Finite Group Databasebeta

← #29508524

Subgroups — #29508524

Order 15424.

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

normal subgroups only

This group has 258 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#21440689C211yesmaximal, characteristic, complemented
2#21440688C211yesmaximal, characteristic, complemented
3#21440674C212yesmaximal, complemented
4#21440672C211yesmaximal, characteristic, complemented
5#21440667C211yesmaximal, characteristic, complemented
6#21440666C212yesmaximal, complemented
7#21440663C212yesmaximal, complemented
8C2×#16838759C211yesmaximal, characteristic, complemented
9SD16×D241C212yesmaximal, complemented
10#21440648C211yesmaximal, characteristic, complemented
11#135×C241C211yesmaximal, characteristic, complemented
12#16838771C2212yes—
13#16838770—21no—
14#16838770—21no—
15#16838770C2211yescharacteristic
16#16838769—22no—
17#16838769C2212yescomplemented
18C2×Dic482—21no—
19C2×Dic482C2211yescharacteristic, complemented
20Q8×D241C2212yescomplemented
21C2×D964—21no—
22C2×D964C2211yescharacteristic
23D4×D241—22no—
24D4×D241C2212yes—
25#16838760C2212yescomplemented
26#16838759C2212yescomplemented
27#16838759C2212yescomplemented
28#16838758C2212yescomplemented
29#16838757C2212yescomplemented
30#16838755C2212yescomplemented
31#16838754C2211yescharacteristic, complemented
32#16838753C2212yescomplemented
33C8×D241C2212yescomplemented
34#41×C241C2211yescharacteristic
35#41×C241C2211yescharacteristic, complemented
36Q16×C241C2211yescharacteristic, complemented
37SD16×C241C2211yescharacteristic, complemented
38SD16×C241C2211yescharacteristic, complemented
39D8×C241C2211yescharacteristic, complemented
40C2×C8×C241C2211yescharacteristic, complemented
41C22×D241—21no—
42C22×D241—22no—
43Dic482—21no—
44Dic482—22no—
45Dic482D411yescharacteristic
46Dic482C2312yes—
47D964—22no—
48D964—21no—
49D964D411yescharacteristic
50D964C2312yes—
51C22⋊D241—22no—
52C22⋊D241—22no—
53C22⋊D241D412yes—
54C2×Dic241—21no—
55C2×Dic241—22no—
56C4×D241—22no—
57C4×D241—22no—
58C4×D241C2312yes—
59#14515349C2312yes—
60Q8×C241C2311yescharacteristic
61Q8×C241C2311yescharacteristic
62D4×C241—21no—
63D4×C241—21no—
64D4×C241C2311yescharacteristic
65D4×C241C2311yescharacteristic
66C2×C4×C241—21no—
67C2×C4×C241—21no—
68C2×C4×C241C2311yescharacteristic
69C8×C241C2311yescharacteristic
70C8×C241C2311yescharacteristic
71C22×C241—21no—
72C22×C241—21no—
73C22×C241C2×D411yescharacteristic
74C2×D241—22no—
75C2×D241—82no—
76C2×D241—81no—
77C2×D241C2×D412yes—
78C4×C241—21no—
79C4×C241—21no—
80C4×C241C2×D411yescharacteristic
81C4×C241C2411yescharacteristic
82Dic241—22no—
83Dic241C2×D412yes—
84C2×C241—21no—
85C2×C241—41no—
86C2×C241—41no—
87C2×C241C22×D411yescharacteristic
88D241—22no—
89D241—42no—
90C241#57711yescharacteristic, complemented, Sylow
91#577—2411nomaximal, Sylow
922−1+4—2411no—
93H(5,𝔽2)—2411no—
94#137—2411no—
95#137—2412no—
96AGL(1,ℤ/8)—2411no—
97AGL(1,ℤ/8)—2412no—
98#135—2412no—
99#135D24111yescharacteristic, complemented
100C2×SD16—2412no—
