Finite Group Databasebeta

← #16848332

Subgroups — #16848332

Order 3968.

600 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 229 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×#11758469C211yesmaximal, characteristic
2C2×#11758449C211yesmaximal, characteristic
3C2×C4⋊C4×D31C211yesmaximal, characteristic
4#14520944C214yesmaximal, complemented
5#14520920C214yesmaximal
6#14520918C211yesmaximal, characteristic, complemented
7#14520915C211yesmaximal, characteristic, complemented
8#14520914C211yesmaximal, characteristic, complemented
9C31×#384C211yesmaximal, characteristic, complemented
10C22×C4×D31C2212yes—
11C22×C4×D31C2211yescharacteristic
12C2×C4⋊C4×C31C2211yescharacteristic
13C2×C4⋊C4×C31C2211yescharacteristic
14C2×C4⋊C4×C31C2214yes—
15C2×C42×C31C2211yescharacteristic
16C2×(C2×D31)⋊C4C2212yes—
17C2×(C2×D31)⋊C4C2212yes—
18C2×C4⋊Dic31C2211yescharacteristic
19C2×C4⋊Dic31C2211yescharacteristic
20C2×Dic31⋊C4C2212yes—
21C2×Dic31⋊C4C2214yes—
22C2×Dic31⋊C4C2212yes—
23C2×C4×Dic31C2212yes—
24C2×C4×Dic31C2211yescharacteristic
25#11758469—24no—
26C4⋊C4×D31—24no—
27#11758449C418yescomplemented
28C31⋊#95C2214yes—
29C31⋊#95C2214yes—
30C23×D31C2311yescharacteristic
31C22×C4×C31C2312yes—
32C22×C4×C31C2312yes—
33C22×C4×C31C2312yes—
34C22×C4×C31C2311yescharacteristic
35C22×Dic31C2312yes—
36C22×Dic31C2312yes—
37C22×Dic31C2312yes—
38C22×Dic31C2311yescharacteristic
39C2×C4×D31—24no—
40C2×C4×D31—44no—
41C2×C4×D31C2×C414yes—
42C2×C4×D31D414yes—
43C2×C4×D31Q814yes—
44C4⋊C4×C31—22no—
45C4⋊C4×C31—22no—
46C4⋊C4×C31—44no—
47C42×C31C2×C414yescomplemented
48(C2×D31)⋊C4—24no—
49(C2×D31)⋊C4C2×C418yes—
50C4⋊Dic31—22no—
51C4⋊Dic31—22no—
52Dic31⋊C4—24no—
53Dic31⋊C4—44no—
54Dic31⋊C4C2×C418yescomplemented
55C4×Dic31—24no—
56C4×Dic31C2×C414yescomplemented
57C23×C31C2411yescharacteristic
58C22×D31—22no—
59C22×D31—22no—
60C22×D31C22×C412yes—
61C22×D31C2×D412yes—
62C22×D31C2×Q812yes—
63C2×C4×C31—22no—
64C2×C4×C31—22no—
65C2×C4×C31—22no—
66C2×C4×C31—22no—
67C2×C4×C31—24no—
68C2×C4×C31—24no—
69C2×C4×C31C22×C414yes—
70C2×C4×C31C22×C412yes—
71C2×C4×C31C2×D412yes—
72C2×C4×C31C2×Q812yes—
73C2×Dic31—22no—
74C2×Dic31—22no—
75C2×Dic31—22no—
76C2×Dic31—22no—
77C2×Dic31—24no—
78C2×Dic31—24no—
79C2×Dic31C22×C414yes—
80C2×Dic31C22×C412yes—
81C2×Dic31C2×D412yes—
82C2×Dic31C2×Q812yes—
83C4×D31—48no—
84C4×D31C4⋊C4116yescomplemented
85#2317—311nomaximal, Sylow
86C22×C31C23×C411yescharacteristic
87C22×C31C22×D411yescharacteristic
88C22×C31C22×Q811yescharacteristic
89C22×C31H(5,𝔽2)12yes—
90C22×C312−1+412yes—
91C2×D31—24no—
92C2×D31—22no—
93C2×D31—22no—
94C2×D31—24no—
95C2×D31C2×C4⋊C414yes—
96C4×C31—24no—
97C4×C31—42no—
98C4×C31—42no—
99C4×C31C2×C4⋊C414yes—
100Dic31—24no—
101Dic31—42no—
102Dic31—42no—
103Dic31C2×C4⋊C414yes—
104C2×#117—311no—
105C2×#117—311no—
106C22×C4⋊C4—311no—
107#389—314no—
108#385—314no—
