Finite Group Databasebeta

← #16856163

Subgroups — #16856163

Order 3968.

656 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 173 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×#11758485C214yesmaximal, complemented
2#14521599C211yesmaximal, characteristic, complemented
3C2×#11758451C211yesmaximal, characteristic, complemented
4#14521438C214yesmaximal, complemented
5#14521426C212yesmaximal, complemented
6#14521422C212yesmaximal, complemented
7C4×M16×C31C211yesmaximal, characteristic, complemented
8C2×#92495C2212yes—
9C22×D124—22no—
10C22×D124C2212yes—
11C2×M16×C31C2212yes—
12C2×C42×C31C2211yescharacteristic
13#11758507—22no—
14#11758485—216no—
15C2×D248C2218yescomplemented
16C2×#92470C2218yescomplemented
17#11758452C2212yescomplemented
18#11758451—22no—
19#11758451C2212yes—
20#11758451C2212yes—
21#11758451C2211yescharacteristic, complemented
22C4×D124—22no—
23#11758446C2211yescharacteristic, complemented
24C31×#97C2212yescomplemented
25C4×C8×C31C2212yescomplemented
26C23×D31—22no—
27C22×C4×C31D412yes—
28C22×C4×C31C2311yescharacteristic
29C2×C22⋊D31—22no—
30#92495—28no—
31C2×D124—28no—
32C2×D124—22no—
33C2×D124—22no—
34C2×D124—24no—
35C2×D124—24no—
36C2×D124—48no—
37C2×D124C2314yes—
38C2×D124C2312yes—
39C2×C4×D31—22no—
40C2×Dic62C2312yes—
41M16×C31D418yescomplemented
42C2×C8×C31C2314yes—
43C42×C31D412yes—
44C42×C31C2311yescharacteristic
45C42×C31C2311yescharacteristic
46(C2×D31)⋊C4—24no—
47C4⋊Dic31—22no—
48D248—216no—
49#92470—216no—
50C23×C31C2×D411yescharacteristic
51C22×D31—22no—
52C22×D31—24no—
53C22×D31—48no—
54C22×D31—44no—
55C2×C4×C31—24no—
56C2×C4×C31C2×D412yes—
57C2×C4×C31C2×D411yescharacteristic
58C2×C4×C31C2×D414yes—
59C2×C4×C31C2×D412yes—
60C2×C4×C31C2411yescharacteristic
61C22⋊D31—44no—
62C2×Dic31—22no—
63D124—28no—
64D124—24no—
65D124—48no—
66D124—44no—
67D124—44no—
68D124—48no—
69C4×D31—44no—
70Dic62—24no—
71C8×C31C2×D418yescomplemented
72#3151—311nomaximal, Sylow
73C22×C31—22no—
74C22×C31#12712yes—
75C22×C31C22×D411yescharacteristic
76C2×D31—42no—
77C2×D31—44no—
78C2×D31—48no—
79C2×D31—44no—
80C2×D31—88no—
81C4×C31—22no—
82C4×C31#12712yes—
83C4×C31AGL(1,ℤ/8)14yescomplemented
84C4×C31C22×D412yes—
85Dic31—42no—
86C2×AGL(1,ℤ/8)—314no—
87#532—311no—
88C2×#127—311no—
89#496—314no—
90#493—312no—
91#492—312no—
92C4×M16D3111yescharacteristic, complemented
93C2×C31—22no—
94C2×C31C2×#12711yescharacteristic
95C2×C31C2×AGL(1,ℤ/8)12yes—
96D31—82no—
97D31—84no—
98C2×#41—312no—
99C22×D4—312no—
100C22×D4—622no—
101AGL(1,ℤ/8)—6216no—
102C2×SD16—318no—
103C2×D8—318no—
104C2×M16C2×D3112yes—
105C4⋊Q8—311no—
106#127—312no—
107#127—312no—
108#127—311no—
109#127—622no—
110#124—312no—
111#121—622no—
112C4×D4—622no—
113C2×C42C2×D3111yescharacteristic
114#97C2×D3112yescomplemented
115C4×C8C2×D3112yescomplemented
116C31#315111yescharacteristic, complemented, Sylow
117C24—622no—
118#41—628no—
119C2×Q8—312no—
120C2×D4—312no—
121C2×D4—314no—
122C2×D4—622no—
123C2×D4—628no—
124C2×D4—622no—
125C2×D4—624no—
126C2×D4—624no—
127C2×D4—622no—
128C2×D4—1248no—
129C22×C4—622no—
130C22×C4D12412yes—
131C22×C4C22×D3111yescharacteristic
132SD16—6216no—
133D8—6216no—
134M16D12418yescomplemented
135C2×C8C22×D3114yes—
136C4⋊C4—622no—
137C22⋊C4—624no—
138C42D12412yes—
139C42C22×D3111yescharacteristic
140C42C22×D3111yescharacteristic
141C23—622no—
142C23—624no—
143C23—1248no—
144C23—1244no—
145C23C2×D12411yescharacteristic
146Q8—624no—
147D4—624no—
148D4—628no—
149D4—1248no—
150D4—1244no—
151D4—1244no—
152D4—1244no—
153D4—1248no—
154C2×C4—24no—
155C2×C4—622no—
156C2×C4—1244no—
157C2×C4C2×D12412yes—
158C2×C4C2×D12411yescharacteristic
159C2×C4C2×D12414yes—
160C2×C4C2×D12412yes—
161C2×C4C23×D3111yescharacteristic
162C8C2×D12418yescomplemented
163C22—22no—
164C22—1248no—
165C22—1244no—
166C22—1242no—
167C22—1244no—
168C22—2488no—
169C22#1175845112yes—
170C22C22×D12411yescharacteristic
171C4—22no—
172C4—1242no—
173C4#1175845112yes—
174C4#1175848514yescomplemented
175C4C22×D12412yes—
176C2—22no—
177C2—2484no—
178C2—2482no—
179C2C2×#1175845111yescharacteristic
180C2C2×#1175848512yes—
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC4×M16×C31C2
socleC22×C31C22×D4
hypercenterC4×M16D31
hypocenterC31#3151
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#11758485C2semidirect
2. #14521599C2semidirect
3. C2×#11758451C2semidirect
4. #14521438C2semidirect
5. #14521426C2semidirect
6. #14521422C2semidirect
7. C4×M16×C31C2semidirect
15. C2×D248C22semidirect
16. C2×#92470C22semidirect
17. #11758452C22semidirect
21. #11758451C22semidirect
23. #11758446C22semidirect
24. C31×#97C22semidirect
25. C4×C8×C31C22semidirect
41. M16×C31D4semidirect
71. C8×C31C2×D4semidirect
83. C4×C31AGL(1,ℤ/8)semidirect
92. C4×M16D31semidirect
114. #97C2×D31semidirect
115. C4×C8C2×D31semidirect
116. C31#3151semidirect
134. M16D124semidirect
162. C8C2×D124semidirect
174. C4#11758485semidirect

Maximal subgroups

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

subgroupindex
C2×#117584852
#145215992
C2×#117584512
#145214382
#145214262
#145214222
C4×M16×C312
#315131