Finite Group Databasebeta

← #28417074

Subgroups — #28417074

Order 14912.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16836593C211yesmaximal, characteristic, complemented
2C2×D4×D233C211yesmaximal, characteristic, complemented
3#21418521C211yesmaximal, characteristic, complemented
4#21418520C211yesmaximal, characteristic, complemented
5#21418516C211yesmaximal, characteristic, complemented
6#21418513C211yesmaximal, characteristic, complemented
7#21418508C211yesmaximal, characteristic, complemented
8#21418507C211yesmaximal, characteristic, complemented
9#21418503C211yesmaximal, characteristic, complemented
10#21418503C211yesmaximal, characteristic, complemented
11C4×C22⋊D233C211yesmaximal, characteristic, complemented
12#21418495C211yesmaximal, characteristic, complemented
13C2×#16836571C211yesmaximal, characteristic, complemented
14C22⋊C4×D233C211yesmaximal, characteristic, complemented
15#121×C233C211yesmaximal, characteristic, complemented
16C23×D233C2211yescharacteristic
17C23×D233C2211yescharacteristic
18#16836593—24no—
19C2×Dic466C2211yescharacteristic, complemented
20C2×D932C2211yescharacteristic
21C2×C22⋊D233C2211yescharacteristic
22C2×C22⋊D233C2211yescharacteristic
23C2×C22⋊D233C2211yescharacteristic
24C2×C22⋊D233C2211yescharacteristic, complemented
25C2×C22⋊D233C2211yescharacteristic, complemented
26D4×D233—24no—
27C22×Dic233C2211yescharacteristic, complemented
28C22×Dic233C2211yescharacteristic, complemented
29C2×C4×D233C2211yescharacteristic
30C2×C4×D233C2211yescharacteristic, complemented
31C2×C4×D233C2211yescharacteristic, complemented
32#16836573C2211yescharacteristic, complemented
33#16836573C2211yescharacteristic, complemented
34#16836573C2211yescharacteristic, complemented
35C22⋊Dic233C2211yescharacteristic, complemented
36C22⋊Dic233C2211yescharacteristic, complemented
37C22⋊Dic233C2211yescharacteristic, complemented
38#16836571C2211yescharacteristic
39#16836571C2211yescharacteristic
40#16836571C2211yescharacteristic
41#16836571C2211yescharacteristic, complemented
42#16836571C2211yescharacteristic, complemented
43#16836571C2211yescharacteristic, complemented
44#16836571C2211yescharacteristic, complemented
45C4×Dic233C2211yescharacteristic, complemented
46C2×D4×C233C2211yescharacteristic
47C2×D4×C233C2211yescharacteristic, complemented
48C2×D4×C233C2211yescharacteristic, complemented
49C22×C4×C233C2211yescharacteristic, complemented
50C4⋊C4×C233C2211yescharacteristic, complemented
51C22⋊C4×C233C2211yescharacteristic, complemented
52C22⋊C4×C233C2211yescharacteristic, complemented
53C22×D233—22no—
54C22×D233—22no—
55C22×D233—22no—
56C22×D233—22no—
57C22×D233—41no—
58C22×D233—41no—
59C22×D233C2311yescharacteristic
60C22×D233C2311yescharacteristic
61C22×D233C2311yescharacteristic
62Dic466—22no—
63D932—22no—
64C22⋊D233—22no—
65C22⋊D233—22no—
66C22⋊D233—22no—
67C22⋊D233—41no—
68C22⋊D233D414yescomplemented
69C2×Dic233—22no—
70C2×Dic233—22no—
71C2×Dic233C2311yescharacteristic
72C2×Dic233C2311yescharacteristic
73C2×Dic233C2311yescharacteristic
74C2×Dic233C2311yescharacteristic
75C2×Dic233C2311yescharacteristic
76C4×D233—22no—
77C4×D233—22no—
78C4×D233—41no—
79C23×C233C2311yescharacteristic
80C23×C233C2311yescharacteristic
81C23×C233C2311yescharacteristic
82D4×C233—22no—
83D4×C233—22no—
84D4×C233—41no—
85C2×C4×C233—41no—
86C2×C4×C233C2311yescharacteristic
87C2×C4×C233C2311yescharacteristic
88C2×C4×C233C2311yescharacteristic
89C2×C4×C233C2311yescharacteristic
90C22×C233—21no—
91C22×C233—21no—
92C22×C233—21no—
93C22×C233—21no—
94C22×C233—21no—
95C22×C233—21no—
96C22×C233—21no—
97C22×C233—21no—
98C22×C233C2×D412yes—
99C22×C233C2411yescharacteristic
100C2×D233—21no—
101C2×D233—21no—
102C2×D233—21no—
103C2×D233—21no—
104C2×D233—21no—
105C2×D233—21no—
106C2×D233—21no—
107C2×D233—44no—
108C2×D233—44no—
109C2×D233C2×D412yes—
110C2×D233#4112yescomplemented
111C4×C233—21no—
112C4×C233—21no—
113C4×C233—21no—
114C4×C233—21no—
115Dic233—21no—
116Dic233—21no—
117Dic233—21no—
118Dic233—21no—
119Dic233C2×D412yes—
120C2×C233—22no—
121C2×C233—41no—
122C2×C233—41no—
123C2×C233C22×D411yescharacteristic
124C2×C233C2×#4111yescharacteristic
125C2×C233H(5,𝔽2)11yescharacteristic
126D233—22no—
127D233—22no—
128D233—41no—
129C233#54611yescharacteristic, complemented, Sylow
