Finite Group Databasebeta

← #16854995

Subgroups — #16854995

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×#11758509C211yesmaximal, characteristic, complemented
3C22×#92480C211yesmaximal, characteristic, complemented
4#14521192C214yesmaximal, complemented
5#14521187C214yesmaximal, complemented
6#14521057C211yesmaximal, characteristic, complemented
7#14521043C211yesmaximal, characteristic, complemented
8#14521016C211yesmaximal, characteristic, complemented
9C31×#535C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C22×D4×C31C2211yescharacteristic
12#11758567—24no—
13#11758567—24no—
14C2×Q8×D31—21no—
15C2×#92497—21no—
16C2×#92497—22no—
17C2×#92495C2211yescharacteristic
18C2×#92495C2212yescomplemented
19C22×Dic62C2211yescharacteristic
20C22×Dic62C2212yescomplemented
21C31×#127—21no—
22C31×#124—21no—
23C31×#121C2214yes—
24C31×#120—22no—
25C31×#117C2211yescharacteristic
26C2×#92480—24no—
27C2×#92480C2212yes—
28C2×#92480C2214yes—
29#11758509—24no—
30C2×#92479C2212yes—
31C2×#92474C2211yescharacteristic
32C22×C31⋊C8C2211yescharacteristic
33C22⋊(C31⋊C8)C2214yescomplemented
34#11758398C2212yescomplemented
35#11758398C2214yescomplemented
36#11758397C2212yescomplemented
37C24×C31—21no—
38#41×C31—22no—
39#41×C31—22no—
40C2×Q8×C31D411yescharacteristic
41C2×D4×C31—24no—
42C2×D4×C31—22no—
43C2×D4×C31—22no—
44C2×D4×C31—21no—
45C2×D4×C31D412yes—
46C2×D4×C31D411yescharacteristic
47C2×D4×C31C2312yes—
48C22×C4×C31C2312yes—
49C22×C4×C31C2311yescharacteristic
50C2×C22⋊D31—21no—
51C2×C22⋊D31—22no—
52C22×Dic31—21no—
53C22×Dic31—22no—
54Q8×D31—28no—
55#92497—216no—
56#92497—28no—
57#92495—22no—
58#92495—24no—
59#92495—24no—
60#92495D414yescomplemented
61C2×D124C2311yescharacteristic
62C2×C4×D31—21no—
63C2×C4×D31—22no—
64C2×Dic62—21no—
65C2×Dic62—22no—
66C2×Dic62—22no—
67C2×Dic62—24no—
68C2×Dic62—24no—
69C2×Dic62D414yescomplemented
70C2×Dic62C2312yes—
71C2×Dic62C2311yescharacteristic
72C4⋊C4×C31C2312yes—
73C22⋊C4×C31—21no—
74C22⋊C4×C31—24no—
75C42×C31—21no—
76#92480—24no—
77#92480—28no—
78#92479—24no—
79#92474—22no—
80C2×C31⋊C8—22no—
81C2×C31⋊C8C2312yes—
82C2×C31⋊C8C2312yes—
83C23×C31—22no—
84C23×C31—42no—
85C23×C31—82no—
86C23×C31C2×D412yes—
87C23×C31C2×D411yescharacteristic
88C22×D31—21no—
89Q8×C31—22no—
90D4×C31—24no—
91D4×C31—22no—
92D4×C31—22no—
93D4×C31—22no—
94D4×C31—42no—
95D4×C31—84no—
96C2×C4×C31—22no—
97C2×C4×C31—22no—
98C2×C4×C31—22no—
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2×D412yes—
101C2×C4×C31C2×D412yes—
102C2×C4×C31C2×D411yescharacteristic
103C2×C4×C31C2411yescharacteristic
104C22⋊D31—28no—
105C22⋊D31—24no—
106C2×Dic31—28no—
107C2×Dic31—24no—
108C2×Dic31—22no—
109C2×Dic31—21no—
110D124—21no—
111D124C2×D412yes—
112C4×D31—28no—
113C4×D31—24no—
114Dic62—28no—
115Dic62—24no—
116Dic62—22no—
117Dic62—21no—
118Dic62C2×D412yes—
119Dic62C2×D414yes—
120C31⋊C8—22no—
121C31⋊C8—22no—
122#3010—311nomaximal, Sylow
123C22×C31—22no—
124C22×C31—22no—
125C22×C31—24no—
126C22×C31—42no—
127C22×C31—44no—
128C22×C31—84no—
129C22×C31#12012yes—
130C22×C31C22×D411yescharacteristic
131C2×D31—22no—
132C2×D31—42no—
