Finite Group Databasebeta

← #16847696

Subgroups — #16847696

Order 3968.

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

normal subgroups only

This group has 354 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#14521681C211yesmaximal, characteristic, complemented
2#14521237C211yesmaximal, characteristic, complemented
3#14521235C211yesmaximal, characteristic, complemented
4#14521219C211yesmaximal, characteristic, complemented
5#14520841C211yesmaximal, characteristic, complemented
6#14520837C211yesmaximal, characteristic, complemented
7C31×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8C22×C22⋊D31—21no—
9#11758561C2211yescharacteristic, complemented
10C2×D4×D31—21no—
11C31×#120C2211yescharacteristic
12#11758530—21no—
13#11758517C2211yescharacteristic
14#11758516—21no—
15#11758514C2211yescharacteristic
16#11758485—21no—
17#11758461—21no—
18#11758459—21no—
19#11758459—21no—
20(C22⋊D31)⋊C4—21no—
21C31×#100C2211yescharacteristic, complemented
22C31×#99C2211yescharacteristic, complemented
23#11758422C2211yescharacteristic, complemented
24#11758413—21no—
25#11758394—21no—
26#11758394—21no—
27C24×C31D411yescharacteristic, complemented
28C23×D31—21no—
29C23×D31D411yescharacteristic, complemented
30C2×D4×C31—21no—
31C2×D4×C31C2311yescharacteristic
32C2×C22⋊D31—21no—
33C2×C22⋊D31—21no—
34C2×C22⋊D31—21no—
35C2×C22⋊D31—21no—
36C2×C22⋊D31—21no—
37C2×C22⋊D31—21no—
38C2×C22⋊D31—21no—
39C2×C22⋊D31—44no—
40C2×C22⋊D31—41no—
41C22×Dic31—21no—
42#92497—41no—
43D4×D31—41no—
44D4×D31—44no—
45#92495—21no—
46C2×D124—21no—
47C2×D124—41no—
48C2×D124D411yescharacteristic, complemented
49C2×C4×D31—21no—
50M16×C31—21no—
51C22⋊C4×C31—21no—
52C22⋊C4×C31—21no—
53C22⋊C4×C31D411yescharacteristic, complemented
54C22⋊Dic31—21no—
55C22⋊Dic31—41no—
56C22⋊Dic31D411yescharacteristic, complemented
57(C2×D31)⋊C4—21no—
58(C2×D31)⋊C4—21no—
59(C2×D31)⋊C4—41no—
60Dic31⋊C4—21no—
61C4×Dic31D411yescharacteristic, complemented
62D248—41no—
63#92470—41no—
64C23×C31—21no—
65C23×C31—41no—
66C23×C31—42no—
67C23×C31C2×D411yescharacteristic
68C23×C31C2×D411yescharacteristic
69C22×D31—21no—
70C22×D31—21no—
71C22×D31—21no—
72C22×D31—21no—
73C22×D31—42no—
74C22×D31—42no—
75C22×D31—41no—
76C22×D31—42no—
77C22×D31—81no—
78D4×C31—41no—
79D4×C31—42no—
80C2×C4×C31—21no—
81C2×C4×C31—41no—
82C2×C4×C31C2×D411yescharacteristic
83C22⋊D31—24no—
84C22⋊D31—42no—
85C22⋊D31—42no—
86C22⋊D31—42no—
87C22⋊D31—41no—
88C22⋊D31—41no—
89C22⋊D31—42no—
90C22⋊D31—41no—
91C22⋊D31—82no—
92C2×Dic31—21no—
93C2×Dic31—21no—
94C2×Dic31—21no—
95C2×Dic31—21no—
96C2×Dic31—42no—
97C2×Dic31—41no—
98D124—21no—
99D124—41no—
100D124—42no—
101D124—82no—
102C4×D31—42no—
103C4×D31—41no—
104Dic62—21no—
105C8×C31—41no—
106#2196—311nomaximal, Sylow
107C22×C31—22no—
108C22×C31—21no—
109C22×C31—21no—
110C22×C31—41no—
111C22×C31—41no—
112C22×C31—41no—
113C22×C31—42no—
114C22×C31—41no—
115C22×C31—81no—
116C22×C31#12011yescharacteristic
117C2×D31—22no—
118C2×D31—41no—
119C2×D31—41no—
120C2×D31—42no—
121C2×D31—41no—
122C2×D31—41no—
123C2×D31—41no—
124C2×D31—41no—
125C2×D31—81no—
126C2×D31—84no—
127C2×D31—81no—
128C2×D31—81no—
129C4×C31—21no—
130C4×C31—41no—
131C4×C31—81no—
132Dic31—22no—
133Dic31—41no—
134Dic31—41no—
135Dic31—81no—
136D42—311no—
137B(4,𝔽2)—311no—
138B(4,𝔽2)—311no—
139#453—311no—
140#353—311no—
141AΣL(1,𝔽16)—311no—
142AΣL(1,𝔽16)D3111yescharacteristic, complemented
