Finite Group Databasebeta

← #626890

Subgroups — #626890

Order 512.

3794 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic, complemented
2C22×#385C211yesmaximal, characteristic
3#15296C218yesmaximal, complemented
4#15292C218yesmaximal, complemented
5#14822C212yesmaximal, complemented
6#14734C212yesmaximal
7#14680C212yesmaximal
8#14339C211yesmaximal, characteristic, complemented
9#14275C212yesmaximal, complemented
10#14128C212yesmaximal, complemented
11#13868C212yesmaximal
12C25×C4C2211yescharacteristic
13C22×#117C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C2×#400C2214yescomplemented
19C2×#399C2214yes—
20#2316—28no—
21C2×#389C2214yescomplemented
22C2×#389C2214yescomplemented
23C2×#385—28no—
24C2×#385C2218yes—
25C2×#385C2214yes—
26C2×#381C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2214yes—
31C2×#380C2214yes—
32C2×#379C2214yes—
33C22×#95C2211yescharacteristic
34C22×#95C2211yescharacteristic
35#1444C2214yescomplemented
36#1444C2214yescomplemented
37#1442C2214yescomplemented
38#1442C2218yescomplemented
39#1442C2214yescomplemented
40#1441C2218yescomplemented
41#1441C2214yescomplemented
42#1441C2218yescomplemented
43#1439C2214yes—
44#1439C2214yes—
45#1439C2218yes—
46#1439C2214yes—
47#1437C2218yes—
48#1436C2218yescomplemented
49C22⋊#95C2212yes—
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2218yes—
53C26C2311yescharacteristic
54C24×C4—24no—
55C24×C4—28no—
56C24×C4—44no—
57C24×C4—44no—
58C24×C4C2311yescharacteristic
59C24×C4C2314yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C2×#117—28no—
66C2×#117—48no—
67C22×C4⋊C4—28no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—44no—
70C22×C4⋊C4D418yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2318yes—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—44no—
85C22×C22⋊C4Q818yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2318yes—
91C22×C22⋊C4C2318yes—
92C22×C42C2314yes—
93#400—48no—
94#399—48no—
95#389—216no—
96#389—216no—
97#385—232no—
98#385—48no—
99#381—28no—
100#380—28no—
101#380—28no—
102#380—28no—
103#380—48no—
104#380—48no—
105#379—48no—
106C2×#95—24no—
107C2×#95—24no—
108C2×#95C2318yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2318yes—
118C2×#95C2314yes—
119C2×#95C2318yes—
120C2×#95C2318yes—
121C2×#95C2318yes—
122C25—22no—
123C25—22no—
124C25—24no—
125C25—24no—
126C25—24no—
127C25—24no—
128C25—42no—
129C25—42no—
130C25C2×Q814yes—
131C25C2412yes—
132C25C2411yescharacteristic
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—28no—
139C23×C4—24no—
140C23×C4—28no—
141C23×C4—24no—
142C23×C4—28no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—28no—
147C23×C4—28no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—416no—
160C23×C4—464no—
161C23×C4—416no—
162C23×C4—416no—
163C23×C4C2×D414yes—
164C23×C4C2×D414yes—
165C23×C4C2×D414yes—
166C23×C4C2×Q814yes—
167C23×C4C2×Q814yes—
168C23×C4#4114yes—
169C23×C4#4118yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2418yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179#117—432no—
180C2×C4⋊C4—232no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—416no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C2×D4116yes—
206C2×C22⋊C4—232no—
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—216no—
218C2×C22⋊C4—216no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—416no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—48no—
224C2×C22⋊C4—48no—
225C2×C22⋊C4C2×Q8116yes—
226C2×C42—28no—
227C2×C42—44no—
228C2×C42—44no—
229#95—28no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—216no—
237#95—48no—
238#95—48no—
239#95—48no—
240#95—44no—
241#95—48no—
242#95—44no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—28no—
248C24—28no—
249C24—216no—
250C24—28no—
251C24—216no—
252C24—22no—
253C24—22no—
254C24—24no—
255C24—24no—
256C24—24no—
257C24—22no—
258C24—22no—
259C24—24no—
260C24—216no—
261C24—48no—
262C24—416no—
263C24—416no—
264C24—416no—
265C24—416no—
266C24—44no—
267C24—48no—
268C24—44no—
269C24—44no—
270C24—44no—
271C24—48no—
272C24C22×D412yes—
273C24C22×Q814yes—
274C24C22×Q812yes—
275C24C2×#4112yes—
276C24C2×#4114yes—
