Finite Group Databasebeta

← #16847713

Subgroups — #16847713

Order 3968.

270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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#14521792C211yesmaximal, characteristic, complemented
2#14521237C211yesmaximal, characteristic, complemented
3#14521235C211yesmaximal, characteristic, complemented
4#14521229C211yesmaximal, characteristic, complemented
5#14520843C211yesmaximal, characteristic, complemented
6#14520837C211yesmaximal, characteristic, complemented
7C31×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8#11758561C2211yescharacteristic
9C31×#120C2211yescharacteristic
10#11758530—21no—
11#11758530—21no—
12#11758514—21no—
13#11758514C2211yescharacteristic
14#11758513—21no—
15#11758513C2211yescharacteristic
16#11758486—21no—
17#11758462—21no—
18#11758462—21no—
19#11758462—21no—
20#11758459—21no—
21C31×#100C2211yescharacteristic
22C31×#99C2211yescharacteristic
23#11758422C2211yescharacteristic
24#11758413—21no—
25#11758394—21no—
26#11758394—21no—
27C24×C31D411yescharacteristic, complemented
28C23×D31D411yescharacteristic, complemented
29C2×D4×C31—21no—
30C2×D4×C31C2311yescharacteristic
31C2×C22⋊D31—21no—
32C2×C22⋊D31—21no—
33C2×C22⋊D31—21no—
34C2×C22⋊D31—21no—
35C2×C22⋊D31—21no—
36C2×C22⋊D31—41no—
37#92497—41no—
38D4×D31—41no—
39#92495—21no—
40C2×D124—41no—
41C2×Dic62—21no—
42C2×Dic62D411yescharacteristic, complemented
43M16×C31—21no—
44C22⋊C4×C31—21no—
45C22⋊C4×C31—21no—
46C22⋊C4×C31D411yescharacteristic, complemented
47C22⋊Dic31—21no—
48C22⋊Dic31—21no—
49C22⋊Dic31—21no—
50C22⋊Dic31—21no—
51C22⋊Dic31—21no—
52C22⋊Dic31—41no—
53C22⋊Dic31D411yescharacteristic, complemented
54(C2×D31)⋊C4—21no—
55(C2×D31)⋊C4—21no—
56(C2×D31)⋊C4—21no—
57(C2×D31)⋊C4—41no—
58C4×Dic31—21no—
59C4×Dic31D411yescharacteristic, complemented
60Dic124—41no—
61#92470—41no—
62C23×C31—21no—
63C23×C31—41no—
64C23×C31—81no—
65C23×C31C2×D411yescharacteristic
66C23×C31C2×D411yescharacteristic
67C22×D31—21no—
68C22×D31—21no—
69C22×D31—21no—
70C22×D31—41no—
71C22×D31—81no—
72D4×C31—41no—
73D4×C31—81no—
74C2×C4×C31—21no—
75C2×C4×C31—41no—
76C2×C4×C31C2×D411yescharacteristic
77C22⋊D31—41no—
78C22⋊D31—41no—
79C22⋊D31—41no—
80C22⋊D31—81no—
81C22⋊D31—81no—
82C22⋊D31—81no—
83C22⋊D31—82no—
84C2×Dic31—21no—
85C2×Dic31—21no—
86C2×Dic31—21no—
87C2×Dic31—21no—
88C2×Dic31—21no—
89C2×Dic31—41no—
90D124—21no—
91D124—82no—
92C4×D31—41no—
93Dic62—21no—
94Dic62—41no—
95Dic62—81no—
96C8×C31—41no—
97#2199—311nomaximal, Sylow
98C22×C31—21no—
99C22×C31—21no—
100C22×C31—41no—
101C22×C31—41no—
102C22×C31—41no—
103C22×C31—41no—
104C22×C31—42no—
105C22×C31—41no—
106C22×C31—81no—
107C22×C31#12011yescharacteristic
108C2×D31—41no—
109C2×D31—41no—
110C2×D31—42no—
111C2×D31—41no—
112C2×D31—41no—
113C2×D31—81no—
114C2×D31—81no—
115C2×D31—81no—
116C4×C31—21no—
117C4×C31—41no—
118C4×C31—81no—
119Dic31—41no—
120Dic31—41no—
121Dic31—41no—
122Dic31—41no—
123Dic31—81no—
124H(3,𝔽4)—311no—
125B(4,𝔽2)—311no—
126B(4,𝔽2)—311no—
127#455—311no—
128#354—311no—
129AΣL(1,𝔽16)—311no—
