Finite Group Databasebeta

← #29508516

Subgroups — #29508516

Order 15424.

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

normal subgroups only

This group has 262 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
1C2×#16838771C211yesmaximal, characteristic, complemented
2#41×D241C211yesmaximal, characteristic, complemented
3#21440674C212yesmaximal, complemented
4#21440673C211yesmaximal, characteristic, complemented
5#21440670C211yesmaximal, characteristic, complemented
6#21440669C212yesmaximal, complemented
7#21440667C211yesmaximal, characteristic, complemented
8#21440665C212yesmaximal, complemented
9#21440663C212yesmaximal, complemented
10M16×D241C211yesmaximal, characteristic, complemented
11#137×C241C211yesmaximal, characteristic, complemented
12#16838771—21no—
13#16838771—21no—
14#16838771C2212yes—
15#16838771C2212yescomplemented
16#16838771C2211yescharacteristic, complemented
17#16838770—21no—
18#16838770C2211yescharacteristic
19#16838769—21no—
20#16838769C2211yescharacteristic, complemented
21Q8×D241C2211yescharacteristic, complemented
22C2×D964—21no—
23C2×D964C2211yescharacteristic, complemented
24D4×D241—21no—
25D4×D241C2211yescharacteristic, complemented
26C2×C4×D241—21no—
27C2×C4×D241—21no—
28C2×C4×D241C2211yescharacteristic
29#16838760C2212yescomplemented
30#16838759C2212yescomplemented
31#16838757C2212yescomplemented
32#16838757C2212yescomplemented
33D1928C2212yescomplemented
34#16838755C2212yescomplemented
35#16838754C2211yescharacteristic, complemented
36#16838753C2212yescomplemented
37C8×D241C2212yescomplemented
38#41×C241C2211yescharacteristic
39C2×Q8×C241C2211yescharacteristic, complemented
40Q16×C241C2212yescomplemented
41SD16×C241C2212yescomplemented
42M16×C241C2211yescharacteristic, complemented
43C22×D241—21no—
44C22×D241—21no—
45C22×D241D411yescharacteristic
46Dic482—21no—
47Dic482C2311yescharacteristic
48D964—21no—
49D964—22no—
50D964—22no—
51D964—21no—
52D964C2312yes—
53D964C2311yescharacteristic
54C22⋊D241—21no—
55C22⋊D241—21no—
56C22⋊D241—21no—
57C2×Dic241—21no—
58C2×Dic241D411yescharacteristic
59C4×D241—21no—
60C4×D241—21no—
61C4×D241—21no—
62C4×D241—22no—
63C4×D241—22no—
64C4×D241D411yescharacteristic
65C4×D241D411yescharacteristic
66C4×D241C2311yescharacteristic
67C4×D241C2311yescharacteristic
68#14515349C2312yes—
69Q8×C241—21no—
70Q8×C241C2312yes—
71Q8×C241C2311yescharacteristic
72D4×C241—21no—
73D4×C241C2311yescharacteristic
74C2×C4×C241—21no—
75C2×C4×C241—21no—
76C2×C4×C241C2311yescharacteristic
77C8×C241C2312yes—
78C22×C241—21no—
79C22×C241C2×D411yescharacteristic
80C2×D241—21no—
81C2×D241—22no—
82C2×D241—41no—
83C2×D241—42no—
84C2×D241—42no—
85C2×D241C2×D411yescharacteristic
86C2×D241C2×D411yescharacteristic
87C4×C241—21no—
88C4×C241—22no—
89C4×C241C2×D411yescharacteristic
90C4×C241C2411yescharacteristic
91Dic241—21no—
92Dic241C2×D411yescharacteristic
93Dic241C2×D411yescharacteristic
94C2×C241—21no—
95C2×C241—41no—
96C2×C241C22×D411yescharacteristic
97D241—21no—
98D241—21no—
99D241—41no—
100D241—42no—
101C241#57511yescharacteristic, complemented, Sylow
102#575—2411nomaximal, Sylow
103C2×#41—2411no—
104C2×#41—2411no—
105#137—2411no—
