Finite Group Databasebeta

← #16859102

Subgroups — #16859102

Order 3968.

1040 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521672C212yesmaximal, complemented
2#14521671C212yesmaximal, complemented
3#14521660C211yesmaximal, characteristic, complemented
4#14521655C211yesmaximal, characteristic, complemented
5#14521654C211yesmaximal, characteristic, complemented
6#14521647C214yesmaximal, complemented
7#14521638C214yesmaximal, complemented
8#14521605C211yesmaximal, characteristic, complemented
9#14521600C211yesmaximal, characteristic, complemented
10#14521584C214yesmaximal, complemented
11#14521582C214yesmaximal, complemented
12#14521562C212yesmaximal, complemented
13C4×#92499C212yesmaximal, complemented
14#117×D31C211yesmaximal, characteristic, complemented
15C31×#541C211yesmaximal, characteristic, complemented
16C2×#92499C2212yes—
17C22×C4×D31C2211yescharacteristic
18C31×C4⋊Q8C2211yescharacteristic, complemented
19C31×#126C2214yes—
20C31×#125C2211yescharacteristic, complemented
21C31×#125C2214yescomplemented
22C31×C22⋊Q8C2212yes—
23C4×Q8×C31C2212yescomplemented
24C31×#117C2211yescharacteristic
25#11758524C2212yescomplemented
26#11758523C2212yes—
27Q8×Dic31C2212yescomplemented
28#11758507C2212yes—
29#11758503C2211yescharacteristic
30#11758475C2214yes—
31#11758475C2214yescomplemented
32#11758474C2214yescomplemented
33#11758473C2214yes—
34#11758473C2214yescomplemented
35#11758472C2214yes—
36#11758472C2214yes—
37#11758472C2212yes—
38#11758471C2214yes—
39#11758471C2214yes—
40#11758471C2214yescomplemented
41#11758471C2214yescomplemented
42#11758470C2214yes—
43#11758470C2214yes—
44#11758470C2214yescomplemented
45#11758470C2212yescomplemented
46#11758469C2212yes—
47#11758469C2212yes—
48#11758469C2214yescomplemented
49C4⋊C4×D31C2212yes—
50#11758466C2214yescomplemented
51#11758465C2212yescomplemented
52#11758464C2214yescomplemented
53#11758462C2214yes—
54#11758461C2214yes—
55#11758460C2214yes—
56#11758459C2214yes—
57(C22⋊D31)⋊C4C2214yes—
58C22⋊C4×D31C2212yes—
59#11758454C2212yes—
60#11758453C2214yescomplemented
61#11758452C2214yes—
62#11758451C2211yescharacteristic, complemented
63C4×D124C2214yes—
64C4×D124C2212yescomplemented
65#11758449C2211yescharacteristic
66#11758449C2211yescharacteristic
67C42×D31C2211yescharacteristic
68C42×D31C2211yescharacteristic
69C42×D31C2214yescomplemented
70#11758447C2211yescharacteristic, complemented
71C23×D31C2311yescharacteristic
72C2×Q8×C31C2312yes—
73C22×C4×C31C2311yescharacteristic
74C2×C22⋊D31C2314yes—
75C22×Dic31C2311yescharacteristic
76#92499—28no—
77C2×D124C2314yes—
78C2×D124C2312yes—
79C2×D124C2314yes—
80C2×D124C2314yes—
81C2×C4×D31—24no—
82C2×C4×D31C2314yes—
83C2×C4×D31C2311yescharacteristic
84C2×C4×D31C2311yescharacteristic
85C2×C4×D31C2311yescharacteristic
86C2×C4×D31C2312yes—
87C2×C4×D31C2312yes—
88C2×C4×D31C2314yes—
89C2×C4×D31C2314yes—
90C2×C4×D31C2311yescharacteristic
91C2×Dic62C2314yes—
92C4⋊C4×C31C2314yes—
93C4⋊C4×C31C2312yes—
94C4⋊C4×C31C2314yes—
95C4⋊C4×C31C2312yes—
96C4⋊C4×C31C2314yes—
97C4⋊C4×C31C2314yes—
98C22⋊C4×C31C2312yes—
99C22⋊C4×C31C2314yes—
100C42×C31C2311yescharacteristic
101C42×C31C2311yescharacteristic
102C42×C31C2314yes—
103C22⋊Dic31C2312yes—
104(C2×D31)⋊C4C2312yes—
105(C2×D31)⋊C4C2312yes—
106(C2×D31)⋊C4C2314yes—
107(C2×D31)⋊C4C2314yes—
108(C2×D31)⋊C4C2314yes—
109(C2×D31)⋊C4C2314yes—