101C2×SD16—2411no—
102#131—2411no—
103#41—4821no—
104#41—4821no—
105#41—4822no—
106#41—2412no—
107#41—2412no—
108#41—2411no—
109#41C2×D24111yescharacteristic
110#41C2×D24111yescharacteristic, complemented
111C2×Q8—4821no—
112C2×Q8—2412no—
113C2×Q8—2411no—
114C2×D4—4821no—
115C2×D4—4822no—
116C2×D4—2412no—
117C2×D4—2411no—
118Q16—2412no—
119Q16C2×D24111yescharacteristic, complemented
120SD16—2412no—
121SD16—2412no—
122SD16—2412no—
123SD16—2412no—
124SD16C2×D24111yescharacteristic, complemented
125SD16C2×D24111yescharacteristic, complemented
126D8—2412no—
127D8C2×D24111yescharacteristic, complemented
128M16—2411no—
129M16—2412no—
130C2×C8—2412no—
131C2×C8C2×D24111yescharacteristic, complemented
132C23—4822no—
133C23—4821no—
134Q8—4822no—
135Q8—4821no—
136Q8—2412no—
137Q8—2411no—
138Q8C22×D24111yescharacteristic
139Q8C22×D24111yescharacteristic
140D4—21no—
141D4—21no—
142D4—4822no—
143D4—4822no—
144D4—4822no—
145D4—4821no—
146D4—2412no—
147D4—2411no—
148D4—2412no—
149D4C22×D24111yescharacteristic
150D4C22×D24111yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—4822no—
154C2×C4—4822no—
155C2×C4—4821no—
156C2×C4—4822no—
157C2×C4—2412no—
158C2×C4C22×D24111yescharacteristic
159C8—2412no—
160C8C22×D24111yescharacteristic
161C8C22×D24111yescharacteristic
162C22—21no—
163C22—21no—
164C22—19281no—
165C22—19282no—
166C22—4822no—
167C22—2412no—
168C22D4×D24111yescharacteristic
169C4—21no—
170C4—21no—
171C4—4822no—
172C4—2412no—
173C4D4×D24111yescharacteristic
174C4C23×D24111yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—9642no—
179C2—4822no—
180C2C2×D4×D24111yescharacteristic
subgroupG/H
centerC2C2×D4×D241
commutator, cosocleC4×C241C24
FrattiniC4C23×D241
Fitting#135×C241C2
socleC2×C241C22×D4
hypercenter#135D241
hypocenterC241#577
LCSG ▷ C4×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #21440689C2semidirect
2. #21440688C2semidirect
3. #21440674C2semidirect
4. #21440672C2semidirect
5. #21440667C2semidirect
6. #21440666C2semidirect
7. #21440663C2semidirect
8. C2×#16838759C2semidirect
9. SD16×D241C2semidirect
10. #21440648C2semidirect
11. #135×C241C2semidirect
17. #16838769C22semidirect
19. C2×Dic482C22semidirect
20. Q8×D241C22semidirect
25. #16838760C22semidirect
26. #16838759C22semidirect
27. #16838759C22semidirect
28. #16838758C22semidirect
29. #16838757C22semidirect
30. #16838755C22semidirect
31. #16838754C22semidirect
32. #16838753C22semidirect
33. C8×D241C22semidirect
35. #41×C241C22semidirect
36. Q16×C241C22semidirect
37. SD16×C241C22semidirect
38. SD16×C241C22semidirect
39. D8×C241C22semidirect
40. C2×C8×C241C22semidirect
90. C241#577semidirect
99. #135D241semidirect
110. #41C2×D241semidirect
119. Q16C2×D241semidirect
124. SD16C2×D241semidirect
125. SD16C2×D241semidirect
127. D8C2×D241semidirect
131. C2×C8C2×D241semidirect

Maximal subgroups

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

subgroupindex
#214406892
#214406882
#214406742
#214406722
#214406672
#214406662
#214406632
C2×#168387592
SD16×D2412
#214406482
#135×C2412
#577241