Finite Group Databasebeta

← #16854997

Subgroups — #16854997

Order 3968.

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

normal subgroups only

This group has 732 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×#11758526C211yesmaximal, characteristic, complemented
2C22×#92481C211yesmaximal, characteristic, complemented
3#14521613C211yesmaximal, characteristic, complemented
4#14521194C214yesmaximal, complemented
5#14521188C214yesmaximal, complemented
6#14521058C211yesmaximal, characteristic, complemented
7#14521045C211yesmaximal, characteristic, complemented
8#14521016C211yesmaximal, characteristic, complemented
9C2×C31×2−1+4C211yesmaximal, characteristic, complemented
10C31×2−1+4—24no—
11C31×2−1+4—24no—
12C2×#41×C31—22no—
13C2×#41×C31—21no—
14C2×#41×C31C2211yescharacteristic
15C2×#41×C31C2211yescharacteristic
16C2×#41×C31C2212yescomplemented
17C22×Q8×C31—21no—
18C22×Q8×C31C2211yescharacteristic
19C22×Q8×C31C2212yescomplemented
20C22×D124C2211yescharacteristic
21#11758526—24no—
22#11758524—21no—
23C2×#92481—24no—
24C2×#92481C2212yes—
25C2×#92481C2214yes—
26#11758517—21no—
27#11758514—22no—
28C2×#92479C2212yes—
29#11758507C2214yes—
30#11758503C2211yescharacteristic
31C2×#92474C2211yescharacteristic
32C22×C31⋊C8C2211yescharacteristic
33#11758423C2212yescomplemented
34#11758423C2214yescomplemented
35#11758420C2212yescomplemented
36C22⋊(C31⋊C8)C2214yescomplemented
37C23×D31—21no—
38#41×C31—216no—
39#41×C31—28no—
40#41×C31—24no—
41#41×C31—24no—
42#41×C31—22no—
43#41×C31—22no—
44#41×C31—22no—
45#41×C31D414yescomplemented
46C2×Q8×C31—28no—
47C2×Q8×C31—24no—
48C2×Q8×C31—24no—
49C2×Q8×C31—21no—
50C2×Q8×C31—22no—
51C2×Q8×C31—22no—
52C2×Q8×C31D411yescharacteristic
53C2×Q8×C31D414yescomplemented
54C2×Q8×C31C2312yes—
55C2×Q8×C31C2311yescharacteristic
56C2×D4×C31—21no—
57C2×D4×C31—22no—
58C2×D4×C31D412yes—
59C2×D4×C31D411yescharacteristic
60C2×D4×C31C2311yescharacteristic
61C22×C4×C31—21no—
62C22×C4×C31—22no—
63C22×C4×C31—21no—
64C22×C4×C31—22no—
65C22×C4×C31C2312yes—
66C22×C4×C31C2311yescharacteristic
67C2×C22⋊D31—24no—
68C2×D124—22no—
69C2×D124—22no—
70C2×D124—21no—
71C2×D124C2312yes—
72C22⋊Dic31—21no—
73#92481—28no—
74#92481—24no—
75#92479—24no—
76(C2×D31)⋊C4—24no—
77C4⋊Dic31C2312yes—
78C4×Dic31—21no—
79#92474—22no—
80C2×C31⋊C8—22no—
81C2×C31⋊C8C2312yes—
82C2×C31⋊C8C2312yes—
83C23×C31—21no—
84C23×C31C2×D412yes—
85C23×C31C2×D411yescharacteristic
86C22×D31—22no—
87C22×D31—42no—
88C22×D31—82no—
89Q8×C31—28no—
90Q8×C31—22no—
91Q8×C31—24no—
92Q8×C31—21no—
93Q8×C31—22no—
94Q8×C31C2×D412yes—
95Q8×C31C2×D414yes—
96D4×C31—28no—
97D4×C31—24no—
98D4×C31—24no—
99D4×C31—22no—
100D4×C31—21no—
101D4×C31C2×D412yes—
102C2×C4×C31—28no—
103C2×C4×C31—28no—
104C2×C4×C31—24no—
105C2×C4×C31—22no—
106C2×C4×C31—21no—
107C2×C4×C31—24no—
108C2×C4×C31—22no—
109C2×C4×C31—22no—
110C2×C4×C31C2×D412yes—
111C2×C4×C31C2×D411yescharacteristic
112C2×C4×C31C2×D412yes—
113C2×C4×C31C2×D412yes—
114C2×C4×C31C2411yescharacteristic
115C22⋊D31—84no—
116C2×Dic31—22no—
117D124—22no—
118D124—22no—
119D124—42no—
120C31⋊C8—22no—
121C31⋊C8—22no—
122#3010—311nomaximal, Sylow
123C22×C31—22no—
124C22×C31—22no—
125C22×C31—22no—
126C22×C31—24no—
127C22×C31—42no—
128C22×C31#12012yes—
129C22×C31C22×D411yescharacteristic
130C2×D31—42no—
131C2×D31—44no—
132C2×D31—84no—
133C4×C31—24no—
134C4×C31—22no—
135C4×C31—22no—
136C4×C31#12012yes—
137C4×C31C22×D411yescharacteristic
138C4×C31C22×D411yescharacteristic