277C24H(5,𝔽2)14yes—
278C24H(5,𝔽2)14yes—
279C242−1+414yes—
280C24C2511yescharacteristic
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—216no—
284C22×C4—216no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—232no—
291C22×C4—28no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—28no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—24no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—416no—
328C22×C4—48no—
329C22×C4—416no—
330C22×C4—416no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—416no—
334C22×C4—416no—
335C22×C4—416no—
336C22×C4—416no—
337C22×C4—44no—
338C22×C4—48no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—4128no—
342C22×C4—44no—
343C22×C4—416no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4C22⋊Q8116yes—
350C22×C4#123116yes—
351C22×C4C22×D414yes—
352C22×C4C22×D414yes—
353C22×C4C22×D414yes—
354C22×C4C22×Q814yes—
355C22×C4C22×Q814yes—
356C22×C4C2×#4114yes—
357C22×C4C2×#4114yes—
358C22×C4C2×#4114yes—
359C22×C4H(5,𝔽2)14yes—
360C4⋊C4—232no—
361C4⋊C4—416no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—88no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C4⋊C4—816no—
370C22⋊C4—232no—
371C22⋊C4—416no—
372C22⋊C4—416no—
373C22⋊C4—416no—
374C22⋊C4—816no—
375C22⋊C4—816no—
376C42—416no—
377C23—24no—
378C23—24no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—28no—
384C23—24no—
385C23—28no—
386C23—24no—
387C23—28no—
388C23—28no—
389C23—232no—
390C23—24no—
391C23—24no—
392C23—28no—
393C23—24no—
394C23—24no—
395C23—28no—
396C23—24no—
397C23—24no—
398C23—28no—
399C23—28no—
400C23—28no—
401C23—28no—
402C23—464no—
403C23—416no—
404C23—464no—
405C23—416no—
406C23—416no—
407C23—416no—
408C23—416no—
409C23—42no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—416no—
418C23—44no—
419C23—44no—
420C23C2×C22⋊Q814yes—
421C23C2×#12314yes—
422C23#53514yes—
423C23#54418yes—
424C23#55318yes—
425C23#55418yes—
426C23C23×D411yescharacteristic
427C23C23×Q811yescharacteristic
428C23C22×#4111yescharacteristic
429C23C22×#4112yes—
430C23C2×H(5,𝔽2)12yes—
431C23C2×H(5,𝔽2)12yes—
432C23C2×H(5,𝔽2)12yes—
433C23C2×2−1+412yes—
434C23#58512yes—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—232no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—416no—
452C2×C4—464no—
453C2×C4—416no—
454C2×C4—416no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—48no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—88no—
484C2×C4—88no—
485C2×C4C2×C22⋊Q818yes—
486C2×C4C2×#12318yes—
487C22—24no—
488C22—216no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—216no—
495C22—28no—
496C22—216no—
497C22—464no—
498C22—44no—
499C22—48no—
500C22—48no—
501C22—44no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22—48no—
507C22#248718yes—
508C22C22×C22⋊Q811yescharacteristic
509C22C22×#12311yescharacteristic
510C22C2×#53511yescharacteristic
511C22C2×#54412yes—
512C22#346012yes—
513C22C2×#55312yes—
514C22C2×#55412yes—
515C22#348411yescharacteristic
516C22#348411yescharacteristic
517C22#348611yescharacteristic
518C22#348711yescharacteristic
519C22#349014yes—
520C22#349212yes—
521C22#349514yes—
522C22#350112yes—
523C22#353414yes—
524C22#353614yes—
525C4—28no—
526C4—28no—
527C4—44no—
528C4—44no—
529C4—44no—
530C4—88no—
531C4—84no—
532C4—84no—
533C4—82no—
534C2—28no—
535C2—48no—
536C2C2×#248711yescharacteristic
537C2#3517411yescharacteristic
538C2#3522411yescharacteristic
539C2#3535011yescharacteristic
540C2#3567211yescharacteristic
541C2#3670711yescharacteristic
542C2#3671011yescharacteristic
543C2#3787314yes—
544C2#3794114yes—
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×#2316C2semidirect
3. #15296C2semidirect
4. #15292C2semidirect
5. #14822C2semidirect
8. #14339C2semidirect
9. #14275C2semidirect
10. #14128C2semidirect
18. C2×#400C22semidirect
21. C2×#389C22semidirect
22. C2×#389C22semidirect
35. #1444C22semidirect
36. #1444C22semidirect
37. #1442C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1441C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
48. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C22×#3852
#152962
#152922
#148222
#147342
#146802
#143392
#142752
#141282
#138682