106#137—2412no—
107#137D24111yescharacteristic, complemented
108AGL(1,ℤ/8)—2411no—
109AGL(1,ℤ/8)—2412no—
110AGL(1,ℤ/8)—2411no—
111#135—2412no—
112#135—2412no—
113C2×M16—2411no—
114#41—4821no—
115#41—4821no—
116#41—4821no—
117#41—4821no—
118#41—2412no—
119#41—2411no—
120#41—2412no—
121#41—2411no—
122#41—2411no—
123#41C2×D24111yescharacteristic
124C2×Q8—2411no—
125C2×Q8C2×D24111yescharacteristic, complemented
126C2×D4—4821no—
127C2×D4—4821no—
128C2×D4—2411no—
129C2×D4—2411no—
130C22×C4—4821no—
131C22×C4—4821no—
132C22×C4—2411no—
133Q16—2412no—
134Q16C2×D24112yescomplemented
135SD16—2412no—
136SD16—2412no—
137SD16—2412no—
138SD16C2×D24112yescomplemented
139D8—2412no—
140D8—2412no—
141M16—2411no—
142M16—2412no—
143M16C2×D24111yescharacteristic, complemented
144C2×C8—2412no—
145C23—4821no—
146C23—4821no—
147C23—2411no—
148Q8—21no—
149Q8—4821no—
150Q8—2411no—
151Q8C22×D24111yescharacteristic
152Q8C22×D24112yes—
153D4—21no—
154D4—4821no—
155D4—4822no—
156D4—4822no—
157D4—4821no—
158D4—4821no—
159D4—4821no—
160D4—4821no—
161D4—2412no—
162D4—2411no—
163D4C22×D24111yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—4821no—
167C2×C4—4821no—
168C2×C4—4822no—
169C2×C4—4822no—
170C2×C4—4821no—
171C2×C4—4821no—
172C2×C4—2411no—
173C2×C4—2411no—
174C2×C4—2411no—
175C2×C4—2411no—
176C2×C4—2411no—
177C2×C4C22×D24111yescharacteristic
178C8—2412no—
179C8C22×D24112yes—
180C22—21no—
181C22—9642no—
182C22—9641no—
183C22—9642no—
184C22—4822no—
185C22—4821no—
186C22—2411no—
187C22—2411no—
188C22D4×D24111yescharacteristic
189C4—21no—
190C4—22no—
191C4—4821no—
192C4—2411no—
193C4—2411no—
194C4D4×D24111yescharacteristic
195C4C23×D24111yescharacteristic
196C2—21no—
197C2—41no—
198C2—9642no—
199C2—9641no—
200C2—4821no—
201C2—4821no—
202C2C2×D4×D24111yescharacteristic
subgroupG/H
centerC2C2×D4×D241
commutator, cosocleC4×C241C24
FrattiniC4C23×D241
Fitting#137×C241C2
socleC2×C241C22×D4
hypercenter#137D241
hypocenterC241#575
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. C2×#16838771C2semidirect
2. #41×D241C2semidirect
3. #21440674C2semidirect
4. #21440673C2semidirect
5. #21440670C2semidirect
6. #21440669C2semidirect
7. #21440667C2semidirect
8. #21440665C2semidirect
9. #21440663C2semidirect
10. M16×D241C2semidirect
11. #137×C241C2semidirect
15. #16838771C22semidirect
16. #16838771C22semidirect
20. #16838769C22semidirect
21. Q8×D241C22semidirect
23. C2×D964C22semidirect
25. D4×D241C22semidirect
29. #16838760C22semidirect
30. #16838759C22semidirect
31. #16838757C22semidirect
32. #16838757C22semidirect
33. D1928C22semidirect
34. #16838755C22semidirect
35. #16838754C22semidirect
36. #16838753C22semidirect
37. C8×D241C22semidirect
39. C2×Q8×C241C22semidirect
40. Q16×C241C22semidirect
41. SD16×C241C22semidirect
42. M16×C241C22semidirect
101. C241#575semidirect
107. #137D241semidirect
125. C2×Q8C2×D241semidirect
134. Q16C2×D241semidirect
138. SD16C2×D241semidirect
143. M16C2×D241semidirect

Maximal subgroups

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

subgroupindex
C2×#168387712
#41×D2412
#214406742
#214406732
#214406702
#214406692
#214406672
#214406652
#214406632
M16×D2412
#137×C2412
#575241