Finite Group Databasebeta

← #4868587

Subgroups — #4868587

Order 512.

3514 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2405C212yesmaximal
2#17186C214yesmaximal, complemented
3#16675C214yesmaximal, complemented
4#16657C214yesmaximal, complemented
5#16426C214yesmaximal, complemented
6#16391C214yesmaximal, complemented
7#16014C211yesmaximal, characteristic, complemented
8#15949C211yesmaximal, characteristic, complemented
9#15463C212yesmaximal, complemented
10#14727C214yesmaximal, complemented
11#14335C211yesmaximal, characteristic
12C24×D4C2211yescharacteristic
13C22×#123C2212yes—
14C22×#123C2212yes—
15C23×C22⋊C4C2212yes—
16#2405—216no—
17C2×#397C2214yes—
18C2×#397C2214yescomplemented
19C2×#394C2214yes—
20C2×#394C2214yes—
21C2×#386C2212yes—
22C2×#386C2212yes—
23C2×#385C2214yes—
24C2×#385C2214yescomplemented
25C2×#381C2212yes—
26C2×#379C2212yes—
27C2×#379C2214yes—
28C2×#379C2214yes—
29#1444C2214yescomplemented
30#1444C2214yescomplemented
31#1443C2214yescomplemented
32#1443C2214yescomplemented
33#1442C2214yescomplemented
34#1442C2214yescomplemented
35#1441C2214yescomplemented
36#1441C2214yescomplemented
37#1440C2214yes—
38#1440C2214yes—
39#1440C2214yescomplemented
40#1440C2214yescomplemented
41#1440C2214yescomplemented
42#1440C2214yescomplemented
43#1440C2214yescomplemented
44#1440C2214yescomplemented
45#1439C2214yes—
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1438C2214yes—
52#1438C2212yes—
53#1438C2211yescharacteristic
54#1438C2211yescharacteristic
55C22⋊#95C2214yescomplemented
56C22⋊#95C2214yescomplemented
57C22⋊#95C2214yescomplemented
58C26C2312yes—
59C23×D4—44no—
60C23×D4—48no—
61C23×D4D418yes—
62C23×D4C2311yescharacteristic
63C23×D4C2312yes—
64C23×D4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C24×C4C2312yes—
68C2×#123—28no—
69C2×#123—28no—
70C2×#123—48no—
71C2×#123—48no—
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⋊C4C2312yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94#397—48no—
95#397—48no—
96#394—216no—
97#394—216no—
98#386—28no—
99#386—28no—
100#385—216no—
101#385—48no—
102#381—44no—
103#379—216no—
104#379—28no—
105#379—48no—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C25—24no—
126C25—24no—
127C25—28no—
128C25—44no—
129C25—44no—
130C25—44no—
131C25—44no—
132C25—44no—
133C25C2×D418yes—
134C25C2412yes—
135C25C2411yescharacteristic
136C25C2412yes—
137C22×D4—28no—
138C22×D4—28no—
139C22×D4—416no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4—432no—
147C22×D4—464no—
148C22×D4C2×D414yes—
149C22×D4C2×D418yes—
150C22×D4C2×D414yes—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—28no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4C2×D414yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2411yescharacteristic
171C23×C4C2411yescharacteristic
172#123—432no—
173#123—432no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
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×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
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—432no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
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—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230#95—28no—
231#95—28no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—44no—
240#95—42no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249C24—24no—
250C24—24no—
251C24—22no—
252C24—22no—
253C24—24no—
254C24—24no—
255C24—24no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—22no—
261C24—22no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—216no—
268C24—216no—
269C24—48no—
270C24—416no—
271C24—416no—
272C24—416no—
273C24—416no—
274C24—48no—
275C24—416no—
276C24—48no—
