Finite Group Databasebeta

← #16855000

Subgroups — #16855000

Order 3968.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11758567C211yesmaximal, characteristic, complemented
2C2×#11758526C211yesmaximal, characteristic, complemented
3C22×#92470C211yesmaximal, characteristic, complemented
4#14521618C211yesmaximal, characteristic, complemented
5#14521195C214yesmaximal, complemented
6#14521189C214yesmaximal, complemented
7#14521057C211yesmaximal, characteristic, complemented
8#14521015C211yesmaximal, characteristic, complemented
9C31×#416C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11#11758567—24no—
12#11758567—24no—
13C2×Q8×D31C2212yescomplemented
14C2×#92497—22no—
15C2×#92497C2212yescomplemented
16C2×#92495—21no—
17C2×#92495C2211yescharacteristic
18C22×D124C2211yescharacteristic
19C22×Dic62—21no—
20C22×Dic62C2211yescharacteristic
21C22×C8×C31C2211yescharacteristic
22C31×#117C2211yescharacteristic
23#11758526—24no—
24C2×#92481C2212yes—
25C2×#92479C2212yes—
26C2×#92474C2211yescharacteristic
27C2×#92470—24no—
28C2×#92470C2214yes—
29#11758472C2214yes—
30#11758459—22no—
31#11758452—21no—
32#11758451—21no—
33C31×#103C2212yescomplemented
34C31×#102C2212yescomplemented
35(C2×D31)⋊C8C2214yescomplemented
36#11758398C2214yescomplemented
37C23×D31—21no—
38#41×C31—22no—
39#41×C31D414yescomplemented
40C2×Q8×C31C2311yescharacteristic
41C2×D4×C31—21no—
42C2×D4×C31C2311yescharacteristic
43C22×C4×C31—21no—
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—22no—
46C2×C22⋊D31D412yes—
47C22×Dic31—21no—
48C22×Dic31—22no—
49Q8×D31—24no—
50Q8×D31—24no—
51#92497—216no—
52#92497—24no—
53#92497—24no—
54#92495—28no—
55#92495—22no—
56#92495—22no—
57C2×D124—22no—
58C2×D124—22no—
59C2×D124—21no—
60C2×D124—24no—
61C2×D124D411yescharacteristic
62C2×D124C2312yes—
63C2×C4×D31—22no—
64C2×C4×D31C2312yes—
65C2×Dic62—28no—
66C2×Dic62—21no—
67C2×Dic62—22no—
68C2×Dic62—22no—
69C2×Dic62D411yescharacteristic
70C2×Dic62D414yescomplemented
71C2×Dic62C2312yes—
72C2×C8×C31—22no—
73C2×C8×C31C2312yes—
74C4⋊C4×C31C2312yes—
75C22⋊C4×C31—21no—
76C42×C31—21no—
77#92481—24no—
78#92479—24no—
79(C2×D31)⋊C4—24no—
80#92474—22no—
81C2×C31⋊C8C2312yes—
82#92470—28no—
83C23×C31—21no—
84C23×C31C2×D411yescharacteristic
85C22×D31—22no—
86C22×D31—42no—
87C22×D31—82no—
88C22×D31C2×D412yes—
89Q8×C31—21no—
90Q8×C31C2×D412yes—
91D4×C31—24no—
92D4×C31—21no—
93D4×C31C2×D412yes—
94C2×C4×C31—21no—
95C2×C4×C31—24no—
96C2×C4×C31—22no—
97C2×C4×C31C2×D412yes—
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2411yescharacteristic
101C22⋊D31—28no—
102C22⋊D31—24no—
103C2×Dic31—28no—
104C2×Dic31—24no—
105C2×Dic31—22no—
106C2×Dic31C2×D412yes—
107D124—22no—
108D124—22no—
109D124—22no—
110D124—42no—
111D124—84no—
112C4×D31—28no—
113C4×D31—22no—
114C4×D31—22no—
115Dic62—28no—
116Dic62—22no—
117Dic62—24no—
118Dic62—22no—
119Dic62C2×D414yes—
120C8×C31—22no—
121C31⋊C8—22no—
122#3010—311nomaximal, Sylow
123C22×C31—22no—
124C22×C31—22no—
125C22×C31—42no—
126C22×C31#12012yes—
127C22×C31C22×D411yescharacteristic
128C2×D31—22no—
129C2×D31—24no—
130C2×D31—42no—
131C2×D31—44no—
132C2×D31—84no—
133C4×C31—22no—
134C4×C31—42no—
135C4×C31#12012yes—
136C4×C31C22×D411yescharacteristic
137C4×C31C22×D411yescharacteristic
138Dic31—22no—
139Dic31—24no—
140C2×2−1+4—311no—
141C2×AGL(1,ℤ/8)—311no—
142C22×SD16—311no—
143#535—311no—
144#449—314no—
145#448—314no—
146#419—311no—
147#416D3111yescharacteristic, complemented
148#408—311no—
149C2×C31—22no—
150C2×C31—42no—
