Finite Group Databasebeta

← #16855321

Subgroups — #16855321

Order 3968.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C22×#92469C211yesmaximal, characteristic, complemented
3#14521561C212yesmaximal, complemented
4#14521308C212yesmaximal, complemented
5#14521303C212yesmaximal, complemented
6#14521296C214yesmaximal, complemented
7#14521043C212yesmaximal, complemented
8C31×#427C211yesmaximal, characteristic, complemented
9C2×#92495C2212yes—
10C22×C4×D31C2211yescharacteristic
11C22×C8×C31C2211yescharacteristic
12C31×#117C2212yes—
13C22×C31⋊C8C2211yescharacteristic
14C2×Dic124C2211yescharacteristic, complemented
15#11758481—24no—
16C2×D248C2211yescharacteristic, complemented
17C2×#92470C2212yescomplemented
18C2×#92469—24no—
19C2×#92469C2212yes—
20C2×#92469C2212yes—
21#11758474C2214yes—
22#11758472C2214yes—
23#11758463—22no—
24#11758459—22no—
25C4×D124—24no—
26C31×#107C2212yescomplemented
27C31×#106C2212yescomplemented
28#11758398C2214yescomplemented
29#11758397C2214yescomplemented
30C23×D31D411yescharacteristic
31C22×C4×C31C2311yescharacteristic
32C2×C22⋊D31—22no—
33C22×Dic31D411yescharacteristic
34#92495—24no—
35#92495—24no—
36C2×D124—24no—
37C2×D124C2312yes—
38C2×C4×D31—22no—
39C2×C4×D31—22no—
40C2×C4×D31—22no—
41C2×C4×D31D412yes—
42C2×C4×D31C2312yes—
43C2×Dic62C2312yes—
44C2×C8×C31D414yescomplemented
45C2×C8×C31C2311yescharacteristic
46C2×C8×C31C2311yescharacteristic
47C4⋊C4×C31C2314yes—
48C22⋊C4×C31—22no—
49C42×C31—22no—
50(C2×D31)⋊C4—24no—
51(C2×D31)⋊C4—24no—
52C4⋊Dic31—24no—
53C2×C31⋊C8—22no—
54C2×C31⋊C8C2312yes—
55Dic124—22no—
56D248—22no—
57#92470—24no—
58#92469—24no—
59#92469—24no—
60C23×C31C2×D411yescharacteristic
61C22×D31—22no—
62C22×D31—22no—
63C22×D31—42no—
64C22×D31C2×D412yes—
65C2×C4×C31—24no—
66C2×C4×C31C2×D412yes—
67C2×C4×C31C2×D411yescharacteristic
68C2×C4×C31#4112yes—
69C2×C4×C31C2411yescharacteristic
70C22⋊D31—44no—
71C2×Dic31—22no—
72C2×Dic31—22no—
73C2×Dic31C2×D412yes—
74D124—22no—
75D124—22no—
76D124—48no—
77C4×D31—24no—
78C4×D31—22no—
79C4×D31—22no—
80C4×D31—44no—
81Dic62—22no—
82Dic62—22no—
83C8×C31C2×D412yes—
84C8×C31C2×D412yes—
85C31⋊C8—22no—
86#3054—311nomaximal, Sylow
87C22×C31—22no—
88C22×C31#12112yes—
89C22×C31C22×D411yescharacteristic
90C2×D31—22no—
91C2×D31—24no—
92C2×D31—44no—
93C2×D31—42no—
94C2×D31—44no—
95C4×C31—44no—
96C4×C31#12112yes—
97C4×C31C22×D411yescharacteristic
98C4×C31C2×#4111yescharacteristic
99Dic31—22no—
100Dic31—42no—
101C2×#135—311no—
102C22×M16—311no—
103#525—312no—
104#470—312no—
105#469—312no—
106#468—314no—
107#427D3111yescharacteristic, complemented
108#416—312no—
109C2×C31—22no—
110C2×C31C2×#12111yescharacteristic
111C2×C31#57512yes—
112D31—42no—
113D31—82no—
114C2×#41—312no—
115C23×C4—311no—
