Finite Group Databasebeta

← #16856187

Subgroups — #16856187

Order 3968.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11758486C211yesmaximal, characteristic, complemented
2C2×#11758485C211yesmaximal, characteristic, complemented
3C2×#11758481C211yesmaximal, characteristic, complemented
4C22×#92470C211yesmaximal, characteristic, complemented
5#14521625C211yesmaximal, characteristic, complemented
6#14521618C211yesmaximal, characteristic, complemented
7#14521443C212yesmaximal, complemented
8#14521438C212yesmaximal, complemented
9#14521433C212yesmaximal, complemented
10#14521422C212yesmaximal, complemented
11C31×#405C211yesmaximal, characteristic, complemented
12C2×#92495C2211yescharacteristic
13C2×#92495C2211yescharacteristic
14C22×D124C2211yescharacteristic
15C22×Dic62C2211yescharacteristic
16C2×M16×C31C2211yescharacteristic
17C22×C8×C31C2211yescharacteristic
18C31×#117C2211yescharacteristic
19#11758486—24no—
20#11758485—24no—
21C2×Dic124C2212yescomplemented
22C2×Dic124C2211yescharacteristic, complemented
23#11758481—24no—
24C2×D248C2212yes—
25C2×D248C2211yescharacteristic, complemented
26C2×#92470—24no—
27C2×#92470C2212yes—
28C2×#92470C2212yes—
29C2×#92470C2212yes—
30C2×#92470C2211yescharacteristic, complemented
31C2×#92470C2212yescomplemented
32C2×#92470C2211yescharacteristic, complemented
33#11758474—22no—
34#11758472—22no—
35#11758463—22no—
36#11758459—22no—
37#11758452C2212yes—
38#11758452C2212yescomplemented
39#11758451C2212yes—
40#11758446C2212yescomplemented
41C31×#97C2212yescomplemented
42C4×C8×C31C2212yescomplemented
43C23×D31—21no—
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—21no—
46C2×C22⋊D31—21no—
47C22×Dic31—21no—
48#92495—22no—
49#92495—22no—
50#92495—22no—
51#92495—22no—
52C2×D124—22no—
53C2×D124—22no—
54C2×D124—21no—
55C2×D124—24no—
56C2×D124C2311yescharacteristic
57C2×D124C2312yes—
58C2×D124C2311yescharacteristic
59C2×C4×D31—21no—
60C2×C4×D31—21no—
61C2×Dic62—22no—
62C2×Dic62—22no—
63C2×Dic62—21no—
64C2×Dic62C2311yescharacteristic
65C2×Dic62C2312yes—
66C2×Dic62C2311yescharacteristic
67M16×C31D414yescomplemented
68C2×C8×C31D414yescomplemented
69C2×C8×C31C2311yescharacteristic
70C2×C8×C31C2312yes—
71C2×C8×C31C2311yescharacteristic
72C4⋊C4×C31D412yes—
73C22⋊C4×C31D412yes—
74C42×C31C2312yes—
75(C2×D31)⋊C4—24no—
76(C2×D31)⋊C4—24no—
77C4⋊Dic31—24no—
78Dic124—22no—
79Dic124—24no—
80D248—24no—
81D248—22no—
82#92470—24no—
83#92470—24no—
84#92470—24no—
85#92470—22no—
86#92470—22no—
87#92470—24no—
88C23×C31C2×D411yescharacteristic
89C22×D31—21no—
90C22×D31—22no—
91C22×D31—21no—
92C22×D31—42no—
93C22×D31—82no—
94C2×C4×C31C2×D414yes—
95C2×C4×C31C2×D411yescharacteristic
96C2×C4×C31C2×D412yes—
97C2×C4×C31C2×D412yes—
98C2×C4×C31C2411yescharacteristic
99C22⋊D31—42no—
100C22⋊D31—42no—
101C2×Dic31—21no—
102C2×Dic31—22no—
103C2×Dic31—21no—
104C2×Dic31—42no—
105D124—22no—
106D124—21no—
107D124—22no—
108D124—21no—
109D124—21no—
110D124—21no—
111D124—42no—
112D124—84no—
113C4×D31—42no—
114C4×D31—42no—
115Dic62—22no—
116Dic62—21no—
117Dic62—22no—
118Dic62—21no—
119Dic62—21no—
120Dic62—21no—
121Dic62—42no—
122C8×C31C2×D412yes—
123C8×C31C2×D412yes—
124C8×C31C2×D414yes—
125#3154—311nomaximal, Sylow
126C22×C31—22no—
127C22×C31#12712yes—
128C22×C31C22×D411yescharacteristic
129C2×D31—41no—
130C2×D31—41no—
131C2×D31—42no—
132C2×D31—42no—
133C2×D31—44no—
134C2×D31—42no—
135C2×D31—84no—
136C4×C31—24no—
137C4×C31#12712yes—
138C4×C31C22×D411yescharacteristic
139C4×C31C22×D411yescharacteristic
140Dic31—42no—
141Dic31—41no—
142Dic31—41no—
143C2×#137—311no—
144C2×AGL(1,ℤ/8)—311no—
145C2×#135—311no—
146C22×SD16—311no—
147#536—311no—
148#535—311no—
149#497—312no—
150#496—312no—
151#495—312no—
152#492—312no—
153#405D3111yescharacteristic, complemented
154C2×C31—22no—