151C2×C31C2×#12011yescharacteristic
152C2×C31#57712yes—
153D31—42no—
154D31—82no—
1552−1+4—624no—
1562−1+4—624no—
157C2×#41—311no—
158C2×#41—312no—
159C2×#41—621no—
160C2×#41—622no—
161C2×#41C2×D3111yescharacteristic
162C22×Q8—312no—
163C22×Q8—311no—
164C22×Q8—621no—
165C22×D4—311no—
166AGL(1,ℤ/8)—624no—
167C2×SD16—312no—
168C2×SD16—314no—
169C2×SD16—624no—
170C2×D8—312no—
171C2×M16—311no—
172C22×C8C2×D3111yescharacteristic
173#127—621no—
174#124—621no—
175#121—314no—
176#120—622no—
177#117C2×D3111yescharacteristic
178#103—314no—
179#103C2×D3112yescomplemented
180#102C2×D3112yescomplemented
181C22⋊C8—314no—
182C31#301011yescharacteristic, complemented, Sylow
183C24—621no—
184#41—22no—
185#41—628no—
186#41—624no—
187#41—6216no—
188#41—624no—
189#41—622no—
190#41—622no—
191#41D12414yescomplemented
192C2×Q8—311no—
193C2×Q8—314no—
194C2×Q8—312no—
195C2×Q8—622no—
196C2×Q8—628no—
197C2×Q8—624no—
198C2×Q8—624no—
199C2×Q8—622no—
200C2×Q8—621no—
201C2×Q8C22×D3111yescharacteristic
202C2×D4—21no—
203C2×D4—312no—
204C2×D4—311no—
205C2×D4—312no—
206C2×D4—624no—
207C2×D4—622no—
208C2×D4—622no—
209C2×D4—622no—
210C2×D4—621no—
211C2×D4C22×D3111yescharacteristic
212C22×C4—21no—
213C22×C4—312no—
214C22×C4—622no—
215C22×C4—622no—
216C22×C4—621no—
217C22×C4C22×D3111yescharacteristic
218SD16—628no—
219SD16—624no—
220D8—624no—
221M16—622no—
222C2×C8—22no—
223C2×C8—312no—
224C2×C8C22×D3112yes—
225C4⋊C4C22×D3112yes—
226C22⋊C4—21no—
227C22⋊C4—624no—
228C42—21no—
229C23—21no—
230C23—312no—
231C23—622no—
232C23—1242no—
233C23—2482no—
234C23D4×D3111yescharacteristic
235Q8—21no—
236Q8—314no—
237Q8—628no—
238Q8—622no—
239Q8—624no—
240Q8—622no—
241Q8C2×D12412yes—
242D4—24no—
243D4—21no—
244D4—628no—
245D4—624no—
246D4—622no—
247D4—622no—
248D4—622no—
249D4—1242no—
250D4—2484no—
251D4C2×D12412yes—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—312no—
256C2×C4—628no—
257C2×C4—628no—
258C2×C4—624no—
259C2×C4—622no—
260C2×C4—622no—
261C2×C4—622no—
262C2×C4C2×D12412yes—
263C2×C4D4×D3111yescharacteristic
264C2×C4D4×D3112yes—
265C2×C4C23×D3111yescharacteristic
266C8—22no—
267C8—622no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—624no—
272C22—622no—
273C22—1244no—
274C22—1242no—
275C22—2484no—
276C22#1175845912yes—
277C22C2×D4×D3111yescharacteristic
278C4—22no—
279C4—42no—
280C4—622no—
281C4—624no—
282C4#1175845912yes—
283C4C22×D12411yescharacteristic
284C4C2×D4×D3111yescharacteristic
285C2—22no—
286C2—42no—
287C2—1242no—
288C2—2482no—
289C2C2×#1175845911yescharacteristic
290C2#1452185612yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#416C2
socleC22×C31C22×D4
hypercenter#416D31
hypocenterC31#3010
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#11758567C2semidirect
2. C2×#11758526C2semidirect
3. C22×#92470C2semidirect
4. #14521618C2semidirect
5. #14521195C2semidirect
6. #14521189C2semidirect
7. #14521057C2semidirect
8. #14521015C2semidirect
9. C31×#416C2semidirect
13. C2×Q8×D31C22semidirect
15. C2×#92497C22semidirect
33. C31×#103C22semidirect
34. C31×#102C22semidirect
35. (C2×D31)⋊C8C22semidirect
36. #11758398C22semidirect
39. #41×C31D4semidirect
70. C2×Dic62D4semidirect
147. #416D31semidirect
179. #103C2×D31semidirect
180. #102C2×D31semidirect
182. C31#3010semidirect
191. #41D124semidirect

Maximal subgroups

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

subgroupindex
C2×#117585672
C2×#117585262
C22×#924702
#145216182
#145211952
#145211892
#145210572
#145210152
C31×#4162
#301031