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#535D3111yescharacteristic, complemented
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—622no—
160C2×#41—621no—
161C2×#41C2×D3111yescharacteristic
162C22×Q8—311no—
163C22×Q8—312no—
164C22×Q8—621no—
165C22×D4C2×D3111yescharacteristic
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—21no—
174#124—21no—
175#121C2×D3114yes—
176#120—22no—
177#117C2×D3111yescharacteristic
178#103—312no—
179#103—314no—
180#102—312no—
181C22⋊C8—314no—
182C31#301011yescharacteristic, complemented, Sylow
183C24—21no—
184#41—22no—
185#41—22no—
186#41—314no—
187#41—624no—
188#41—624no—
189#41—6216no—
190#41—628no—
191#41—622no—
192C2×Q8—314no—
193C2×Q8—312no—
194C2×Q8—311no—
195C2×Q8—624no—
196C2×Q8—624no—
197C2×Q8—628no—
198C2×Q8—622no—
199C2×Q8—622no—
200C2×Q8—621no—
201C2×Q8C22⋊D3111yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—311no—
207C2×D4—622no—
208C2×D4—621no—
209C2×D4C22⋊D3112yes—
210C2×D4C22⋊D3111yescharacteristic
211C2×D4C22×D3112yes—
212C22×C4—622no—
213C22×C4—621no—
214C22×C4—622no—
215C22×C4—621no—
216C22×C4C22×D3112yes—
217C22×C4C22×D3111yescharacteristic
218SD16—628no—
219SD16—624no—
220D8—624no—
221M16—622no—
222C2×C8—312no—
223C2×C8—312no—
224C2×C8—622no—
225C4⋊C4C22×D3112yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—621no—
233C23C2×C22⋊D3112yes—
234C23C2×C22⋊D3111yescharacteristic
235Q8—22no—
236Q8—314no—
237Q8—312no—
238Q8—628no—
239Q8—624no—
240Q8—622no—
241Q8—621no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—312no—
249D4—628no—
250D4—624no—
251D4—621no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—628no—
256C2×C4—628no—
257C2×C4—624no—
258C2×C4—622no—
259C2×C4—621no—
260C2×C4—624no—
261C2×C4D4×D3112yes—
262C2×C4D4×D3112yes—
263C2×C4C2×C22⋊D3112yes—
264C2×C4C2×C22⋊D3111yescharacteristic
265C2×C4C23×D3111yescharacteristic
266C8—622no—
267C8—622no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—622no—
275C22—1242no—
276C22#1175851412yes—
277C22C22×C22⋊D3111yescharacteristic
278C4—22no—
279C4—42no—
280C4—624no—
281C4—622no—
282C4#1175851412yes—
283C4C2×D4×D3111yescharacteristic
284C4C2×D4×D3111yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—1242no—
289C2C2×#1175851411yescharacteristic
290C2#1452186012yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#535C2
socleC22×C31C22×D4
hypercenter#535D31
hypocenterC31#3010
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#11758567C2semidirect
2. C2×#11758509C2semidirect
3. C22×#92480C2semidirect
4. #14521192C2semidirect
5. #14521187C2semidirect
6. #14521057C2semidirect
7. #14521043C2semidirect
8. #14521016C2semidirect
9. C31×#535C2semidirect
18. C2×#92495C22semidirect
20. C22×Dic62C22semidirect
33. C22⋊(C31⋊C8)C22semidirect
34. #11758398C22semidirect
35. #11758398C22semidirect
36. #11758397C22semidirect
60. #92495D4semidirect
69. C2×Dic62D4semidirect
143. #535D31semidirect
182. C31#3010semidirect

Maximal subgroups

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

subgroupindex
C2×#117585672
C2×#117585092
C22×#924802
#145211922
#145211872
#145210572
#145210432
#145210162
C31×#5352
#301031