Finite Group Databasebeta

← #16848423

Subgroups — #16848423

Order 3968.

612 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 205 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×#11758513C211yesmaximal, characteristic
2C2×C4×Dic62C211yesmaximal, characteristic, complemented
3#14520954C211yesmaximal, characteristic, complemented
4#14520952C211yesmaximal, characteristic
5#14520937C214yesmaximal, complemented
6#14520922C214yesmaximal, complemented
7#14520918C211yesmaximal, characteristic, complemented
8#14520895C211yesmaximal, characteristic, complemented
9C2×C4×D4×C31C211yesmaximal, characteristic
10C22×D4×C31C2211yescharacteristic
11C23×C4×C31C2212yes—
12C22×Dic62C2211yescharacteristic
13C4×D4×C31—24no—
14C2×C4⋊C4×C31C2211yescharacteristic
15C2×C22⋊C4×C31C2212yes—
16C2×C42×C31C2211yescharacteristic
17C2×C22⋊Dic31C2214yes—
18C2×C22⋊Dic31C2214yes—
19#11758513C418yescomplemented
20C2×C4⋊Dic31C2211yescharacteristic
21C2×Dic31⋊C4C2212yes—
22C2×Dic31⋊C4C2214yes—
23C2×C4×Dic31C2211yescharacteristic
24C2×C4×Dic31C2212yes—
25C2×C4×Dic31C2211yescharacteristic
26C4×Dic62—24no—
27C31⋊#95C2212yes—
28C31⋊#95C2212yes—
29C31⋊#95C2214yes—
30C24×C31C2312yes—
31C2×D4×C31—22no—
32C2×D4×C31—22no—
33C2×D4×C31C2×C414yes—
34C22×C4×C31—24no—
35C22×C4×C31—44no—
36C22×C4×C31C2312yes—
37C22×C4×C31C2311yescharacteristic
38C22×C4×C31C2311yescharacteristic
39C22×C4×C31C2311yescharacteristic
40C22×Dic31C2312yes—
41C22×Dic31C2312yes—
42C22×Dic31C2312yes—
43C22×Dic31C2312yes—
44C2×Dic62—22no—
45C2×Dic62—22no—
46C2×Dic62C2×C414yescomplemented
47C4⋊C4×C31—22no—
48C22⋊C4×C31—24no—
49C42×C31D414yes—
50C22⋊Dic31—44no—
51C22⋊Dic31C2×C4116yescomplemented
52C4⋊Dic31—22no—
53Dic31⋊C4—24no—
54Dic31⋊C4—44no—
55C4×Dic31—24no—
56C4×Dic31—22no—
57C4×Dic31C2×C414yescomplemented
58C23×C31—22no—
59C23×C31—22no—
60C23×C31—24no—
61C23×C31—24no—
62C23×C31C22×C414yes—
63C23×C31C2411yescharacteristic
64D4×C31—28no—
65C2×C4×C31—22no—
66C2×C4×C31—22no—
67C2×C4×C31—22no—
68C2×C4×C31—24no—
69C2×C4×C31—48no—
70C2×C4×C31C22×C412yes—
71C2×C4×C31C2×D412yes—
72C2×C4×C31C2×D412yes—
73C2×C4×C31C2×D412yes—
74C2×C4×C31#4112yes—
75C2×Dic31—22no—
76C2×Dic31—22no—
77C2×Dic31—22no—
78C2×Dic31—22no—
79C2×Dic31—22no—
80C2×Dic31—22no—
81C2×Dic31—24no—
82C2×Dic31—22no—
83C2×Dic31—22no—
84C2×Dic31C22×C414yes—
85C2×Dic31C22×C414yes—
86Dic62—28no—
87#2326—311nomaximal, Sylow
88C22×C31—24no—
89C22×C31—28no—
90C22×C31—42no—
91C22×C31—42no—
92C22×C31—44no—
93C22×C31C23×C411yescharacteristic
94C22×C31C22×D411yescharacteristic
95C22×C31C2×#4111yescharacteristic
96C22×C31H(5,𝔽2)12yes—
97C22×C312−1+412yes—
98C4×C31—22no—
99C4×C31—22no—
100C4×C31—42no—
101C4×C31C4×D414yes—
102Dic31—24no—
103Dic31—24no—
104Dic31—42no—
105Dic31—42no—
106C2×#124—311no—
107C2×C4×Q8—311no—
108C2×C4×D4D3111yescharacteristic
109#391—311no—
110#390—311no—
111#388—314no—
112#385—314no—
113#384—311no—
