Finite Group Databasebeta

← #27164607

Subgroups — #27164607

Order 12736.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20263950C211yesmaximal, characteristic, complemented
2#20263943C211yesmaximal, characteristic, complemented
3#20263941C211yesmaximal, characteristic, complemented
4#20263940C211yesmaximal, characteristic, complemented
5#20263937C211yesmaximal, characteristic, complemented
6#20263933C211yesmaximal, characteristic, complemented
7#20263930C211yesmaximal, characteristic, complemented
8#20263930C211yesmaximal, characteristic, complemented
9#20263928C211yesmaximal, characteristic, complemented
10#20263925C211yesmaximal, characteristic, complemented
11#20263925C211yesmaximal, characteristic, complemented
12C4×D796C211yesmaximal, characteristic, complemented
13#20263917C211yesmaximal, characteristic, complemented
14C22⋊C4×D199C211yesmaximal, characteristic, complemented
15#126×C199C211yesmaximal, characteristic, complemented
16C23×D199C2211yescharacteristic
17C2×Dic398C2211yescharacteristic, complemented
18C2×D796C2211yescharacteristic
19C2×D796C2211yescharacteristic
20C2×D796C2211yescharacteristic
21C2×D796C2211yescharacteristic
22C2×D796C2211yescharacteristic, complemented
23C2×C22⋊D199C2211yescharacteristic
24C2×C22⋊D199C2211yescharacteristic
25C2×C4×D199C2211yescharacteristic
26C2×C4×D199C2211yescharacteristic
27C2×C4×D199C2211yescharacteristic, complemented
28C2×C4×D199C2211yescharacteristic, complemented
29C4⋊Dic199C2211yescharacteristic, complemented
30Dic199⋊C4C2211yescharacteristic, complemented
31Dic199⋊C4C2211yescharacteristic, complemented
32C22⋊Dic199C2211yescharacteristic
33(C2×D199)⋊C4C2211yescharacteristic
34(C2×D199)⋊C4C2211yescharacteristic
35(C2×D199)⋊C4C2211yescharacteristic
36(C2×D199)⋊C4C2211yescharacteristic
37(C2×D199)⋊C4C2211yescharacteristic
38(C2×D199)⋊C4C2211yescharacteristic
39(C2×D199)⋊C4C2211yescharacteristic, complemented
40(C2×D199)⋊C4C2211yescharacteristic, complemented
41(C2×D199)⋊C4C2211yescharacteristic, complemented
42(C2×D199)⋊C4C2211yescharacteristic, complemented
43C4×Dic199C2211yescharacteristic, complemented
44C4⋊C4×C199C2211yescharacteristic, complemented
45C4⋊C4×C199C2211yescharacteristic, complemented
46C4⋊C4×C199C2211yescharacteristic, complemented
47C22⋊C4×C199C2211yescharacteristic
48C22⋊C4×C199C2211yescharacteristic
49C22⋊C4×C199C2211yescharacteristic
50C42×C199C2211yescharacteristic, complemented
51C22×D199—22no—
52C22×D199—41no—
53C22×D199—41no—
54C22×D199C2311yescharacteristic
55C22×D199C2311yescharacteristic
56C22×D199C2311yescharacteristic
57C22×D199C2311yescharacteristic
58Dic398—41no—
59D796—22no—
60D796—22no—
61D796—41no—
62D796—41no—
63D796—41no—
64C22⋊D199—41no—
65C22⋊D199—41no—
66C2×Dic199C2311yescharacteristic
67C2×Dic199C2311yescharacteristic
68C2×Dic199C2311yescharacteristic
69C2×Dic199C2311yescharacteristic
70C4×D199—41no—
71C4×D199—41no—
72C4×D199—41no—
73C4×D199—41no—
74C23×C199C2311yescharacteristic
75C2×C4×C199C2311yescharacteristic
76C2×C4×C199C2311yescharacteristic
77C2×C4×C199C2311yescharacteristic
78C2×C4×C199C2311yescharacteristic
79C2×C4×C199C2311yescharacteristic
80C2×C4×C199C2311yescharacteristic
81C22×C199—21no—
82C22×C199—21no—
83C22×C199—21no—
84C22×C199C2411yescharacteristic
85C2×D199—21no—
86C2×D199—21no—
87C2×D199—21no—
88C2×D199—21no—
89C2×D199—21no—
90C2×D199—21no—
91C2×D199—21no—
92C2×D199—21no—
93C2×D199—21no—
94C2×D199—21no—
95C2×D199—21no—
96C2×D199—44no—
97C2×D199#4112yes—
98C4×C199—21no—
99C4×C199—21no—
100C4×C199—21no—
101C4×C199—21no—
102C4×C199—21no—
103C4×C199—21no—
104Dic199—21no—
105Dic199—21no—
106Dic199—21no—
107Dic199—21no—
108C2×C199—41no—
109C2×C199C2×#4111yescharacteristic
110C2×C199H(5,𝔽2)11yescharacteristic