130AΣL(1,𝔽16)D3111yescharacteristic, complemented
131C2×C31—21no—
132C2×C31—41no—
133C2×C31—41no—
134C2×C31—81no—
135C2×C31B(4,𝔽2)11yescharacteristic
136D31—81no—
137D31—81no—
138H(5,𝔽2)—311no—
139#137—621no—
140#124—311no—
141#124—621no—
142#124—621no—
143#124—621no—
144#124—621no—
145#120—311no—
146#120—621no—
147#120—621no—
148#120—621no—
149#120—621no—
150#120C2×D3111yescharacteristic
151#104—621no—
152#100C2×D3111yescharacteristic
153#99—311no—
154#99—621no—
155#99—621no—
156#99C2×D3111yescharacteristic
157C31#219911yescharacteristic, complemented, Sylow
158C24—311no—
159C24D12411yescharacteristic, complemented
160#41—621no—
161#41—1241no—
162C2×Q8—311no—
163C2×Q8—621no—
164C2×D4—21no—
165C2×D4—621no—
166C2×D4—621no—
167C2×D4—621no—
168C2×D4—621no—
169C2×D4—621no—
170C2×D4—1241no—
171C2×D4—1241no—
172C2×D4—1241no—
173C2×D4C22×D3111yescharacteristic
174Q16—1241no—
175SD16—1241no—
176M16—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—311no—
180C22⋊C4—621no—
181C22⋊C4—621no—
182C22⋊C4—621no—
183C22⋊C4—621no—
184C22⋊C4—621no—
185C22⋊C4—621no—
186C22⋊C4—621no—
187C22⋊C4—621no—
188C22⋊C4—1241no—
189C22⋊C4—1241no—
190C22⋊C4D12411yescharacteristic, complemented
191C42—311no—
192C42—621no—
193C23—21no—
194C23—41no—
195C23—81no—
196C23—621no—
197C23—621no—
198C23—621no—
199C23—1241no—
200C23—2481no—
201C23C2×D12411yescharacteristic
202C23D4×D3111yescharacteristic
203Q8—621no—
204Q8—1241no—
205Q8—2481no—
206D4—41no—
207D4—81no—
208D4—621no—
209D4—1241no—
210D4—1241no—
211D4—1241no—
212D4—2482no—
213D4—2481no—
214D4—2481no—
215D4—2481no—
216D4—2482no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—621no—
220C2×C4—621no—
221C2×C4—621no—
222C2×C4—621no—
223C2×C4—621no—
224C2×C4—1241no—
225C2×C4—1241no—
226C2×C4D4×D3111yescharacteristic
227C8—41no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—81no—
237C22—1241no—
238C22—1241no—
239C22—1242no—
240C22—1241no—
241C22—1241no—
242C22—2481no—
243C22—2481no—
244C22—2481no—
245C22#1175845911yescharacteristic
246C4—21no—
247C4—41no—
248C4—81no—
249C4—1241no—
250C4—1241no—
251C4—1241no—
252C4—1241no—
253C4—2481no—
254C2—21no—
255C2—41no—
256C2—41no—
257C2—81no—
258C2—2481no—
259C2—2481no—
260C2#1452123711yescharacteristic
subgroupG/H
centerC2#14521237
commutator, cosocleC2×D4×C31C23
FrattiniC2×D4C22×D31
FittingC31×AΣL(1,𝔽16)C2
socleC2×C31B(4,𝔽2)
hypercenterAΣL(1,𝔽16)D31
hypocenterC31#2199
LCSG ▷ C2×D4×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
1. #14521792C2semidirect
2. #14521237C2semidirect
3. #14521235C2semidirect
4. #14521229C2semidirect
5. #14520843C2semidirect
6. #14520837C2semidirect
7. C31×AΣL(1,𝔽16)C2semidirect
27. C24×C31D4semidirect
28. C23×D31D4semidirect
42. C2×Dic62D4semidirect
46. C22⋊C4×C31D4semidirect
53. C22⋊Dic31D4semidirect
59. C4×Dic31D4semidirect
130. AΣL(1,𝔽16)D31semidirect
157. C31#2199semidirect
159. C24D124semidirect
190. C22⋊C4D124semidirect

Maximal subgroups

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

subgroupindex
#145217922
#145212372
#145212352
#145212292
#145208432
#145208372
C31×AΣL(1,𝔽16)2
#219931