Finite Group Databasebeta

← #16856296

Subgroups — #16856296

Order 3968.

368 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#14521604C211yesmaximal, characteristic
2#14521604C211yesmaximal, characteristic
3#14521452C211yesmaximal, characteristic, complemented
4#14521428C211yesmaximal, characteristic, complemented
5#14521400C211yesmaximal, characteristic, complemented
6#14521289C214yesmaximal
7#14521141C214yesmaximal, complemented
8C4×C8×D31C211yesmaximal, characteristic
9C31×#487C211yesmaximal, characteristic, complemented
10C31×C4⋊Q8C2211yescharacteristic
11C31×C4⋊Q8C2211yescharacteristic
12#11758523—21no—
13#11758523C2212yes—
14Q8×Dic31—21no—
15Q8×Dic31C2212yes—
16C2×#92482C2214yes—
17C2×C8×D31C2212yes—
18#11758474—21no—
19#11758474C2212yes—
20#11758469—21no—
21#11758469—21no—
22#11758467—21no—
23#11758467—21no—
24#11758464—21no—
25#11758464C2212yes—
26C42×D31C2211yescharacteristic
27#11758446C2211yescharacteristic
28#11758446C2211yescharacteristic
29C31×#103C2214yes—
30C4×C8×C31C2211yescharacteristic
31#11758404C2214yes—
32C8×Dic31C2212yes—
33#11758395C2214yes—
34C4×C31⋊C8C2211yescharacteristic
35C2×Q8×C31—21no—
36C2×Q8×C31C2312yes—
37C2×C4×D31D411yescharacteristic
38C2×C4×D31D411yescharacteristic
39C2×C4×D31C2311yescharacteristic
40C2×Dic62—21no—
41C2×Dic62C2312yes—
42C2×C8×C31C2312yes—
43C4⋊C4×C31—21no—
44C4⋊C4×C31—21no—
45C4⋊C4×C31—21no—
46C4⋊C4×C31C2312yes—
47C42×C31C2311yescharacteristic
48#92482—28no—
49(C2×D31)⋊C4—22no—
50(C2×D31)⋊C4—22no—
51C4⋊Dic31—21no—
52C4⋊Dic31—21no—
53C4⋊Dic31—21no—
54C4⋊Dic31—22no—
55C4⋊Dic31—22no—
56C4⋊Dic31C2312yes—
57Dic31⋊C4—22no—
58Dic31⋊C4—22no—
59C4×Dic31—22no—
60C4×Dic31—22no—
61C4×Dic31D411yescharacteristic
62C4×Dic31D411yescharacteristic
63C4×Dic31C2311yescharacteristic
64C2×C31⋊C8C2312yes—
65C8×D31—24no—
66C22×D31C2×D411yescharacteristic
67Q8×C31—24no—
68Q8×C31—42no—
69C2×C4×C31—22no—
70C2×C4×C31—22no—
71C2×C4×C31C2×D411yescharacteristic
72C2×C4×C31C2×D411yescharacteristic
73C2×C4×C31C2411yescharacteristic
74C2×Dic31—22no—
75C2×Dic31—22no—
76C2×Dic31C2×D411yescharacteristic
77C2×Dic31C2×D411yescharacteristic
78C2×Dic31C2×D411yescharacteristic
79C4×D31—22no—
80C4×D31—22no—
81C4×D31Q1614yescomplemented
82Dic62—24no—
83Dic62—42no—
84C8×C31—22no—
85C31⋊C8#4114yes—
86#3170—311nomaximal, Sylow
87C22×C31C22×D411yescharacteristic
88C2×D31—21no—
89C2×D31—21no—
90C2×D31C2×Q1612yes—
91C4×C31—42no—
92C4×C31—42no—
93C4×C31C2×Q1612yes—
94C4×C31#13512yes—
95C4×C31C2×#4112yes—
96Dic31—21no—
97Dic31—21no—
98Dic31—42no—
99Dic31—42no—
100Dic31C2×Q1612yes—
101#533—311no—
102#533—311no—
103#500—311no—
104#494—311no—
105#487—311no—
106#487D3111yescharacteristic, complemented
107#467—314no—
108C4×Q16—314no—
109C2×C4×C8—311no—
110C2×C31#53211yescharacteristic
111C2×C31C22×Q1611yescharacteristic
112C2×C31C2×#13511yescharacteristic
113D31—22no—
114C2×Q16—314no—
115C22×C8—312no—
116C4⋊Q8—311no—
117C4⋊Q8—311no—
118C4⋊Q8C2×D3111yescharacteristic
119C4⋊Q8C2×D3111yescharacteristic
120#125—621no—
121#125—621no—
122C22⋊Q8—312no—
123C22⋊Q8—312no—
124C22⋊Q8—621no—
125C22⋊Q8—621no—
126C4×Q8—312no—
127C4×Q8—312no—
128C4×Q8—621no—
129C4×Q8—621no—
130#117—621no—
131#117—621no—
132C2×C42—311no—
133#107—314no—
134#103—314no—
135#103C2×D3114yes—
136C4×C8—312no—
137C4×C8—311no—
138C4×C8C2×D3111yescharacteristic
139C31#317011yescharacteristic, complemented, Sylow
140C2×Q8—21no—
141C2×Q8—312no—
142C2×Q8—621no—
143C2×Q8C22×D3112yes—
144C22×C4—311no—
145C22×C4—311no—
146C22×C4—311no—
147Q16—628no—
148C2×C8—312no—
149C2×C8—624no—
150C2×C8C22×D3112yes—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—312no—
155C4⋊C4—622no—
156C4⋊C4—622no—
157C4⋊C4—622no—
158C4⋊C4—622no—
159C4⋊C4—621no—
160C4⋊C4—621no—
161C4⋊C4—621no—
162C4⋊C4C22×D3112yes—
163C22⋊C4—622no—
164C22⋊C4—622no—
165C42—311no—
166C42—311no—
167C42—311no—
168C42—622no—
169C42—622no—
170C42C22×D3111yescharacteristic
171C23—311no—
172Q8—24no—
173Q8—42no—
174Q8—624no—
175Q8—1242no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—311no—
179C2×C4—314no—
180C2×C4—311no—
181C2×C4—311no—
182C2×C4—622no—
183C2×C4—622no—
184C2×C4—622no—
185C2×C4—622no—
186C2×C4D4×D3111yescharacteristic
187C2×C4D4×D3111yescharacteristic
188C2×C4C23×D3111yescharacteristic
189C8—22no—
190C8—314no—
191C22—312no—
192C22—621no—
193C22—621no—
194C22C2×D4×D3111yescharacteristic
195C4—42no—
196C4—42no—
197C4—312no—
198C4—621no—
199C4—621no—
200C4—1242no—
201C4—1242no—
202C4#1175849312yes—
203C4Q16×D3112yes—
204C4#41×D3112yes—
205C2—622no—
206C2#1452159811yescharacteristic
207C2C2×Q16×D3111yescharacteristic
208C2C2×#1175849311yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#487C2
socleC22×C31C22×D4
prime radicalC22×D31C2×D4
hypercenter#487D31
hypocenterC31#3170
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #14521452C2semidirect
4. #14521428C2semidirect
5. #14521400C2semidirect
7. #14521141C2semidirect
9. C31×#487C2semidirect
81. C4×D31Q16semidirect
106. #487D31semidirect
139. C31#3170semidirect

Maximal subgroups

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

subgroupindex
#145216042
#145214522
#145214282
#145214002
#145212892
#145211412
C4×C8×D312
C31×#4872
#317031