Finite Group Databasebeta

← #16848425

Subgroups — #16848425

Order 3968.

612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 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×#11758524C211yesmaximal, characteristic
2C2×Q8×Dic31C211yesmaximal, characteristic, complemented
3C2×#11758470C211yesmaximal, characteristic
4#14520951C211yesmaximal, characteristic
5#14520938C214yesmaximal, complemented
6#14520923C214yesmaximal, complemented
7#14520918C211yesmaximal, characteristic, complemented
8#14520895C211yesmaximal, characteristic, complemented
9C31×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C22×D124C2211yescharacteristic
12C22×C4×D31C2212yes—
13C2×C4⋊C4×C31C2211yescharacteristic
14C2×C42×C31C2211yescharacteristic
15#11758524C418yescomplemented
16Q8×Dic31—24no—
17C2×(C2×D31)⋊C4C2212yes—
18C2×(C2×D31)⋊C4C2214yes—
19C2×(C2×D31)⋊C4C2214yes—
20C2×C4⋊Dic31C2211yescharacteristic
21C2×C4⋊Dic31C2212yes—
22C2×Dic31⋊C4C2214yes—
23C2×C4×Dic31C2212yes—
24C2×C4×Dic31C2211yescharacteristic
25C2×C4×Dic31C2211yescharacteristic
26#11758470—24no—
27C31×#95C2214yes—
28C31⋊#95C2212yes—
29C31⋊#95C2212yes—
30C23×D31C2312yes—
31C2×Q8×C31—22no—
32C2×Q8×C31—22no—
33C2×Q8×C31C2×C414yescomplemented
34C22×C4×C31C2312yes—
35C22×C4×C31C2312yes—
36C22×C4×C31C2312yes—
37C22×C4×C31C2311yescharacteristic
38C22×Dic31C2312yes—
39C22×Dic31C2312yes—
40C22×Dic31C2311yescharacteristic
41C22×Dic31C2311yescharacteristic
42C2×D124—22no—
43C2×D124—22no—
44C2×D124C2×C414yes—
45C2×C4×D31—24no—
46C2×C4×D31—44no—
47C4⋊C4×C31—22no—
48C42×C31—22no—
49(C2×D31)⋊C4—24no—
50(C2×D31)⋊C4—44no—
51(C2×D31)⋊C4C2×C4116yescomplemented
52C4⋊Dic31—22no—
53C4⋊Dic31—24no—
54Dic31⋊C4—44no—
55C4×Dic31—24no—
56C4×Dic31C2×C414yescomplemented
57C4×Dic31D414yes—
58C23×C31C2411yescharacteristic
59C22×D31—22no—
60C22×D31—22no—
61C22×D31—24no—
62C22×D31—24no—
63C22×D31C22×C414yes—
64Q8×C31—28no—
65C2×C4×C31—22no—
66C2×C4×C31—22no—
67C2×C4×C31—22no—
68C2×C4×C31—22no—
69C2×C4×C31—24no—
70C2×C4×C31—22no—
71C2×C4×C31—22no—
72C2×C4×C31C22×C414yes—
73C2×C4×C31C22×C412yes—
74C2×C4×C31C2×D412yes—
75C2×C4×C31#4112yes—
76C2×Dic31—22no—
77C2×Dic31—22no—
78C2×Dic31—22no—
79C2×Dic31—22no—
80C2×Dic31—22no—
81C2×Dic31—24no—
82C2×Dic31C22×C414yes—
83C2×Dic31C2×D412yes—
84C2×Dic31C2×D412yes—
85D124—28no—
86C4×D31—48no—
87#2326—311nomaximal, Sylow
88C22×C31C23×C411yescharacteristic
89C22×C31C22×D411yescharacteristic
90C22×C31C2×#4111yescharacteristic
91C22×C31H(5,𝔽2)12yes—
92C22×C312−1+412yes—
93C2×D31—24no—
94C2×D31—28no—
95C2×D31—42no—
96C2×D31—42no—
97C2×D31—44no—
98C4×C31—24no—
99C4×C31—42no—
100C4×C31—42no—
101C4×C31C4×D414yes—
102Dic31—24no—
103Dic31—22no—
104Dic31—22no—
105Dic31—42no—
106C2×#124—311no—
107C2×C4×Q8—311no—
108C2×C4×D4—311no—
109#391D3111yescharacteristic, complemented
110#390—311no—
111#388—314no—
