Finite Group Databasebeta

← #29508523

Subgroups — #29508523

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#21440687C211yesmaximal, characteristic, complemented
3#21440673C211yesmaximal, characteristic, complemented
4#21440671C212yesmaximal, complemented
5#21440669C212yesmaximal, complemented
6#21440668C211yesmaximal, characteristic, complemented
7#21440664C211yesmaximal, characteristic, complemented
8#21440663C212yesmaximal, complemented
9SD16×D241C212yesmaximal, complemented
10#21440648C211yesmaximal, characteristic, complemented
11C2×SD16×C241C211yesmaximal, characteristic, complemented
12#16838771—21no—
13#16838771C2212yescomplemented
14#16838770—22no—
15#16838770C2211yescharacteristic
16#16838770C2211yescharacteristic
17#16838770C2211yescharacteristic, complemented
18#16838769—21no—
19#16838769C2212yes—
20Q8×D241—21no—
21Q8×D241C2212yescomplemented
22C2×C22⋊D241—22no—
23D4×D241—21no—
24D4×D241C2212yes—
25Dic964C2211yescharacteristic, complemented
26#16838760C2212yescomplemented
27#16838759C2211yescharacteristic, complemented
28#16838759C2211yescharacteristic, complemented
29#16838758C2212yescomplemented
30#16838757C2212yescomplemented
31D1928C2211yescharacteristic, complemented
32#16838755C2212yescomplemented
33#16838754C2211yescharacteristic, complemented
34#16838753C2212yescomplemented
35C8×D241C2212yescomplemented
36C2×Q8×C241C2211yescharacteristic, complemented
37C2×D4×C241C2211yescharacteristic
38SD16×C241C2212yescomplemented
39SD16×C241C2212yescomplemented
40C2×C8×C241C2211yescharacteristic, complemented
41C22×D241—22no—
42Dic482—22no—
43Dic482D411yescharacteristic
44Dic482C2311yescharacteristic
45Dic482C2311yescharacteristic
46D964—22no—
47D964D411yescharacteristic
48D964C2311yescharacteristic
49D964C2311yescharacteristic
50C22⋊D241—21no—
51C22⋊D241—21no—
52C22⋊D241—22no—
53C22⋊D241—22no—
54C22⋊D241D412yes—
55C2×Dic241—22no—
56C4×D241—22no—
57C4×D241—21no—
58C4×D241—21no—
59C4×D241—22no—
60C4×D241C2312yes—
61#14515349C2312yes—
62C23×C241—21no—
63Q8×C241—21no—
64Q8×C241C2312yes—
65D4×C241—21no—
66D4×C241C2312yes—
67C2×C4×C241—21no—
68C2×C4×C241C2311yescharacteristic
69C8×C241C2311yescharacteristic
70C8×C241C2311yescharacteristic
71C22×C241—22no—
72C22×C241—81no—
73C22×C241C2×D411yescharacteristic
74C2×D241—21no—
75C2×D241—21no—
76C2×D241—82no—
77C2×D241C2×D412yes—
78C4×C241—22no—
79C4×C241C2×D411yescharacteristic
80C4×C241C2411yescharacteristic
81Dic241—21no—
82Dic241—21no—
83Dic241C2×D412yes—
84C2×C241—21no—
85C2×C241—42no—
86C2×C241C22×D411yescharacteristic
87D241—22no—
88D241—41no—
89D241—41no—
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#135—2411no—
100C2×SD16—2412no—
101C2×SD16D24111yescharacteristic, complemented
102#131—2411no—
103#41—4822no—
104#41—4821no—
105#41—4821no—
106#41—2412no—
107#41—2412no—
108#41—2411no—
109#41—2411no—
110#41—2411no—
111C2×Q8—4821no—
112C2×Q8—2412no—
113C2×Q8C2×D24111yescharacteristic, complemented
114C2×D4—4821no—
115C2×D4—4822no—
116C2×D4—2412no—
117C2×D4C2×D24111yescharacteristic
118Q16—2412no—
119Q16—2411no—
120SD16—2412no—
121SD16—2412no—
122SD16—2411no—
123SD16—2411no—
124SD16C2×D24112yescomplemented
125SD16C2×D24112yescomplemented
126D8—2412no—
127D8—2411no—
128M16—2411no—
129M16—2412no—
130C2×C8—2412no—
131C2×C8C2×D24111yescharacteristic, complemented
132C23—21no—
133C23—4822no—
134Q8—21no—
135Q8—4822no—
136Q8—2411no—
137Q8—2411no—
138Q8—2411no—
139Q8C22×D24112yes—
140D4—21no—
141D4—4822no—
142D4—4822no—
143D4—4821no—
144D4—4821no—
145D4—4822no—
146D4—2412no—
147D4—2411no—
148D4—2411no—
149D4—2411no—
150D4C22×D24112yes—
151C2×C4—21no—
152C2×C4—4822no—
153C2×C4—4822no—
154C2×C4—4822no—
155C2×C4—4821no—
156C2×C4—4821no—
157C2×C4—2412no—
158C2×C4C22×D24111yescharacteristic
159C8—2412no—
160C8C22×D24111yescharacteristic
161C8C22×D24111yescharacteristic
162C22—22no—
163C22—81no—
164C22—19282no—
165C22—4821no—
166C22—4821no—
167C22—2412no—
168C22D4×D24111yescharacteristic
169C4—22no—
170C4—4821no—
171C4—4821no—
172C4—2412no—
173C4D4×D24111yescharacteristic
174C4C23×D24111yescharacteristic
175C2—21no—
176C2—42no—
177C2—9641no—
178C2—9641no—
179C2—4822no—
180C2C2×D4×D24111yescharacteristic
subgroupG/H
centerC2C2×D4×D241
commutator, cosocleC4×C241C24
FrattiniC4C23×D241
FittingC2×SD16×C241C2
socleC2×C241C22×D4
hypercenterC2×SD16D241
hypocenterC241#577
LCSG ▷ C4×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #21440690C2semidirect
2. #21440687C2semidirect
3. #21440673C2semidirect
4. #21440671C2semidirect
5. #21440669C2semidirect
6. #21440668C2semidirect
7. #21440664C2semidirect
8. #21440663C2semidirect
9. SD16×D241C2semidirect
10. #21440648C2semidirect
11. C2×SD16×C241C2semidirect
13. #16838771C22semidirect
17. #16838770C22semidirect
21. Q8×D241C22semidirect
25. Dic964C22semidirect
26. #16838760C22semidirect
27. #16838759C22semidirect
28. #16838759C22semidirect
29. #16838758C22semidirect
30. #16838757C22semidirect
31. D1928C22semidirect
32. #16838755C22semidirect
33. #16838754C22semidirect
34. #16838753C22semidirect
35. C8×D241C22semidirect
36. C2×Q8×C241C22semidirect
38. SD16×C241C22semidirect
39. SD16×C241C22semidirect
40. C2×C8×C241C22semidirect
90. C241#577semidirect
101. C2×SD16D241semidirect
113. C2×Q8C2×D241semidirect
124. SD16C2×D241semidirect
125. SD16C2×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
#214406902
#214406872
#214406732
#214406712
#214406692
#214406682
#214406642
#214406632
SD16×D2412
#214406482
C2×SD16×C2412
#577241