Finite Group Databasebeta

← #29508527

Subgroups — #29508527

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#21440687C211yesmaximal, characteristic, complemented
3#21440675C211yesmaximal, characteristic, complemented
4#21440671C212yesmaximal, complemented
5#21440666C212yesmaximal, complemented
6#21440662C211yesmaximal, characteristic, complemented
7#21440661C212yesmaximal, complemented
8C2×#16838758C211yesmaximal, characteristic, complemented
9SD16×D241C212yesmaximal, complemented
10#21440650C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C241C211yesmaximal, characteristic, complemented
12#16838770—21no—
13#16838770C2211yescharacteristic
14#16838770C2211yescharacteristic
15#16838769—21no—
16#16838769—22no—
17#16838769C2212yes—
18#16838769C2212yescomplemented
19C2×Dic482—21no—
20C2×Dic482C2211yescharacteristic, complemented
21Q8×D241C2212yescomplemented
22C2×C22⋊D241—22no—
23D4×D241—21no—
24D4×D241C2212yes—
25Dic964C2212yescomplemented
26#16838760C2211yescharacteristic, complemented
27#16838759C2212yescomplemented
28#16838758C2212yescomplemented
29#16838758C2212yescomplemented
30#16838758C2211yescharacteristic, complemented
31#16838757C2211yescharacteristic, complemented
32#16838755C2211yescharacteristic, complemented
33#16838753C2212yescomplemented
34C2×#14515349C2211yescharacteristic, complemented
35C8×D241C2212yescomplemented
36#41×C241C2211yescharacteristic, complemented
37C2×D4×C241C2211yescharacteristic
38SD16×C241C2212yescomplemented
39D8×C241C2212yescomplemented
40M16×C241C2211yescharacteristic, complemented
41C22×D241—22no—
42Dic482—21no—
43Dic482—22no—
44Dic482D411yescharacteristic
45Dic482C2312yes—
46Dic482C2311yescharacteristic
47D964D411yescharacteristic
48D964C2311yescharacteristic
49C22⋊D241—21no—
50C22⋊D241—22no—
51C22⋊D241—22no—
52C22⋊D241—22no—
53C22⋊D241D412yes—
54C2×Dic241—21no—
55C2×Dic241—22no—
56C2×Dic241—22no—
57C4×D241—22no—
58C4×D241—21no—
59C4×D241C2312yes—
60#14515349C2311yescharacteristic
61#14515349C2311yescharacteristic
62C23×C241—21no—
63Q8×C241C2311yescharacteristic
64D4×C241—21no—
65D4×C241—21no—
66D4×C241C2311yescharacteristic
67D4×C241C2312yes—
68C2×C4×C241—21no—
69C2×C4×C241C2311yescharacteristic
70C8×C241C2312yes—
71C22×C241—21no—
72C22×C241—22no—
73C22×C241—81no—
74C22×C241C2×D411yescharacteristic
75C2×D241—21no—
76C2×D241—82no—
77C2×D241C2×D412yes—
78C4×C241—21no—
79C4×C241C2×D411yescharacteristic
80C4×C241C2411yescharacteristic
81Dic241—21no—
82Dic241—22no—
83Dic241C2×D412yes—
84C2×C241—21no—
85C2×C241—42no—
86C2×C241—41no—
87C2×C241C22×D411yescharacteristic
88D241—22no—
89D241—41no—
90C241#57711yescharacteristic, complemented, Sylow
91#577—2411nomaximal, Sylow
922−1+4—2411no—
93H(5,𝔽2)—2411no—
94#137—2412no—
95#137—2411no—
96AGL(1,ℤ/8)—2412no—
97AGL(1,ℤ/8)D24111yescharacteristic, complemented
98#135—2411no—
99#135—2412no—
100C2×SD16—2411no—
101C2×SD16—2412no—
102#131—2411no—
103#41—4821no—
104#41—4821no—
105#41—4822no—
106#41—2412no—
107#41—2412no—
108#41—2411no—
109#41—2411no—
110#41C2×D24111yescharacteristic, complemented
111C2×Q8—4821no—
112C2×Q8—2412no—
113C2×Q8—2411no—
114C2×D4—4821no—
115C2×D4—4822no—
116C2×D4—2412no—
117C2×D4C2×D24111yescharacteristic
118Q16—2411no—
119Q16—2412no—
120SD16—2411no—
121SD16—2412no—
122SD16—2412no—
123SD16—2411no—
124SD16—2412no—
125SD16C2×D24112yescomplemented
126D8—2411no—
127D8C2×D24112yescomplemented
128M16—2412no—
129M16C2×D24111yescharacteristic, complemented
130C2×C8—2411no—
131C2×C8—2412no—
132C23—21no—
133C23—4822no—
134Q8—4822no—
135Q8—4821no—
136Q8—2412no—
137Q8—2411no—
138Q8—2411no—
139Q8C22×D24111yescharacteristic
140D4—21no—
141D4—21no—
142D4—4822no—
143D4—4822no—
144D4—4821no—
145D4—4822no—
146D4—2412no—
147D4—2411no—
148D4—2411no—
149D4C22×D24111yescharacteristic
150D4C22×D24112yes—
151C2×C4—21no—
152C2×C4—4822no—
153C2×C4—4821no—
154C2×C4—4822no—
155C2×C4—4822no—
156C2×C4—4821no—
157C2×C4—2412no—
158C2×C4C22×D24111yescharacteristic
159C8—2411no—
160C8—2411no—
161C8C22×D24112yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—19282no—
166C22—4821no—
167C22—2412no—
168C22D4×D24111yescharacteristic
169C4—21no—
170C4—4821no—
171C4—4822no—
172C4—2412no—
173C4D4×D24111yescharacteristic
174C4C23×D24111yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—9641no—
179C2—4822no—
180C2C2×D4×D24111yescharacteristic
subgroupG/H
centerC2C2×D4×D241
commutator, cosocleC4×C241C24
FrattiniC4C23×D241
FittingAGL(1,ℤ/8)×C241C2
socleC2×C241C22×D4
hypercenterAGL(1,ℤ/8)D241
hypocenterC241#577
LCSG ▷ C4×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #21440689C2semidirect
2. #21440687C2semidirect
3. #21440675C2semidirect
4. #21440671C2semidirect
5. #21440666C2semidirect
6. #21440662C2semidirect
7. #21440661C2semidirect
8. C2×#16838758C2semidirect
9. SD16×D241C2semidirect
10. #21440650C2semidirect
11. AGL(1,ℤ/8)×C241C2semidirect
18. #16838769C22semidirect
20. C2×Dic482C22semidirect
21. Q8×D241C22semidirect
25. Dic964C22semidirect
26. #16838760C22semidirect
27. #16838759C22semidirect
28. #16838758C22semidirect
29. #16838758C22semidirect
30. #16838758C22semidirect
31. #16838757C22semidirect
32. #16838755C22semidirect
33. #16838753C22semidirect
34. C2×#14515349C22semidirect
35. C8×D241C22semidirect
36. #41×C241C22semidirect
38. SD16×C241C22semidirect
39. D8×C241C22semidirect
40. M16×C241C22semidirect
90. C241#577semidirect
97. AGL(1,ℤ/8)D241semidirect
110. #41C2×D241semidirect
125. SD16C2×D241semidirect
127. D8C2×D241semidirect
129. M16C2×D241semidirect

Maximal subgroups

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

subgroupindex
#214406892
#214406872
#214406752
#214406712
#214406662
#214406622
#214406612
C2×#168387582
SD16×D2412
#214406502
AGL(1,ℤ/8)×C2412
#577241