116#135—624no—
117C2×Q16—311no—
118C2×SD16—312no—
119C2×D8—311no—
120C2×M16—312no—
121C2×M16—312no—
122C2×M16—624no—
123C22×C8—311no—
124C22×C8C2×D3111yescharacteristic
125#123—622no—
126C22⋊Q8—314no—
127#121—314no—
128#120—622no—
129C4×D4—624no—
130#117C2×D3112yes—
131#107C2×D3112yescomplemented
132#106C2×D3112yescomplemented
133#103—314no—
134#102—314no—
135C31#305411yescharacteristic, complemented, Sylow
136C24—311no—
137#41—624no—
138#41—624no—
139C2×Q8—312no—
140C2×D4—312no—
141C2×D4—624no—
142C2×D4—622no—
143C22×C4—311no—
144C22×C4—312no—
145C22×C4—312no—
146C22×C4—622no—
147C22×C4—622no—
148C22×C4—622no—
149C22×C4C22×D3111yescharacteristic
150Q16—622no—
151SD16—624no—
152D8—622no—
153M16—624no—
154M16—624no—
155C2×C8—312no—
156C2×C8—622no—
157C2×C8D12414yescomplemented
158C2×C8C22×D3111yescharacteristic
159C2×C8C22×D3111yescharacteristic
160C4⋊C4—624no—
161C4⋊C4C22×D3114yes—
162C22⋊C4—22no—
163C22⋊C4—624no—
164C22⋊C4—624no—
165C42—22no—
166C23—312no—
167C23—622no—
168C23—622no—
169C23—1242no—
170C23D4×D3111yescharacteristic
171Q8—622no—
172Q8—622no—
173D4—622no—
174D4—622no—
175D4—1248no—
176D4—1244no—
177C2×C4—24no—
178C2×C4—312no—
179C2×C4—622no—
180C2×C4—624no—
181C2×C4—622no—
182C2×C4—622no—
183C2×C4—622no—
184C2×C4—1244no—
185C2×C4C2×D12412yes—
186C2×C4D4×D3111yescharacteristic
187C2×C4#9249912yes—
188C2×C4C23×D3111yescharacteristic
189C8—622no—
190C8C2×D12412yes—
191C8C2×D12412yes—
192C22—22no—
193C22—622no—
194C22—624no—
195C22—1244no—
196C22—1242no—
197C22—1244no—
198C22#1175847212yes—
199C22C2×D4×D3111yescharacteristic
200C4—44no—
201C4—622no—
202C4—1242no—
203C4#1175847212yes—
204C4C22×D12411yescharacteristic
205C4C2×#9249911yescharacteristic
206C2—22no—
207C2—1242no—
208C2—2482no—
209C2C2×#1175847211yescharacteristic
210C2#1452184612yes—
subgroupG/H
centerC2×C4D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#427C2
socleC22×C31C22×D4
hypercenter#427D31
hypocenterC31#3054
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. C2×#11758481C2semidirect
2. C22×#92469C2semidirect
3. #14521561C2semidirect
4. #14521308C2semidirect
5. #14521303C2semidirect
6. #14521296C2semidirect
7. #14521043C2semidirect
8. C31×#427C2semidirect
14. C2×Dic124C22semidirect
16. C2×D248C22semidirect
17. C2×#92470C22semidirect
26. C31×#107C22semidirect
27. C31×#106C22semidirect
28. #11758398C22semidirect
29. #11758397C22semidirect
44. C2×C8×C31D4semidirect
107. #427D31semidirect
131. #107C2×D31semidirect
132. #106C2×D31semidirect
135. C31#3054semidirect
157. C2×C8D124semidirect

Maximal subgroups

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

subgroupindex
C2×#117584812
C22×#924692
#145215612
#145213082
#145213032
#145212962
#145210432
C31×#4272
#305431