143C2×C31—21no—
144C2×C31—41no—
145C2×C31—41no—
146C2×C31—42no—
147C2×C31B(4,𝔽2)11yescharacteristic
148D31—42no—
149D31—81no—
150D31—81no—
151H(5,𝔽2)—311no—
152C22×D4—621no—
153C22×D4—621no—
154AGL(1,ℤ/8)—621no—
155#127—311no—
156#121—621no—
157#121—621no—
158#120—311no—
159#120—621no—
160#120—621no—
161#120—621no—
162#120C2×D3111yescharacteristic
163C4×D4—621no—
164#104—621no—
165#100C2×D3111yescharacteristic, complemented
166#99—311no—
167#99—621no—
168#99—621no—
169#99C2×D3111yescharacteristic, complemented
170C31#219611yescharacteristic, complemented, Sylow
171C24—311no—
172C24—621no—
173C24D12411yescharacteristic, complemented
174#41—621no—
175#41—1241no—
176C2×D4—21no—
177C2×D4—311no—
178C2×D4—621no—
179C2×D4—621no—
180C2×D4—621no—
181C2×D4—621no—
182C2×D4—621no—
183C2×D4—621no—
184C2×D4—621no—
185C2×D4—621no—
186C2×D4—1241no—
187C2×D4—1241no—
188C2×D4—1244no—
189C2×D4—1244no—
190C2×D4—1241no—
191C2×D4C22×D3111yescharacteristic
192C22×C4—621no—
193C22×C4—621no—
194SD16—1241no—
195D8—1241no—
196M16—21no—
197C4⋊C4—621no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—311no—
201C22⋊C4—621no—
202C22⋊C4—621no—
203C22⋊C4—621no—
204C22⋊C4—1241no—
205C22⋊C4—1241no—
206C22⋊C4D12411yescharacteristic, complemented
207C42—311no—
208C23—21no—
209C23—41no—
210C23—42no—
211C23—621no—
212C23—621no—
213C23—621no—
214C23—621no—
215C23—1241no—
216C23—1242no—
217C23—1242no—
218C23—1242no—
219C23—2481no—
220C23C2×D12411yescharacteristic
221C23D4×D3111yescharacteristic
222Q8—621no—
223D4—41no—
224D4—42no—
225D4—621no—
226D4—624no—
227D4—1241no—
228D4—1242no—
229D4—1242no—
230D4—1242no—
231D4—1242no—
232D4—1241no—
233D4—1241no—
234D4—1242no—
235D4—1241no—
236D4—2482no—
237D4—2482no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—621no—
241C2×C4—621no—
242C2×C4—621no—
243C2×C4—621no—
244C2×C4—1241no—
245C2×C4—1242no—
246C2×C4—1242no—
247C2×C4—1241no—
248C2×C4D4×D3111yescharacteristic
249C8—41no—
250C22—22no—
251C22—21no—
252C22—21no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—41no—
258C22—81no—
259C22—622no—
260C22—1241no—
261C22—1241no—
262C22—1241no—
263C22—1242no—
264C22—1241no—
265C22—1241no—
266C22—1241no—
267C22—2481no—
268C22—2484no—
269C22—2481no—
270C22—2481no—
271C22#1175845911yescharacteristic
272C4—21no—
273C4—41no—
274C4—81no—
275C4—622no—
276C4—1241no—
277C4—1241no—
278C4—2481no—
279C2—21no—
280C2—41no—
281C2—41no—
282C2—42no—
283C2—1242no—
284C2—2481no—
285C2—2481no—
286C2#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#2196
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. #14521681C2semidirect
2. #14521237C2semidirect
3. #14521235C2semidirect
4. #14521219C2semidirect
5. #14520841C2semidirect
6. #14520837C2semidirect
7. C31×AΣL(1,𝔽16)C2semidirect
9. #11758561C22semidirect
21. C31×#100C22semidirect
22. C31×#99C22semidirect
23. #11758422C22semidirect
27. C24×C31D4semidirect
29. C23×D31D4semidirect
48. C2×D124D4semidirect
53. C22⋊C4×C31D4semidirect
56. C22⋊Dic31D4semidirect
61. C4×Dic31D4semidirect
142. AΣL(1,𝔽16)D31semidirect
165. #100C2×D31semidirect
169. #99C2×D31semidirect
170. C31#2196semidirect
173. C24D124semidirect
206. C22⋊C4D124semidirect

Maximal subgroups

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

subgroupindex
#145216812
#145212372
#145212352
#145212192
#145208412
#145208372
C31×AΣL(1,𝔽16)2
#219631