
381C22⋊C4—416no—
382C22⋊C4—816no—
383C22⋊C4—816no—
384C23—24no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—24no—
392C23—28no—
393C23—28no—
394C23—28no—
395C23—28no—
396C23—232no—
397C23—24no—
398C23—28no—
399C23—28no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—28no—
406C23—28no—
407C23—28no—
408C23—28no—
409C23—464no—
410C23—416no—
411C23—416no—
412C23—416no—
413C23—416no—
414C23—416no—
415C23—464no—
416C23—416no—
417C23—42no—
418C23—42no—
419C23—44no—
420C23—42no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23C2×C22⋊Q814yes—
428C23#53514yes—
429C23#53614yes—
430C23#54218yes—
431C23#54418yes—
432C23#54418yes—
433C23C23×D411yescharacteristic
434C23C23×Q811yescharacteristic
435C23C22×#4111yescharacteristic
436C23C2×H(5,𝔽2)12yes—
437C23C2×H(5,𝔽2)12yes—
438C23C2×H(5,𝔽2)12yes—
439C23C2×H(5,𝔽2)12yes—
440C23C2×2−1+412yes—
441C23C2×2−1+412yes—
442Q8—816no—
443C2×C4—232no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—416no—
461C2×C4—464no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—88no—
490C2×C4—88no—
491C2×C4—88no—
492C2×C4—88no—
493C2×C4C2×C22⋊Q818yes—
494C2×C4#54318yes—
495C22—24no—
496C22—24no—
497C22—216no—
498C22—24no—
499C22—24no—
500C22—216no—
501C22—24no—
502C22—24no—
503C22—216no—
504C22—28no—
505C22—464no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—48no—
510C22—44no—
511C22—44no—
512C22—48no—
513C22—48no—
514C22—44no—
515C22#262818yes—
516C22C22×C22⋊Q811yescharacteristic
517C22C2×#53511yescharacteristic
518C22C2×#53611yescharacteristic
519C22C2×#54212yes—
520C22C2×#54312yes—
521C22C2×#54412yes—
522C22C2×#54412yes—
523C22#348411yescharacteristic
524C22#348411yescharacteristic
525C22#348411yescharacteristic
526C22#348611yescharacteristic
527C22#349012yes—
528C22#349212yes—
529C22#353614yes—
530C22#353614yes—
531C22#353714yes—
532C22#353814yes—
533C4—28no—
534C4—28no—
535C4—44no—
536C4—44no—
537C4—84no—
538C4—82no—
539C4—84no—
540C4—82no—
541C4—88no—
542C2—28no—
543C2—48no—
544C2C2×#262811yescharacteristic
545C2#3498211yescharacteristic
546C2#3522411yescharacteristic
547C2#3670711yescharacteristic
548C2#3670811yescharacteristic
549C2#3671711yescharacteristic
550C2#3671811yescharacteristic
551C2#3672914yes—
552C2#3747714yes—
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×#2659C2semidirect
3. #16304C2semidirect
4. #16303C2semidirect
5. #14822C2semidirect
7. #14720C2semidirect
10. #14347C2semidirect
11. #14275C2semidirect
19. C2×#400C22semidirect
21. C2×#398C22semidirect
22. C2×#395C22semidirect
23. C2×#395C22semidirect
36. #1444C22semidirect
37. #1444C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
59. C24×C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
C22×#3802
#163042
#163032
#148222
#147342
#147202
#147172
#146802
#143472
#142752