Finite Group Databasebeta

← #16848896

Subgroups — #16848896

Order 3968.

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

normal subgroups only

This group has 408 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×#11758447C211yesmaximal, characteristic
2#14520946C212yesmaximal
3#14520944C212yesmaximal
4#14520936C212yesmaximal
5#14520934C212yesmaximal
6#14520932C212yesmaximal
7#14520930C212yesmaximal
8#14520915C211yesmaximal, characteristic
9C31×#382C211yesmaximal, characteristic
10C2×C4⋊C4×C31C2211yescharacteristic
11C2×C4⋊C4×C31C2211yescharacteristic
12C2×C42×C31C2211yescharacteristic
13C2×C4⋊Dic31C2211yescharacteristic
14C2×C4⋊Dic31C2211yescharacteristic
15C2×C4⋊Dic31C2211yescharacteristic
16C2×C4⋊Dic31C2211yescharacteristic
17C2×Dic31⋊C4C2212yes—
18C2×Dic31⋊C4C2212yes—
19C2×Dic31⋊C4C2212yes—
20C2×Dic31⋊C4C2212yes—
21C2×C4×Dic31C2212yes—
22C2×C4×Dic31C2212yes—
23#11758447C418yescomplemented
24C31×#95C2212yes—
25C31×#95C2212yes—
26C31⋊#95C2212yes—
27C31⋊#95C2212yes—
28C31⋊#95C2212yes—
29C31⋊#95C2212yes—
30C31⋊#95C2212yes—
31C31⋊#95C2212yes—
32C22×C4×C31C2312yes—
33C22×C4×C31C2312yes—
34C22×C4×C31C2311yescharacteristic
35C22×C4×C31C2311yescharacteristic
36C22×C4×C31C2311yescharacteristic
37C22×Dic31C2312yes—
38C22×Dic31C2312yes—
39C22×Dic31C2312yes—
40C22×Dic31C2312yes—
41C4⋊C4×C31—41no—
42C4⋊C4×C31—41no—
43C42×C31C2×C414yes—
44C4⋊Dic31—41no—
45C4⋊Dic31—41no—
46C4⋊Dic31C2×C414yes—
47C4⋊Dic31C2×C414yes—
48Dic31⋊C4—42no—
49Dic31⋊C4—42no—
50Dic31⋊C4C2×C418yes—
51Dic31⋊C4C2×C418yes—
52C4×Dic31—42no—
53C4×Dic31—42no—
54C23×C31C2411yescharacteristic
55C2×C4×C31—21no—
56C2×C4×C31—21no—
57C2×C4×C31—21no—
58C2×C4×C31—21no—
59C2×C4×C31—21no—
60C2×C4×C31—21no—
61C2×C4×C31—22no—
62C2×C4×C31—22no—
63C2×C4×C31—22no—
64C2×C4×C31—22no—
65C2×C4×C31—22no—
66C2×C4×C31—22no—
67C2×C4×C31C22×C412yes—
68C2×C4×C31C22×C412yes—
69C2×C4×C31C22×C412yes—
70C2×Dic31—22no—
71C2×Dic31—22no—
72C2×Dic31—22no—
73C2×Dic31—22no—
74C2×Dic31—22no—
75C2×Dic31—22no—
76C2×Dic31—22no—
77C2×Dic31—22no—
78C2×Dic31—22no—
79C2×Dic31—22no—
80C2×Dic31C22×C414yes—
81C2×Dic31C22×C414yes—
82#2382—311nomaximal, Sylow
83C22×C31C23×C411yescharacteristic
84C22×C31H(5,𝔽2)11yescharacteristic
85C22×C31H(5,𝔽2)11yescharacteristic
86C22×C312−1+411yescharacteristic
87C22×C312−1+411yescharacteristic
88C22×C312−1+411yescharacteristic
89C22×C312−1+411yescharacteristic
90C4×C31—22no—
91C4×C31—22no—
92C4×C31—22no—
93C4×C31—42no—
94C4×C31—42no—
95Dic31—24no—
96Dic31—24no—
97Dic31—42no—
98Dic31—42no—
99C2×#125—311no—
100#389—312no—
101#389—312no—
102#387—312no—
103#387—312no—
104#387—312no—
105#387—312no—
106#382—311no—
107#382D3111yescharacteristic
108C2×C31#51911yescharacteristic
109C2×C31#52011yescharacteristic
110C2×C31#52011yescharacteristic
111C2×C31#56211yescharacteristic
112C2×C31#56311yescharacteristic
113C2×C31#56311yescharacteristic
114C2×C31#56411yescharacteristic
115#125—318no—
116C2×C4⋊C4—311no—
117C2×C4⋊C4—311no—
118C2×C4⋊C4—312no—
119C2×C4⋊C4—312no—
120C2×C4⋊C4—311no—
121C2×C4⋊C4—311no—
122C2×C4⋊C4—312no—
123C2×C4⋊C4—312no—
124C2×C4⋊C4C2×D3111yescharacteristic
125C2×C4⋊C4C2×D3111yescharacteristic
126C2×C42—312no—
127C2×C42—312no—
128C2×C42C2×D3111yescharacteristic
129#95—312no—
130#95—312no—
131#95—312no—
132#95—312no—
133#95—312no—
134#95—312no—
135#95C2×D3112yes—
136#95C2×D3112yes—
137C31#238211yescharacteristic, complemented, Sylow
138C22×C4—312no—
139C22×C4—312no—
140C22×C4—312no—
141C22×C4—312no—
142C22×C4C22×D3112yes—
143C22×C4C22×D3112yes—
144C22×C4C22×D3111yescharacteristic
145C22×C4C22×D3111yescharacteristic
146C22×C4C22×D3111yescharacteristic
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—318no—
150C4⋊C4—318no—
151C4⋊C4—314no—
152C4⋊C4—314no—
153C4⋊C4—1242no—
154C4⋊C4—1242no—
155C4⋊C4—1241no—
156C4⋊C4—1241no—
157C42—1242no—
158C42—1242no—
159C42C4×D3114yes—
160C23C23×D3111yescharacteristic
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—314no—
174C2×C4—314no—
175C2×C4—622no—
176C2×C4—622no—
177C2×C4—622no—
178C2×C4—622no—
179C2×C4—622no—
180C2×C4—622no—
181C2×C4—622no—
182C2×C4—622no—
183C2×C4—622no—
184C2×C4—622no—
185C2×C4C2×C4×D3112yes—
186C2×C4C2×C4×D3112yes—
187C2×C4C2×C4×D3112yes—
188C22C22×C4×D3111yescharacteristic
189C22#1175856111yescharacteristic
190C22#1175856411yescharacteristic
191C22#1175856611yescharacteristic
192C22#1175856711yescharacteristic
193C22#1175856711yescharacteristic
194C22#1175856711yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—624no—
201C4—624no—
202C4—1242no—
203C4—1242no—
204C2#1452152611yescharacteristic
205C2#1452152911yescharacteristic
206C2#1452153011yescharacteristic
207C2#1452179311yescharacteristic
208C2#1452179911yescharacteristic
209C2#1452180311yescharacteristic
210C2C31⋊#56411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#382C2
socle, prime radical, cosocleC23×C31C24
hypercenter#382D31
hypocenterC31#2382
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
23. #11758447C4semidirect
137. C31#2382semidirect

Maximal subgroups

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

subgroupindex
C2×#117584472
#145209462
#145209442
#145209362
#145209342
#145209322
#145209302
#145209152
C31×#3822
#238231