Finite Group Databasebeta

← #16848968

Subgroups — #16848968

Order 3968.

560 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 157 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×C22⋊D31C211yesmaximal, characteristic
2#14520987C211yesmaximal, characteristic, complemented
3#14520984C211yesmaximal, characteristic, complemented
4#14520980C211yesmaximal, characteristic, complemented
5#14520977C212yesmaximal, complemented
6#14520975C211yesmaximal, characteristic, complemented
7#14520964C212yesmaximal
8C4×Dic31⋊C4C212yesmaximal, complemented
9C4×(C2×D31)⋊C4C212yesmaximal, complemented
10C4×C22⋊Dic31C211yesmaximal, characteristic, complemented
11C2×C31×#117C211yesmaximal, characteristic, complemented
12C23×C4×C31C2211yescharacteristic
13C22×C22⋊D31C2211yescharacteristic
14C22×C4×D31C2211yescharacteristic
15C31×#117—24no—
16C2×C4⋊C4×C31C2211yescharacteristic
17C2×C4⋊C4×C31C2211yescharacteristic
18C2×C22⋊C4×C31C2212yes—
19C2×C42×C31C2212yes—
20C2×C22⋊Dic31C2212yes—
21C2×C22⋊Dic31C2211yescharacteristic
22C4×C22⋊D31—24no—
23C2×(C2×D31)⋊C4C2212yes—
24C2×(C2×D31)⋊C4C2212yes—
25C2×(C2×D31)⋊C4C2211yescharacteristic
26C2×Dic31⋊C4C2212yes—
27C2×Dic31⋊C4C2212yes—
28C2×Dic31⋊C4C2211yescharacteristic
29C2×C4×Dic31C2212yes—
30C2×C4×Dic31C2211yescharacteristic
31C31⋊#95C2211yescharacteristic
32C31⋊#95C2211yescharacteristic
33C31⋊#95C2212yes—
34C31⋊#95C2212yes—
35C31⋊#95C2212yes—
36C31⋊#95C2212yes—
37C24×C31C2311yescharacteristic
38C23×D31C2311yescharacteristic
39C22×C4×C31—24no—
40C22×C4×C31D414yes—
41C22×C4×C31C2312yes—
42C22×C4×C31C2312yes—
43C22×C4×C31C2311yescharacteristic
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—22no—
46C2×C22⋊D31—22no—
47C2×C22⋊D31—22no—
48C22×Dic31C2312yes—
49C22×Dic31C2311yescharacteristic
50C22×Dic31C2311yescharacteristic
51C22×Dic31C2311yescharacteristic
52C22×Dic31C2312yes—
53C2×C4×D31—22no—
54C2×C4×D31—24no—
55C4⋊C4×C31—22no—
56C4⋊C4×C31—22no—
57C22⋊C4×C31—24no—
58C42×C31—24no—
59C22⋊Dic31—22no—
60C22⋊Dic31—24no—
61(C2×D31)⋊C4—24no—
62(C2×D31)⋊C4—22no—
63(C2×D31)⋊C4—24no—
64Dic31⋊C4—22no—
65Dic31⋊C4—24no—
66Dic31⋊C4—24no—
67C4×Dic31—22no—
68C4×Dic31—24no—
69C23×C31—22no—
70C23×C31—21no—
71C23×C31—21no—
72C23×C31—22no—
73C23×C31C2×D412yes—
74C23×C31C2411yescharacteristic
75C22×D31—21no—
76C22×D31—22no—
77C22×D31—22no—
78C22×D31—21no—
79C22×D31#4112yes—
80C2×C4×C31—22no—
81C2×C4×C31—28no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31—22no—
88C2×C4×C31C2×D412yes—
89C2×C4×C31C2×D412yes—
90C2×C4×C31#4114yes—
91C22⋊D31—28no—
92C2×Dic31—21no—
93C2×Dic31—22no—
94C2×Dic31—22no—
95C2×Dic31—21no—
96C2×Dic31—21no—
97C2×Dic31—21no—
98C2×Dic31—22no—
99C2×Dic31—22no—
100C2×Dic31—22no—
101C2×Dic31—22no—
102C2×Dic31—22no—
103C2×Dic31#4112yes—
104C2×Dic31#4114yes—
105C4×D31—44no—
106#2395—311nomaximal, Sylow
107C22×C31—22no—
108C22×C31—22no—
109C22×C31—22no—
110C22×C31—42no—
111C22×C31—42no—
112C22×C31C22×D411yescharacteristic
113C22×C31C2×#4112yes—
114C22×C31C2×#4111yescharacteristic
115C22×C31C2×#4111yescharacteristic
116C22×C31C2×#4112yes—
117C2×D31—22no—
118C2×D31—24no—
119C2×D31—41no—
120C2×D31—42no—
121C2×D31—41no—
122C4×C31—22no—
123C4×C31—42no—
124C4×C31—42no—
125C4×C31#12314yes—
126Dic31—22no—
127Dic31—24no—
128Dic31—41no—
129Dic31—42no—
130Dic31—41no—
131C2×C4×D4—311no—
