Finite Group Databasebeta

← #16848280

Subgroups — #16848280

Order 3968.

588 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 237 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×#11758521C211yesmaximal, characteristic
2C2×Q8×Dic31C211yesmaximal, characteristic
3C2×#11758464C211yesmaximal, characteristic
4#14520955C211yesmaximal, characteristic
5#14520946C214yesmaximal
6#14520918C211yesmaximal, characteristic
7#14520900C214yesmaximal
8C42×Dic31C211yesmaximal, characteristic
9C31×#391C211yesmaximal, characteristic
10C22×Q8×C31C2211yescharacteristic
11C22×Dic62C2211yescharacteristic
12C2×C4⋊C4×C31C2211yescharacteristic
13C2×C42×C31C2211yescharacteristic
14Q8×Dic31—24no—
15#11758521C418yescomplemented
16C2×C4⋊Dic31C2211yescharacteristic
17C2×C4⋊Dic31C2212yes—
18C2×Dic31⋊C4C2214yes—
19C2×Dic31⋊C4C2212yes—
20C2×Dic31⋊C4C2214yes—
21C2×C4×Dic31C2212yes—
22C2×C4×Dic31C2212yes—
23C2×C4×Dic31C2212yes—
24C2×C4×Dic31C2211yescharacteristic
25C2×C4×Dic31C2211yescharacteristic
26C2×C4×Dic31C2212yes—
27#11758464—24no—
28C31×#95C2214yes—
29C31⋊#95C2214yes—
30C2×Q8×C31—22no—
31C2×Q8×C31—22no—
32C2×Q8×C31C2×C414yes—
33C22×C4×C31C2312yes—
34C22×C4×C31C2312yes—
35C22×C4×C31C2312yes—
36C22×C4×C31C2311yescharacteristic
37C22×Dic31C2312yes—
38C22×Dic31C2312yes—
39C22×Dic31C2311yescharacteristic
40C22×Dic31C2312yes—
41C22×Dic31C2311yescharacteristic
42C2×Dic62—22no—
43C2×Dic62—22no—
44C2×Dic62C2×C414yes—
45C4⋊C4×C31—22no—
46C42×C31—22no—
47C4⋊Dic31—22no—
48C4⋊Dic31—24no—
49Dic31⋊C4—24no—
50Dic31⋊C4—28no—
51Dic31⋊C4C2×C4116yes—
52C4×Dic31—24no—
53C4×Dic31—24no—
54C4×Dic31—24no—
55C4×Dic31C2×C414yes—
56C4×Dic31D414yes—
57C4×Dic31Q818yescomplemented
58C23×C31C2411yescharacteristic
59Q8×C31—28no—
60C2×C4×C31—22no—
61C2×C4×C31—22no—
62C2×C4×C31—22no—
63C2×C4×C31—22no—
64C2×C4×C31—22no—
65C2×C4×C31—24no—
66C2×C4×C31C22×C414yes—
67C2×C4×C31C22×C412yes—
68C2×C4×C31C2×D412yes—
69C2×C4×C31C2×Q814yes—
70C2×C4×C31#4112yes—
71C2×Dic31—22no—
72C2×Dic31—22no—
73C2×Dic31—22no—
74C2×Dic31—22no—
75C2×Dic31—24no—
76C2×Dic31C22×C414yes—
77C2×Dic31C22×C414yes—
78C2×Dic31C2×D412yes—
79C2×Dic31C2×D412yes—
80C2×Dic31C2×Q814yes—
81C2×Dic31C2×Q814yes—
82C2×Dic31#4114yes—
83Dic62—28no—
84C4×C4⋊Q8—311nomaximal, Sylow
85C22×C31C23×C411yescharacteristic
86C22×C31C22×D411yescharacteristic
87C22×C31C22×Q812yes—
88C22×C31C2×#4111yescharacteristic
89C22×C31C2×#4112yes—
90C4×C31—24no—
91C4×C31—24no—
92C4×C31—42no—
93C4×C31C4×D414yes—
94Dic31—24no—
95Dic31—22no—
96Dic31—42no—
97Dic31C4×Q818yescomplemented
98Dic31C4⋊Q814yescomplemented
99C2×C4⋊Q8—311no—
