Finite Group Databasebeta

← #16855320

Subgroups — #16855320

Order 3968.

500 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11758481C211yesmaximal, characteristic, complemented
2#14521560C212yesmaximal, complemented
3#14521307C212yesmaximal, complemented
4#14521302C212yesmaximal, complemented
5#14521295C214yesmaximal, complemented
6#14521087C211yesmaximal, characteristic, complemented
7#14521046C212yesmaximal, complemented
8C22×M16×C31C211yesmaximal, characteristic, complemented
9C23×C4×C31C2211yescharacteristic
10C2×#92495C2212yes—
11C2×M16×C31—24no—
12C2×M16×C31C2212yes—
13C2×M16×C31C2212yes—
14C22×C8×C31C2211yescharacteristic
15C22×C8×C31C2211yescharacteristic
16#11758530—22no—
17#11758507C2214yes—
18#11758506—22no—
19C4×C22⋊D31—24no—
20#11758503C2212yes—
21#11758501C2214yes—
22C2×Dic124C2211yescharacteristic, complemented
23#11758481—24no—
24C2×D248C2211yescharacteristic, complemented
25C2×#92470C2212yescomplemented
26#11758409C2214yescomplemented
27#11758406C2212yescomplemented
28#11758405C2212yescomplemented
29#11758404C2214yescomplemented
30C24×C31D411yescharacteristic
31C22×C4×C31—22no—
32C22×C4×C31—22no—
33C22×C4×C31D412yes—
34C22×C4×C31D411yescharacteristic
35C22×C4×C31C2312yes—
36C22×C4×C31C2311yescharacteristic
37C2×C22⋊D31—24no—
38C2×C22⋊D31—22no—
39#92495—24no—
40#92495—24no—
41C2×D124C2312yes—
42C2×C4×D31—22no—
43C2×Dic62C2312yes—
44M16×C31—24no—
45M16×C31—24no—
46C2×C8×C31—22no—
47C2×C8×C31D414yescomplemented
48C2×C8×C31C2312yes—
49C2×C8×C31C2311yescharacteristic
50C2×C8×C31C2311yescharacteristic
51C22⋊Dic31—24no—
52C22⋊Dic31—22no—
53(C2×D31)⋊C4—24no—
54C4⋊Dic31C2314yes—
55Dic31⋊C4—24no—
56C4×Dic31—22no—
57Dic124—22no—
58D248—22no—
59#92470—24no—
60C23×C31—22no—
61C23×C31—42no—
62C23×C31C2×D412yes—
63C23×C31C2×D411yescharacteristic
64C22×D31—22no—
65C2×C4×C31—22no—
66C2×C4×C31—24no—
67C2×C4×C31—22no—
68C2×C4×C31—22no—
69C2×C4×C31C2×D412yes—
70C2×C4×C31C2×D411yescharacteristic
71C2×C4×C31C2×D412yes—
72C2×C4×C31#4112yes—
73C2×C4×C31C2411yescharacteristic
74C22⋊D31—48no—
75C22⋊D31—44no—
76C2×Dic31—24no—
77C2×Dic31—22no—
78D124—22no—
79D124—22no—
80C4×D31—44no—
81Dic62—22no—
82Dic62—22no—
83C8×C31—22no—
84C8×C31C2×D412yes—
85C8×C31C2×D412yes—
86#3054—311nomaximal, Sylow
87C22×C31—22no—
88C22×C31—22no—
89C22×C31—24no—
90C22×C31—44no—
91C22×C31#12112yes—
92C22×C31C22×D411yescharacteristic
93C2×D31—42no—
94C2×D31—44no—
95C4×C31—22no—
96C4×C31#12112yes—
97C4×C31C22×D411yescharacteristic
98C4×C31C2×#4111yescharacteristic
99Dic31—42no—
100Dic31—44no—
101C2×#135—311no—
102C22×M16D3111yescharacteristic, complemented
103#525—312no—
104#470—312no—
105#469—312no—
106#468—314no—
107#427—311no—
108#416—312no—
109C2×C31—22no—
110C2×C31—42no—
111C2×C31C2×#12111yescharacteristic
