Finite Group Databasebeta

← #30657414

Subgroups — #30657414

Order 16448.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#22551587C211yesmaximal, characteristic, complemented
2#22551587C211yesmaximal, characteristic, complemented
3#22551581C211yesmaximal, characteristic, complemented
4#22551577C211yesmaximal, characteristic, complemented
5#22551577C211yesmaximal, characteristic, complemented
6#22551576C211yesmaximal, characteristic, complemented
7#22551568C211yesmaximal, characteristic, complemented
8#22551562C211yesmaximal, characteristic
9#22551554C211yesmaximal, characteristic, complemented
10C4×C22⋊D257C211yesmaximal, characteristic, complemented
11#22551550C211yesmaximal, characteristic, complemented
12#22551548C211yesmaximal, characteristic, complemented
13C2×#16861595C211yesmaximal, characteristic, complemented
14C22⋊C4×D257C211yesmaximal, characteristic, complemented
15C22⋊Q8×C257C211yesmaximal, characteristic, complemented
16C23×D257C2211yescharacteristic
17C2×D1028C2211yescharacteristic
18C2×D1028C2211yescharacteristic
19C2×C22⋊D257C2211yescharacteristic
20C22×Dic257C2211yescharacteristic
21C2×C4×D257C2211yescharacteristic
22C2×C4×D257C2211yescharacteristic
23C2×C4×D257C2211yescharacteristic, complemented
24C4⋊Dic257C2211yescharacteristic, complemented
25C4⋊Dic257C2211yescharacteristic, complemented
26#16861597C2211yescharacteristic, complemented
27#16861597C2211yescharacteristic, complemented
28#16861597C2211yescharacteristic, complemented
29C22⋊Dic257C2211yescharacteristic
30#16861595C2211yescharacteristic
31#16861595C2211yescharacteristic
32#16861595C2211yescharacteristic
33#16861595C2211yescharacteristic
34#16861595C2211yescharacteristic
35#16861595C2211yescharacteristic
36#16861595C2211yescharacteristic
37#16861595C2211yescharacteristic
38#16861595C2211yescharacteristic, complemented
39#16861595C2211yescharacteristic, complemented
40#16861595C2211yescharacteristic, complemented
41C4×Dic257C2211yescharacteristic, complemented
42C4×Dic257C2211yescharacteristic, complemented
43C4×Dic257C2211yescharacteristic, complemented
44C2×Q8×C257C2211yescharacteristic, complemented
45C22×C4×C257C2211yescharacteristic
46C4⋊C4×C257C2211yescharacteristic, complemented
47C4⋊C4×C257C2211yescharacteristic, complemented
48C4⋊C4×C257C2211yescharacteristic, complemented
49C22⋊C4×C257C2211yescharacteristic
50C22⋊C4×C257C2211yescharacteristic
51C22×D257—22no—
52C22×D257—22no—
53C22×D257—41no—
54C22×D257C2311yescharacteristic
55C22×D257C2311yescharacteristic
56C22×D257C2311yescharacteristic
57D1028—22no—
58D1028—41no—
59C22⋊D257—22no—
60C2×Dic257—22no—
61C2×Dic257C2311yescharacteristic
62C2×Dic257C2311yescharacteristic
63C2×Dic257C2311yescharacteristic
64C2×Dic257C2311yescharacteristic
65C2×Dic257C2311yescharacteristic
66C4×D257—22no—
67C4×D257—41no—
68C4×D257—41no—
69C23×C257C2311yescharacteristic
70Q8×C257—41no—
71C2×C4×C257—41no—
72C2×C4×C257C2311yescharacteristic
73C2×C4×C257C2311yescharacteristic
74C2×C4×C257C2311yescharacteristic
75C2×C4×C257C2311yescharacteristic
76C2×C4×C257C2311yescharacteristic
77C2×C4×C257C2311yescharacteristic
78C22×C257—21no—
79C22×C257—21no—
80C22×C257#4112yes—
81C22×C257C2411yescharacteristic
82C2×D257—21no—
83C2×D257—21no—
84C2×D257—21no—
85C2×D257—21no—
86C2×D257—21no—
87C2×D257—21no—
88C2×D257—21no—
89C2×D257—21no—
90C2×D257—44no—
91C2×D257#4112yes—
92C4×C257—21no—
93C4×C257—21no—
94C4×C257—21no—
95C4×C257—21no—
96C4×C257—21no—
97C4×C257—21no—
98Dic257—21no—
99Dic257—21no—
100Dic257—21no—
101Dic257—21no—
102Dic257—21no—
103C2×C257—22no—
104C2×C257C2×#4111yescharacteristic
105C2×C257C2×#4111yescharacteristic
106C2×C257H(5,𝔽2)11yescharacteristic