111C2×C199H(5,𝔽2)11yescharacteristic
112D199—22no—
113D199—41no—
114D199—41no—
115D199—41no—
116C199#53811yescharacteristic, complemented, Sylow
117#538—1991nomaximal, Sylow
118#126—1991no—
119#126D19911yescharacteristic, complemented
120#124—1991no—
121#124—1991no—
122#123—1991no—
123#123—1991no—
124C22⋊Q8—1991no—
125#121—1991no—
126#121—1991no—
127#121—1991no—
128#120—1991no—
129#120—1991no—
130C4×D4—1991no—
131C4×D4—1991no—
132C2×C22⋊C4—1991no—
133C24—1991no—
134C2×Q8—1991no—
135C2×D4—1991no—
136C2×D4—1991no—
137C2×D4—1991no—
138C2×D4—1991no—
139C2×D4—1991no—
140C2×D4—1991no—
141C2×D4—1991no—
142C22×C4—1991no—
143C22×C4—1991no—
144C22×C4—1991no—
145C22×C4—1991no—
146C4⋊C4—1991no—
147C4⋊C4—1991no—
148C4⋊C4—1991no—
149C4⋊C4C2×D19911yescharacteristic, complemented
150C4⋊C4C2×D19911yescharacteristic, complemented
151C4⋊C4C2×D19911yescharacteristic, complemented
152C22⋊C4—1991no—
153C22⋊C4—1991no—
154C22⋊C4—1991no—
155C22⋊C4—1991no—
156C22⋊C4—1991no—
157C22⋊C4—1991no—
158C22⋊C4—1991no—
159C22⋊C4—1991no—
160C22⋊C4—1991no—
161C22⋊C4—1991no—
162C22⋊C4—1991no—
163C22⋊C4C2×D19911yescharacteristic
164C22⋊C4C2×D19911yescharacteristic
165C22⋊C4C2×D19911yescharacteristic
166C42—1991no—
167C42C2×D19911yescharacteristic, complemented
168C23—7961no—
169C23—7961no—
170C23—3982no—
171C23—1991no—
172C23—1991no—
173C23—1991no—
174C23—1991no—
175C23C22×D19911yescharacteristic
176Q8—7961no—
177D4—7961no—
178D4—7961no—
179D4—7961no—
180D4—7961no—
181D4—7961no—
182D4—3982no—
183D4—3982no—
184C2×C4—7961no—
185C2×C4—7961no—
186C2×C4—7961no—
187C2×C4—7961no—
188C2×C4—1991no—
189C2×C4—1991no—
190C2×C4—1991no—
191C2×C4—1991no—
192C2×C4C22×D19911yescharacteristic
193C2×C4C22×D19911yescharacteristic
194C2×C4C22×D19911yescharacteristic
195C2×C4C22×D19911yescharacteristic
196C2×C4C22×D19911yescharacteristic
197C2×C4C22×D19911yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—7964no—
202C22—3981no—
203C22—3981no—
204C22—3981no—
205C22—3981no—
206C22—3981no—
207C22—3981no—
208C22—3981no—
209C22—3981no—
210C22—3981no—
211C22—3981no—
212C22—3981no—
213C22—1992no—
214C22C23×D19911yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—3981no—
222C4—3981no—
223C4—3981no—
224C4—3981no—
225C2—41no—
226C2—7961no—
227C2—7961no—
228C2—7961no—
229C2—3982no—
230C2#41×D19911yescharacteristic
231C2#2026400111yescharacteristic
232C2#2026400111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#126×C199C2
hypercenter#126D199
hypocenterC199#538
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #20263950C2semidirect
2. #20263943C2semidirect
3. #20263941C2semidirect
4. #20263940C2semidirect
5. #20263937C2semidirect
6. #20263933C2semidirect
7. #20263930C2semidirect
8. #20263930C2semidirect
9. #20263928C2semidirect
10. #20263925C2semidirect
11. #20263925C2semidirect
12. C4×D796C2semidirect
13. #20263917C2semidirect
14. C22⋊C4×D199C2semidirect
15. #126×C199C2semidirect
17. C2×Dic398C22semidirect
22. C2×D796C22semidirect
27. C2×C4×D199C22semidirect
28. C2×C4×D199C22semidirect
29. C4⋊Dic199C22semidirect
30. Dic199⋊C4C22semidirect
31. Dic199⋊C4C22semidirect
39. (C2×D199)⋊C4C22semidirect
40. (C2×D199)⋊C4C22semidirect
41. (C2×D199)⋊C4C22semidirect
42. (C2×D199)⋊C4C22semidirect
43. C4×Dic199C22semidirect
44. C4⋊C4×C199C22semidirect
45. C4⋊C4×C199C22semidirect
46. C4⋊C4×C199C22semidirect
50. C42×C199C22semidirect
116. C199#538semidirect
119. #126D199semidirect
149. C4⋊C4C2×D199semidirect
150. C4⋊C4C2×D199semidirect
151. C4⋊C4C2×D199semidirect
167. C42C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202639502
#202639432
#202639412
#202639402
#202639372
#202639332
#202639302
#202639282
#202639252
C4×D7962
#202639172
C22⋊C4×D1992
#126×C1992
#538199