Finite Group Databasebeta

← #16854043

Subgroups — #16854043

Order 3968.

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

normal subgroups only

This group has 688 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×#11758463C212yesmaximal, complemented
2C2×#11758507C211yesmaximal, characteristic
3C22×C4×Dic31C211yesmaximal, characteristic, complemented
4#14520950C212yesmaximal, complemented
5#14520948C212yesmaximal, complemented
6#14520941C214yesmaximal, complemented
7#14520926C212yesmaximal, complemented
8C31×#380C211yesmaximal, characteristic, complemented
9C23×C4×C31C2211yescharacteristic
10C22×C22⋊D31C2212yes—
11C23×Dic31C2211yescharacteristic
12C23×Dic31C2211yescharacteristic
13C22×D124C2211yescharacteristic
14C2×C4⋊C4×C31C2212yes—
15C2×C22⋊C4×C31C2212yes—
16#11758507—24no—
17C2×(C2×D31)⋊C4C2214yes—
18C2×(C2×D31)⋊C4C2212yes—
19C2×(C2×D31)⋊C4C2214yes—
20C2×C4⋊Dic31C2214yes—
21C2×C4⋊Dic31C2211yescharacteristic
22C2×C4×Dic31—24no—
23C2×C4×Dic31C2212yes—
24C2×C4×Dic31C2212yes—
25#11758463—28no—
26C31×#95C2212yes—
27C31⋊#95C2214yes—
28C24×C31C2311yescharacteristic
29C23×D31C2312yes—
30C22×C4×C31—24no—
31C22×C4×C31D414yes—
32C22×C4×C31C2314yes—
33C22×C4×C31C2311yescharacteristic
34C22×C4×C31C2311yescharacteristic
35C2×C22⋊D31—24no—
36C2×C22⋊D31—24no—
37C2×C22⋊D31—24no—
38C22×Dic31—22no—
39C22×Dic31—22no—
40C22×Dic31—22no—
41C22×Dic31—24no—
42C22×Dic31D414yescomplemented
43C22×Dic31C2312yes—
44C22×Dic31C2312yes—
45C22×Dic31C2312yes—
46C2×D124—22no—
47C2×D124—24no—
48C4⋊C4×C31—24no—
49C22⋊C4×C31—24no—
50(C2×D31)⋊C4—24no—
51(C2×D31)⋊C4—28no—
52(C2×D31)⋊C4—28no—
53C4⋊Dic31—22no—
54C4⋊Dic31—28no—
55C4×Dic31—24no—
56C4×Dic31—24no—
57C23×C31—22no—
58C23×C31—22no—
59C23×C31C2×D412yes—
60C23×C31C2×D412yes—
61C23×C31#4112yes—
62C23×C31C2411yescharacteristic
63C22×D31—22no—
64C22×D31—22no—
65C22×D31—24no—
66C22×D31—24no—
67C22×D31—22no—
68C2×C4×C31—22no—
69C2×C4×C31—28no—
70C2×C4×C31—24no—
71C2×C4×C31—24no—
72C2×C4×C31—24no—
73C2×C4×C31C2×D412yes—
74C2×C4×C31C2×D412yes—
75C2×C4×C31#4114yes—
76C22⋊D31—48no—
77C2×Dic31—28no—
78C2×Dic31—28no—
79C2×Dic31—22no—
80C2×Dic31—22no—
81C2×Dic31—22no—
82C2×Dic31—22no—
83C2×Dic31—22no—
84C2×Dic31—22no—
85C2×Dic31—22no—
86C2×Dic31C2×D414yes—
87C2×Dic31#4114yes—
88D124—44no—
89#2854—311nomaximal, Sylow
90C22×C31—22no—
91C22×C31—22no—
92C22×C31—24no—
93C22×C31—24no—
94C22×C31#12114yes—
95C22×C31C22×D411yescharacteristic
96C22×C31C22×D411yescharacteristic
97C22×C31C2×#4112yes—
98C22×C31C2×#4111yescharacteristic
99C22×C31C2×#4112yes—
100C2×D31—42no—
101C2×D31—42no—
102C2×D31—42no—
103C2×D31—44no—
104C2×D31—44no—
105C4×C31—22no—
106C4×C31—44no—
107C4×C31#12314yescomplemented
108Dic31—24no—
109Dic31—24no—
110Dic31—42no—
111C2×#123—312no—
112C2×#121—311no—
113C22×C42—311no—
114#390—312no—
115#389—312no—
116#388—314no—
117#386—312no—
118#380D3111yescharacteristic, complemented
119C2×C31—24no—
