Finite Group Databasebeta

← #30657376

Subgroups — #30657376

Order 16448.

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×#16861617C211yesmaximal, characteristic, complemented
2C2×D4×D257C211yesmaximal, characteristic, complemented
3#22551578C211yesmaximal, characteristic, complemented
4#22551577C211yesmaximal, characteristic, complemented
5#22551573C211yesmaximal, characteristic, complemented
6#22551570C211yesmaximal, characteristic, complemented
7#22551565C211yesmaximal, characteristic, complemented
8#22551564C211yesmaximal, characteristic, complemented
9#22551560C211yesmaximal, characteristic, complemented
10#22551560C211yesmaximal, characteristic, complemented
11C4×C22⋊D257C211yesmaximal, characteristic, complemented
12#22551552C211yesmaximal, characteristic, complemented
13C2×#16861595C211yesmaximal, characteristic, complemented
14C22⋊C4×D257C211yesmaximal, characteristic, complemented
15#121×C257C211yesmaximal, characteristic, complemented
16C23×D257C2211yescharacteristic
17C23×D257C2211yescharacteristic
18#16861617—24no—
19C2×Dic514C2211yescharacteristic, complemented
20C2×D1028C2211yescharacteristic
21C2×C22⋊D257C2211yescharacteristic
22C2×C22⋊D257C2211yescharacteristic
23C2×C22⋊D257C2211yescharacteristic
24C2×C22⋊D257C2211yescharacteristic, complemented
25C2×C22⋊D257C2211yescharacteristic, complemented
26D4×D257—24no—
27C22×Dic257C2211yescharacteristic, complemented
28C22×Dic257C2211yescharacteristic, complemented
29C2×C4×D257C2211yescharacteristic
30C2×C4×D257C2211yescharacteristic, complemented
31C2×C4×D257C2211yescharacteristic, complemented
32#16861597C2211yescharacteristic, complemented
33#16861597C2211yescharacteristic, complemented
34#16861597C2211yescharacteristic, complemented
35C22⋊Dic257C2211yescharacteristic, complemented
36C22⋊Dic257C2211yescharacteristic, complemented
37C22⋊Dic257C2211yescharacteristic, complemented
38#16861595C2211yescharacteristic
39#16861595C2211yescharacteristic
40#16861595C2211yescharacteristic
41#16861595C2211yescharacteristic, complemented
42#16861595C2211yescharacteristic, complemented
43#16861595C2211yescharacteristic, complemented
44#16861595C2211yescharacteristic, complemented
45C4×Dic257C2211yescharacteristic, complemented
46C2×D4×C257C2211yescharacteristic
47C2×D4×C257C2211yescharacteristic, complemented
48C2×D4×C257C2211yescharacteristic, complemented
49C22×C4×C257C2211yescharacteristic, complemented
50C4⋊C4×C257C2211yescharacteristic, complemented
51C22⋊C4×C257C2211yescharacteristic, complemented
52C22⋊C4×C257C2211yescharacteristic, complemented
53C22×D257—22no—
54C22×D257—22no—
55C22×D257—22no—
56C22×D257—22no—
57C22×D257—41no—
58C22×D257—41no—
59C22×D257C2311yescharacteristic
60C22×D257C2311yescharacteristic
61C22×D257C2311yescharacteristic
62Dic514—22no—
63D1028—22no—
64C22⋊D257—22no—
65C22⋊D257—22no—
66C22⋊D257—22no—
67C22⋊D257—41no—
68C22⋊D257D414yescomplemented
69C2×Dic257—22no—
70C2×Dic257—22no—
71C2×Dic257C2311yescharacteristic
72C2×Dic257C2311yescharacteristic
73C2×Dic257C2311yescharacteristic
74C2×Dic257C2311yescharacteristic
75C2×Dic257C2311yescharacteristic
76C4×D257—22no—
77C4×D257—22no—
78C4×D257—41no—
79C23×C257C2311yescharacteristic
80C23×C257C2311yescharacteristic
81C23×C257C2311yescharacteristic
82D4×C257—22no—
83D4×C257—22no—
84D4×C257—41no—
85C2×C4×C257—41no—
86C2×C4×C257C2311yescharacteristic
87C2×C4×C257C2311yescharacteristic
88C2×C4×C257C2311yescharacteristic
89C2×C4×C257C2311yescharacteristic
90C22×C257—21no—
91C22×C257—21no—
92C22×C257—21no—
93C22×C257—21no—
94C22×C257—21no—
95C22×C257—21no—
96C22×C257—21no—
97C22×C257—21no—
98C22×C257C2×D412yes—
99C22×C257C2411yescharacteristic
100C2×D257—21no—
101C2×D257—21no—
102C2×D257—21no—
103C2×D257—21no—
104C2×D257—21no—
105C2×D257—21no—
106C2×D257—21no—
107C2×D257—44no—
108C2×D257—44no—
109C2×D257C2×D412yes—
110C2×D257#4112yescomplemented
111C4×C257—21no—
112C4×C257—21no—
113C4×C257—21no—
114C4×C257—21no—
115Dic257—21no—
116Dic257—21no—
117Dic257—21no—
118Dic257—21no—
119Dic257C2×D412yes—
120C2×C257—22no—
121C2×C257—41no—
122C2×C257—41no—
123C2×C257C22×D411yescharacteristic
124C2×C257C2×#4111yescharacteristic
125C2×C257H(5,𝔽2)11yescharacteristic
126D257—22no—
127D257—22no—
