Finite Group Databasebeta

← #30632360

Subgroups — #30632360

Order 16064.

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#22548710C211yesmaximal, characteristic, complemented
2#22548710C211yesmaximal, characteristic, complemented
3#22548701C211yesmaximal, characteristic, complemented
4#22548699C211yesmaximal, characteristic, complemented
5#22548697C211yesmaximal, characteristic, complemented
6#22548695C211yesmaximal, characteristic, complemented
7#22548692C211yesmaximal, characteristic, complemented
8#22548683C211yesmaximal, characteristic
9D4×Dic251C211yesmaximal, characteristic, complemented
10(C22⋊D251)⋊C4C211yesmaximal, characteristic, complemented
11#22548674C211yesmaximal, characteristic, complemented
12#22548673C211yesmaximal, characteristic, complemented
13C2×(C2×D251)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D251C211yesmaximal, characteristic, complemented
15#123×C251C211yesmaximal, characteristic, complemented
16C23×D251C2211yescharacteristic
17C2×Dic502C2211yescharacteristic, complemented
18C2×C22⋊D251C2211yescharacteristic
19C2×C22⋊D251C2211yescharacteristic
20C22×Dic251C2211yescharacteristic
21C22×Dic251C2211yescharacteristic, complemented
22C2×C4×D251C2211yescharacteristic
23C2×C4×D251C2211yescharacteristic
24C4⋊Dic251C2211yescharacteristic, complemented
25C4⋊Dic251C2211yescharacteristic, complemented
26C4⋊Dic251C2211yescharacteristic, complemented
27C4⋊Dic251C2211yescharacteristic, complemented
28Dic251⋊C4C2211yescharacteristic, complemented
29Dic251⋊C4C2211yescharacteristic, complemented
30C22⋊Dic251C2211yescharacteristic
31C22⋊Dic251C2211yescharacteristic
32C22⋊Dic251C2211yescharacteristic, complemented
33(C2×D251)⋊C4C2211yescharacteristic
34(C2×D251)⋊C4C2211yescharacteristic
35(C2×D251)⋊C4C2211yescharacteristic
36(C2×D251)⋊C4C2211yescharacteristic
37(C2×D251)⋊C4C2211yescharacteristic
38(C2×D251)⋊C4C2211yescharacteristic
39(C2×D251)⋊C4C2211yescharacteristic
40(C2×D251)⋊C4C2211yescharacteristic
41C4×Dic251C2211yescharacteristic, complemented
42C4×Dic251C2211yescharacteristic, complemented
43C4×Dic251C2211yescharacteristic, complemented
44C2×D4×C251C2211yescharacteristic
45C22×C4×C251C2211yescharacteristic
46C4⋊C4×C251C2211yescharacteristic, complemented
47C4⋊C4×C251C2211yescharacteristic, complemented
48C22⋊C4×C251C2211yescharacteristic
49C22⋊C4×C251C2211yescharacteristic, complemented
50C22⋊C4×C251C2211yescharacteristic, complemented
51C22×D251—22no—
52C22×D251—22no—
53C22×D251—41no—
54C22×D251C2311yescharacteristic
55C22×D251C2311yescharacteristic
56Dic502—41no—
57C22⋊D251—22no—
58C22⋊D251—41no—
59C2×Dic251—41no—
60C2×Dic251—41no—
61C2×Dic251C2311yescharacteristic
62C2×Dic251C2311yescharacteristic
63C2×Dic251C2311yescharacteristic
64C2×Dic251C2311yescharacteristic
65C2×Dic251C2311yescharacteristic
66C2×Dic251C2311yescharacteristic
67C4×D251—22no—
68C4×D251—41no—
69C23×C251C2311yescharacteristic
70C23×C251C2311yescharacteristic
71D4×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251C2311yescharacteristic
74C2×C4×C251C2311yescharacteristic
75C2×C4×C251C2311yescharacteristic
76C2×C4×C251C2311yescharacteristic
77C2×C4×C251C2311yescharacteristic
78C22×C251—21no—
79C22×C251—21no—
80C22×C251—21no—
81C22×C251—21no—
82C22×C251—21no—
83C22×C251#4112yes—
84C22×C251C2411yescharacteristic
85C2×D251—21no—
86C2×D251—21no—
87C2×D251—21no—
88C2×D251—21no—
89C2×D251—21no—
90C2×D251—44no—
91C2×D251#4112yes—
92C4×C251—21no—
93C4×C251—21no—
94C4×C251—21no—
95C4×C251—21no—
96C4×C251—21no—
97Dic251—21no—
98Dic251—21no—
99Dic251—21no—
100Dic251—21no—
101Dic251—21no—
102Dic251—21no—
103C2×C251—22no—
104C2×C251—41no—
105C2×C251C2×#4111yescharacteristic
106C2×C251C2×#4111yescharacteristic