130#546—2331nomaximal, Sylow
131C2×#41—2331no—
132C22×D4—2331no—
133#124—2331no—
134#123—2331no—
135#123—2331no—
136C22⋊Q8—2331no—
137#121—2331no—
138#121—2331no—
139#121D23311yescharacteristic, complemented
140#120—2331no—
141#120—2331no—
142C4×D4—2331no—
143C4×D4—2331no—
144C2×C22⋊C4—2331no—
145C2×C22⋊C4—2331no—
146C24—2331no—
147C24—2331no—
148#41—4664no—
149C2×Q8—2331no—
150C2×D4—4664no—
151C2×D4—2331no—
152C2×D4—2331no—
153C2×D4—2331no—
154C2×D4—2331no—
155C2×D4—2331no—
156C2×D4—2331no—
157C2×D4C2×D23311yescharacteristic
158C2×D4C2×D23311yescharacteristic, complemented
159C2×D4C2×D23311yescharacteristic, complemented
160C22×C4—2331no—
161C22×C4—2331no—
162C22×C4—2331no—
163C22×C4—2331no—
164C22×C4—2331no—
165C22×C4C2×D23311yescharacteristic, complemented
166C4⋊C4—2331no—
167C4⋊C4—2331no—
168C4⋊C4—2331no—
169C4⋊C4C2×D23311yescharacteristic, complemented
170C22⋊C4—2331no—
171C22⋊C4—2331no—
172C22⋊C4—2331no—
173C22⋊C4—2331no—
174C22⋊C4—2331no—
175C22⋊C4—2331no—
176C22⋊C4—2331no—
177C22⋊C4—2331no—
178C22⋊C4—2331no—
179C22⋊C4—2331no—
180C22⋊C4C2×D23311yescharacteristic, complemented
181C22⋊C4C2×D23311yescharacteristic, complemented
182C42—2331no—
183C23—9321no—
184C23—9321no—
185C23—4662no—
186C23—4662no—
187C23—4662no—
188C23—4662no—
189C23—2331no—
190C23—2331no—
191C23—2331no—
192C23C22×D23311yescharacteristic
193C23C22×D23311yescharacteristic
194C23C22×D23311yescharacteristic
195Q8—4662no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—9321no—
200D4—4662no—
201D4—4662no—
202D4—4662no—
203D4—4662no—
204D4—2334no—
205C2×C4—41no—
206C2×C4—9321no—
207C2×C4—4662no—
208C2×C4—4662no—
209C2×C4—4662no—
210C2×C4—4662no—
211C2×C4—2331no—
212C2×C4—2331no—
213C2×C4—2331no—
214C2×C4—2331no—
215C2×C4—2331no—
216C2×C4C22×D23311yescharacteristic
217C2×C4C22×D23311yescharacteristic
218C2×C4C22×D23311yescharacteristic
219C2×C4C22×D23311yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—9324no—
229C22—9324no—
230C22—4661no—
231C22—4661no—
232C22—4661no—
233C22—4661no—
234C22—4661no—
235C22—4661no—
236C22—4661no—
237C22—2332no—
238C22—2332no—
239C22D4×D23312yes—
240C22C23×D23311yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—4661no—
246C4—4661no—
247C4—4661no—
248C4—4661no—
249C4—2332no—
250C2—22no—
251C2—41no—
252C2—41no—
253C2—9321no—
254C2—4662no—
255C2—4662no—
256C2C2×D4×D23311yescharacteristic
257C2#41×D23311yescharacteristic
258C2#2141858011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D233
commutator, socle, cosocleC22×C233C24
Fitting#121×C233C2
hypercenter#121D233
hypocenterC233#546
LCSG ▷ C22×C233 ▷ C233
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16836593C2semidirect
2. C2×D4×D233C2semidirect
3. #21418521C2semidirect
4. #21418520C2semidirect
5. #21418516C2semidirect
6. #21418513C2semidirect
7. #21418508C2semidirect
8. #21418507C2semidirect
9. #21418503C2semidirect
10. #21418503C2semidirect
11. C4×C22⋊D233C2semidirect
12. #21418495C2semidirect
13. C2×#16836571C2semidirect
14. C22⋊C4×D233C2semidirect
15. #121×C233C2semidirect
19. C2×Dic466C22semidirect
24. C2×C22⋊D233C22semidirect
25. C2×C22⋊D233C22semidirect
27. C22×Dic233C22semidirect
28. C22×Dic233C22semidirect
30. C2×C4×D233C22semidirect
31. C2×C4×D233C22semidirect
32. #16836573C22semidirect
33. #16836573C22semidirect
34. #16836573C22semidirect
35. C22⋊Dic233C22semidirect
36. C22⋊Dic233C22semidirect
37. C22⋊Dic233C22semidirect
41. #16836571C22semidirect
42. #16836571C22semidirect
43. #16836571C22semidirect
44. #16836571C22semidirect
45. C4×Dic233C22semidirect
47. C2×D4×C233C22semidirect
48. C2×D4×C233C22semidirect
49. C22×C4×C233C22semidirect
50. C4⋊C4×C233C22semidirect
51. C22⋊C4×C233C22semidirect
52. C22⋊C4×C233C22semidirect
68. C22⋊D233D4semidirect
110. C2×D233#41semidirect
129. C233#546semidirect
139. #121D233semidirect
158. C2×D4C2×D233semidirect
159. C2×D4C2×D233semidirect
165. C22×C4C2×D233semidirect
169. C4⋊C4C2×D233semidirect
180. C22⋊C4C2×D233semidirect
181. C22⋊C4C2×D233semidirect

Maximal subgroups

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

subgroupindex
C2×#168365932
C2×D4×D2332
#214185212
#214185202
#214185162
#214185132
#214185082
#214185072
#214185032
C4×C22⋊D2332
#214184952
C2×#168365712
C22⋊C4×D2332
#121×C2332
#546233