Finite Group Databasebeta

← #16847703

Subgroups — #16847703

Order 3968.

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

normal subgroups only

This group has 258 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#14521730C211yesmaximal, characteristic, complemented
2#14521240C211yesmaximal, characteristic, complemented
3#14521235C211yesmaximal, characteristic
4#14521229C211yesmaximal, characteristic, complemented
5#14520843C211yesmaximal, characteristic, complemented
6#14520840C211yesmaximal, characteristic, complemented
7C31×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8#11758561C2211yescharacteristic
9C31×#120C2211yescharacteristic
10#11758530—21no—
11C2×C22⋊Dic31—21no—
12#11758513C2211yescharacteristic
13#11758512—21no—
14#11758512C2211yescharacteristic
15D4×Dic31—21no—
16#11758486—21no—
17#11758463—21no—
18#11758456—21no—
19#11758455—21no—
20#11758454—21no—
21C31×#100C2211yescharacteristic
22C31×#99C2211yescharacteristic
23#11758422C2211yescharacteristic
24#11758413—21no—
25#11758394—21no—
26#11758394—21no—
27C24×C31D411yescharacteristic
28C2×D4×C31—21no—
29C2×D4×C31C2311yescharacteristic
30C2×C22⋊D31—21no—
31C2×C22⋊D31—21no—
32C2×C22⋊D31—41no—
33C22×Dic31—21no—
34C22×Dic31—21no—
35C22×Dic31D411yescharacteristic, complemented
36#92497—41no—
37D4×D31—41no—
38#92495—21no—
39C2×Dic62D411yescharacteristic, complemented
40M16×C31—21no—
41C22⋊C4×C31—21no—
42C22⋊C4×C31—21no—
43C22⋊C4×C31D411yescharacteristic
44C22⋊Dic31—21no—
45C22⋊Dic31—21no—
46C22⋊Dic31—21no—
47C22⋊Dic31—21no—
48C22⋊Dic31—21no—
49C22⋊Dic31—21no—
50C22⋊Dic31—41no—
51C22⋊Dic31D411yescharacteristic, complemented
52(C2×D31)⋊C4—41no—
53C4⋊Dic31—21no—
54C4⋊Dic31—41no—
55Dic31⋊C4—21no—
56Dic31⋊C4—21no—
57Dic31⋊C4—21no—
58C4×Dic31—21no—
59C4×Dic31D411yescharacteristic, complemented
60Dic124—41no—
61#92470—41no—
62C23×C31—21no—
63C23×C31—41no—
64C23×C31—42no—
65C23×C31C2×D411yescharacteristic
66C23×C31C2×D411yescharacteristic
67C22×D31—21no—
68C22×D31—21no—
69D4×C31—41no—
70D4×C31—42no—
71C2×C4×C31—21no—
72C2×C4×C31—41no—
73C2×C4×C31C2×D411yescharacteristic
74C22⋊D31—41no—
75C22⋊D31—41no—
76C22⋊D31—41no—
77C22⋊D31—82no—
78C2×Dic31—21no—
79C2×Dic31—21no—
80C2×Dic31—21no—
81C2×Dic31—21no—
82C2×Dic31—21no—
83C2×Dic31—21no—
84C2×Dic31—42no—
85C2×Dic31—41no—
86C2×Dic31—41no—
87C2×Dic31—81no—
88D124—21no—
89C4×D31—41no—
90Dic62—21no—
91Dic62—41no—
92C8×C31—41no—
93#2197—311nomaximal, Sylow
94C22×C31—22no—
95C22×C31—21no—
96C22×C31—21no—
97C22×C31—41no—
98C22×C31—41no—
99C22×C31—41no—
100C22×C31—42no—
101C22×C31—41no—
102C22×C31—81no—
103C22×C31#12011yescharacteristic
104C2×D31—41no—
105C2×D31—81no—
106C2×D31—81no—
107C4×C31—21no—
108C4×C31—41no—
109C4×C31—81no—
110Dic31—41no—
111Dic31—41no—
112Dic31—41no—
113Dic31—41no—
114Dic31—41no—
115Dic31—81no—
116#551—311no—
117#458—311no—
118B(4,𝔽2)—311no—
