Finite Group Databasebeta

← #16854996

Subgroups — #16854996

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
2C22×#92480C211yesmaximal, characteristic, complemented
3#14521616C211yesmaximal, characteristic, complemented
4#14521191C214yesmaximal, complemented
5#14521186C214yesmaximal, complemented
6#14521059C211yesmaximal, characteristic, complemented
7#14521045C211yesmaximal, characteristic, complemented
8#14521017C211yesmaximal, characteristic, complemented
9C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C22×D4×C31C2211yescharacteristic
12#11758567—24no—
13#11758567—24no—
14C2×Q8×D31C2212yescomplemented
15C2×#92497—22no—
16C2×#92497C2212yescomplemented
17C2×#92495—21no—
18C2×#92495C2211yescharacteristic
19C22×Dic62—21no—
20C22×Dic62C2211yescharacteristic
21C31×AGL(1,ℤ/8)—24no—
22C2×SD16×C31C2212yes—
23C2×D8×C31C2212yes—
24C2×M16×C31C2211yescharacteristic
25#11758530—22no—
26#11758517—21no—
27#11758515C2214yes—
28#11758513—21no—
29C2×#92480—24no—
30C2×#92480C2214yes—
31#11758503C2211yescharacteristic
32C22×C31⋊C8C2211yescharacteristic
33#11758423C2212yescomplemented
34#11758420C2212yescomplemented
35(C2×D31)⋊C8C2214yescomplemented
36#11758404C2214yescomplemented
37C24×C31—21no—
38#41×C31—22no—
39#41×C31D414yescomplemented
40C2×Q8×C31C2311yescharacteristic
41C2×D4×C31—21no—
42C2×D4×C31—22no—
43C2×D4×C31—22no—
44C2×D4×C31—21no—
45C2×D4×C31C2312yes—
46C2×D4×C31C2311yescharacteristic
47C22×C4×C31—21no—
48C22×C4×C31C2311yescharacteristic
49C2×C22⋊D31—22no—
50C2×C22⋊D31—24no—
51C2×C22⋊D31D412yes—
52C22×Dic31—21no—
53C22×Dic31—22no—
54Q8×D31—24no—
55Q8×D31—24no—
56#92497—216no—
57#92497—24no—
58#92497—24no—
59#92495—28no—
60#92495—22no—
61#92495—22no—
62C2×D124D411yescharacteristic
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—
72SD16×C31—24no—
73D8×C31—24no—
74M16×C31—22no—
75C2×C8×C31C2312yes—
76C22⋊Dic31—21no—
77C22⋊Dic31—24no—
78#92480—28no—
79C4⋊Dic31C2312yes—
80C4×Dic31—21no—
81C2×C31⋊C8—22no—
82C2×C31⋊C8C2312yes—
83C23×C31—21no—
84C23×C31—22no—
85C23×C31—42no—
86C23×C31—82no—
87C23×C31C2×D411yescharacteristic
88C22×D31C2×D412yes—
89Q8×C31—21no—
90Q8×C31C2×D412yes—
91D4×C31—24no—
92D4×C31—21no—
93D4×C31—22no—
94D4×C31—22no—
95D4×C31—42no—
96D4×C31C2×D412yes—
97C2×C4×C31—21no—
98C2×C4×C31—24no—
99C2×C4×C31C2×D411yescharacteristic
100C2×C4×C31C2×D412yes—
101C2×C4×C31C2×D412yes—
102C2×C4×C31C2411yescharacteristic
103C22⋊D31—28no—
104C22⋊D31—24no—
105C22⋊D31—84no—
106C2×Dic31—28no—
107C2×Dic31—24no—
108C2×Dic31—22no—
109C2×Dic31—22no—
110C2×Dic31C2×D412yes—
111D124—22no—
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—42no—
127C22×C31—44no—
128C22×C31—84no—
129C22×C31#12012yes—
130C22×C31C22×D411yescharacteristic
131C2×D31—22no—
132C2×D31—24no—
133C4×C31—22no—
134C4×C31#12012yes—
135C4×C31C22×D411yescharacteristic
136C4×C31C22×D411yescharacteristic
137Dic31—22no—
138Dic31—24no—
139Dic31—42no—
140C2×2−1+4—311no—
141C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
142C22×SD16—311no—
143#535—311no—
144#449—314no—
145#448—314no—
146#419—311no—
147#416—311no—
148#408—311no—
149C2×C31—22no—
150C2×C31—42no—
151C2×C31—82no—