155C2×C31C2×#12711yescharacteristic
156C2×C31#57712yes—
157D31—81no—
158D31—81no—
159D31—82no—
160C2×#41—311no—
161C2×#41—311no—
162C22×Q8—311no—
163C22×D4—311no—
164#137—624no—
165AGL(1,ℤ/8)—624no—
166#135—624no—
167C2×Q16—311no—
168C2×Q16—312no—
169C2×SD16—312no—
170C2×SD16—312no—
171C2×SD16—311no—
172C2×SD16—311no—
173C2×SD16—312no—
174C2×SD16—312no—
175C2×SD16—624no—
176C2×D8—311no—
177C2×D8—312no—
178C2×M16C2×D3111yescharacteristic
179C22×C8C2×D3111yescharacteristic
180C4⋊Q8—312no—
181#127—312no—
182#124—312no—
183#124—312no—
184#123—622no—
185C22⋊Q8—622no—
186#121—622no—
187#120—622no—
188#117C2×D3111yescharacteristic
189#97C2×D3112yescomplemented
190C4×C8C2×D3112yescomplemented
191C31#315411yescharacteristic, complemented, Sylow
192C24—621no—
193#41—622no—
194#41—622no—
195#41—622no—
196#41—622no—
197C2×Q8—311no—
198C2×Q8—312no—
199C2×Q8—311no—
200C2×Q8—621no—
201C2×Q8—622no—
202C2×Q8—622no—
203C2×D4—311no—
204C2×D4—312no—
205C2×D4—311no—
206C2×D4—621no—
207C2×D4—621no—
208C2×D4—622no—
209C2×D4—622no—
210C2×D4—621no—
211C2×D4—624no—
212C22×C4—621no—
213C22×C4—621no—
214C22×C4—621no—
215C22×C4C22×D3111yescharacteristic
216Q16—622no—
217Q16—624no—
218SD16—624no—
219SD16—624no—
220SD16—622no—
221SD16—622no—
222SD16—624no—
223SD16—624no—
224D8—622no—
225D8—624no—
226M16D12414yescomplemented
227C2×C8D12414yescomplemented
228C2×C8C22×D3111yescharacteristic
229C2×C8C22×D3112yes—
230C2×C8C22×D3111yescharacteristic
231C4⋊C4—624no—
232C4⋊C4D12412yes—
233C22⋊C4—624no—
234C22⋊C4—624no—
235C22⋊C4D12412yes—
236C42C22×D3112yes—
237C23—621no—
238C23—622no—
239C23—621no—
240C23—1242no—
241C23—2482no—
242C23C2×D12411yescharacteristic
243Q8—622no—
244Q8—621no—
245Q8—622no—
246Q8—621no—
247Q8—621no—
248Q8—621no—
249Q8—1242no—
250D4—621no—
251D4—622no—
252D4—621no—
253D4—621no—
254D4—622no—
255D4—621no—
256D4—1242no—
257D4—1242no—
258D4—1242no—
259D4—2484no—
260C2×C4—621no—
261C2×C4—622no—
262C2×C4—621no—
263C2×C4—1242no—
264C2×C4—1242no—
265C2×C4—1242no—
266C2×C4C2×D12414yes—
267C2×C4C2×D12411yescharacteristic
268C2×C4C2×D12412yes—
269C2×C4C2×D12412yes—
270C2×C4C23×D3111yescharacteristic
271C8C2×D12412yes—
272C8C2×D12412yes—
273C8C2×D12414yes—
274C22—22no—
275C22—1242no—
276C22—1244no—
277C22—1242no—
278C22—1241no—
279C22—1241no—
280C22—1242no—
281C22—2484no—
282C22#1175845112yes—
283C22C22×D12411yescharacteristic
284C4—24no—
285C4—1241no—
286C4—1241no—
287C4—1242no—
288C4#1175845112yes—
289C4C22×D12411yescharacteristic
290C4C22×D12411yescharacteristic
291C2—22no—
292C2—2482no—
293C2—2481no—
294C2—2481no—
295C2C2×#1175845111yescharacteristic
296C2#1452186112yes—
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#405C2
socleC22×C31C22×D4
hypercenter#405D31
hypocenterC31#3154
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#11758486C2semidirect
2. C2×#11758485C2semidirect
3. C2×#11758481C2semidirect
4. C22×#92470C2semidirect
5. #14521625C2semidirect
6. #14521618C2semidirect
7. #14521443C2semidirect
8. #14521438C2semidirect
9. #14521433C2semidirect
10. #14521422C2semidirect
11. C31×#405C2semidirect
21. C2×Dic124C22semidirect
22. C2×Dic124C22semidirect
25. C2×D248C22semidirect
30. C2×#92470C22semidirect
31. C2×#92470C22semidirect
32. C2×#92470C22semidirect
38. #11758452C22semidirect
40. #11758446C22semidirect
41. C31×#97C22semidirect
42. C4×C8×C31C22semidirect
67. M16×C31D4semidirect
68. C2×C8×C31D4semidirect
153. #405D31semidirect
189. #97C2×D31semidirect
190. C4×C8C2×D31semidirect
191. C31#3154semidirect
226. M16D124semidirect
227. C2×C8D124semidirect

Maximal subgroups

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

subgroupindex
C2×#117584862
C2×#117584852
C2×#117584812
C22×#924702
#145216252
#145216182
#145214432
#145214382
#145214332
#145214222
C31×#4052
#315431