Finite Group Databasebeta

← #29508394

Subgroups — #29508394

Order 15424.

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#21440637C211yesmaximal, characteristic, complemented
2#21440637C211yesmaximal, characteristic, complemented
3#21440631C211yesmaximal, characteristic, complemented
4#21440627C211yesmaximal, characteristic, complemented
5#21440627C211yesmaximal, characteristic, complemented
6#21440626C211yesmaximal, characteristic, complemented
7#21440618C211yesmaximal, characteristic, complemented
8#21440612C211yesmaximal, characteristic
9#21440604C211yesmaximal, characteristic, complemented
10C4×C22⋊D241C211yesmaximal, characteristic, complemented
11#21440600C211yesmaximal, characteristic, complemented
12#21440598C211yesmaximal, characteristic, complemented
13C2×#16838747C211yesmaximal, characteristic, complemented
14C22⋊C4×D241C211yesmaximal, characteristic, complemented
15C22⋊Q8×C241C211yesmaximal, characteristic, complemented
16C23×D241C2211yescharacteristic
17C2×D964C2211yescharacteristic
18C2×D964C2211yescharacteristic
19C2×C22⋊D241C2211yescharacteristic
20C22×Dic241C2211yescharacteristic
21C2×C4×D241C2211yescharacteristic
22C2×C4×D241C2211yescharacteristic
23C2×C4×D241C2211yescharacteristic, complemented
24C4⋊Dic241C2211yescharacteristic, complemented
25C4⋊Dic241C2211yescharacteristic, complemented
26#16838749C2211yescharacteristic, complemented
27#16838749C2211yescharacteristic, complemented
28#16838749C2211yescharacteristic, complemented
29C22⋊Dic241C2211yescharacteristic
30#16838747C2211yescharacteristic
31#16838747C2211yescharacteristic
32#16838747C2211yescharacteristic
33#16838747C2211yescharacteristic
34#16838747C2211yescharacteristic
35#16838747C2211yescharacteristic
36#16838747C2211yescharacteristic
37#16838747C2211yescharacteristic
38#16838747C2211yescharacteristic, complemented
39#16838747C2211yescharacteristic, complemented
40#16838747C2211yescharacteristic, complemented
41C4×Dic241C2211yescharacteristic, complemented
42C4×Dic241C2211yescharacteristic, complemented
43C4×Dic241C2211yescharacteristic, complemented
44C2×Q8×C241C2211yescharacteristic, complemented
45C22×C4×C241C2211yescharacteristic
46C4⋊C4×C241C2211yescharacteristic, complemented
47C4⋊C4×C241C2211yescharacteristic, complemented
48C4⋊C4×C241C2211yescharacteristic, complemented
49C22⋊C4×C241C2211yescharacteristic
50C22⋊C4×C241C2211yescharacteristic
51C22×D241—22no—
52C22×D241—22no—
53C22×D241—41no—
54C22×D241C2311yescharacteristic
55C22×D241C2311yescharacteristic
56C22×D241C2311yescharacteristic
57D964—22no—
58D964—41no—
59C22⋊D241—22no—
60C2×Dic241—22no—
61C2×Dic241C2311yescharacteristic
62C2×Dic241C2311yescharacteristic
63C2×Dic241C2311yescharacteristic
64C2×Dic241C2311yescharacteristic
65C2×Dic241C2311yescharacteristic
66C4×D241—22no—
67C4×D241—41no—
68C4×D241—41no—
69C23×C241C2311yescharacteristic
70Q8×C241—41no—
71C2×C4×C241—41no—
72C2×C4×C241C2311yescharacteristic
73C2×C4×C241C2311yescharacteristic
74C2×C4×C241C2311yescharacteristic
75C2×C4×C241C2311yescharacteristic
76C2×C4×C241C2311yescharacteristic
77C2×C4×C241C2311yescharacteristic
78C22×C241—21no—
79C22×C241—21no—
80C22×C241#4112yes—
81C22×C241C2411yescharacteristic
82C2×D241—21no—
83C2×D241—21no—
84C2×D241—21no—
85C2×D241—21no—
86C2×D241—21no—
87C2×D241—21no—
88C2×D241—21no—
89C2×D241—21no—
90C2×D241—44no—
91C2×D241#4112yes—
92C4×C241—21no—
93C4×C241—21no—
94C4×C241—21no—
95C4×C241—21no—
96C4×C241—21no—
97C4×C241—21no—
98Dic241—21no—
99Dic241—21no—
100Dic241—21no—
