Finite Group Databasebeta

← #4894600

Subgroups — #4894600

Order 512.

3362 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17272C212yesmaximal, complemented
2#17271C211yesmaximal, characteristic, complemented
3#17135C211yesmaximal, characteristic, complemented
4#16667C214yesmaximal, complemented
5#16562C214yesmaximal, complemented
6#16319C214yesmaximal, complemented
7#16298C214yesmaximal, complemented
8#15327C214yesmaximal, complemented
9#14726C214yesmaximal, complemented
10#14335C211yesmaximal, characteristic
11#13867C212yesmaximal
12C24×D4C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14C2×#399C2214yescomplemented
15C2×#396C2214yes—
16C2×#396C2214yescomplemented
17C2×#396C2214yescomplemented
18C2×#386C2214yes—
19C2×#386C2214yes—
20C2×#385C2214yescomplemented
21C2×#385C2214yescomplemented
22C2×#381C2212yes—
23C2×#380C2214yescomplemented
24C2×#379C2214yes—
25C2×#379C2212yes—
26C2×#379C2214yes—
27#1445C2214yescomplemented
28#1444C2214yescomplemented
29#1443C2214yescomplemented
30#1442C2214yescomplemented
31#1442C2214yescomplemented
32#1441C2214yescomplemented
33#1441C2214yescomplemented
34#1441C2214yescomplemented
35#1439C2214yes—
36#1439C2214yes—
37#1439C2214yes—
38#1439C2214yescomplemented
39#1439C2214yescomplemented
40#1439C2214yescomplemented
41#1439C2214yescomplemented
42#1439C2214yescomplemented
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1439C2214yescomplemented
46#1439C2214yescomplemented
47#1438C2212yes—
48#1438C2212yes—
49#1438C2212yes—
50#1438C2212yes—
51#1438C2211yescharacteristic
52#1438C2211yescharacteristic
53#1438C2214yes—
54#1438C2212yes—
55C22⋊#95C2214yes—
56C22⋊#95C2214yescomplemented
57C26C2312yes—
58C23×D4—44no—
59C23×D4—48no—
60C23×D4Q818yes—
61C23×D4C2311yescharacteristic
62C23×D4C2312yes—
63C23×D4C2311yescharacteristic
64C24×C4C2314yes—
65C24×C4C2311yescharacteristic
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C22⋊C4—48no—
73C22×C22⋊C4—44no—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90#399—48no—
91#396—48no—
92#396—48no—
93#396—48no—
94#386—48no—
95#386—48no—
96#385—48no—
97#385—48no—
98#381—44no—
99#380—48no—
100#379—48no—
101#379—48no—
102#379—44no—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2312yes—
115C2×#95C2312yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C25—24no—
121C25—24no—
122C25—28no—
123C25—44no—
124C25—44no—
125C25—44no—
126C25—44no—
127C25—44no—
128C25C2×Q818yes—
129C25C2412yes—
130C25C2411yescharacteristic
131C25C2412yes—
132C22×D4—28no—
133C22×D4—28no—
134C22×D4—416no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4—432no—
142C22×D4—464no—
143C22×D4C2×Q814yes—
144C22×D4C2×Q818yes—
145C22×D4C2×Q814yes—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—28no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4C2×Q814yes—
156C23×C4C2414yes—
157C23×C4C2414yes—
158C23×C4C2414yes—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2411yescharacteristic
162C23×C4C2411yescharacteristic
163C23×C4C2414yes—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—432no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227#95—28no—
228#95—28no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—42no—
233#95—44no—
234#95—42no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244C24—24no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—22no—
249C24—24no—
250C24—22no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—24no—
255C24—24no—
256C24—24no—
257C24—22no—
258C24—22no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—216no—
263C24—216no—
264C24—48no—
265C24—416no—
266C24—416no—
267C24—416no—
268C24—416no—
269C24—48no—
270C24—48no—
