Finite Group Databasebeta

← #4128993

Subgroups — #4128993

Order 512.

2747 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17198C211yesmaximal, characteristic, complemented
2#16211C211yesmaximal, characteristic, complemented
3#15824C214yesmaximal, complemented
4#15727C214yesmaximal, complemented
5#15527C214yesmaximal, complemented
6#15259C212yesmaximal
7#14232C212yesmaximal, complemented
8#14134C211yesmaximal, characteristic
9#14098C214yesmaximal, complemented
10#13971C214yesmaximal, complemented
11#13947C214yesmaximal, complemented
12C22×#124C2212yes—
13C22×#120C2211yescharacteristic
14C23×C22⋊C4C2212yes—
15C2×#397C2214yescomplemented
16C2×#393C2212yes—
17C2×#393C2212yes—
18C2×#391C2214yescomplemented
19C2×#388C2214yes—
20C2×#388C2214yes—
21C2×#386C2214yes—
22C2×#385C2214yes—
23C2×#379C2214yes—
24C2×C4×C22⋊C4C2214yes—
25#1446C2214yescomplemented
26#1443C2214yescomplemented
27#1443C2214yescomplemented
28#1442C2214yescomplemented
29#1441C2214yescomplemented
30#1441C2214yescomplemented
31#1440C2214yes—
32#1440C2211yescharacteristic
33#1440C2214yes—
34#1440C2211yescharacteristic
35#1440C2214yes—
36#1439C2214yes—
37#1439C2214yes—
38#1439C2214yescomplemented
39#1438C2214yes—
40#1438C2212yes—
41#1438C2214yes—
42#1438C2212yes—
43#1437C2212yes—
44#1437C2214yes—
45#1437C2212yes—
46#1437C2214yes—
47#1437C2214yescomplemented
48#1437C2214yescomplemented
49#1434C2214yescomplemented
50#1434C2214yescomplemented
51#1434C2214yescomplemented
52#1434C2214yescomplemented
53C22⋊#95C2214yes—
54C22⋊#95C2214yes—
55C22⋊#95C2214yes—
56C4×#95C2214yescomplemented
57C26C2311yescharacteristic
58C23×Q8C2312yes—
59C23×D4C2312yes—
60C23×D4C2311yescharacteristic
61C24×C4C2314yes—
62C2×#124—216no—
63C2×#124—44no—
64C2×#120—24no—
65C2×#120—44no—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C22⋊C4—48no—
69C22×C22⋊C4—44no—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2311yescharacteristic
84C22×C42C2312yes—
85C22×C42C2314yes—
86C22×C42C2314yes—
87#397—48no—
88#393—28no—
89#393—44no—
90#391—48no—
91#388—48no—
92#388—48no—
93#386—48no—
94#385—48no—
95#379—216no—
96C4×C22⋊C4—216no—
97C2×#95C2314yes—
98C2×#95C2312yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2312yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C25—22no—
119C25—24no—
120C25—28no—
121C25—44no—
122C25—44no—
123C25C2411yescharacteristic
124C25C2411yescharacteristic
125C25C2412yes—
126C22×Q8—28no—
127C22×Q8—24no—
128C22×Q8—42no—
129C22×Q8—44no—
130C22×Q8—42no—
131C22×D4—28no—
132C22×D4—24no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—44no—
138C22×D4—44no—
139C23×C4—28no—
140C23×C4—28no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4C2414yes—
147C23×C4C2414yes—
148C23×C4C2414yes—
149C23×C4C2411yescharacteristic
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2412yes—
153C23×C4C2414yes—
154#124—432no—
155#120—416no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—432no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4#41116yes—
201C2×C22⋊C4#4114yes—
202C2×C42—28no—
203C2×C42—28no—
204C2×C42—28no—
205C2×C42—42no—
206C2×C42—44no—
207C2×C42—44no—
208C2×C42—44no—
209C2×C42—44no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—24no—
214#95—28no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—44no—
231C24—24no—
232C24—22no—
233C24—24no—
234C24—24no—
235C24—22no—
236C24—24no—
237C24—24no—
238C24—28no—
239C24—24no—
240C24—24no—
241C24—22no—
242C24—24no—