139Dic31—42no—
140C2×2−1+4D3111yescharacteristic, complemented
141C2×AGL(1,ℤ/8)—311no—
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—42no—
152C2×C31C2×#12011yescharacteristic
153C2×C31#57712yes—
154D31—82no—
1552−1+4—24no—
1562−1+4—24no—
157C2×#41—22no—
158C2×#41—21no—
159C2×#41C2×D3111yescharacteristic
160C2×#41C2×D3111yescharacteristic
161C2×#41C2×D3112yescomplemented
162C22×Q8—21no—
163C22×Q8C2×D3111yescharacteristic
164C22×Q8C2×D3112yescomplemented
165C22×D4—311no—
166AGL(1,ℤ/8)—624no—
167C2×SD16—314no—
168C2×SD16—312no—
169C2×SD16—624no—
170C2×D8—312no—
171C2×M16—311no—
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—621no—
184#41—216no—
185#41—28no—
186#41—24no—
187#41—24no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41C22⋊D3114yescomplemented
192C2×Q8—28no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8C22⋊D3111yescharacteristic
199C2×Q8C22⋊D3114yescomplemented
200C2×Q8C22×D3112yes—
201C2×Q8C22×D3111yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—312no—
205C2×D4—624no—
206C2×D4—622no—
207C2×D4—622no—
208C2×D4—621no—
209C2×D4C22⋊D3112yes—
210C2×D4C22⋊D3111yescharacteristic
211C2×D4C22×D3111yescharacteristic
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4C22×D3112yes—
217C22×C4C22×D3111yescharacteristic
218SD16—628no—
219SD16—624no—
220D8—624no—
221M16—622no—
222C2×C8—312no—
223C2×C8—312no—
224C2×C8—622no—
225C4⋊C4—312no—
226C22⋊C4—621no—
227C22⋊C4—624no—
228C42—621no—
229C23—21no—
230C23—622no—
231C23—1242no—
232C23—2482no—
233C23C2×C22⋊D3112yes—
234C23C2×C22⋊D3111yescharacteristic
235Q8—28no—
236Q8—22no—
237Q8—24no—
238Q8—21no—
239Q8—22no—
240Q8C2×C22⋊D3112yes—
241Q8C2×C22⋊D3114yes—
242D4—28no—
243D4—24no—
244D4—24no—
245D4—22no—
246D4—21no—
247D4—622no—
248D4—622no—
249D4—1242no—
250D4—2484no—
251D4C2×C22⋊D3112yes—
252C2×C4—28no—
253C2×C4—28no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—622no—
261C2×C4C2×C22⋊D3112yes—
262C2×C4C2×C22⋊D3111yescharacteristic
263C2×C4C2×C22⋊D3112yes—
264C2×C4C2×C22⋊D3112yes—
265C2×C4C23×D3111yescharacteristic
266C8—622no—
267C8—622no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—42no—
273C22—1244no—
274C22—1242no—
275C22—2484no—
276C22#1175853012yes—
277C22C22×C22⋊D3111yescharacteristic
278C4—24no—
279C4—22no—
280C4—22no—
281C4—1242no—
282C4#1175853012yes—
283C4C22×C22⋊D3111yescharacteristic
284C4C22×C22⋊D3111yescharacteristic
285C2—22no—
286C2—42no—
287C2—42no—
288C2—2482no—
289C2C2×#1175853011yescharacteristic
290C2#1452185712yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×2−1+4C2
socleC22×C31C22×D4
hypercenterC2×2−1+4D31
hypocenterC31#3010
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#11758526C2semidirect
2. C22×#92481C2semidirect
3. #14521613C2semidirect
4. #14521194C2semidirect
5. #14521188C2semidirect
6. #14521058C2semidirect
7. #14521045C2semidirect
8. #14521016C2semidirect
9. C2×C31×2−1+4C2semidirect
16. C2×#41×C31C22semidirect
19. C22×Q8×C31C22semidirect
33. #11758423C22semidirect
34. #11758423C22semidirect
35. #11758420C22semidirect
36. C22⋊(C31⋊C8)C22semidirect
45. #41×C31D4semidirect
53. C2×Q8×C31D4semidirect
140. C2×2−1+4D31semidirect
161. C2×#41C2×D31semidirect
164. C22×Q8C2×D31semidirect
182. C31#3010semidirect
191. #41C22⋊D31semidirect
199. C2×Q8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585262
C22×#924812
#145216132
#145211942
#145211882
#145210582
#145210452
#145210162
C2×C31×2−1+42
#301031