271C24—416no—
272C24—84no—
273C24—84no—
274C24—88no—
275C24—88no—
276C24—88no—
277C24—88no—
278C24—88no—
279C24—88no—
280C24—88no—
281C24—88no—
282C24—88no—
283C24C22×Q814yes—
284C24C22×Q812yes—
285C24C22×Q814yes—
286C24H(5,𝔽2)14yes—
287C24H(5,𝔽2)18yes—
288C24C2511yescharacteristic
289C2×D4—216no—
290C2×D4—216no—
291C2×D4—44no—
292C2×D4—48no—
293C2×D4—44no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—416no—
297C2×D4—416no—
298C2×D4—416no—
299C2×D4—4128no—
300C2×D4—416no—
301C22×C4—28no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
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—24no—
330C22×C4—24no—
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—48no—
348C22×C4—416no—
349C22×C4—416no—
350C22×C4—416no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4C22×Q812yes—
357C22×C4C22×Q812yes—
358C22×C4H(5,𝔽2)18yes—
359C4⋊C4—88no—
360C4⋊C4—88no—
361C4⋊C4—88no—
362C4⋊C4—88no—
363C4⋊C4—88no—
364C4⋊C4—88no—
365C22⋊C4—48no—
366C22⋊C4—416no—
367C22⋊C4—416no—
368C22⋊C4—416no—
369C22⋊C4—48no—
370C22⋊C4—88no—
371C22⋊C4—88no—
372C22⋊C4—88no—
373C22⋊C4—88no—
374C22⋊C4—88no—
375C22⋊C4—88no—
376C22⋊C4—84no—
377C22⋊C4—84no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—28no—
385C23—28no—
386C23—28no—
387C23—24no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—24no—
392C23—28no—
393C23—28no—
394C23—28no—
395C23—28no—
396C23—28no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—28no—
401C23—28no—
402C23—432no—
403C23—432no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—816no—
435C23—832no—
436C23—832no—
437C23—832no—
438C23—832no—
439C23—832no—
440C23—832no—
441C23#54314yes—
442C23#54318yes—
443C23C23×Q811yescharacteristic
444C23C2×H(5,𝔽2)12yes—
445C23C2×H(5,𝔽2)14yes—
446C23C2×H(5,𝔽2)14yes—
447C23#58514yes—
448D4—416no—
449D4—432no—
450D4—416no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—416no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
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—832no—
501C22—24no—
502C22—28no—
503C22—28no—
504C22—216no—
505C22—44no—
506C22—44no—
507C22—48no—
508C22—48no—
509C22—48no—
510C22—48no—
511C22—48no—
512C22—48no—
513C22—48no—
514C22—48no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—48no—
519C22—48no—
520C22—864no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22C2×#54311yescharacteristic
526C22C2×#54312yes—
527C22#350114yes—
528C22#353214yes—
529C22#353212yes—
530C22#353312yes—
531C22#353414yes—
532C22#353714yes—
533C22#354214yes—
534C22#354214yes—
535C22#356014yes—
536C4—42no—
537C4—42no—
538C4—48no—
539C4—84no—
540C4—84no—
541C4—84no—
542C4—84no—
543C4—84no—
544C2—48no—
545C2—44no—
546C2—84no—
547C2#3674712yes—
548C2#3747511yescharacteristic
549C2#4060914yes—
550C2#5525114yes—
551C2#5573714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17272C2semidirect
2. #17271C2semidirect
3. #17135C2semidirect
4. #16667C2semidirect
5. #16562C2semidirect
6. #16319C2semidirect
7. #16298C2semidirect
8. #15327C2semidirect
9. #14726C2semidirect
14. C2×#399C22semidirect
16. C2×#396C22semidirect
17. C2×#396C22semidirect
20. C2×#385C22semidirect
21. C2×#385C22semidirect
23. C2×#380C22semidirect
27. #1445C22semidirect
28. #1444C22semidirect
29. #1443C22semidirect
30. #1442C22semidirect
31. #1442C22semidirect
32. #1441C22semidirect
33. #1441C22semidirect
34. #1441C22semidirect
38. #1439C22semidirect
39. #1439C22semidirect
40. #1439C22semidirect
41. #1439C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
56. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
#172722
#172712
#171352
#166672
#165622
#163192
#162982
#153272
#147262
#143352
#138672