107C2×C251H(5,𝔽2)11yescharacteristic
108D251—22no—
109D251—41no—
110C251#55111yescharacteristic, complemented, Sylow
111#551—2511nomaximal, Sylow
112#126—2511no—
113#126—2511no—
114#124—2511no—
115#123—2511no—
116#123—2511no—
117#123D25111yescharacteristic, complemented
118C22⋊Q8—2511no—
119C22⋊Q8—2511no—
120#120—2511no—
121C4×D4—2511no—
122C4×D4—2511no—
123#117—2511no—
124#117—2511no—
125C2×C22⋊C4—2511no—
126C2×C22⋊C4—2511no—
127C24—2511no—
128C2×Q8—2511no—
129C2×D4—2511no—
130C2×D4—2511no—
131C2×D4C2×D25111yescharacteristic
132C22×C4—2511no—
133C22×C4—2511no—
134C22×C4—2511no—
135C22×C4—2511no—
136C22×C4C2×D25111yescharacteristic
137C4⋊C4—2511no—
138C4⋊C4—2511no—
139C4⋊C4—2511no—
140C4⋊C4—2511no—
141C4⋊C4—2511no—
142C4⋊C4—2511no—
143C4⋊C4C2×D25111yescharacteristic, complemented
144C4⋊C4C2×D25111yescharacteristic, complemented
145C22⋊C4—2511no—
146C22⋊C4—2511no—
147C22⋊C4—2511no—
148C22⋊C4—2511no—
149C22⋊C4—2511no—
150C22⋊C4—2511no—
151C22⋊C4—2511no—
152C22⋊C4—2511no—
153C22⋊C4—2511no—
154C22⋊C4—2511no—
155C22⋊C4—2511no—
156C22⋊C4C2×D25111yescharacteristic
157C22⋊C4C2×D25111yescharacteristic, complemented
158C22⋊C4C2×D25111yescharacteristic, complemented
159C42—2511no—
160C42—2511no—
161C42—2511no—
162C23—10041no—
163C23—5022no—
164C23—5022no—
165C23—2511no—
166C23—2511no—
167C23C22×D25111yescharacteristic
168C23C22×D25111yescharacteristic
169Q8—10041no—
170D4—22no—
171D4—10041no—
172D4—5022no—
173C2×C4—22no—
174C2×C4—10041no—
175C2×C4—10041no—
176C2×C4—10041no—
177C2×C4—5022no—
178C2×C4—2511no—
179C2×C4—2511no—
180C2×C4—2511no—
181C2×C4—2511no—
182C2×C4—2511no—
183C2×C4—2511no—
184C2×C4C22×D25111yescharacteristic
185C2×C4C22×D25111yescharacteristic
186C2×C4C22×D25111yescharacteristic
187C2×C4C22×D25111yescharacteristic
188C2×C4C22×D25111yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—10044no—
195C22—5021no—
196C22—5021no—
197C22—5021no—
198C22—5021no—
199C22—5021no—
200C22—2512no—
201C22#1686080912yes—
202C22C23×D25111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—5021no—
209C4—5021no—
210C4—5021no—
211C4—5021no—
212C4—5021no—
213C4—5021no—
214C2—22no—
215C2—41no—
216C2—10041no—
217C2—5022no—
218C2#41×D25111yescharacteristic
219C2C2×#1686080911yescharacteristic
220C2#2254876111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D251
commutator, socle, cosocleC22×C251C24
Fitting#123×C251C2
prime radical#22548683C2
hypercenter#123D251
hypocenterC251#551
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #22548710C2semidirect
2. #22548710C2semidirect
3. #22548701C2semidirect
4. #22548699C2semidirect
5. #22548697C2semidirect
6. #22548695C2semidirect
7. #22548692C2semidirect
9. D4×Dic251C2semidirect
10. (C22⋊D251)⋊C4C2semidirect
11. #22548674C2semidirect
12. #22548673C2semidirect
13. C2×(C2×D251)⋊C4C2semidirect
14. C22⋊C4×D251C2semidirect
15. #123×C251C2semidirect
17. C2×Dic502C22semidirect
21. C22×Dic251C22semidirect
24. C4⋊Dic251C22semidirect
25. C4⋊Dic251C22semidirect
26. C4⋊Dic251C22semidirect
27. C4⋊Dic251C22semidirect
28. Dic251⋊C4C22semidirect
29. Dic251⋊C4C22semidirect
32. C22⋊Dic251C22semidirect
41. C4×Dic251C22semidirect
42. C4×Dic251C22semidirect
43. C4×Dic251C22semidirect
46. C4⋊C4×C251C22semidirect
47. C4⋊C4×C251C22semidirect
49. C22⋊C4×C251C22semidirect
50. C22⋊C4×C251C22semidirect
110. C251#551semidirect
117. #123D251semidirect
143. C4⋊C4C2×D251semidirect
144. C4⋊C4C2×D251semidirect
157. C22⋊C4C2×D251semidirect
158. C22⋊C4C2×D251semidirect

Maximal subgroups

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

subgroupindex
#225487102
#225487012
#225486992
#225486972
#225486952
#225486922
#225486832
D4×Dic2512
(C22⋊D251)⋊C42
#225486742
#225486732
C2×(C2×D251)⋊C42
C22⋊C4×D2512
#123×C2512
#551251