107D257—22no—
108D257—41no—
109D257—41no—
110C257#55111yescharacteristic, complemented, Sylow
111#551—2571nomaximal, Sylow
112#126—2571no—
113#126—2571no—
114#124—2571no—
115#123—2571no—
116#123—2571no—
117#123—2571no—
118C22⋊Q8—2571no—
119C22⋊Q8D25711yescharacteristic, complemented
120#120—2571no—
121C4×D4—2571no—
122C4×D4—2571no—
123#117—2571no—
124#117—2571no—
125C2×C22⋊C4—2571no—
126C2×C22⋊C4—2571no—
127C24—2571no—
128C2×Q8C2×D25711yescharacteristic, complemented
129C2×D4—2571no—
130C2×D4—2571no—
131C2×D4—2571no—
132C22×C4—2571no—
133C22×C4—2571no—
134C22×C4—2571no—
135C22×C4—2571no—
136C22×C4C2×D25711yescharacteristic
137C4⋊C4—2571no—
138C4⋊C4—2571no—
139C4⋊C4—2571no—
140C4⋊C4—2571no—
141C4⋊C4—2571no—
142C4⋊C4C2×D25711yescharacteristic, complemented
143C4⋊C4C2×D25711yescharacteristic, complemented
144C4⋊C4C2×D25711yescharacteristic, complemented
145C22⋊C4—2571no—
146C22⋊C4—2571no—
147C22⋊C4—2571no—
148C22⋊C4—2571no—
149C22⋊C4—2571no—
150C22⋊C4—2571no—
151C22⋊C4—2571no—
152C22⋊C4—2571no—
153C22⋊C4—2571no—
154C22⋊C4—2571no—
155C22⋊C4—2571no—
156C22⋊C4—2571no—
157C22⋊C4C2×D25711yescharacteristic
158C22⋊C4C2×D25711yescharacteristic
159C42—2571no—
160C42—2571no—
161C42—2571no—
162C23—2571no—
163C23—2571no—
164C23—2571no—
165C23—5142no—
166C23—5142no—
167C23—10281no—
168C23C22×D25711yescharacteristic
169Q8—41no—
170D4—5142no—
171D4—5142no—
172D4—10281no—
173C2×C4—2571no—
174C2×C4—2571no—
175C2×C4—2571no—
176C2×C4—2571no—
177C2×C4—2571no—
178C2×C4—5142no—
179C2×C4—5142no—
180C2×C4—41no—
181C2×C4—10281no—
182C2×C4—10281no—
183C2×C4C22×D25711yescharacteristic
184C2×C4C22×D25711yescharacteristic
185C2×C4C22×D25711yescharacteristic
186C2×C4C22×D25711yescharacteristic
187C2×C4C22×D25711yescharacteristic
188C2×C4C22×D25711yescharacteristic
189C22—2572no—
190C22—21no—
191C22—21no—
192C22—5141no—
193C22—5141no—
194C22—5141no—
195C22—5141no—
196C22—5141no—
197C22—5141no—
198C22—5141no—
199C22—5141no—
200C22—10284no—
201C22#1686161912yes—
202C22C23×D25711yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—5141no—
210C4—5141no—
211C4—5141no—
212C4—5141no—
213C4—5141no—
214C2—22no—
215C2—5142no—
216C2—10281no—
217C2—10281no—
218C2#41×D25711yescharacteristic
219C2C2×#1686161911yescharacteristic
220C2#2255163711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
FittingC22⋊Q8×C257C2
prime radical#22551562C2
hypercenterC22⋊Q8D257
hypocenterC257#551
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22551587C2semidirect
2. #22551587C2semidirect
3. #22551581C2semidirect
4. #22551577C2semidirect
5. #22551577C2semidirect
6. #22551576C2semidirect
7. #22551568C2semidirect
9. #22551554C2semidirect
10. C4×C22⋊D257C2semidirect
11. #22551550C2semidirect
12. #22551548C2semidirect
13. C2×#16861595C2semidirect
14. C22⋊C4×D257C2semidirect
15. C22⋊Q8×C257C2semidirect
23. C2×C4×D257C22semidirect
24. C4⋊Dic257C22semidirect
25. C4⋊Dic257C22semidirect
26. #16861597C22semidirect
27. #16861597C22semidirect
28. #16861597C22semidirect
38. #16861595C22semidirect
39. #16861595C22semidirect
40. #16861595C22semidirect
41. C4×Dic257C22semidirect
42. C4×Dic257C22semidirect
43. C4×Dic257C22semidirect
44. C2×Q8×C257C22semidirect
46. C4⋊C4×C257C22semidirect
47. C4⋊C4×C257C22semidirect
48. C4⋊C4×C257C22semidirect
110. C257#551semidirect
119. C22⋊Q8D257semidirect
128. C2×Q8C2×D257semidirect
142. C4⋊C4C2×D257semidirect
143. C4⋊C4C2×D257semidirect
144. C4⋊C4C2×D257semidirect

Maximal subgroups

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

subgroupindex
#225515872
#225515812
#225515772
#225515762
#225515682
#225515622
#225515542
C4×C22⋊D2572
#225515502
#225515482
C2×#168615952
C22⋊C4×D2572
C22⋊Q8×C2572
#551257