114#376—311no—
115C2×C31—44no—
116C2×C31C2×C4×D411yescharacteristic
117C2×C31#52012yes—
118C2×C31#53511yescharacteristic
119C2×C31#53611yescharacteristic
120C2×C31#54212yes—
121C22×Q8—311no—
122C22×D4C2×D3111yescharacteristic
123C23×C4C2×D3112yes—
124#124—318no—
125C4×Q8—624no—
126C4×D4—24no—
127C2×C4⋊C4—311no—
128C2×C4⋊C4—312no—
129C2×C4⋊C4—314no—
130C2×C4⋊C4C2×D3111yescharacteristic
131C2×C22⋊C4—314no—
132C2×C22⋊C4—314no—
133C2×C22⋊C4C2×D3112yes—
134C2×C42—311no—
135C2×C42—312no—
136C2×C42—311no—
137C2×C42C2×D3111yescharacteristic
138#95—312no—
139#95—312no—
140#95—314no—
141C31#232611yescharacteristic, complemented, Sylow
142C24C22×D3112yes—
143C2×Q8—314no—
144C2×Q8—622no—
145C2×Q8—622no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4C4×D3114yes—
149C22×C4—24no—
150C22×C4—44no—
151C22×C4—312no—
152C22×C4—312no—
153C22×C4—312no—
154C22×C4—312no—
155C22×C4C22×D3112yes—
156C22×C4C22×D3111yescharacteristic
157C22×C4C22×D3111yescharacteristic
158C22×C4C22×D3111yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—624no—
161C4⋊C4—622no—
162C4⋊C4—1244no—
163C22⋊C4—24no—
164C22⋊C4—3116no—
165C22⋊C4—1244no—
166C42—314no—
167C42—622no—
168C42—624no—
169C42C22⋊D3114yes—
170C23—22no—
171C23—24no—
172C23—24no—
173C23—22no—
174C23C2×C4×D3114yes—
175C23C23×D3111yescharacteristic
176Q8—628no—
177D4—28no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—48no—
183C2×C4—314no—
184C2×C4—314no—
185C2×C4—622no—
186C2×C4—622no—
187C2×C4—622no—
188C2×C4—622no—
189C2×C4—622no—
190C2×C4—622no—
191C2×C4—624no—
192C2×C4—622no—
193C2×C4—622no—
194C2×C4C2×C4×D3112yes—
195C2×C4#9249512yes—
196C2×C4C2×C22⋊D3112yes—
197C2×C4C2×C22⋊D3112yes—
198C2×C4C2×C22⋊D3112yes—
199C22—24no—
200C22—28no—
201C22—42no—
202C22—42no—
203C22—44no—
204C22C22×C4×D3111yescharacteristic
205C22C2×#9249511yescharacteristic
206C22#1175856112yes—
207C22#1175856712yes—
208C22C22×C22⋊D3111yescharacteristic
209C4—22no—
210C4—22no—
211C4—42no—
212C4—624no—
213C4—624no—
214C4—1242no—
215C4—1242no—
216C4C4×C22⋊D3114yes—
217C2—44no—
218C2C2×C4×C22⋊D3111yescharacteristic
219C2#1452152812yes—
220C2#1452161611yescharacteristic
221C2#1452162011yescharacteristic
222C2#1452166412yes—
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC2×C4×D4×C31C2
socle, cosocleC23×C31C24
prime radicalC22×D4×C31C22
hypercenterC2×C4×D4D31
hypocenterC31#2326
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic62C2semidirect
3. #14520954C2semidirect
5. #14520937C2semidirect
6. #14520922C2semidirect
7. #14520918C2semidirect
8. #14520895C2semidirect
19. #11758513C4semidirect
46. C2×Dic62C2×C4semidirect
51. C22⋊Dic31C2×C4semidirect
57. C4×Dic31C2×C4semidirect
141. C31#2326semidirect

Maximal subgroups

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

subgroupindex
C2×#117585132
C2×C4×Dic622
#145209542
#145209522
#145209372
#145209222
#145209182
#145208952
C2×C4×D4×C312
#232631