Finite Group Databasebeta

← #29508521

Subgroups — #29508521

Order 15424.

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

normal subgroups only

This group has 268 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#21440688C211yesmaximal, characteristic, complemented
2#21440687C211yesmaximal, characteristic, complemented
3#21440670C211yesmaximal, characteristic, complemented
4#21440669C212yesmaximal, complemented
5#21440666C212yesmaximal, complemented
6#21440663C212yesmaximal, complemented
7#21440662C211yesmaximal, characteristic, complemented
8C2×#16838755C211yesmaximal, characteristic, complemented
9D8×D241C212yesmaximal, complemented
10#21440650C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C241C211yesmaximal, characteristic, complemented
12#16838771C2212yescomplemented
13#16838770—21no—
14#16838770C2211yescharacteristic
15#16838770C2211yescharacteristic, complemented
16#16838769—21no—
17#16838769C2212yescomplemented
18C2×D964—21no—
19C2×D964C2211yescharacteristic, complemented
20C2×C22⋊D241—22no—
21D4×D241—21no—
22D4×D241—22no—
23D4×D241C2212yescomplemented
24D4×D241C2212yescomplemented
25#16838760C2211yescharacteristic, complemented
26#16838759C2212yescomplemented
27#16838758C2211yescharacteristic, complemented
28#16838757C2211yescharacteristic, complemented
29D1928C2212yescomplemented
30#16838755C2211yescharacteristic, complemented
31#16838755C2212yescomplemented
32#16838755C2212yescomplemented
33#16838753C2212yescomplemented
34C2×#14515349C2211yescharacteristic, complemented
35C8×D241C2212yescomplemented
36#41×C241C2211yescharacteristic, complemented
37C2×D4×C241C2211yescharacteristic, complemented
38SD16×C241C2212yescomplemented
39D8×C241C2212yescomplemented
40M16×C241C2211yescharacteristic, complemented
41C22×D241—21no—
42C22×D241—22no—
43C22×D241—22no—
44Dic482D411yescharacteristic
45Dic482C2311yescharacteristic
46D964—22no—
47D964—21no—
48D964D411yescharacteristic
49D964C2312yes—
50D964C2311yescharacteristic
51C22⋊D241—21no—
52C22⋊D241—22no—
53C22⋊D241—22no—
54C22⋊D241—22no—
55C22⋊D241D412yes—
56C2×Dic241—22no—
57C4×D241—21no—
58C4×D241—22no—
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—22no—
77C2×D241—82no—
78C2×D241—82no—
79C2×D241—81no—
80C2×D241C2×D412yes—
81C4×C241—21no—
82C4×C241C2×D411yescharacteristic
83C4×C241C2411yescharacteristic
84Dic241—21no—
85Dic241C2×D412yes—
86C2×C241—21no—
87C2×C241—41no—
88C2×C241—42no—
89C2×C241C22×D411yescharacteristic
90D241—22no—
91D241—41no—
92D241—42no—
93C241#57611yescharacteristic, complemented, Sylow
94#576—2411nomaximal, Sylow
95H(5,𝔽2)—2411no—
96H(5,𝔽2)—2411no—
97AGL(1,ℤ/8)—2412no—
98AGL(1,ℤ/8)—2412no—
99AGL(1,ℤ/8)—2411no—
100AGL(1,ℤ/8)D24111yescharacteristic, complemented
101#135—2411no—
102#135—2412no—
103C2×D8—2411no—
104C2×D8—2412no—
105#131—2411no—
106#41—4821no—
107#41—4821no—
108#41—2412no—
109#41—2412no—
110#41—2411no—
111#41—2411no—
112#41C2×D24111yescharacteristic, complemented
113C2×D4—4821no—
114C2×D4—4822no—
115C2×D4—4821no—
116C2×D4—4822no—
117C2×D4—2412no—
118C2×D4—2412no—
119C2×D4—2411no—
120C2×D4C2×D24111yescharacteristic, complemented
121Q16—2411no—
122SD16—2411no—
123SD16—2411no—
124SD16—2412no—
125SD16C2×D24112yescomplemented
126D8—2412no—
127D8—2411no—
128D8—2412no—
129D8—2412no—
130D8C2×D24112yescomplemented
131M16—2412no—
132M16C2×D24111yescharacteristic, complemented
133C2×C8—2411no—
134C2×C8—2412no—
135C23—21no—
136C23—4822no—
137C23—4822no—
138C23—4821no—
139Q8—2411no—
140Q8—2411no—
141Q8C22×D24111yescharacteristic
142D4—21no—
143D4—21no—
144D4—4822no—
145D4—4822no—
146D4—4821no—
147D4—4822no—
148D4—4822no—
149D4—4821no—
150D4—2412no—
151D4—2411no—
152D4—2412no—
153D4—2411no—
154D4C22×D24112yes—
155D4C22×D24111yescharacteristic
156C2×C4—21no—
157C2×C4—4822no—
158C2×C4—4822no—
159C2×C4—4821no—
160C2×C4—2412no—
161C2×C4C22×D24111yescharacteristic
162C8—2411no—
163C8—2411no—
164C8C22×D24112yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—19281no—
169C22—19282no—
170C22—19282no—
171C22—4822no—
172C22—4821no—
173C22—2412no—
174C22D4×D24111yescharacteristic
175C4—21no—
176C4—4821no—
177C4—2412no—
178C4D4×D24111yescharacteristic
179C4C23×D24111yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—9642no—
184C2—9641no—
185C2—4822no—
186C2C2×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#576
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. #21440688C2semidirect
2. #21440687C2semidirect
3. #21440670C2semidirect
4. #21440669C2semidirect
5. #21440666C2semidirect
6. #21440663C2semidirect
7. #21440662C2semidirect
8. C2×#16838755C2semidirect
9. D8×D241C2semidirect
10. #21440650C2semidirect
11. AGL(1,ℤ/8)×C241C2semidirect
12. #16838771C22semidirect
15. #16838770C22semidirect
17. #16838769C22semidirect
19. C2×D964C22semidirect
23. D4×D241C22semidirect
24. D4×D241C22semidirect
25. #16838760C22semidirect
26. #16838759C22semidirect
27. #16838758C22semidirect
28. #16838757C22semidirect
29. D1928C22semidirect
30. #16838755C22semidirect
31. #16838755C22semidirect
32. #16838755C22semidirect
33. #16838753C22semidirect
34. C2×#14515349C22semidirect
35. C8×D241C22semidirect
36. #41×C241C22semidirect
37. C2×D4×C241C22semidirect
38. SD16×C241C22semidirect
39. D8×C241C22semidirect
40. M16×C241C22semidirect
93. C241#576semidirect
100. AGL(1,ℤ/8)D241semidirect
112. #41C2×D241semidirect
120. C2×D4C2×D241semidirect
125. SD16C2×D241semidirect
130. D8C2×D241semidirect
132. M16C2×D241semidirect

Maximal subgroups

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

subgroupindex
#214406882
#214406872
#214406702
#214406692
#214406662
#214406632
#214406622
C2×#168387552
D8×D2412
#214406502
AGL(1,ℤ/8)×C2412
#576241