Finite Group Databasebeta

← #27164655

Subgroups — #27164655

Order 12736.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#15691885C211yesmaximal, characteristic, complemented
2C22×C22⋊D199C211yesmaximal, characteristic, complemented
3#20263941C211yesmaximal, characteristic, complemented
4#20263939C211yesmaximal, characteristic, complemented
5#20263937C211yesmaximal, characteristic, complemented
6#20263932C211yesmaximal, characteristic, complemented
7#20263927C211yesmaximal, characteristic, complemented
8#20263927C211yesmaximal, characteristic, complemented
9#20263926C211yesmaximal, characteristic, complemented
10#20263923C211yesmaximal, characteristic, complemented
11D4×Dic199C211yesmaximal, characteristic, complemented
12(C22⋊D199)⋊C4C211yesmaximal, characteristic, complemented
13C2×C22⋊Dic199C211yesmaximal, characteristic, complemented
14C22⋊C4×D199C211yesmaximal, characteristic, complemented
15#120×C199C211yesmaximal, characteristic, complemented
16C23×D199C2211yescharacteristic
17#15691885—24no—
18C2×Dic398C2211yescharacteristic, complemented
19C2×C22⋊D199—24no—
20C2×C22⋊D199C2211yescharacteristic
21C2×C22⋊D199C2211yescharacteristic
22C2×C22⋊D199C2211yescharacteristic
23C2×C22⋊D199C2211yescharacteristic
24C2×C22⋊D199C2211yescharacteristic, complemented
25C2×C22⋊D199C2211yescharacteristic, complemented
26C22×Dic199C2211yescharacteristic
27C22×Dic199C2211yescharacteristic, complemented
28C22×Dic199C2211yescharacteristic, complemented
29C22×Dic199C2211yescharacteristic, complemented
30C2×C4×D199C2211yescharacteristic, complemented
31C2×C4×D199C2211yescharacteristic, complemented
32C4⋊Dic199C2211yescharacteristic, complemented
33C4⋊Dic199C2211yescharacteristic, complemented
34Dic199⋊C4C2211yescharacteristic, complemented
35Dic199⋊C4C2211yescharacteristic, complemented
36C22⋊Dic199C2211yescharacteristic
37C22⋊Dic199C2211yescharacteristic
38C22⋊Dic199C2211yescharacteristic, complemented
39C22⋊Dic199C2211yescharacteristic, complemented
40C22⋊Dic199C2211yescharacteristic, complemented
41(C2×D199)⋊C4C2211yescharacteristic, complemented
42(C2×D199)⋊C4C2211yescharacteristic, complemented
43(C2×D199)⋊C4C2211yescharacteristic, complemented
44(C2×D199)⋊C4C2211yescharacteristic, complemented
45C4×Dic199C2211yescharacteristic, complemented
46C24×C199C2211yescharacteristic
47C2×D4×C199C2211yescharacteristic
48C2×D4×C199C2211yescharacteristic, complemented
49C2×D4×C199C2211yescharacteristic, complemented
50C22⋊C4×C199C2211yescharacteristic
51C22⋊C4×C199C2211yescharacteristic, complemented
52C22⋊C4×C199C2211yescharacteristic, complemented
53C22×D199—22no—
54C22×D199—22no—
55C22×D199—41no—
56C22×D199C2311yescharacteristic
57C22×D199C2311yescharacteristic
58Dic398—22no—
59C22⋊D199—22no—
60C22⋊D199—22no—
61C22⋊D199—22no—
62C22⋊D199—22no—
63C22⋊D199—41no—
64C22⋊D199D414yescomplemented
65C2×Dic199—22no—
66C2×Dic199—22no—
67C2×Dic199—22no—
68C2×Dic199—41no—
69C2×Dic199C2311yescharacteristic
70C2×Dic199C2311yescharacteristic
71C2×Dic199C2311yescharacteristic
72C2×Dic199C2311yescharacteristic
73C2×Dic199C2311yescharacteristic
74C2×Dic199C2311yescharacteristic
75C4×D199—22no—
76C4×D199—41no—
77C23×C199—22no—
78C23×C199—22no—
79C23×C199—41no—
80C23×C199C2311yescharacteristic
81C23×C199C2311yescharacteristic
82C23×C199C2311yescharacteristic
83C23×C199C2311yescharacteristic
84D4×C199—22no—
85D4×C199—22no—
86D4×C199—41no—
87C2×C4×C199C2311yescharacteristic
88C2×C4×C199C2311yescharacteristic
89C2×C4×C199C2311yescharacteristic
90C22×C199—21no—
91C22×C199—21no—
92C22×C199—21no—
93C22×C199—21no—
94C22×C199—21no—
95C22×C199—21no—
96C22×C199—21no—
97C22×C199—21no—
98C22×C199—21no—
99C22×C199—21no—
100C22×C199—44no—
101C22×C199C2×D412yes—
102C22×C199#4112yescomplemented
103C22×C199C2411yescharacteristic
104C2×D199—21no—
105C2×D199—21no—
106C2×D199—21no—
107C2×D199—21no—
108C2×D199—21no—
109C2×D199—44no—
110C2×D199C2×D412yes—
111C4×C199—21no—
112C4×C199—21no—
113C4×C199—21no—
114Dic199—21no—
115Dic199—21no—
116Dic199—21no—
117Dic199—21no—
118Dic199—21no—
119Dic199C2×D412yes—
120C2×C199—22no—
121C2×C199—22no—
122C2×C199—41no—
123C2×C199—41no—
124C2×C199C22×D411yescharacteristic
125C2×C199C2×#4111yescharacteristic
126C2×C199H(5,𝔽2)11yescharacteristic
