Finite Group Databasebeta

← #16848990

Subgroups — #16848990

Order 3968.

840 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 177 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×C4×D124C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D31C214yesmaximal, complemented
3#14520985C212yesmaximal, complemented
4#14520984C212yesmaximal, complemented
5#14520958C212yesmaximal, complemented
6#14520957C212yesmaximal, complemented
7#14520895C211yesmaximal, characteristic, complemented
8C2×C4×D4×C31C211yesmaximal, characteristic, complemented
9C22×D4×C31C2211yescharacteristic
10C23×C4×C31C2212yes—
11C22×C22⋊D31C2214yes—
12C22×D124C2211yescharacteristic
13C22×C4×D31C2212yes—
14C4×D4×C31—24no—
15C2×C4⋊C4×C31C2211yescharacteristic
16C2×C22⋊C4×C31C2212yes—
17C2×C42×C31C2211yescharacteristic
18C2×C22⋊Dic31C2214yes—
19C4×C22⋊D31—216no—
20C2×(C2×D31)⋊C4C2212yes—
21C2×(C2×D31)⋊C4C2214yes—
22C2×C4⋊Dic31C2211yescharacteristic
23C2×Dic31⋊C4C2214yes—
24C2×C4×Dic31C2212yes—
25C4×D124—24no—
26C31⋊#95C2212yes—
27C31⋊#95C2212yes—
28C24×C31C2312yes—
29C23×D31C2312yes—
30C2×D4×C31—24no—
31C2×D4×C31—22no—
32C22×C4×C31—28no—
33C22×C4×C31—24no—
34C22×C4×C31C2312yes—
35C22×C4×C31C2311yescharacteristic
36C22×C4×C31C2311yescharacteristic
37C22×C4×C31C2311yescharacteristic
38C2×C22⋊D31—28no—
39C2×C22⋊D31—28no—
40C2×C22⋊D31—28no—
41C22×Dic31C2312yes—
42C22×Dic31C2314yes—
43C2×D124—24no—
44C2×D124—22no—
45C2×C4×D31—24no—
46C2×C4×D31—28no—
47C4⋊C4×C31—22no—
48C22⋊C4×C31—24no—
49C42×C31D414yescomplemented
50C22⋊Dic31—28no—
51(C2×D31)⋊C4—24no—
52(C2×D31)⋊C4—28no—
53C4⋊Dic31—22no—
54Dic31⋊C4—28no—
55C4×Dic31D418yescomplemented
56C23×C31—24no—
57C23×C31—22no—
58C23×C31—24no—
59C23×C31—22no—
60C23×C31#4114yes—
61C23×C31C2411yescharacteristic
62C22×D31—22no—
63C22×D31—24no—
64C22×D31—22no—
65C22×D31—24no—
66C22×D31#4114yes—
67D4×C31—28no—
68C2×C4×C31—22no—
69C2×C4×C31—22no—
70C2×C4×C31—24no—
71C2×C4×C31—22no—
72C2×C4×C31—48no—
73C2×C4×C31C2×D414yes—
74C2×C4×C31C2×D412yes—
75C2×C4×C31C2×D412yes—
76C2×C4×C31C2×D412yes—
77C22⋊D31—232no—
78C2×Dic31—24no—
79C2×Dic31—24no—
80C2×Dic31—24no—
81C2×Dic31—22no—
82C2×Dic31C2×D418yes—
83D124—28no—
84C4×D31—48no—
85#2397—311nomaximal, Sylow
86C22×C31—28no—
87C22×C31—24no—
88C22×C31—42no—
89C22×C31—44no—
90C22×C31—42no—
91C22×C31C22×D412yes—
92C22×C31C22×D411yescharacteristic
93C22×C31C2×#4112yes—
94C22×C31C2×#4112yes—
95C2×D31—24no—
96C2×D31—28no—
97C2×D31—44no—
98C2×D31—42no—
99C2×D31—42no—
100C4×C31—22no—
101C4×C31—22no—
102C4×C31—42no—
103C4×C31#12714yes—
104Dic31—28no—
105Dic31—42no—
106C2×C4×D4—314no—
107C2×C4×D4—311no—
108C2×C4×D4D3111yescharacteristic, complemented
109#399—312no—
110#399—312no—
111#392—312no—
112#392—312no—
113#376—311no—
114C2×C31—44no—