112C2×C31#57512yes—
113D31—82no—
114C2×#41—312no—
115C23×C4C2×D3111yescharacteristic
116#135—624no—
117C2×Q16—311no—
118C2×SD16—312no—
119C2×D8—311no—
120C2×M16—24no—
121C2×M16C2×D3112yes—
122C2×M16C2×D3112yes—
123C22×C8C2×D3111yescharacteristic
124C22×C8C2×D3111yescharacteristic
125#123—622no—
126C22⋊Q8—314no—
127#121—314no—
128#120—622no—
129C4×D4—624no—
130#117—312no—
131#107—312no—
132#106—312no—
133#103—314no—
134#102—314no—
135C31#305411yescharacteristic, complemented, Sylow
136C24D12411yescharacteristic
137#41—624no—
138#41—624no—
139C2×Q8—312no—
140C2×D4—312no—
141C2×D4—624no—
142C2×D4—622no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—622no—
146C22×C4D12411yescharacteristic
147C22×C4D12412yes—
148C22×C4C22×D3112yes—
149C22×C4C22×D3111yescharacteristic
150Q16—622no—
151SD16—624no—
152D8—622no—
153M16—24no—
154M16—24no—
155C2×C8—22no—
156C2×C8C22⋊D3114yescomplemented
157C2×C8C22×D3112yes—
158C2×C8C22×D3111yescharacteristic
159C2×C8C22×D3111yescharacteristic
160C4⋊C4—314no—
161C4⋊C4—624no—
162C22⋊C4—622no—
163C22⋊C4—624no—
164C22⋊C4—624no—
165C42—622no—
166C23—22no—
167C23—42no—
168C23—622no—
169C23C2×D12412yes—
170C23C2×D12411yescharacteristic
171Q8—622no—
172Q8—622no—
173D4—622no—
174D4—622no—
175D4—1248no—
176D4—1244no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—624no—
182C2×C4—622no—
183C2×C4—1244no—
184C2×C4C2×D12412yes—
185C2×C4C2×D12411yescharacteristic
186C2×C4#9249512yes—
187C2×C4C2×C22⋊D3112yes—
188C2×C4C23×D3111yescharacteristic
189C8—22no—
190C8C2×C22⋊D3112yes—
191C8C2×C22⋊D3112yes—
192C22—22no—
193C22—22no—
194C22—24no—
195C22—44no—
196C22—1242no—
197C22—1244no—
198C22#1175850712yes—
199C22C22×D12411yescharacteristic
200C4—22no—
201C4—1242no—
202C4—1244no—
203C4#1175850712yes—
204C4C2×#9249511yescharacteristic
205C4C22×C22⋊D3111yescharacteristic
206C2—22no—
207C2—42no—
208C2—2482no—
209C2C2×#1175850711yescharacteristic
210C2#1452184912yes—
subgroupG/H
centerC2×C4C2×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC22×M16×C31C2
socleC22×C31C22×D4
hypercenterC22×M16D31
hypocenterC31#3054
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C2×#11758481C2semidirect
2. #14521560C2semidirect
3. #14521307C2semidirect
4. #14521302C2semidirect
5. #14521295C2semidirect
6. #14521087C2semidirect
7. #14521046C2semidirect
8. C22×M16×C31C2semidirect
22. C2×Dic124C22semidirect
24. C2×D248C22semidirect
25. C2×#92470C22semidirect
26. #11758409C22semidirect
27. #11758406C22semidirect
28. #11758405C22semidirect
29. #11758404C22semidirect
47. C2×C8×C31D4semidirect
102. C22×M16D31semidirect
135. C31#3054semidirect
156. C2×C8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584812
#145215602
#145213072
#145213022
#145212952
#145210872
#145210462
C22×M16×C312
#305431