Finite Group Databasebeta

← #6660406

Subgroups — #6660406

Order 512.

2420 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35305C212yesmaximal
2#32714C214yesmaximal, complemented
3#29900C214yesmaximal, complemented
4#29743C214yesmaximal, complemented
5#29605C214yesmaximal, complemented
6#27609C211yesmaximal, characteristic, complemented
7#24561C214yesmaximal, complemented
8#24386C214yesmaximal, complemented
9#24016C212yesmaximal, complemented
10#23928C211yesmaximal, characteristic, complemented
11#20580C211yesmaximal, characteristic
12C23×#41C2211yescharacteristic
13C2×#536C2212yes—
14C2×#536C2212yes—
15C22×C22⋊Q8C2212yes—
16C2×#485C2214yescomplemented
17C2×#470C2214yes—
18C2×#451C2214yescomplemented
19C2×#407C2212yes—
20C2×#407C2214yes—
21#2848C2214yes—
22#2848C2214yes—
23#2616C2214yes—
24#2616C2214yes—
25#2435—22no—
26#2434—22no—
27#2432—22no—
28#2432—22no—
29#2406—22no—
30#2406—22no—
31#2405—22no—
32#2405—22no—
33#2311C2212yes—
34#2311C2212yes—
35#2280C2212yes—
36#2077C2214yescomplemented
37#2077C2214yescomplemented
38#2077C2214yescomplemented
39#2076C2214yescomplemented
40#2036C2214yescomplemented
41#2036C2214yescomplemented
42#2036C2214yescomplemented
43#2035C2214yescomplemented
44#2035C2214yescomplemented
45#2035C2214yescomplemented
46#2035C2214yescomplemented
47#2035C2214yescomplemented
48#2001C2214yescomplemented
49#2001C2214yescomplemented
50#2001C2214yescomplemented
51#2001C2214yescomplemented
52#1894C2214yes—
53#1894C2214yes—
54#1873C2214yescomplemented
55#1872C2214yescomplemented
56#1854C2214yescomplemented
57#1853C2214yescomplemented
58#1825C2214yescomplemented
59#1809C2214yes—
60#1787C2214yescomplemented
61#1780C2211yescharacteristic
62#1780C2214yes—
63#1780C2212yes—
64#1780C2211yescharacteristic
65C22×#41—44no—
66C22×#41—48no—
67C22×#41D411yescharacteristic
68C22×#41D412yes—
69C22×#41D411yescharacteristic
70C22×#41C2311yescharacteristic
71C22×#41C2312yes—
72C22×#41C2311yescharacteristic
73C23×Q8D411yescharacteristic
74C23×Q8C2312yes—
75C23×D4D411yescharacteristic
76C23×D4D412yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C22×Q16C2314yes—
80C22×M16C2314yes—
81#536—48no—
82#536—48no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—22no—
85C2×#124—22no—
86C2×#124—22no—
87C2×#123—22no—
88C2×#123—22no—
89C2×#123—22no—
90C2×#123—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—44no—
94C2×C22⋊Q8—44no—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2314yes—
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2314yes—
101C2×#117C2312yes—
102C2×#117C2312yes—
103C22×C4⋊C4—22no—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—22no—
106#485—48no—
107#470—48no—
108#451—48no—
109C2×#107C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118#407—48no—
119#407—44no—
120C2×C22⋊C8C2314yes—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2312yes—
127#395—24no—
128#395—24no—
129#394—24no—
130#394—24no—
131#394—24no—
132#394—24no—
133#394—24no—
134#394—24no—
135#391—22no—
136#391—22no—
137#390—22no—
138#390—22no—
139#386—22no—
140#386—22no—
141#386—22no—
142#386—22no—
143#385—24no—
144#385—24no—
145#385—24no—
146#385—24no—
147#385—24no—
148#385—24no—
149#381—22no—
150#380C2314yes—
151#380C2314yes—
152#380C2314yes—
153#380C2314yes—
154#380C2314yes—
155#380C2314yes—
156#379—24no—
157#379—24no—
158#379—22no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—22no—
161#340C2314yes—
162#340C2314yes—
163#340C2314yes—
164#340C2314yes—
165#340C2314yes—
166#340C2314yes—
167C25C2×D412yes—
168C25C2×D411yescharacteristic
169C2×#41—28no—
170C2×#41—28no—
171C2×#41—42no—
172C2×#41—44no—
173C2×#41—42no—
174C2×#41—42no—
175C2×#41—42no—
176C2×#41—42no—
177C2×#41—44no—
178C2×#41—42no—
179C2×#41—44no—
180C2×#41—44no—
181C2×#41—44no—
182C2×#41—44no—
183C2×#41—44no—
184C2×#41—464no—
185C22×Q8—22no—
186C22×Q8—28no—
187C22×Q8—42no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—44no—
192C22×Q8C2×D411yescharacteristic
193C22×Q8C2×D412yes—
194C22×Q8C2×D411yescharacteristic
195C22×Q8C2414yes—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4C2×D411yescharacteristic
202C22×D4C2×D412yes—