112#385—314no—
113#384—311no—
114#376—311no—
115C2×C31C2×C4×D411yescharacteristic
116C2×C31#52012yes—
117C2×C31#53511yescharacteristic
118C2×C31#53611yescharacteristic
119C2×C31#54212yes—
120D31—44no—
121C22×Q8C2×D3111yescharacteristic
122C22×D4—311no—
123C23×C4—312no—
124#124—318no—
125C4×Q8—624no—
126C4×D4—624no—
127C2×C4⋊C4—311no—
128C2×C4⋊C4—312no—
129C2×C4⋊C4—314no—
130C2×C4⋊C4C2×D3111yescharacteristic
131C2×C22⋊C4—314no—
132C2×C22⋊C4—312no—
133C2×C22⋊C4—314no—
134C2×C42—311no—
135C2×C42—311no—
136C2×C42—312no—
137C2×C42C2×D3111yescharacteristic
138#95—312no—
139#95—312no—
140#95C2×D3114yes—
141C31#232611yescharacteristic, complemented, Sylow
142C24—312no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8C4×D3114yescomplemented
146C2×D4—314no—
147C2×D4—622no—
148C2×D4—622no—
149C22×C4—311no—
150C22×C4—312no—
151C22×C4—311no—
152C22×C4—312no—
153C22×C4—624no—
154C22×C4—1244no—
155C22×C4C22×D3112yes—
156C22×C4C22×D3112yes—
157C22×C4C22×D3112yes—
158C22×C4C22×D3111yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—624no—
161C4⋊C4—622no—
162C4⋊C4—1244no—
163C22⋊C4—3116no—
164C22⋊C4—624no—
165C22⋊C4—1244no—
166C42—22no—
167C42—314no—
168C42—314no—
169C42—624no—
170C23—314no—
171C23—622no—
172C23—622no—
173C23—624no—
174C23—624no—
175C23C23×D3111yescharacteristic
176Q8—28no—
177D4—628no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—314no—
186C2×C4—312no—
187C2×C4—312no—
188C2×C4—622no—
189C2×C4—622no—
190C2×C4—622no—
191C2×C4—622no—
192C2×C4—622no—
193C2×C4—624no—
194C2×C4—1248no—
195C2×C4C2×C4×D3114yes—
196C2×C4C2×C4×D3112yes—
197C2×C4D4×D3112yes—
198C2×C4#9249712yes—
199C22—624no—
200C22—628no—
201C22—1242no—
202C22—1242no—
203C22—1244no—
204C22C22×C4×D3111yescharacteristic
205C22C2×D4×D3111yescharacteristic
206C22C2×#9249711yescharacteristic
207C22#1175856412yes—
208C22#1175856612yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—622no—
213C4—622no—
214C4—624no—
215C4—1242no—
216C4(C22⋊D31)⋊C414yes—
217C2—1244no—
218C2C2×(C22⋊D31)⋊C411yescharacteristic
219C2#1452153112yes—
220C2#1452161711yescharacteristic
221C2#1452162411yescharacteristic
222C2#1452167212yes—
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#391C2
socle, cosocleC23×C31C24
prime radicalC22×D124C22
hypercenter#391D31
hypocenterC31#2326
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic31C2semidirect
5. #14520938C2semidirect
6. #14520923C2semidirect
7. #14520918C2semidirect
8. #14520895C2semidirect
9. C31×#391C2semidirect
15. #11758524C4semidirect
33. C2×Q8×C31C2×C4semidirect
51. (C2×D31)⋊C4C2×C4semidirect
56. C4×Dic31C2×C4semidirect
109. #391D31semidirect
141. C31#2326semidirect
145. C2×Q8C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585242
C2×Q8×Dic312
C2×#117584702
#145209512
#145209382
#145209232
#145209182
#145208952
C31×#3912
#232631