119#455—311no—
120#354—311no—
121#352—311no—
122AΣL(1,𝔽16)D3111yescharacteristic, complemented
123C2×C31—21no—
124C2×C31—41no—
125C2×C31—41no—
126C2×C31—42no—
127C2×C31B(4,𝔽2)11yescharacteristic
128D31—81no—
129H(5,𝔽2)—311no—
130#137—621no—
131#126—621no—
132#124—311no—
133#123—311no—
134#123—621no—
135#123—621no—
136C22⋊Q8—621no—
137#120—621no—
138#120C2×D3111yescharacteristic
139C4×D4—621no—
140#117—621no—
141C2×C22⋊C4—621no—
142#104—621no—
143#100C2×D3111yescharacteristic
144#99—311no—
145#99—621no—
146#99—621no—
147#99C2×D3111yescharacteristic
148C31#219711yescharacteristic, complemented, Sylow
149C24D12411yescharacteristic
150#41—621no—
151#41—1241no—
152C2×Q8—311no—
153C2×D4—21no—
154C2×D4—621no—
155C2×D4—621no—
156C2×D4—1241no—
157C2×D4—1241no—
158C2×D4C22×D3111yescharacteristic
159C22×C4—311no—
160C22×C4—621no—
161C22×C4—621no—
162Q16—1241no—
163SD16—1241no—
164M16—21no—
165C4⋊C4—621no—
166C4⋊C4—621no—
167C4⋊C4—621no—
168C4⋊C4—621no—
169C4⋊C4—1241no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—311no—
173C22⋊C4—621no—
174C22⋊C4—621no—
175C22⋊C4—621no—
176C22⋊C4—621no—
177C22⋊C4—621no—
178C22⋊C4—621no—
179C22⋊C4—1241no—
180C22⋊C4—1241no—
181C22⋊C4D12411yescharacteristic
182C42—311no—
183C42—621no—
184C23—21no—
185C23—41no—
186C23—42no—
187C23—621no—
188C23—621no—
189C23C2×D12411yescharacteristic
190C23D4×D3111yescharacteristic
191Q8—621no—
192Q8—1241no—
193D4—41no—
194D4—42no—
195D4—621no—
196D4—1241no—
197D4—1241no—
198D4—1241no—
199D4—2482no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—621no—
203C2×C4—621no—
204C2×C4—621no—
205C2×C4—621no—
206C2×C4—621no—
207C2×C4—621no—
208C2×C4—1241no—
209C2×C4—1242no—
210C2×C4—1241no—
211C2×C4—1241no—
212C2×C4—2481no—
213C2×C4D4×D3111yescharacteristic
214C8—41no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—1241no—
225C22—2481no—
226C22—2481no—
227C22#1175845911yescharacteristic
228C4—21no—
229C4—41no—
230C4—81no—
231C4—1241no—
232C4—1241no—
233C4—1241no—
234C4—1241no—
235C4—1241no—
236C4—2481no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—2481no—
242C2#1452123711yescharacteristic
subgroupG/H
centerC2#14521237
commutator, cosocleC2×D4×C31C23
FrattiniC2×D4C22×D31
FittingC31×AΣL(1,𝔽16)C2
socleC2×C31B(4,𝔽2)
prime radical#14521235C2
hypercenterAΣL(1,𝔽16)D31
hypocenterC31#2197
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. #14521730C2semidirect
2. #14521240C2semidirect
4. #14521229C2semidirect
5. #14520843C2semidirect
6. #14520840C2semidirect
7. C31×AΣL(1,𝔽16)C2semidirect
35. C22×Dic31D4semidirect
39. C2×Dic62D4semidirect
51. C22⋊Dic31D4semidirect
59. C4×Dic31D4semidirect
122. AΣL(1,𝔽16)D31semidirect
148. C31#2197semidirect

Maximal subgroups

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

subgroupindex
#145217302
#145212402
#145212352
#145212292
#145208432
#145208402
C31×AΣL(1,𝔽16)2
#219731