127D199—22no—
128D199—41no—
129C199#54611yescharacteristic, complemented, Sylow
130#546—1991nomaximal, Sylow
131C2×#41—1991no—
132C22×D4—1991no—
133#124—1991no—
134#123—1991no—
135#123—1991no—
136C22⋊Q8—1991no—
137#121—1991no—
138#121—1991no—
139#121—1991no—
140#120—1991no—
141#120D19911yescharacteristic, complemented
142C4×D4—1991no—
143C4×D4—1991no—
144C2×C22⋊C4—1991no—
145C2×C22⋊C4—1991no—
146C24—1991no—
147C24C2×D19911yescharacteristic
148#41—3984no—
149C2×Q8—1991no—
150C2×D4—3984no—
151C2×D4—1991no—
152C2×D4—1991no—
153C2×D4—1991no—
154C2×D4—1991no—
155C2×D4—1991no—
156C2×D4—1991no—
157C2×D4C2×D19911yescharacteristic
158C2×D4C2×D19911yescharacteristic, complemented
159C2×D4C2×D19911yescharacteristic, complemented
160C22×C4—1991no—
161C22×C4—1991no—
162C22×C4—1991no—
163C22×C4—1991no—
164C22×C4—1991no—
165C22×C4—1991no—
166C4⋊C4—1991no—
167C4⋊C4—1991no—
168C4⋊C4—1991no—
169C4⋊C4—1991no—
170C22⋊C4—1991no—
171C22⋊C4—1991no—
172C22⋊C4—1991no—
173C22⋊C4—1991no—
174C22⋊C4—1991no—
175C22⋊C4—1991no—
176C22⋊C4—1991no—
177C22⋊C4—1991no—
178C22⋊C4—1991no—
179C22⋊C4C2×D19911yescharacteristic
180C22⋊C4C2×D19911yescharacteristic, complemented
181C22⋊C4C2×D19911yescharacteristic, complemented
182C42—1991no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—7961no—
187C23—3982no—
188C23—3982no—
189C23—1991no—
190C23—1991no—
191C23C22×D19911yescharacteristic
192C23C22×D19911yescharacteristic
193C23C22×D19911yescharacteristic
194C23C22×D19911yescharacteristic
195Q8—3982no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—7961no—
200D4—3982no—
201D4—3982no—
202D4—3982no—
203D4—3982no—
204D4—1994no—
205C2×C4—7961no—
206C2×C4—7961no—
207C2×C4—3982no—
208C2×C4—3982no—
209C2×C4—3982no—
210C2×C4—3982no—
211C2×C4—1991no—
212C2×C4—1991no—
213C2×C4—1991no—
214C2×C4—1991no—
215C2×C4—1991no—
216C2×C4—1991no—
217C2×C4C22×D19911yescharacteristic
218C2×C4C22×D19911yescharacteristic
219C2×C4C22×D19911yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—44no—
231C22—7964no—
232C22—3981no—
233C22—3981no—
234C22—3981no—
235C22—3981no—
236C22—3981no—
237C22—1992no—
238C22D4×D19912yes—
239C22#1569188512yescomplemented
240C22C23×D19911yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—3981no—
245C4—3981no—
246C4—3981no—
247C4—3981no—
248C4—3981no—
249C4—1992no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—41no—
254C2—7961no—
255C2—3982no—
256C2C2×D4×D19911yescharacteristic
257C2C2×#1569188511yescharacteristic
258C2#2026400011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#120×C199C2
hypercenter#120D199
hypocenterC199#546
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#15691885C2semidirect
2. C22×C22⋊D199C2semidirect
3. #20263941C2semidirect
4. #20263939C2semidirect
5. #20263937C2semidirect
6. #20263932C2semidirect
7. #20263927C2semidirect
8. #20263927C2semidirect
9. #20263926C2semidirect
10. #20263923C2semidirect
11. D4×Dic199C2semidirect
12. (C22⋊D199)⋊C4C2semidirect
13. C2×C22⋊Dic199C2semidirect
14. C22⋊C4×D199C2semidirect
15. #120×C199C2semidirect
18. C2×Dic398C22semidirect
24. C2×C22⋊D199C22semidirect
25. C2×C22⋊D199C22semidirect
27. C22×Dic199C22semidirect
28. C22×Dic199C22semidirect
29. C22×Dic199C22semidirect
30. C2×C4×D199C22semidirect
31. C2×C4×D199C22semidirect
32. C4⋊Dic199C22semidirect
33. C4⋊Dic199C22semidirect
34. Dic199⋊C4C22semidirect
35. Dic199⋊C4C22semidirect
38. C22⋊Dic199C22semidirect
39. C22⋊Dic199C22semidirect
40. C22⋊Dic199C22semidirect
41. (C2×D199)⋊C4C22semidirect
42. (C2×D199)⋊C4C22semidirect
43. (C2×D199)⋊C4C22semidirect
44. (C2×D199)⋊C4C22semidirect
45. C4×Dic199C22semidirect
48. C2×D4×C199C22semidirect
49. C2×D4×C199C22semidirect
51. C22⋊C4×C199C22semidirect
52. C22⋊C4×C199C22semidirect
64. C22⋊D199D4semidirect
102. C22×C199#41semidirect
129. C199#546semidirect
141. #120D199semidirect
158. C2×D4C2×D199semidirect
159. C2×D4C2×D199semidirect
180. C22⋊C4C2×D199semidirect
181. C22⋊C4C2×D199semidirect
239. C22#15691885semidirect

Maximal subgroups

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

subgroupindex
C2×#156918852
C22×C22⋊D1992
#202639412
#202639392
#202639372
#202639322
#202639272
#202639262
#202639232
D4×Dic1992
(C22⋊D199)⋊C42
C2×C22⋊Dic1992
C22⋊C4×D1992
#120×C1992
#546199