277C24—88no—
278C24—84no—
279C24—88no—
280C24—84no—
281C24—88no—
282C24—88no—
283C24—88no—
284C24—88no—
285C24—88no—
286C24—88no—
287C24—88no—
288C24C22×D414yes—
289C24C22×D412yes—
290C24C22×D414yes—
291C24H(5,𝔽2)14yes—
292C24H(5,𝔽2)18yes—
293C24C2511yescharacteristic
294C2×D4—216no—
295C2×D4—216no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—44no—
299C2×D4—48no—
300C2×D4—44no—
301C2×D4—416no—
302C2×D4—416no—
303C2×D4—416no—
304C2×D4—4128no—
305C2×D4—416no—
306C22×C4—28no—
307C22×C4—22no—
308C22×C4—22no—
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—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—48no—
354C22×C4—416no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—88no—
364C22×C4—84no—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C4⋊C4—416no—
368C4⋊C4—416no—
369C4⋊C4—88no—
370C4⋊C4—88no—
371C22⋊C4—416no—
372C22⋊C4—416no—
373C22⋊C4—416no—
374C22⋊C4—48no—
375C22⋊C4—416no—
376C22⋊C4—416no—
377C22⋊C4—48no—
378C22⋊C4—416no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C22⋊C4—88no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C22⋊C4—88no—
385C22⋊C4—88no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—28no—
390C23—28no—
391C23—28no—
392C23—28no—
393C23—28no—
394C23—28no—
395C23—28no—
396C23—24no—
397C23—28no—
398C23—24no—
399C23—28no—
400C23—28no—
401C23—28no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—28no—
406C23—28no—
407C23—432no—
408C23—432no—
409C23—42no—
410C23—44no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—44no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—816no—
440C23—832no—
441C23—832no—
442C23—832no—
443C23—832no—
444C23—832no—
445C23—832no—
446C23#53414yes—
447C23#53418yes—
448C23C23×D411yescharacteristic
449C23C2×H(5,𝔽2)12yes—
450C23C2×H(5,𝔽2)14yes—
451C23#58514yes—
452C23#58514yes—
453D4—416no—
454D4—432no—
455D4—416no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—416no—
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—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—816no—
506C2×C4—816no—
507C22—24no—
508C22—28no—
509C22—28no—
510C22—216no—
511C22—48no—
512C22—48no—
513C22—48no—
514C22—44no—
515C22—44no—
516C22—48no—
517C22—48no—
518C22—48no—
519C22—48no—
520C22—48no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—48no—
525C22—48no—
526C22—864no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22—84no—
531C22C2×#53411yescharacteristic
532C22C2×#53412yes—
533C22#348514yes—
534C22#353214yes—
535C22#353312yes—
536C22#353314yes—
537C22#353314yes—
538C22#353312yes—
539C22#354514yes—
540C22#356014yes—
541C22#356014yes—
542C4—42no—
543C4—42no—
544C4—84no—
545C4—84no—
546C4—84no—
547C4—84no—
548C4—84no—
549C4—84no—
550C2—48no—
551C2—44no—
552C2—84no—
553C2#3696512yes—
554C2#3743611yescharacteristic
555C2#4140414yes—
556C2#5525714yes—
557C2#5612614yes—
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
2. #17186C2semidirect
3. #16675C2semidirect
4. #16657C2semidirect
5. #16426C2semidirect
6. #16391C2semidirect
7. #16014C2semidirect
8. #15949C2semidirect
9. #15463C2semidirect
10. #14727C2semidirect
18. C2×#397C22semidirect
24. C2×#385C22semidirect
29. #1444C22semidirect
30. #1444C22semidirect
31. #1443C22semidirect
32. #1443C22semidirect
33. #1442C22semidirect
34. #1442C22semidirect
35. #1441C22semidirect
36. #1441C22semidirect
39. #1440C22semidirect
40. #1440C22semidirect
41. #1440C22semidirect
42. #1440C22semidirect
43. #1440C22semidirect
44. #1440C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
55. C22⋊#95C22semidirect
56. C22⋊#95C22semidirect
57. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24052
#171862
#166752
#166572
#164262
#163912
#160142
#159492
#154632
#147272
#143352