109#384—311no—
110#384D3111yescharacteristic, complemented
111#382—311no—
112#382—311no—
113C2×C31C22×C4⋊C411yescharacteristic
114C2×C31#52012yes—
115C2×C31#53511yescharacteristic
116C2×C31#53611yescharacteristic
117C2×C31#54412yes—
118D31—24no—
119C23×C4—311no—
120C23×C4—312no—
121#117—318no—
122#117—624no—
123C2×C4⋊C4—312no—
124C2×C4⋊C4—311no—
125C2×C4⋊C4—312no—
126C2×C4⋊C4—311no—
127C2×C4⋊C4—314no—
128C2×C4⋊C4—624no—
129C2×C4⋊C4C2×D3111yescharacteristic
130C2×C4⋊C4C2×D3111yescharacteristic
131C2×C4⋊C4C2×D3114yes—
132C2×C22⋊C4—312no—
133C2×C22⋊C4—312no—
134C2×C42—311no—
135C2×C42—312no—
136C2×C42C2×D3111yescharacteristic
137#95—314no—
138#95—314no—
139C31#231711yescharacteristic, complemented, Sylow
140C24—311no—
141C22×C4—311no—
142C22×C4—314no—
143C22×C4—314no—
144C22×C4—314no—
145C22×C4—312no—
146C22×C4—312no—
147C22×C4—312no—
148C22×C4—624no—
149C22×C4—1244no—
150C22×C4C22×D3112yes—
151C22×C4C22×D3112yes—
152C22×C4C22×D3112yes—
153C22×C4C22×D3111yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—44no—
157C4⋊C4—318no—
158C4⋊C4—624no—
159C4⋊C4—622no—
160C4⋊C4—622no—
161C4⋊C4—1244no—
162C22⋊C4—318no—
163C22⋊C4—624no—
164C42—314no—
165C42—624no—
166C42C4×D3114yescomplemented
167C23—312no—
168C23—312no—
169C23—312no—
170C23—622no—
171C23—622no—
172C23C23×D3111yescharacteristic
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—312no—
180C2×C4—312no—
181C2×C4—312no—
182C2×C4—3116no—
183C2×C4—314no—
184C2×C4—622no—
185C2×C4—622no—
186C2×C4—622no—
187C2×C4—622no—
188C2×C4—624no—
189C2×C4—624no—
190C2×C4—1248no—
191C2×C4C2×C4×D3114yes—
192C2×C4C2×C4×D3112yes—
193C2×C4D4×D3112yes—
194C2×C4Q8×D3112yes—
195C22—314no—
196C22—622no—
197C22—622no—
198C22—624no—
199C22—624no—
200C22C22×C4×D3111yescharacteristic
201C22C2×D4×D3111yescharacteristic
202C22#1175856112yes—
203C22C2×Q8×D3111yescharacteristic
204C22#1175856412yes—
205C4—24no—
206C4—42no—
207C4—42no—
208C4—314no—
209C4—624no—
210C4—1242no—
211C4—1242no—
212C4C4⋊C4×D3114yes—
213C2—624no—
214C2C2×C4⋊C4×D3111yescharacteristic
215C2#1452153012yes—
216C2#1452161211yescharacteristic
217C2#1452162411yescharacteristic
218C2#1452167612yes—
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#384C2
socle, cosocleC23×C31C24
prime radicalC23×D31C23
hypercenter#384D31
hypocenterC31#2317
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
4. #14520944C2semidirect
6. #14520918C2semidirect
7. #14520915C2semidirect
8. #14520914C2semidirect
9. C31×#384C2semidirect
27. #11758449C4semidirect
47. C42×C31C2×C4semidirect
54. Dic31⋊C4C2×C4semidirect
56. C4×Dic31C2×C4semidirect
84. C4×D31C4⋊C4semidirect
110. #384D31semidirect
139. C31#2317semidirect
166. C42C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584692
C2×#117584492
C2×C4⋊C4×D312
#145209442
#145209202
#145209182
#145209152
#145209142
C31×#3842
#231731