128D257—41no—
129C257#54611yescharacteristic, complemented, Sylow
130#546—2571nomaximal, Sylow
131C2×#41—2571no—
132C22×D4—2571no—
133#124—2571no—
134#123—2571no—
135#123—2571no—
136C22⋊Q8—2571no—
137#121—2571no—
138#121—2571no—
139#121D25711yescharacteristic, complemented
140#120—2571no—
141#120—2571no—
142C4×D4—2571no—
143C4×D4—2571no—
144C2×C22⋊C4—2571no—
145C2×C22⋊C4—2571no—
146C24—2571no—
147C24—2571no—
148#41—5144no—
149C2×Q8—2571no—
150C2×D4—2571no—
151C2×D4—2571no—
152C2×D4—2571no—
153C2×D4—2571no—
154C2×D4—2571no—
155C2×D4—2571no—
156C2×D4—5144no—
157C2×D4C2×D25711yescharacteristic
158C2×D4C2×D25711yescharacteristic, complemented
159C2×D4C2×D25711yescharacteristic, complemented
160C22×C4—2571no—
161C22×C4—2571no—
162C22×C4—2571no—
163C22×C4—2571no—
164C22×C4—2571no—
165C22×C4C2×D25711yescharacteristic, complemented
166C4⋊C4—2571no—
167C4⋊C4—2571no—
168C4⋊C4—2571no—
169C4⋊C4C2×D25711yescharacteristic, complemented
170C22⋊C4—2571no—
171C22⋊C4—2571no—
172C22⋊C4—2571no—
173C22⋊C4—2571no—
174C22⋊C4—2571no—
175C22⋊C4—2571no—
176C22⋊C4—2571no—
177C22⋊C4—2571no—
178C22⋊C4—2571no—
179C22⋊C4—2571no—
180C22⋊C4C2×D25711yescharacteristic, complemented
181C22⋊C4C2×D25711yescharacteristic, complemented
182C42—2571no—
183C23—2571no—
184C23—2571no—
185C23—2571no—
186C23—5142no—
187C23—5142no—
188C23—5142no—
189C23—5142no—
190C23—10281no—
191C23—10281no—
192C23C22×D25711yescharacteristic
193C23C22×D25711yescharacteristic
194C23C22×D25711yescharacteristic
195Q8—5142no—
196D4—2574no—
197D4—22no—
198D4—22no—
199D4—5142no—
200D4—5142no—
201D4—5142no—
202D4—5142no—
203D4—41no—
204D4—10281no—
205C2×C4—2571no—
206C2×C4—2571no—
207C2×C4—2571no—
208C2×C4—2571no—
209C2×C4—2571no—
210C2×C4—5142no—
211C2×C4—5142no—
212C2×C4—5142no—
213C2×C4—5142no—
214C2×C4—41no—
215C2×C4—10281no—
216C2×C4C22×D25711yescharacteristic
217C2×C4C22×D25711yescharacteristic
218C2×C4C22×D25711yescharacteristic
219C2×C4C22×D25711yescharacteristic
220C22—2572no—
221C22—2572no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—5141no—
231C22—5141no—
232C22—5141no—
233C22—5141no—
234C22—5141no—
235C22—5141no—
236C22—5141no—
237C22—10284no—
238C22—10284no—
239C22D4×D25712yes—
240C22C23×D25711yescharacteristic
241C4—2572no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—5141no—
247C4—5141no—
248C4—5141no—
249C4—5141no—
250C2—22no—
251C2—5142no—
252C2—5142no—
253C2—41no—
254C2—41no—
255C2—10281no—
256C2C2×D4×D25711yescharacteristic
257C2#41×D25711yescharacteristic
258C2#2255163711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
Fitting#121×C257C2
hypercenter#121D257
hypocenterC257#546
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16861617C2semidirect
2. C2×D4×D257C2semidirect
3. #22551578C2semidirect
4. #22551577C2semidirect
5. #22551573C2semidirect
6. #22551570C2semidirect
7. #22551565C2semidirect
8. #22551564C2semidirect
9. #22551560C2semidirect
10. #22551560C2semidirect
11. C4×C22⋊D257C2semidirect
12. #22551552C2semidirect
13. C2×#16861595C2semidirect
14. C22⋊C4×D257C2semidirect
15. #121×C257C2semidirect
19. C2×Dic514C22semidirect
24. C2×C22⋊D257C22semidirect
25. C2×C22⋊D257C22semidirect
27. C22×Dic257C22semidirect
28. C22×Dic257C22semidirect
30. C2×C4×D257C22semidirect
31. C2×C4×D257C22semidirect
32. #16861597C22semidirect
33. #16861597C22semidirect
34. #16861597C22semidirect
35. C22⋊Dic257C22semidirect
36. C22⋊Dic257C22semidirect
37. C22⋊Dic257C22semidirect
41. #16861595C22semidirect
42. #16861595C22semidirect
43. #16861595C22semidirect
44. #16861595C22semidirect
45. C4×Dic257C22semidirect
47. C2×D4×C257C22semidirect
48. C2×D4×C257C22semidirect
49. C22×C4×C257C22semidirect
50. C4⋊C4×C257C22semidirect
51. C22⋊C4×C257C22semidirect
52. C22⋊C4×C257C22semidirect
68. C22⋊D257D4semidirect
110. C2×D257#41semidirect
129. C257#546semidirect
139. #121D257semidirect
158. C2×D4C2×D257semidirect
159. C2×D4C2×D257semidirect
165. C22×C4C2×D257semidirect
169. C4⋊C4C2×D257semidirect
180. C22⋊C4C2×D257semidirect
181. C22⋊C4C2×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#168616172
C2×D4×D2572
#225515782
#225515772
#225515732
#225515702
#225515652
#225515642
#225515602
C4×C22⋊D2572
#225515522
C2×#168615952
C22⋊C4×D2572
#121×C2572
#546257