Finite Group Databasebeta

← #29508359

Subgroups — #29508359

Order 15424.

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×#16838770C211yesmaximal, characteristic, complemented
2C2×D4×D241C211yesmaximal, characteristic, complemented
3#21440628C211yesmaximal, characteristic, complemented
4#21440627C211yesmaximal, characteristic, complemented
5#21440620C211yesmaximal, characteristic, complemented
6#21440617C211yesmaximal, characteristic, complemented
7#21440615C211yesmaximal, characteristic, complemented
8#21440614C211yesmaximal, characteristic, complemented
9#21440612C211yesmaximal, characteristic, complemented
10#21440612C211yesmaximal, characteristic, complemented
11#21440604C211yesmaximal, characteristic, complemented
12#21440602C211yesmaximal, characteristic, complemented
13C2×#16838747C211yesmaximal, characteristic, complemented
14C22⋊C4×D241C211yesmaximal, characteristic, complemented
15#123×C241C211yesmaximal, characteristic, complemented
16C23×D241C2211yescharacteristic
17C23×D241C2211yescharacteristic
18#16838770—24no—
19C2×Dic482C2211yescharacteristic, complemented
20C2×D964C2211yescharacteristic
21C2×D964C2211yescharacteristic
22C2×D964C2211yescharacteristic, complemented
23C2×D964C2211yescharacteristic, complemented
24C2×C22⋊D241C2211yescharacteristic
25C2×C22⋊D241C2211yescharacteristic
26C2×C22⋊D241C2211yescharacteristic, complemented
27C2×C22⋊D241C2211yescharacteristic, complemented
28D4×D241—24no—
29C22×Dic241C2211yescharacteristic, complemented
30C2×C4×D241C2211yescharacteristic
31C2×C4×D241C2211yescharacteristic, complemented
32C2×C4×D241C2211yescharacteristic, complemented
33C2×C4×D241C2211yescharacteristic, complemented
34#16838749C2211yescharacteristic, complemented
35#16838749C2211yescharacteristic, complemented
36C22⋊Dic241C2211yescharacteristic, complemented
37#16838747C2211yescharacteristic
38#16838747C2211yescharacteristic
39#16838747C2211yescharacteristic
40#16838747C2211yescharacteristic, complemented
41#16838747C2211yescharacteristic, complemented
42#16838747C2211yescharacteristic, complemented
43#16838747C2211yescharacteristic, complemented
44#16838747C2211yescharacteristic, complemented
45C4×Dic241C2211yescharacteristic, complemented
46C2×D4×C241C2211yescharacteristic
47C22×C4×C241C2211yescharacteristic, complemented
48C4⋊C4×C241C2211yescharacteristic, complemented
49C4⋊C4×C241C2211yescharacteristic, complemented
50C22⋊C4×C241C2211yescharacteristic, complemented
51C22⋊C4×C241C2211yescharacteristic, complemented
52C22⋊C4×C241C2211yescharacteristic, complemented
53C22×D241—22no—
54C22×D241—22no—
55C22×D241—22no—
56C22×D241—22no—
57C22×D241—41no—
58C22×D241—41no—
59C22×D241C2311yescharacteristic
60C22×D241C2311yescharacteristic
61C22×D241C2311yescharacteristic
62C22×D241C2311yescharacteristic
63Dic482—22no—
64D964—22no—
65D964—22no—
66D964—22no—
67D964—41no—
68C22⋊D241—22no—
69C22⋊D241—22no—
70C22⋊D241—41no—
71C22⋊D241D414yescomplemented
72C2×Dic241—41no—
73C2×Dic241C2311yescharacteristic
74C2×Dic241C2311yescharacteristic
75C2×Dic241C2311yescharacteristic
76C2×Dic241C2311yescharacteristic
77C4×D241—22no—
78C4×D241—22no—
79C4×D241—22no—
80C4×D241—41no—
81C23×C241C2311yescharacteristic
82C23×C241C2311yescharacteristic
83D4×C241—22no—
84C2×C4×C241—22no—
85C2×C4×C241C2311yescharacteristic
86C2×C4×C241C2311yescharacteristic
87C2×C4×C241C2311yescharacteristic
88C2×C4×C241C2311yescharacteristic
89C2×C4×C241C2311yescharacteristic
90C22×C241—21no—
91C22×C241—21no—
92C22×C241—21no—
93C22×C241—21no—
94C22×C241—21no—
95C22×C241C2×D412yes—
96C22×C241C2411yescharacteristic
97C2×D241—21no—
98C2×D241—21no—
99C2×D241—21no—
100C2×D241—21no—
101C2×D241—21no—
102C2×D241—21no—
103C2×D241—21no—
104C2×D241—21no—
105C2×D241—21no—
106C2×D241—21no—
107C2×D241—44no—
108C2×D241—44no—
109C2×D241C2×D412yes—
110C2×D241#4112yescomplemented
111C4×C241—21no—
112C4×C241—21no—
113C4×C241—21no—
114C4×C241—21no—
115C4×C241—21no—
116Dic241—21no—
117Dic241—21no—
118Dic241—21no—
119Dic241C2×D412yes—
120C2×C241—22no—
121C2×C241—41no—
122C2×C241C22×D411yescharacteristic
123C2×C241C2×#4111yescharacteristic
124C2×C241H(5,𝔽2)11yescharacteristic
125D241—22no—
126D241—22no—
127D241—41no—
128D241—41no—