152C2×C31C2×#12011yescharacteristic
153C2×C31#57712yes—
154D31—42no—
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×D4C2×D3111yescharacteristic
166AGL(1,ℤ/8)—24no—
167C2×SD16—314no—
168C2×SD16—624no—
169C2×SD16C2×D3112yes—
170C2×D8C2×D3112yes—
171C2×M16C2×D3111yescharacteristic
172C22×C8—311no—
173#127—621no—
174#124—621no—
175#121—314no—
176#120—622no—
177#117—311no—
178#103—312no—
179#103—314no—
180#102—312no—
181C22⋊C8—314no—
182C31#301011yescharacteristic, complemented, Sylow
183C24—21no—
184#41—22no—
185#41—628no—
186#41—624no—
187#41—6216no—
188#41—624no—
189#41—622no—
190#41—622no—
191#41C22⋊D3114yescomplemented
192C2×Q8—311no—
193C2×Q8—314no—
194C2×Q8—312no—
195C2×Q8—622no—
196C2×Q8—628no—
197C2×Q8—624no—
198C2×Q8—624no—
199C2×Q8—621no—
200C2×Q8—622no—
201C2×Q8C22×D3111yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—312no—
207C2×D4—311no—
208C2×D4—622no—
209C2×D4—624no—
210C2×D4C22×D3112yes—
211C2×D4C22×D3111yescharacteristic
212C22×C4—21no—
213C22×C4—312no—
214C22×C4—622no—
215C22×C4—622no—
216C22×C4—621no—
217C22×C4C22×D3111yescharacteristic
218SD16—24no—
219SD16—628no—
220D8—24no—
221M16—22no—
222C2×C8—312no—
223C2×C8—622no—
224C2×C8C22×D3112yes—
225C4⋊C4—312no—
226C22⋊C4—624no—
227C22⋊C4—621no—
228C42—621no—
229C23—21no—
230C23—22no—
231C23—42no—
232C23—82no—
233C23—312no—
234C23D4×D3111yescharacteristic
235Q8—21no—
236Q8—314no—
237Q8—628no—
238Q8—622no—
239Q8—624no—
240Q8—622no—
241Q8C2×C22⋊D3112yes—
242D4—24no—
243D4—21no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—628no—
248D4—624no—
249D4—622no—
250D4—2484no—
251D4C2×C22⋊D3112yes—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—312no—
255C2×C4—628no—
256C2×C4—628no—
257C2×C4—624no—
258C2×C4—622no—
259C2×C4—622no—
260C2×C4—622no—
261C2×C4—622no—
262C2×C4D4×D3111yescharacteristic
263C2×C4D4×D3112yes—
264C2×C4C2×C22⋊D3112yes—
265C2×C4C23×D3111yescharacteristic
266C8—22no—
267C8—622no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—624no—
275C22—622no—
276C22#1175851412yes—
277C22C2×D4×D3111yescharacteristic
278C4—22no—
279C4—622no—
280C4—624no—
281C4—1242no—
282C4#1175851412yes—
283C4C2×D4×D3111yescharacteristic
284C4C22×C22⋊D3111yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—1242no—
289C2C2×#1175851411yescharacteristic
290C2#1452185912yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC22×C31C22×D4
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#3010
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11758567C2semidirect
2. C22×#92480C2semidirect
3. #14521616C2semidirect
4. #14521191C2semidirect
5. #14521186C2semidirect
6. #14521059C2semidirect
7. #14521045C2semidirect
8. #14521017C2semidirect
9. C2×C31×AGL(1,ℤ/8)C2semidirect
14. C2×Q8×D31C22semidirect
16. C2×#92497C22semidirect
33. #11758423C22semidirect
34. #11758420C22semidirect
35. (C2×D31)⋊C8C22semidirect
36. #11758404C22semidirect
39. #41×C31D4semidirect
70. C2×Dic62D4semidirect
141. C2×AGL(1,ℤ/8)D31semidirect
182. C31#3010semidirect
191. #41C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585672
C22×#924802
#145216162
#145211912
#145211862
#145210592
#145210452
#145210172
C2×C31×AGL(1,ℤ/8)2
#301031