110(C2×D31)⋊C4C2314yes—
111(C2×D31)⋊C4C2314yes—
112(C2×D31)⋊C4C2314yes—
113(C2×D31)⋊C4C2314yes—
114C4⋊Dic31C2312yes—
115C4⋊Dic31C2314yes—
116C4⋊Dic31C2312yes—
117Dic31⋊C4C2314yes—
118Dic31⋊C4C2312yes—
119Dic31⋊C4C2312yes—
120Dic31⋊C4C2314yes—
121Dic31⋊C4C2314yes—
122C4×Dic31C2312yes—
123C4×Dic31C2311yescharacteristic
124C4×Dic31C2311yescharacteristic
125C4×Dic31C2312yes—
126C4×Dic31C2314yes—
127C4×Dic31C2314yes—
128C23×C31C2411yescharacteristic
129C22×D31—22no—
130C22×D31—42no—
131C22×D31C2411yescharacteristic
132C22×D31C2414yes—
133C22×D31C2411yescharacteristic
134Q8×C31—24no—
135C2×C4×C31—22no—
136C2×C4×C31C2412yes—
137C2×C4×C31C2414yes—
138C2×C4×C31C2412yes—
139C2×C4×C31C2414yes—
140C2×C4×C31C2411yescharacteristic
141C2×C4×C31C2411yescharacteristic
142C22⋊D31—44no—
143C2×Dic31—22no—
144C2×Dic31C2412yes—
145C2×Dic31C2414yes—
146C2×Dic31C2412yes—
147C2×Dic31C2411yescharacteristic
148C2×Dic31C2411yescharacteristic
149D124—28no—
150D124—24no—
151D124—44no—
152D124—44no—
153C4×D31—22no—
154C4×D31—22no—
155C4×D31—22no—
156C4×D31—28no—
157C4×D31—42no—
158C4×D31—44no—
159C4×D31—44no—
160C4×D31—42no—
161C4×D31#4114yes—
162Dic62—44no—
163#3455—311nomaximal, Sylow
164C22×C31—21no—
165C22×C31—22no—
166C22×C31C2511yescharacteristic
167C2×D31—22no—
168C2×D31—21no—
169C2×D31—24no—
170C2×D31—24no—
171C2×D31—24no—
172C2×D31—22no—
173C2×D31—44no—
174C2×D31C2×#4112yes—
175C4×C31—21no—
176C4×C31—24no—
177C4×C31—24no—
178C4×C31—22no—
179C4×C31—22no—
180C4×C31C2×#4112yes—
181Dic31—21no—
182Dic31—22no—
183Dic31—24no—
184Dic31—22no—
185Dic31C2×#4112yes—
186#542—312no—
187#542—312no—
188#541—311no—
189#541D3111yescharacteristic, complemented
190#540—311no—
191#540—311no—
192#539—314no—
193#538—314no—
194#533—311no—
195#532—311no—
196#529—314no—
197#529—314no—
198#525—312no—
199C4×#41—312no—
200C2×#117—311no—
201C2×C31—41no—
202C2×C31C22×#4111yescharacteristic
203C2×C31#58512yes—
204D31—22no—
205D31—44no—
206D31—41no—
207C2×#41—312no—
208C23×C4—311no—
209C4⋊Q8—312no—
210C4⋊Q8C2×D3111yescharacteristic, complemented
211#127—311no—
212#126—314no—
213#126—314no—
214#126—314no—
215#126C2×D3114yes—
216#125—314no—
217#125—311no—
218#125C2×D3114yescomplemented
219#125C2×D3111yescharacteristic, complemented
220#124—312no—
221#124—314no—
222#124—314no—
223#123—314no—
224#123—314no—
225#123—314no—
226#123—314no—
227#123—314no—
228C22⋊Q8—312no—
229C22⋊Q8—314no—
230C22⋊Q8—314no—
231C22⋊Q8—314no—
232C22⋊Q8C2×D3112yes—
233#121—312no—
234#121—312no—
235#121—314no—
236#121—314no—
237#121—314no—
238#120—314no—
239C4×Q8—312no—
240C4×Q8—314no—
241C4×Q8C2×D3112yescomplemented
242C4×D4—314no—
243C4×D4—312no—
244C4×D4—314no—
245C4×D4—314no—
246C4×D4—314no—
247C4×D4—314no—
248C4×D4—312no—
249#117—311no—
250#117—312no—
251#117—312no—
252#117—314no—
253#117—312no—
254#117—311no—
255#117—311no—
256#117C2×D3111yescharacteristic
257C2×C4⋊C4—312no—
258C2×C22⋊C4—312no—
259C2×C42—311no—
260C2×C42—311no—
261C2×C42—314no—
262C31#345511yescharacteristic, complemented, Sylow
263C24—311no—
264#41—628no—
265C2×Q8—314no—
266C2×Q8C22×D3112yes—
267C2×D4—314no—
268C2×D4—314no—
269C2×D4—312no—
270C2×D4—314no—
271C2×D4—314no—
272C22×C4—311no—
273C22×C4—311no—
274C22×C4—314no—
275C22×C4—311no—
276C22×C4—311no—
277C22×C4—311no—
278C22×C4—312no—
279C22×C4—312no—
280C22×C4—314no—
281C22×C4—314no—
282C22×C4—624no—