132C2×#117D3111yescharacteristic, complemented
133#400—311no—
134#399—311no—
135#398—311no—
136#397—311no—
137#397—312no—
138#394—312no—
139C4×C4⋊C4—312no—
140C4×C22⋊C4—311no—
141C4×C22⋊C4—312no—
142C2×C31—42no—
143C2×C31C2×#12311yescharacteristic
144C2×C31#52511yescharacteristic
145C2×C31#52912yes—
146C2×C31#52912yes—
147C2×C31#53211yescharacteristic
148D31—42no—
149C22×D4—311no—
150C23×C4—311no—
151C23×C4C2×D3111yescharacteristic
152C4×D4—624no—
153#117—24no—
154C2×C4⋊C4—311no—
155C2×C4⋊C4—312no—
156C2×C4⋊C4—312no—
157C2×C4⋊C4C2×D3111yescharacteristic
158C2×C4⋊C4C2×D3111yescharacteristic
159C2×C22⋊C4—311no—
160C2×C22⋊C4—312no—
161C2×C22⋊C4—311no—
162C2×C22⋊C4—312no—
163C2×C22⋊C4—312no—
164C2×C22⋊C4C2×D3112yes—
165C2×C42—311no—
166C2×C42—312no—
167C2×C42C2×D3112yes—
168#95—312no—
169#95—311no—
170#95—312no—
171#95—311no—
172#95—312no—
173#95—312no—
174C31#239511yescharacteristic, complemented, Sylow
175C24—311no—
176C24C22×D3111yescharacteristic
177C2×D4—622no—
178C2×D4—622no—
179C2×D4—622no—
180C22×C4—24no—
181C22×C4—312no—
182C22×C4—311no—
183C22×C4—312no—
184C22×C4—311no—
185C22×C4—311no—
186C22×C4—624no—
187C22×C4—622no—
188C22×C4C22⋊D3114yes—
189C22×C4C22×D3112yes—
190C22×C4C22×D3112yes—
191C22×C4C22×D3111yescharacteristic
192C22×C4C22×D3111yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—624no—
196C4⋊C4—624no—
197C4⋊C4—622no—
198C22⋊C4—24no—
199C22⋊C4—624no—
200C22⋊C4—624no—
201C22⋊C4—624no—
202C22⋊C4—622no—
203C22⋊C4—622no—
204C42—24no—
205C42—624no—
206C42—622no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—312no—
212C23—622no—
213C23—622no—
214C23—621no—
215C23—621no—
216C23C2×C22⋊D3112yes—
217C23C23×D3111yescharacteristic
218D4—628no—
219C2×C4—22no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—314no—
228C2×C4—312no—
229C2×C4—622no—
230C2×C4—622no—
231C2×C4—621no—
232C2×C4—621no—
233C2×C4—622no—
234C2×C4—622no—
235C2×C4—622no—
236C2×C4—622no—
237C2×C4—622no—
238C2×C4—621no—
239C2×C4—621no—
240C2×C4—1244no—
241C2×C4#9249514yes—
242C2×C4C2×C22⋊D3112yes—
243C2×C4C2×C22⋊D3112yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—622no—
250C22—624no—
251C22—1241no—
252C22—1242no—
253C22—1241no—
254C22C2×#9249512yes—
255C22#41×D3111yescharacteristic
256C22#41×D3111yescharacteristic
257C22#41×D3112yes—
258C22C22×C22⋊D3111yescharacteristic
259C4—22no—
260C4—42no—
261C4—42no—
262C4—622no—
263C4—624no—
264C4—1241no—
265C4—1242no—
266C4—1241no—
267C4#1175850614yes—
268C2—42no—
269C2—1242no—
270C2C2×#1175850611yescharacteristic
271C2#1452155611yescharacteristic
272C2#1452157712yes—
273C2#1452157712yes—
274C2#1452159511yescharacteristic
subgroupG/H
centerC22×C4C22×D31
commutator, socle, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC2×C31×#117C2
prime radicalC22×C22⋊D31C22
hypercenterC2×#117D31
hypocenterC31#2395
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #14520987C2semidirect
3. #14520984C2semidirect
4. #14520980C2semidirect
5. #14520977C2semidirect
6. #14520975C2semidirect
8. C4×Dic31⋊C4C2semidirect
9. C4×(C2×D31)⋊C4C2semidirect
10. C4×C22⋊Dic31C2semidirect
11. C2×C31×#117C2semidirect
132. C2×#117D31semidirect
174. C31#2395semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D312
#145209872
#145209842
#145209802
#145209772
#145209752
#145209642
C4×Dic31⋊C42
C4×(C2×D31)⋊C42
C4×C22⋊Dic312
C2×C31×#1172
#239531