Finite Group Databasebeta

← #16855118

Subgroups — #16855118

Order 3968.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#14521885C211yesmaximal, characteristic
2#14521625C212yesmaximal, complemented
3#14521620C211yesmaximal, characteristic, complemented
4#14521240C218yesmaximal, complemented
5#14521022C212yesmaximal, complemented
6C31×#410C211yesmaximal, characteristic, complemented
7C2×#41×C31C2211yescharacteristic
8#11758567—22no—
9#11758566—22no—
10#41×D31—24no—
11#11758564—21no—
12#11758561—21no—
13#11758561C2214yes—
14C2×#92495—22no—
15C2×#92495C2212yes—
16C22×Dic62C2211yescharacteristic, complemented
17C31×#117C2212yescomplemented
18#11758521—21no—
19#11758513—21no—
20#11758512C2214yescomplemented
21#11758503C2211yescharacteristic, complemented
22#11758501—22no—
23#11758474—24no—
24#11758463C2218yescomplemented
25#11758452—22no—
26#11758446—22no—
27C31×#99C2214yescomplemented
28#11758394C2218yescomplemented
29#41×C31—22no—
30#41×C31—22no—
31C2×Q8×C31D411yescharacteristic
32C2×D4×C31D411yescharacteristic
33C2×D4×C31C2312yes—
34C22×C4×C31D412yes—
35C22×C4×C31C2311yescharacteristic
36C2×C22⋊D31—24no—
37C2×C22⋊D31C2314yes—
38C22×Dic31C2312yes—
39#92499—24no—
40Q8×D31—24no—
41#92497—24no—
42#92497—28no—
43D4×D31—24no—
44D4×D31—28no—
45#92495—24no—
46#92495—24no—
47#92495—22no—
48#92495—24no—
49#92495—22no—
50#92495—22no—
51#92495—22no—
52C2×D124—22no—
53C2×D124D412yes—
54C2×C4×D31—24no—
55C2×C4×D31D414yes—
56C2×Dic62—22no—
57C2×Dic62—22no—
58C2×Dic62—22no—
59C2×Dic62—21no—
60C2×Dic62—21no—
61C2×Dic62D412yes—
62C4⋊C4×C31—22no—
63C22⋊C4×C31C2314yes—
64C42×C31—22no—
65C22⋊Dic31—24no—
66C22⋊Dic31C2312yes—
67(C2×D31)⋊C4—28no—
68C4⋊Dic31—21no—
69C4⋊Dic31—28no—
70Dic31⋊C4—24no—
71C4×Dic31—21no—
72C23×C31C2×D412yes—
73C23×C31C2411yescharacteristic
74C22×D31—24no—
75C22×D31C2×D414yes—
76Q8×C31—22no—
77D4×C31—24no—
78D4×C31—22no—
79C2×C4×C31—22no—
80C2×C4×C31—22no—
81C2×C4×C31—24no—
82C2×C4×C31C2×D412yes—
83C2×C4×C31C2×D411yescharacteristic
84C2×C4×C31C2×D411yescharacteristic
85C2×C4×C31C2×D412yes—
86C2×C4×C31C2×D412yes—
87C22⋊D31—28no—
88C22⋊D31—24no—
89C22⋊D31—24no—
90C2×Dic31—24no—
91C2×Dic31—22no—
92C2×Dic31—22no—
93C2×Dic31—24no—
94C2×Dic31C2×D414yes—
95D124—24no—
96D124—22no—
97D124—22no—
98C4×D31—28no—
99C4×D31—24no—
100C4×D31—24no—
101Dic62—24no—
102Dic62—22no—
103Dic62—22no—
104Dic62—41no—
105Dic62—42no—
106Dic62—41no—
107#3028—311nomaximal, Sylow
108C22×C31—22no—
109C22×C31—41no—
110C22×C31—42no—
111C22×C31C22×D411yescharacteristic
112C22×C31C22×D412yes—
113C2×D31—24no—
114C2×D31—44no—
115C2×D31—84no—
116C4×C31—22no—
117C4×C31—44no—
118C4×C31#12011yescharacteristic
119C4×C31#12012yes—
120C4×C31#12011yescharacteristic
121Dic31—24no—
122Dic31—42no—
123Dic31—42no—
124#585—311no—
125#536—311no—
126#536—312no—
127#458—318no—
128#410—312no—
129#410D3111yescharacteristic, complemented
130C2×C31—21no—
131C2×C31—22no—
132C2×C31—42no—
133C2×C31C2×#12011yescharacteristic