243C24—24no—
244C24—21no—
245C24—24no—
246C24—22no—
247C24—24no—
248C24—416no—
249C24—416no—
250C24—416no—
251C24—48no—
252C24—48no—
253C24—88no—
254C24—88no—
255C24—88no—
256C24—84no—
257C24—88no—
258C24C2×#4112yes—
259C24C2×#4118yes—
260C24H(5,𝔽2)14yes—
261C24C2511yescharacteristic
262C2×Q8—48no—
263C2×Q8—416no—
264C2×Q8—84no—
265C2×Q8—88no—
266C2×Q8—84no—
267C2×D4—48no—
268C2×D4—416no—
269C2×D4—44no—
270C2×D4—48no—
271C2×D4—88no—
272C2×D4—84no—
273C2×D4—84no—
274C2×D4—88no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
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—416no—
322C22×C4—416no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4C2×#4114yes—
329C22×C4C2×#4118yes—
330C22×C4C2×#4118yes—
331C22×C4H(5,𝔽2)18yes—
332C4⋊C4—88no—
333C4⋊C4—88no—
334C22⋊C4—232no—
335C22⋊C4—44no—
336C22⋊C4—48no—
337C22⋊C4—416no—
338C22⋊C4—416no—
339C22⋊C4—416no—
340C22⋊C4—416no—
341C22⋊C4—416no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—88no—
345C22⋊C4—88no—
346C22⋊C4—88no—
347C22⋊C4—88no—
348C42—48no—
349C42—88no—
350C42—88no—
351C23—28no—
352C23—28no—
353C23—28no—
354C23—24no—
355C23—28no—
356C23—28no—
357C23—28no—
358C23—28no—
359C23—24no—
360C23—28no—
361C23—28no—
362C23—22no—
363C23—24no—
364C23—432no—
365C23—42no—
366C23—44no—
367C23—44no—
368C23—42no—
369C23—42no—
370C23—44no—
371C23—42no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—42no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—42no—
381C23—44no—
382C23—44no—
383C23—42no—
384C23—44no—
385C23—44no—
386C23—41no—
387C23—44no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—42no—
392C23—44no—
393C23—44no—
394C23—832no—
395C23—88no—
396C23—832no—
397C23—832no—
398C23#55118yes—
399C23C22×#4114yes—
400C23C22×#4111yescharacteristic
401C23C2×H(5,𝔽2)12yes—
402C23C2×H(5,𝔽2)14yes—
403C23#58514yes—
404Q8—816no—
405D4—816no—
406D4—88no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—24no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—832no—
458C22—216no—
459C22—28no—
460C22—44no—
461C22—48no—
462C22—48no—
463C22—48no—
464C22—48no—
465C22—48no—
466C22—44no—
467C22—42no—
468C22—48no—
469C22—44no—
470C22—832no—
471C22—84no—
472C22—84no—
473C22—81no—
474C22—84no—
475C22—84no—
476C22—84no—
477C22—82no—
478C22C2×#55112yes—
479C22#347514yes—
480C22#348114yes—
481C22#349011yescharacteristic
482C22#349414yes—
483C22#351012yes—
484C22#351414yes—
485C22#351914yes—
486C22#353212yes—
487C22#355814yes—
488C22#356014yes—
489C4—48no—
490C4—48no—
491C4—44no—
492C4—48no—
493C4—81no—
494C4—84no—
495C4—84no—
496C4—84no—
497C2—48no—
498C2—82no—
499C2—161no—
500C2#4645711yescharacteristic
501C2#4652314yes—
502C2#5926914yes—
503C2#5937012yes—
504C2#5961314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17198C2semidirect
2. #16211C2semidirect
3. #15824C2semidirect
4. #15727C2semidirect
5. #15527C2semidirect
7. #14232C2semidirect
9. #14098C2semidirect
10. #13971C2semidirect
11. #13947C2semidirect
15. C2×#397C22semidirect
18. C2×#391C22semidirect
25. #1446C22semidirect
26. #1443C22semidirect
27. #1443C22semidirect
28. #1442C22semidirect
29. #1441C22semidirect
30. #1441C22semidirect
38. #1439C22semidirect
47. #1437C22semidirect
48. #1437C22semidirect
49. #1434C22semidirect
50. #1434C22semidirect
51. #1434C22semidirect
52. #1434C22semidirect
56. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
#171982
#162112
#158242
#157272
#155272
#152592
#142322
#141342
#140982
#139712
#139472