120C2×C31C2×#12111yescharacteristic
121C2×C31C2×#12311yescharacteristic
122C2×C31#52511yescharacteristic
123C2×C31#52912yes—
124C2×C31#53212yes—
125D31—82no—
126C22×D4—312no—
127C22×D4—311no—
128C23×C4—311no—
129C23×C4—311no—
130C23×C4C2×D3111yescharacteristic
131#123—628no—
132#121—624no—
133C2×C4⋊C4—311no—
134C2×C4⋊C4—314no—
135C2×C4⋊C4C2×D3112yes—
136C2×C22⋊C4—312no—
137C2×C22⋊C4—314no—
138C2×C22⋊C4—314no—
139C2×C22⋊C4C2×D3112yes—
140C2×C42—312no—
141C2×C42—312no—
142C2×C42—624no—
143#95—314no—
144#95C2×D3112yes—
145C31#285411yescharacteristic, complemented, Sylow
146C24—312no—
147C24C22×D3111yescharacteristic
148C2×D4—624no—
149C2×D4—624no—
150C2×D4—624no—
151C2×D4—622no—
152C2×D4—624no—
153C22×C4—24no—
154C22×C4—314no—
155C22×C4—312no—
156C22×C4—312no—
157C22×C4—312no—
158C22×C4—624no—
159C22×C4—622no—
160C22×C4—622no—
161C22×C4—622no—
162C22×C4D12414yes—
163C22×C4C22×D3114yes—
164C22×C4C22×D3111yescharacteristic
165C22×C4C22×D3111yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—628no—
168C4⋊C4—622no—
169C22⋊C4—24no—
170C22⋊C4—628no—
171C22⋊C4—628no—
172C22⋊C4—624no—
173C42—624no—
174C42—624no—
175C23—22no—
176C23—22no—
177C23—622no—
178C23—622no—
179C23—624no—
180C23—622no—
181C23—624no—
182C23C2×D12412yes—
183C23D4×D3112yes—
184C23#9249912yes—
185C23C23×D3111yescharacteristic
186D4—1248no—
187D4—1244no—
188C2×C4—22no—
189C2×C4—28no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—314no—
194C2×C4—314no—
195C2×C4—628no—
196C2×C4—628no—
197C2×C4—622no—
198C2×C4—622no—
199C2×C4—622no—
200C2×C4—622no—
201C2×C4—622no—
202C2×C4—622no—
203C2×C4—622no—
204C2×C4C2×D12412yes—
205C2×C4C2×D12412yes—
206C2×C4#9249714yes—
207C22—22no—
208C22—22no—
209C22—24no—
210C22—24no—
211C22—1242no—
212C22—1242no—
213C22—1242no—
214C22—1244no—
215C22—1244no—
216C22#1175847214yes—
217C22C22×D12411yescharacteristic
218C22C2×D4×D3111yescharacteristic
219C22C2×#9249712yes—
220C22C2×#9249911yescharacteristic
221C22#41×D3112yes—
222C4—22no—
223C4—44no—
224C4—624no—
225C4—624no—
226C4—1242no—
227C4#1175846314yescomplemented
228C2—24no—
229C2—2482no—
230C2C2×#1175847211yescharacteristic
231C2C2×#1175846311yescharacteristic
232C2#1452156111yescharacteristic
233C2#1452158512yes—
234C2#1452159412yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#380C2
prime radicalC2×#11758507C2
hypercenter#380D31
hypocenterC31#2854
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×#11758463C2semidirect
3. C22×C4×Dic31C2semidirect
4. #14520950C2semidirect
5. #14520948C2semidirect
6. #14520941C2semidirect
7. #14520926C2semidirect
8. C31×#380C2semidirect
42. C22×Dic31D4semidirect
107. C4×C31#123semidirect
118. #380D31semidirect
145. C31#2854semidirect
227. C4#11758463semidirect

Maximal subgroups

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

subgroupindex
C2×#117584632
C2×#117585072
C22×C4×Dic312
#145209502
#145209482
#145209412
#145209262
C31×#3802
#285431