203C22×D4C2×D412yes—
204C22×D4C2×D412yes—
205C22×D4C2×D412yes—
206C22×D4C2×D411yescharacteristic
207C22×D4C2×D412yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C2×Q16—44no—
228C2×Q16—44no—
229C2×Q16—44no—
230C2×M16—28no—
231C2×M16—44no—
232C2×M16—44no—
233C22×C8C2414yes—
234C22×C8C2414yes—
235C4⋊Q8—48no—
236C4⋊Q8—48no—
237#124—48no—
238#124—48no—
239#123—48no—
240#123—48no—
241#123—48no—
242#123—48no—
243C22⋊Q8—28no—
244C22⋊Q8—28no—
245C22⋊Q8—28no—
246C22⋊Q8—28no—
247C22⋊Q8—44no—
248C22⋊Q8—44no—
249C22⋊Q8—44no—
250C22⋊Q8—44no—
251C22⋊Q8—44no—
252C22⋊Q8—44no—
253C22⋊Q8—44no—
254C22⋊Q8—44no—
255C22⋊Q8—48no—
256C22⋊Q8—48no—
257#117—28no—
258#117—44no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2414yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—48no—
287C2×C42—22no—
288C2×C42—22no—
289#107—44no—
290#106—44no—
291#106—44no—
292#103—44no—
293#103—44no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298C22⋊C8—28no—
299C22⋊C8—28no—
300C22⋊C8—44no—
301C22⋊C8—44no—
302C22⋊C8—42no—
303C22⋊C8—44no—
304C22⋊C8—42no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—42no—
315C24—42no—
316C24—44no—
317C24—44no—
318C24—44no—
319C24C22×D412yes—
320C24C22×D411yescharacteristic
321C24C22×D411yescharacteristic
322C24C22×D411yescharacteristic
323C24C22×D412yes—
324#41—416no—
325#41—416no—
326#41—416no—
327#41—416no—
328#41—432no—
329#41—416no—
330#41—416no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—42no—
340C2×Q8—41no—
341C2×Q8—42no—
342C2×Q8—41no—
343C2×Q8—42no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—416no—
347C2×Q8—44no—
348C2×Q8—88no—
349C2×Q8—88no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—432no—
370C2×D4—416no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—84no—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4H(5,𝔽2)18yes—
416C22×C4H(5,𝔽2)14yes—
417C22×C4C2511yescharacteristic
418Q16—88no—
419M16—88no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4—48no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C42—44no—
454C42—44no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—48no—
473C23—44no—
474C23—48no—
475C23—84no—
476C23—84no—
477C23—88no—
478C23—88no—
479C23—88no—
480C23#53412yes—
481C23#53414yes—
482C23C23×D411yescharacteristic
483Q8—44no—
484Q8—48no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—44no—
489Q8—44no—
490Q8—88no—
491D4—48no—
492D4—48no—
493D4—44no—
494D4—48no—
495D4—48no—
496D4—48no—
497D4—44no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—88no—
534C2×C4—88no—
535C2×C4#53414yes—
536C2×C4#53412yes—
537C2×C4C2×H(5,𝔽2)12yes—
538C2×C4C2×H(5,𝔽2)14yes—
539C8—44no—
540C8—44no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—28no—
545C22—42no—
546C22—42no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—41no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—816no—
561C22—88no—
562C22C2×#53411yescharacteristic
563C22C2×#53412yes—
564C22#358614yes—
565C4—22no—
566C4—24no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—84no—
571C4—84no—
572C4—84no—
573C4—84no—
574C4#353214yes—
575C2—44no—
576C2—42no—
577C2—82no—
578C2—81no—
579C2—81no—
580C2#3743611yescharacteristic
581C2#6075412yes—
582C2#6077314yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #32714C2semidirect
3. #29900C2semidirect
4. #29743C2semidirect
5. #29605C2semidirect
6. #27609C2semidirect
7. #24561C2semidirect
8. #24386C2semidirect
9. #24016C2semidirect
10. #23928C2semidirect
16. C2×#485C22semidirect
18. C2×#451C22semidirect
36. #2077C22semidirect
37. #2077C22semidirect
38. #2077C22semidirect
39. #2076C22semidirect
40. #2036C22semidirect
41. #2036C22semidirect
42. #2036C22semidirect
43. #2035C22semidirect
44. #2035C22semidirect
45. #2035C22semidirect
46. #2035C22semidirect
47. #2035C22semidirect
48. #2001C22semidirect
49. #2001C22semidirect
50. #2001C22semidirect
51. #2001C22semidirect
54. #1873C22semidirect
55. #1872C22semidirect
56. #1854C22semidirect
57. #1853C22semidirect
58. #1825C22semidirect
60. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#353052
#327142
#299002
#297432
#296052
#276092
#245612
#243862
#240162
#239282
#205802