Finite Group Databasebeta

← #29508393

Subgroups — #29508393

Order 15424.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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#21440635C211yesmaximal, characteristic, complemented
2#21440635C211yesmaximal, characteristic, complemented
3#21440628C211yesmaximal, characteristic, complemented
4#21440626C211yesmaximal, characteristic, complemented
5#21440625C211yesmaximal, characteristic, complemented
6#21440624C211yesmaximal, characteristic, complemented
7#21440621C211yesmaximal, characteristic, complemented
8#21440619C211yesmaximal, characteristic, complemented
9#21440611C211yesmaximal, characteristic
10C4×C22⋊D241C211yesmaximal, characteristic, complemented
11#21440602C211yesmaximal, characteristic, complemented
12#21440601C211yesmaximal, characteristic, complemented
13#21440598C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic241C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C241C211yesmaximal, characteristic, complemented
16C2×Dic482C2211yescharacteristic, complemented
17C2×C22⋊D241C2211yescharacteristic
18C2×C22⋊D241C2211yescharacteristic
19C2×C22⋊D241C2211yescharacteristic
20C22×Dic241C2211yescharacteristic
21C22×Dic241C2211yescharacteristic
22C2×C4×D241C2211yescharacteristic, complemented
23C4⋊Dic241C2211yescharacteristic, complemented
24C4⋊Dic241C2211yescharacteristic, complemented
25C4⋊Dic241C2211yescharacteristic, complemented
26#16838749C2211yescharacteristic, complemented
27#16838749C2211yescharacteristic, complemented
28#16838749C2211yescharacteristic, complemented
29#16838749C2211yescharacteristic, complemented
30#16838749C2211yescharacteristic, complemented
31C22⋊Dic241C2211yescharacteristic
32C22⋊Dic241C2211yescharacteristic
33C22⋊Dic241C2211yescharacteristic
34C22⋊Dic241C2211yescharacteristic
35C22⋊Dic241C2211yescharacteristic
36C22⋊Dic241C2211yescharacteristic
37C22⋊Dic241C2211yescharacteristic
38#16838747C2211yescharacteristic, complemented
39#16838747C2211yescharacteristic, complemented
40#16838747C2211yescharacteristic, complemented
41C4×Dic241C2211yescharacteristic, complemented
42C4×Dic241C2211yescharacteristic, complemented
43C4×Dic241C2211yescharacteristic, complemented
44C24×C241C2211yescharacteristic
45C22×C4×C241C2211yescharacteristic
46C22×C4×C241C2211yescharacteristic
47C22⋊C4×C241C2211yescharacteristic
48C22⋊C4×C241C2211yescharacteristic
49C22⋊C4×C241C2211yescharacteristic
50C22⋊C4×C241C2211yescharacteristic
51C22×D241C2311yescharacteristic
52Dic482—41no—
53C22⋊D241—22no—
54C22⋊D241—22no—
55C22⋊D241—41no—
56C2×Dic241—22no—
57C2×Dic241—41no—
58C2×Dic241C2311yescharacteristic
59C2×Dic241C2311yescharacteristic
60C2×Dic241C2311yescharacteristic
61C2×Dic241C2311yescharacteristic
62C2×Dic241C2311yescharacteristic
63C2×Dic241C2311yescharacteristic
64C2×Dic241C2311yescharacteristic
65C4×D241—41no—
66C23×C241—22no—
67C23×C241—22no—
68C23×C241—41no—
69C23×C241C2311yescharacteristic
70C23×C241C2311yescharacteristic
71C23×C241C2311yescharacteristic
72C2×C4×C241—22no—
73C2×C4×C241—41no—
74C2×C4×C241C2311yescharacteristic
75C2×C4×C241C2311yescharacteristic
76C2×C4×C241C2311yescharacteristic
77C2×C4×C241C2311yescharacteristic
78C22×C241—21no—
79C22×C241—21no—
80C22×C241—21no—
81C22×C241—21no—
82C22×C241—21no—
83C22×C241—21no—
84C22×C241—21no—
85C22×C241—44no—
86C22×C241#4112yes—
87C22×C241#4112yes—
88C22×C241C2411yescharacteristic
89C2×D241—21no—
90C2×D241—21no—
91C2×D241—21no—
92C4×C241—21no—
93C4×C241—21no—
94C4×C241—21no—
95C4×C241—21no—
96Dic241—21no—
97Dic241—21no—
98Dic241—21no—
99Dic241—21no—
100Dic241—21no—
101Dic241—21no—
102Dic241—21no—