100C2×C4×Q8—311no—
101C2×C4×Q8—311no—
102#391—311no—
103#391D3111yescharacteristic
104#389—314no—
105#384—311no—
106C4×C4⋊C4—314no—
107C43—311no—
108C2×C31C2×C4×D411yescharacteristic
109C2×C31C2×C4×Q812yes—
110C2×C31C2×C4⋊Q811yescharacteristic
111C2×C31#53211yescharacteristic
112C2×C31#53312yes—
113C22×Q8—311no—
114C22×Q8C2×D3111yescharacteristic
115C4⋊Q8—318no—
116C4×Q8—624no—
117C4×Q8—624no—
118C2×C4⋊C4—314no—
119C2×C4⋊C4—311no—
120C2×C4⋊C4—312no—
121C2×C4⋊C4—312no—
122C2×C4⋊C4—314no—
123C2×C4⋊C4C2×D3111yescharacteristic
124C2×C42—311no—
125C2×C42—311no—
126C2×C42—312no—
127C2×C42—312no—
128C2×C42—312no—
129C2×C42—312no—
130C2×C42C2×D3111yescharacteristic
131#95—314no—
132#95C2×D3114yes—
133C31C4×C4⋊Q811yescharacteristic, complemented, Sylow
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—314no—
137C2×Q8—622no—
138C2×Q8—622no—
139C2×Q8C4×D3114yes—
140C22×C4—311no—
141C22×C4—312no—
142C22×C4—312no—
143C22×C4—311no—
144C22×C4—312no—
145C22×C4C22×D3112yes—
146C22×C4C22×D3112yes—
147C22×C4C22×D3112yes—
148C22×C4C22×D3111yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—3116no—
151C4⋊C4—628no—
152C4⋊C4—624no—
153C4⋊C4—624no—
154C4⋊C4—622no—
155C42—22no—
156C42—314no—
157C42—314no—
158C42—318no—
159C42—624no—
160C42—624no—
161C42—624no—
162C23C23×D3111yescharacteristic
163Q8—28no—
164Q8—628no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—314no—
172C2×C4—312no—
173C2×C4—314no—
174C2×C4—314no—
175C2×C4—314no—
176C2×C4—314no—
177C2×C4—312no—
178C2×C4—622no—
179C2×C4—622no—
180C2×C4—622no—
181C2×C4—622no—
182C2×C4—624no—
183C2×C4C2×C4×D3114yes—
184C2×C4C2×C4×D3112yes—
185C2×C4D4×D3112yes—
186C2×C4#9249712yes—
187C2×C4Q8×D3114yes—
188C22C22×C4×D3111yescharacteristic
189C22C2×D4×D3111yescharacteristic
190C22C2×#9249711yescharacteristic
191C22C2×Q8×D3112yes—
192C22#41×D3112yes—
193C4—24no—
194C4—24no—
195C4—42no—
196C4—318no—
197C4—314no—
198C4—624no—
199C4—622no—
200C4—1242no—
201C4(C22⋊D31)⋊C414yes—
202C2C2×(C22⋊D31)⋊C411yescharacteristic
203C2C4×Q8×D3112yes—
204C2C4⋊Q8×D3111yescharacteristic
205C2#1452159811yescharacteristic
206C2#1452160112yes—
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#391C2
socle, prime radical, cosocleC23×C31C24
hypercenter#391D31
hypocenterC31C4×C4⋊Q8
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
15. #11758521C4semidirect
57. C4×Dic31Q8semidirect
97. Dic31C4×Q8semidirect
98. Dic31C4⋊Q8semidirect
133. C31C4×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#117585212
C2×Q8×Dic312
C2×#117584642
#145209552
#145209462
#145209182
#145209002
C42×Dic312
C31×#3912
C4×C4⋊Q831