129C241#54611yescharacteristic, complemented, Sylow
130#546—2411nomaximal, Sylow
131C2×#41—2411no—
132C22×D4—2411no—
133#124—2411no—
134#123—2411no—
135#123D24111yescharacteristic, complemented
136C22⋊Q8—2411no—
137#121—2411no—
138#121—2411no—
139#121—2411no—
140#120—2411no—
141#120—2411no—
142C4×D4—2411no—
143C4×D4—2411no—
144C2×C22⋊C4—2411no—
145C2×C22⋊C4—2411no—
146C24—2411no—
147C24—2411no—
148#41—4824no—
149C2×Q8—2411no—
150C2×D4—4824no—
151C2×D4—2411no—
152C2×D4—2411no—
153C2×D4—2411no—
154C2×D4—2411no—
155C2×D4—2411no—
156C2×D4—2411no—
157C2×D4—2411no—
158C2×D4—2411no—
159C2×D4C2×D24111yescharacteristic
160C22×C4—2411no—
161C22×C4—2411no—
162C22×C4—2411no—
163C22×C4—2411no—
164C22×C4—2411no—
165C22×C4C2×D24111yescharacteristic, complemented
166C4⋊C4—2411no—
167C4⋊C4—2411no—
168C4⋊C4C2×D24111yescharacteristic, complemented
169C4⋊C4C2×D24111yescharacteristic, complemented
170C22⋊C4—2411no—
171C22⋊C4—2411no—
172C22⋊C4—2411no—
173C22⋊C4—2411no—
174C22⋊C4—2411no—
175C22⋊C4—2411no—
176C22⋊C4—2411no—
177C22⋊C4—2411no—
178C22⋊C4—2411no—
179C22⋊C4C2×D24111yescharacteristic, complemented
180C22⋊C4C2×D24111yescharacteristic, complemented
181C22⋊C4C2×D24111yescharacteristic, complemented
182C42—2411no—
183C23—9641no—
184C23—9641no—
185C23—4822no—
186C23—4822no—
187C23—4822no—
188C23—4822no—
189C23—2411no—
190C23—2411no—
191C23—2411no—
192C23—2411no—
193C23C22×D24111yescharacteristic
194C23C22×D24111yescharacteristic
195Q8—4822no—
196D4—22no—
197D4—9641no—
198D4—9641no—
199D4—4822no—
200D4—4822no—
201D4—4822no—
202D4—4822no—
203D4—4822no—
204D4—2414no—
205C2×C4—22no—
206C2×C4—9641no—
207C2×C4—9641no—
208C2×C4—4822no—
209C2×C4—4822no—
210C2×C4—4822no—
211C2×C4—2411no—
212C2×C4—2411no—
213C2×C4—2411no—
214C2×C4—2411no—
215C2×C4C22×D24111yescharacteristic
216C2×C4C22×D24111yescharacteristic
217C2×C4C22×D24111yescharacteristic
218C2×C4C22×D24111yescharacteristic
219C2×C4C22×D24111yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—9644no—
226C22—9644no—
227C22—4821no—
228C22—4821no—
229C22—4821no—
230C22—4821no—
231C22—4821no—
232C22—4821no—
233C22—4821no—
234C22—4821no—
235C22—4821no—
236C22—4821no—
237C22—2412no—
238C22—2412no—
239C22D4×D24112yes—
240C22C23×D24111yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—4821no—
247C4—4821no—
248C4—4821no—
249C4—2412no—
250C2—22no—
251C2—41no—
252C2—9641no—
253C2—9641no—
254C2—4822no—
255C2—4822no—
256C2C2×D4×D24111yescharacteristic
257C2#41×D24111yescharacteristic
258C2#2144068811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
Fitting#123×C241C2
hypercenter#123D241
hypocenterC241#546
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#16838770C2semidirect
2. C2×D4×D241C2semidirect
3. #21440628C2semidirect
4. #21440627C2semidirect
5. #21440620C2semidirect
6. #21440617C2semidirect
7. #21440615C2semidirect
8. #21440614C2semidirect
9. #21440612C2semidirect
10. #21440612C2semidirect
11. #21440604C2semidirect
12. #21440602C2semidirect
13. C2×#16838747C2semidirect
14. C22⋊C4×D241C2semidirect
15. #123×C241C2semidirect
19. C2×Dic482C22semidirect
22. C2×D964C22semidirect
23. C2×D964C22semidirect
26. C2×C22⋊D241C22semidirect
27. C2×C22⋊D241C22semidirect
29. C22×Dic241C22semidirect
31. C2×C4×D241C22semidirect
32. C2×C4×D241C22semidirect
33. C2×C4×D241C22semidirect
34. #16838749C22semidirect
35. #16838749C22semidirect
36. C22⋊Dic241C22semidirect
40. #16838747C22semidirect
41. #16838747C22semidirect
42. #16838747C22semidirect
43. #16838747C22semidirect
44. #16838747C22semidirect
45. C4×Dic241C22semidirect
47. C22×C4×C241C22semidirect
48. C4⋊C4×C241C22semidirect
49. C4⋊C4×C241C22semidirect
50. C22⋊C4×C241C22semidirect
51. C22⋊C4×C241C22semidirect
52. C22⋊C4×C241C22semidirect
71. C22⋊D241D4semidirect
110. C2×D241#41semidirect
129. C241#546semidirect
135. #123D241semidirect
165. C22×C4C2×D241semidirect
168. C4⋊C4C2×D241semidirect
169. C4⋊C4C2×D241semidirect
179. C22⋊C4C2×D241semidirect
180. C22⋊C4C2×D241semidirect
181. C22⋊C4C2×D241semidirect

Maximal subgroups

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

subgroupindex
C2×#168387702
C2×D4×D2412
#214406282
#214406272
#214406202
#214406172
#214406152
#214406142
#214406122
#214406042
#214406022
C2×#168387472
C22⋊C4×D2412
#123×C2412
#546241