101Dic241—21no—
102Dic241—21no—
103C2×C241—22no—
104C2×C241C2×#4111yescharacteristic
105C2×C241C2×#4111yescharacteristic
106C2×C241H(5,𝔽2)11yescharacteristic
107D241—22no—
108D241—41no—
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⋊Q8D24111yescharacteristic, complemented
120#120—2411no—
121C4×D4—2411no—
122C4×D4—2411no—
123#117—2411no—
124#117—2411no—
125C2×C22⋊C4—2411no—
126C2×C22⋊C4—2411no—
127C24—2411no—
128C2×Q8C2×D24111yescharacteristic, complemented
129C2×D4—2411no—
130C2×D4—2411no—
131C2×D4—2411no—
132C22×C4—2411no—
133C22×C4—2411no—
134C22×C4—2411no—
135C22×C4—2411no—
136C22×C4C2×D24111yescharacteristic
137C4⋊C4—2411no—
138C4⋊C4—2411no—
139C4⋊C4—2411no—
140C4⋊C4—2411no—
141C4⋊C4—2411no—
142C4⋊C4C2×D24111yescharacteristic, complemented
143C4⋊C4C2×D24111yescharacteristic, complemented
144C4⋊C4C2×D24111yescharacteristic, complemented
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⋊C4—2411no—
156C22⋊C4—2411no—
157C22⋊C4C2×D24111yescharacteristic
158C22⋊C4C2×D24111yescharacteristic
159C42—2411no—
160C42—2411no—
161C42—2411no—
162C23—9641no—
163C23—4822no—
164C23—4822no—
165C23—2411no—
166C23—2411no—
167C23—2411no—
168C23C22×D24111yescharacteristic
169Q8—41no—
170D4—9641no—
171D4—4822no—
172D4—4822no—
173C2×C4—41no—
174C2×C4—9641no—
175C2×C4—9641no—
176C2×C4—4822no—
177C2×C4—4822no—
178C2×C4—2411no—
179C2×C4—2411no—
180C2×C4—2411no—
181C2×C4—2411no—
182C2×C4—2411no—
183C2×C4C22×D24111yescharacteristic
184C2×C4C22×D24111yescharacteristic
185C2×C4C22×D24111yescharacteristic
186C2×C4C22×D24111yescharacteristic
187C2×C4C22×D24111yescharacteristic
188C2×C4C22×D24111yescharacteristic
189C22—21no—
190C22—21no—
191C22—9644no—
192C22—4821no—
193C22—4821no—
194C22—4821no—
195C22—4821no—
196C22—4821no—
197C22—4821no—
198C22—4821no—
199C22—4821no—
200C22—2412no—
201C22#1683877112yes—
202C22C23×D24111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—4821no—
210C4—4821no—
211C4—4821no—
212C4—4821no—
213C4—4821no—
214C2—22no—
215C2—9641no—
216C2—9641no—
217C2—4822no—
218C2#41×D24111yescharacteristic
219C2C2×#1683877111yescharacteristic
220C2#2144068711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
FittingC22⋊Q8×C241C2
prime radical#21440612C2
hypercenterC22⋊Q8D241
hypocenterC241#551
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21440637C2semidirect
2. #21440637C2semidirect
3. #21440631C2semidirect
4. #21440627C2semidirect
5. #21440627C2semidirect
6. #21440626C2semidirect
7. #21440618C2semidirect
9. #21440604C2semidirect
10. C4×C22⋊D241C2semidirect
11. #21440600C2semidirect
12. #21440598C2semidirect
13. C2×#16838747C2semidirect
14. C22⋊C4×D241C2semidirect
15. C22⋊Q8×C241C2semidirect
23. C2×C4×D241C22semidirect
24. C4⋊Dic241C22semidirect
25. C4⋊Dic241C22semidirect
26. #16838749C22semidirect
27. #16838749C22semidirect
28. #16838749C22semidirect
38. #16838747C22semidirect
39. #16838747C22semidirect
40. #16838747C22semidirect
41. C4×Dic241C22semidirect
42. C4×Dic241C22semidirect
43. C4×Dic241C22semidirect
44. C2×Q8×C241C22semidirect
46. C4⋊C4×C241C22semidirect
47. C4⋊C4×C241C22semidirect
48. C4⋊C4×C241C22semidirect
110. C241#551semidirect
119. C22⋊Q8D241semidirect
128. C2×Q8C2×D241semidirect
142. C4⋊C4C2×D241semidirect
143. C4⋊C4C2×D241semidirect
144. C4⋊C4C2×D241semidirect

Maximal subgroups

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

subgroupindex
#214406372
#214406312
#214406272
#214406262
#214406182
#214406122
#214406042
C4×C22⋊D2412
#214406002
#214405982
C2×#168387472
C22⋊C4×D2412
C22⋊Q8×C2412
#551241