103C2×C241—22no—
104C2×C241—22no—
105C2×C241—41no—
106C2×C241C2×#4111yescharacteristic
107C2×C241C2×#4111yescharacteristic
108C2×C241H(5,𝔽2)11yescharacteristic
109D241—41no—
110C241#55111yescharacteristic, complemented, Sylow
111#551—2411nomaximal, Sylow
112#126—2411no—
113#126—2411no—
114#124—2411no—
115#123—2411no—
116#123—2411no—
117#123—2411no—
118C22⋊Q8—2411no—
119C22⋊Q8—2411no—
120#120—2411no—
121C4×D4—2411no—
122C4×D4—2411no—
123#117—2411no—
124#117—2411no—
125C2×C22⋊C4—2411no—
126C2×C22⋊C4D24111yescharacteristic, complemented
127C24C2×D24111yescharacteristic
128C2×Q8—2411no—
129C2×D4—2411no—
130C2×D4—2411no—
131C2×D4—2411no—
132C22×C4—2411no—
133C22×C4—2411no—
134C22×C4—2411no—
135C22×C4C2×D24111yescharacteristic
136C22×C4C2×D24111yescharacteristic
137C4⋊C4—2411no—
138C4⋊C4—2411no—
139C4⋊C4—2411no—
140C4⋊C4—2411no—
141C4⋊C4—2411no—
142C4⋊C4—2411no—
143C4⋊C4—2411no—
144C4⋊C4—2411no—
145C22⋊C4—2411no—
146C22⋊C4—2411no—
147C22⋊C4—2411no—
148C22⋊C4—2411no—
149C22⋊C4—2411no—
150C22⋊C4—2411no—
151C22⋊C4—2411no—
152C22⋊C4—2411no—
153C22⋊C4—2411no—
154C22⋊C4—2411no—
155C22⋊C4C2×D24111yescharacteristic
156C22⋊C4C2×D24111yescharacteristic
157C22⋊C4C2×D24111yescharacteristic
158C22⋊C4C2×D24111yescharacteristic
159C42—2411no—
160C42—2411no—
161C42—2411no—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—2411no—
166C23C22×D24111yescharacteristic
167C23C22×D24111yescharacteristic
168C23C22×D24111yescharacteristic
169Q8—9641no—
170D4—9641no—
171D4—4822no—
172D4—4822no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—9641no—
176C2×C4—9641no—
177C2×C4—4822no—
178C2×C4—2411no—
179C2×C4—2411no—
180C2×C4—2411no—
181C2×C4—2411no—
182C2×C4—2411no—
183C2×C4—2411no—
184C2×C4—2411no—
185C2×C4C22×D24111yescharacteristic
186C2×C4C22×D24111yescharacteristic
187C2×C4C22×D24111yescharacteristic
188C2×C4C22×D24111yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—44no—
197C22—4821no—
198C22—4821no—
199C22—4821no—
200C22#1683876912yes—
201C22#1683877012yes—
202C22C23×D24111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—4821no—
208C4—4821no—
209C4—4821no—
210C4—4821no—
211C4—4821no—
212C4—4821no—
213C4—4821no—
214C2—22no—
215C2—22no—
216C2—41no—
217C2—9641no—
218C2C2×#1683876911yescharacteristic
219C2C2×#1683877011yescharacteristic
220C2#2144068711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
FittingC2×C22⋊C4×C241C2
prime radical#21440611C2
hypercenterC2×C22⋊C4D241
hypocenterC241#551
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #21440635C2semidirect
2. #21440635C2semidirect
3. #21440628C2semidirect
4. #21440626C2semidirect
5. #21440625C2semidirect
6. #21440624C2semidirect
7. #21440621C2semidirect
8. #21440619C2semidirect
10. C4×C22⋊D241C2semidirect
11. #21440602C2semidirect
12. #21440601C2semidirect
13. #21440598C2semidirect
14. C2×C22⋊Dic241C2semidirect
15. C2×C22⋊C4×C241C2semidirect
16. C2×Dic482C22semidirect
22. C2×C4×D241C22semidirect
23. C4⋊Dic241C22semidirect
24. C4⋊Dic241C22semidirect
25. C4⋊Dic241C22semidirect
26. #16838749C22semidirect
27. #16838749C22semidirect
28. #16838749C22semidirect
29. #16838749C22semidirect
30. #16838749C22semidirect
38. #16838747C22semidirect
39. #16838747C22semidirect
40. #16838747C22semidirect
41. C4×Dic241C22semidirect
42. C4×Dic241C22semidirect
43. C4×Dic241C22semidirect
110. C241#551semidirect
126. C2×C22⋊C4D241semidirect

Maximal subgroups

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

subgroupindex
#214406352
#214406282
#214406262
#214406252
#214406242
#214406212
#214406192
#214406112
C4×C22⋊D2412
#214406022
#214406012
#214405982
C2×C22⋊Dic2412
C2×C22⋊C4×C2412
#551241