283C22×C4C22×D3111yescharacteristic
284C4⋊C4—312no—
285C4⋊C4—312no—
286C4⋊C4—314no—
287C4⋊C4—314no—
288C4⋊C4—314no—
289C4⋊C4—312no—
290C4⋊C4—312no—
291C4⋊C4—314no—
292C4⋊C4C22×D3112yes—
293C4⋊C4C22×D3112yes—
294C4⋊C4C22×D3114yes—
295C4⋊C4C22×D3114yes—
296C4⋊C4C22×D3114yes—
297C4⋊C4C22×D3114yes—
298C22⋊C4—312no—
299C22⋊C4—312no—
300C22⋊C4—312no—
301C22⋊C4—314no—
302C22⋊C4—314no—
303C22⋊C4—314no—
304C22⋊C4—314no—
305C22⋊C4—314no—
306C22⋊C4—314no—
307C22⋊C4—314no—
308C22⋊C4—314no—
309C22⋊C4C22×D3112yes—
310C22⋊C4C22×D3114yes—
311C42—312no—
312C42—312no—
313C42—314no—
314C42—311no—
315C42—311no—
316C42—314no—
317C42C22×D3111yescharacteristic
318C42C22×D3111yescharacteristic
319C42C22×D3114yes—
320C23—311no—
321C23—311no—
322C23—314no—
323C23—622no—
324C23—1242no—
325C23C23×D3111yescharacteristic
326Q8—24no—
327Q8—1244no—
328D4—628no—
329D4—624no—
330D4—1244no—
331D4—1244no—
332D4—1244no—
333C2×C4—22no—
334C2×C4—311no—
335C2×C4—314no—
336C2×C4—312no—
337C2×C4—312no—
338C2×C4—314no—
339C2×C4—311no—
340C2×C4—622no—
341C2×C4—628no—
342C2×C4—622no—
343C2×C4—622no—
344C2×C4—622no—
345C2×C4—1242no—
346C2×C4—1244no—
347C2×C4—1242no—
348C2×C4—1244no—
349C2×C4C23×D3112yes—
350C2×C4C23×D3114yes—
351C2×C4C23×D3114yes—
352C2×C4C23×D3112yes—
353C2×C4C23×D3111yescharacteristic
354C2×C4C23×D3111yescharacteristic
355C22—21no—
356C22—22no—
357C22—312no—
358C22—622no—
359C22—621no—
360C22—624no—
361C22—624no—
362C22—624no—
363C22—622no—
364C22—1244no—
365C22C24×D3111yescharacteristic
366C4—21no—
367C4—24no—
368C4—22no—
369C4—22no—
370C4—24no—
371C4—312no—
372C4—621no—
373C4—622no—
374C4—624no—
375C4—622no—
376C4#41×D3112yes—
377C2—41no—
378C2—622no—
379C2—1241no—
380C2—1244no—
381C2C2×#41×D3111yescharacteristic
382C2#1452188612yes—
subgroupG/H
center, FrattiniC22C24×D31
commutator, socle, cosocleC22×C31C25
FittingC31×#541C2
hypercenter#541D31
hypocenterC31#3455
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #14521672C2semidirect
2. #14521671C2semidirect
3. #14521660C2semidirect
4. #14521655C2semidirect
5. #14521654C2semidirect
6. #14521647C2semidirect
7. #14521638C2semidirect
8. #14521605C2semidirect
9. #14521600C2semidirect
10. #14521584C2semidirect
11. #14521582C2semidirect
12. #14521562C2semidirect
13. C4×#92499C2semidirect
14. #117×D31C2semidirect
15. C31×#541C2semidirect
18. C31×C4⋊Q8C22semidirect
20. C31×#125C22semidirect
21. C31×#125C22semidirect
23. C4×Q8×C31C22semidirect
25. #11758524C22semidirect
27. Q8×Dic31C22semidirect
31. #11758475C22semidirect
32. #11758474C22semidirect
34. #11758473C22semidirect
40. #11758471C22semidirect
41. #11758471C22semidirect
44. #11758470C22semidirect
45. #11758470C22semidirect
48. #11758469C22semidirect
50. #11758466C22semidirect
51. #11758465C22semidirect
52. #11758464C22semidirect
60. #11758453C22semidirect
62. #11758451C22semidirect
64. C4×D124C22semidirect
69. C42×D31C22semidirect
70. #11758447C22semidirect
189. #541D31semidirect
210. C4⋊Q8C2×D31semidirect
218. #125C2×D31semidirect
219. #125C2×D31semidirect
241. C4×Q8C2×D31semidirect
262. C31#3455semidirect

Maximal subgroups

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

subgroupindex
#145216722
#145216712
#145216602
#145216552
#145216542
#145216472
#145216382
#145216052
#145216002
#145215842
#145215822
#145215622
C4×#924992
#117×D312
C31×#5412
#345531