Finite Group Databasebeta

← #29508526

Subgroups — #29508526

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#21440690C211yesmaximal, characteristic, complemented
2#21440688C211yesmaximal, characteristic, complemented
3#21440674C212yesmaximal, complemented
4#21440670C211yesmaximal, characteristic, complemented
5#21440669C212yesmaximal, complemented
6#21440665C212yesmaximal, complemented
7#21440662C211yesmaximal, characteristic, complemented
8C2×#16838757C211yesmaximal, characteristic, complemented
9SD16×D241C212yesmaximal, complemented
10#21440650C211yesmaximal, characteristic, complemented
11#137×C241C211yesmaximal, characteristic, complemented
12#16838771—21no—
13#16838771C2212yes—
14#16838771C2212yescomplemented
15#16838770—21no—
16#16838770—22no—
17#16838770C2211yescharacteristic
18#16838770C2211yescharacteristic, complemented
19Q8×D241—21no—
20Q8×D241C2212yescomplemented
21C2×D964—21no—
22C2×D964C2211yescharacteristic
23D4×D241—22no—
24D4×D241C2212yes—
25#16838760C2211yescharacteristic, complemented
26#16838759C2212yescomplemented
27#16838758C2211yescharacteristic, complemented
28#16838757C2212yescomplemented
29#16838757C2212yescomplemented
30#16838757C2211yescharacteristic, complemented
31D1928C2212yescomplemented
32#16838755C2211yescharacteristic, complemented
33#16838753C2212yescomplemented
34C2×#14515349C2211yescharacteristic, complemented
35C8×D241C2212yescomplemented
36#41×C241C2211yescharacteristic
37C2×Q8×C241C2211yescharacteristic, complemented
38Q16×C241C2212yescomplemented
39SD16×C241C2212yescomplemented
40M16×C241C2211yescharacteristic, complemented
41C22×D241—21no—
42C22×D241—22no—
43Dic482—22no—
44Dic482D411yescharacteristic
45Dic482C2311yescharacteristic
46D964—22no—
47D964—22no—
48D964—21no—
49D964D411yescharacteristic
50D964C2311yescharacteristic
51D964C2312yes—
52C22⋊D241—21no—
53C22⋊D241—22no—
54C22⋊D241D412yes—
55C4×D241—22no—
56C4×D241—21no—
57C4×D241—22no—
58C4×D241—22no—
59C4×D241C2312yes—
60#14515349C2311yescharacteristic
61#14515349C2311yescharacteristic
62Q8×C241—21no—
63Q8×C241C2312yes—
64Q8×C241C2311yescharacteristic
65D4×C241—21no—
66D4×C241C2311yescharacteristic
67C2×C4×C241—21no—
68C2×C4×C241—21no—
69C2×C4×C241C2311yescharacteristic
70C8×C241C2312yes—
71C22×C241—21no—
72C22×C241C2×D411yescharacteristic
73C2×D241—21no—
74C2×D241—22no—
75C2×D241—82no—
76C2×D241—81no—
77C2×D241C2×D412yes—
78C4×C241—21no—
79C4×C241—22no—
80C4×C241C2×D411yescharacteristic
81C4×C241C2411yescharacteristic
82Dic241—21no—
83Dic241C2×D412yes—
84C2×C241—21no—
85C2×C241—41no—
86C2×C241C22×D411yescharacteristic
87D241—22no—
88D241—41no—
89D241—42no—
90C241#57711yescharacteristic, complemented, Sylow
91#577—2411nomaximal, Sylow
922−1+4—2411no—
93H(5,𝔽2)—2411no—
94#137—2412no—
95#137D24111yescharacteristic, complemented
96AGL(1,ℤ/8)—2412no—
97AGL(1,ℤ/8)—2411no—
98#135—2411no—
99#135—2412no—
100C2×SD16—2411no—
101C2×SD16—2412no—
102#131—2411no—
103#41—4821no—
104#41—4822no—
105#41—4821no—
106#41—2412no—
107#41—2412no—
108#41—2411no—
109#41—2411no—
110#41C2×D24111yescharacteristic
111C2×Q8—4821no—
112C2×Q8—2412no—
113C2×Q8C2×D24111yescharacteristic, complemented
114C2×D4—4821no—
115C2×D4—4822no—
116C2×D4—2412no—
117C2×D4—2411no—
118Q16—2411no—
119Q16C2×D24112yescomplemented
120SD16—2412no—
121SD16—2412no—
122SD16—2411no—
123SD16—2411no—
124SD16—2412no—
125SD16C2×D24112yescomplemented
126D8—2411no—
127D8—2412no—
128M16—2412no—
129M16C2×D24111yescharacteristic, complemented
130C2×C8—2411no—
131C2×C8—2412no—
132C23—4822no—
133C23—4821no—
134Q8—21no—
135Q8—4822no—
136Q8—2411no—
137Q8—2411no—
138Q8C22×D24112yes—
139Q8C22×D24111yescharacteristic
140D4—21no—
141D4—4822no—
142D4—4822no—
143D4—4821no—
144D4—4822no—
145D4—4821no—
146D4—2412no—
147D4—2411no—
148D4—2411no—
149D4—2412no—
150D4C22×D24111yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—4822no—
154C2×C4—4822no—
155C2×C4—4822no—
156C2×C4—4821no—
157C2×C4—2412no—
158C2×C4C22×D24111yescharacteristic
159C8—2411no—
160C8—2411no—
161C8C22×D24112yes—
162C22—21no—
163C22—19281no—
164C22—19282no—
165C22—4821no—
166C22—4822no—
167C22—2412no—
168C22D4×D24111yescharacteristic
169C4—21no—
170C4—22no—
171C4—4821no—
172C4—2412no—
173C4D4×D24111yescharacteristic
174C4C23×D24111yescharacteristic
175C2—21no—
176C2—41no—
177C2—9642no—
178C2—9641no—
179C2—4822no—
180C2C2×D4×D24111yescharacteristic
subgroupG/H
centerC2C2×D4×D241
commutator, cosocleC4×C241C24
FrattiniC4C23×D241
Fitting#137×C241C2
socleC2×C241C22×D4
hypercenter#137D241
hypocenterC241#577
LCSG ▷ C4×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #21440690C2semidirect
2. #21440688C2semidirect
3. #21440674C2semidirect
4. #21440670C2semidirect
5. #21440669C2semidirect
6. #21440665C2semidirect
7. #21440662C2semidirect
8. C2×#16838757C2semidirect
9. SD16×D241C2semidirect
10. #21440650C2semidirect
11. #137×C241C2semidirect
14. #16838771C22semidirect
18. #16838770C22semidirect
20. Q8×D241C22semidirect
25. #16838760C22semidirect
26. #16838759C22semidirect
27. #16838758C22semidirect
28. #16838757C22semidirect
29. #16838757C22semidirect
30. #16838757C22semidirect
31. D1928C22semidirect
32. #16838755C22semidirect
33. #16838753C22semidirect
34. C2×#14515349C22semidirect
35. C8×D241C22semidirect
37. C2×Q8×C241C22semidirect
38. Q16×C241C22semidirect
39. SD16×C241C22semidirect
40. M16×C241C22semidirect
90. C241#577semidirect
95. #137D241semidirect
113. C2×Q8C2×D241semidirect
119. Q16C2×D241semidirect
125. SD16C2×D241semidirect
129. M16C2×D241semidirect

Maximal subgroups

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

subgroupindex
#214406902
#214406882
#214406742
#214406702
#214406692
#214406652
#214406622
C2×#168387572
SD16×D2412
#214406502
#137×C2412
#577241