134D31—44no—
1352−1+4—622no—
1362−1+4—621no—
137H(5,𝔽2)—314no—
138H(5,𝔽2)—622no—
139H(5,𝔽2)—621no—
140C2×#41—312no—
141C2×#41—624no—
142C2×#41—622no—
143C2×#41C2×D3111yescharacteristic
144C22×Q8—311no—
145C4⋊Q8—622no—
146C4⋊Q8—621no—
147#124—622no—
148#124—621no—
149#123—314no—
150#123—318no—
151C22⋊Q8—624no—
152C22⋊Q8—622no—
153#117—311no—
154#117C2×D3112yescomplemented
155#99—318no—
156#99C2×D3114yescomplemented
157C31#302811yescharacteristic, complemented, Sylow
158#41—22no—
159#41—22no—
160#41—624no—
161#41—624no—
162#41—628no—
163#41—624no—
164#41—624no—
165#41—622no—
166#41—622no—
167#41—624no—
168#41—622no—
169#41—622no—
170C2×Q8—312no—
171C2×Q8—622no—
172C2×Q8—624no—
173C2×Q8—621no—
174C2×Q8—621no—
175C2×Q8—622no—
176C2×Q8—622no—
177C2×Q8D12411yescharacteristic
178C2×D4—314no—
179C2×D4—312no—
180C2×D4—622no—
181C2×D4—628no—
182C2×D4—624no—
183C2×D4—624no—
184C2×D4D12411yescharacteristic
185C2×D4C22×D3112yes—
186C22×C4—314no—
187C22×C4—312no—
188C22×C4—624no—
189C22×C4D12412yes—
190C22×C4C22×D3111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—628no—
193C4⋊C4—624no—
194C4⋊C4—621no—
195C22⋊C4—312no—
196C22⋊C4—628no—
197C22⋊C4—624no—
198C22⋊C4C22×D3114yes—
199C42—22no—
200C42—621no—
201C23—314no—
202C23—624no—
203C23C2×D12412yes—
204C23C23×D3111yescharacteristic
205Q8—22no—
206Q8—624no—
207Q8—622no—
208Q8—622no—
209Q8—1241no—
210Q8—1241no—
211Q8—1242no—
212D4—24no—
213D4—22no—
214D4—624no—
215D4—628no—
216D4—624no—
217D4—624no—
218D4—622no—
219D4—622no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—314no—
224C2×C4—628no—
225C2×C4—624no—
226C2×C4—624no—
227C2×C4—624no—
228C2×C4—622no—
229C2×C4—622no—
230C2×C4—624no—
231C2×C4C2×D12412yes—
232C2×C4C2×D12411yescharacteristic
233C2×C4C2×D12411yescharacteristic
234C2×C4D4×D3112yes—
235C2×C4D4×D3112yes—
236C22—22no—
237C22—41no—
238C22—42no—
239C22—624no—
240C22—1244no—
241C22—2484no—
242C22C22×D12411yescharacteristic
243C22C2×D4×D3112yes—
244C4—22no—
245C4—44no—
246C4—624no—
247C4—1242no—
248C4—1242no—
249C4#1175845911yescharacteristic
250C4#1175845911yescharacteristic
251C4#1175845912yes—
252C2—22no—
253C2—21no—
254C2—42no—
255C2—1244no—
256C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC31×#410C2
socleC2×C31C2×#120
prime radical#14521885C2
hypercenter#410D31
hypocenterC31#3028
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #14521625C2semidirect
3. #14521620C2semidirect
4. #14521240C2semidirect
5. #14521022C2semidirect
6. C31×#410C2semidirect
16. C22×Dic62C22semidirect
17. C31×#117C22semidirect
20. #11758512C22semidirect
21. #11758503C22semidirect
24. #11758463C22semidirect
27. C31×#99C22semidirect
28. #11758394C22semidirect
129. #410D31semidirect
154. #117C2×D31semidirect
156. #99C2×D31semidirect
157. C31#3028semidirect

Maximal subgroups

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

subgroupindex
#145218852
#145216252
#145216202
#145212402
#145210222
C31×#4102
#302831