115C2×C31#52511yescharacteristic
116C2×C31#52514yes—
117C2×C31#52511yescharacteristic
118C2×C31C2×#12711yescharacteristic
119D31—44no—
120C22×D4—314no—
121C22×D4—311no—
122C22×D4C2×D3111yescharacteristic
123C23×C4—312no—
124C23×C4C2×D3112yes—
125C4×D4—24no—
126C4×D4—6216no—
127C4×D4—624no—
128C2×C4⋊C4—314no—
129C2×C4⋊C4—311no—
130C2×C4⋊C4C2×D3111yescharacteristic
131C2×C22⋊C4—314no—
132C2×C22⋊C4—314no—
133C2×C22⋊C4—312no—
134C2×C22⋊C4C2×D3112yes—
135C2×C42—312no—
136C2×C42C2×D3111yescharacteristic
137#95—312no—
138#95—312no—
139C31#239711yescharacteristic, complemented, Sylow
140C24—312no—
141C24C22×D3112yes—
142C2×D4—24no—
143C2×D4—22no—
144C2×D4—624no—
145C2×D4—628no—
146C2×D4—628no—
147C2×D4—628no—
148C2×D4—622no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—314no—
152C22×C4—312no—
153C22×C4—628no—
154C22×C4—624no—
155C22×C4C22×D3112yes—
156C22×C4C22×D3111yescharacteristic
157C22×C4C22×D3111yescharacteristic
158C22×C4C22×D3111yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—628no—
161C4⋊C4—622no—
162C22⋊C4—24no—
163C22⋊C4—628no—
164C22⋊C4—628no—
165C22⋊C4—624no—
166C42—318no—
167C42C22⋊D3114yescomplemented
168C23—22no—
169C23—24no—
170C23—24no—
171C23—22no—
172C23—314no—
173C23—624no—
174C23—624no—
175C23—622no—
176C23—622no—
177C23#9249514yes—
178C23C23×D3111yescharacteristic
179D4—28no—
180D4—628no—
181D4—6232no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—318no—
188C2×C4—624no—
189C2×C4—624no—
190C2×C4—624no—
191C2×C4—622no—
192C2×C4—1248no—
193C2×C4D4×D3114yes—
194C2×C4C2×C22⋊D3112yes—
195C2×C4C2×C22⋊D3112yes—
196C2×C4C2×C22⋊D3112yes—
197C22—28no—
198C22—24no—
199C22—42no—
200C22—44no—
201C22—42no—
202C22—624no—
203C22—628no—
204C22—1244no—
205C22—1242no—
206C22—1242no—
207C22C2×#9249512yes—
208C22C2×D4×D3112yes—
209C22#41×D3112yes—
210C22C22×C22⋊D3111yescharacteristic
211C4—22no—
212C4—22no—
213C4—42no—
214C4—628no—
215C4—1242no—
216C4#1175851714yes—
217C2—44no—
218C2—1244no—
219C2#1452155611yescharacteristic
220C2#1452155814yes—
221C2#1452156011yescharacteristic
222C2C2×#1175851711yescharacteristic
subgroupG/H
centerC22×C4C22×D31
commutator, socle, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC2×C4×D4×C31C2
hypercenterC2×C4×D4D31
hypocenterC31#2397
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D124C2semidirect
2. C2×C4×C22⋊D31C2semidirect
3. #14520985C2semidirect
4. #14520984C2semidirect
5. #14520958C2semidirect
6. #14520957C2semidirect
7. #14520895C2semidirect
8. C2×C4×D4×C31C2semidirect
49. C42×C31D4semidirect
55. C4×Dic31D4semidirect
108. C2×C4×D4D31semidirect
139. C31#2397semidirect
167. C42C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D1242
C2×C4×C22⋊D312
#145209852
#145209842
#145209582
#145209572
#145208952
C2×C4×D4×C312
#239731