Finite Group Databasebeta

← #16854993

Subgroups — #16854993

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×#11758564C211yesmaximal, characteristic, complemented
2C2×#11758509C211yesmaximal, characteristic, complemented
3#14521616C211yesmaximal, characteristic, complemented
4#14521193C214yesmaximal, complemented
5#14521186C214yesmaximal, complemented
6#14521058C211yesmaximal, characteristic, complemented
7#14521046C211yesmaximal, characteristic, complemented
8#14521015C211yesmaximal, characteristic, complemented
9C22×SD16×C31C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C22×D4×C31C2211yescharacteristic
12#11758564—24no—
13#11758564—24no—
14C2×#92499—21no—
15C2×#92499C2212yescomplemented
16C2×Q8×D31—21no—
17C2×Q8×D31C2212yescomplemented
18C2×#92495—22no—
19C2×#92495C2211yescharacteristic
20C2×#92495C2211yescharacteristic
21C2×SD16×C31—24no—
22C2×SD16×C31C2214yes—
23C22×C8×C31C2211yescharacteristic
24#11758530—22no—
25#11758517—21no—
26#11758515C2214yes—
27#11758513—21no—
28C2×#92480C2212yes—
29#11758509—24no—
30C2×#92479C2212yes—
31#11758503C2211yescharacteristic
32C2×#92474C2211yescharacteristic
33#11758423C2214yescomplemented
34#11758409C2212yescomplemented
35(C2×D31)⋊C8C2214yescomplemented
36#11758404C2212yescomplemented
37C24×C31—21no—
38C2×Q8×C31—21no—
39C2×Q8×C31—22no—
40C2×Q8×C31D414yescomplemented
41C2×Q8×C31C2312yes—
42C2×D4×C31—22no—
43C2×D4×C31—22no—
44C2×D4×C31—21no—
45C2×D4×C31C2312yes—
46C22×C4×C31—21no—
47C22×C4×C31C2311yescharacteristic
48C2×C22⋊D31—21no—
49C2×C22⋊D31—24no—
50C2×C22⋊D31D412yes—
51#92499—28no—
52#92499—24no—
53#92499—24no—
54Q8×D31—28no—
55Q8×D31—24no—
56Q8×D31—24no—
57#92495—216no—
58#92495—22no—
59#92495—22no—
60#92495—22no—
61#92495D414yescomplemented
62C2×D124—22no—
63C2×D124D411yescharacteristic
64C2×D124C2311yescharacteristic
65C2×C4×D31—22no—
66C2×C4×D31—21no—
67C2×C4×D31—22no—
68C2×C4×D31C2312yes—
69C2×Dic62—22no—
70C2×Dic62D411yescharacteristic
71C2×Dic62C2311yescharacteristic
72SD16×C31—28no—
73C2×C8×C31—22no—
74C2×C8×C31C2312yes—
75C22⋊Dic31—21no—
76C22⋊Dic31—24no—
77#92480—24no—
78#92479—24no—
79C4⋊Dic31C2312yes—
80C4×Dic31—21no—
81#92474—22no—
82C2×C31⋊C8C2312yes—
83C23×C31—22no—
84C23×C31—42no—
85C23×C31—82no—
86C23×C31C2×D411yescharacteristic
87C22×D31—21no—
88C22×D31C2×D412yes—
89Q8×C31—24no—
90Q8×C31—22no—
91Q8×C31C2×D414yes—
92D4×C31—22no—
93D4×C31—22no—
94D4×C31—42no—
95C2×C4×C31—24no—
96C2×C4×C31—22no—
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31C2×D412yes—
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2411yescharacteristic
101C22⋊D31—24no—
102C22⋊D31—24no—
103C22⋊D31—84no—
104C2×Dic31—21no—
105C2×Dic31—22no—
106C2×Dic31C2×D412yes—
107D124—28no—
108D124—22no—
109D124—21no—
110D124C2×D412yes—
111C4×D31—28no—
112C4×D31—28no—
113C4×D31—24no—
114C4×D31—22no—
115C4×D31—22no—
116Dic62—28no—
117Dic62—22no—
118Dic62—21no—
119Dic62C2×D412yes—
120C8×C31—22no—
121C31⋊C8—22no—
122#3010—311nomaximal, Sylow
123C22×C31—22no—
124C22×C31—42no—
125C22×C31—44no—
126C22×C31—84no—
127C22×C31#12012yes—
128C22×C31C22×D411yescharacteristic
129C2×D31—22no—
130C2×D31—22no—
131C2×D31—24no—
132C2×D31—42no—
133C4×C31—24no—
134C4×C31#12012yes—
135C4×C31C22×D411yescharacteristic
136C4×C31C22×D411yescharacteristic
137Dic31—22no—
138Dic31—22no—
139Dic31—42no—
140C2×2−1+4—311no—
141C2×AGL(1,ℤ/8)—311no—
142C22×SD16D3111yescharacteristic, complemented
143#535—311no—
144#449—314no—
145#448—314no—
146#419—311no—
147#416—311no—
148#408—311no—
149C2×C31—22no—
150C2×C31—82no—
151C2×C31C2×#12011yescharacteristic
152C2×C31#57712yes—
153D31—42no—
154D31—42no—
1552−1+4—624no—
1562−1+4—624no—
157C2×#41—311no—
158C2×#41—312no—
159C2×#41—311no—
160C2×#41—622no—
161C2×#41—621no—
162C22×Q8—312no—
163C22×Q8—621no—
164C22×Q8C2×D3111yescharacteristic
165C22×D4C2×D3111yescharacteristic
166AGL(1,ℤ/8)—624no—
167C2×SD16—24no—
168C2×SD16—312no—
169C2×SD16C2×D3114yes—
170C2×D8—312no—
171C2×M16—311no—
172C22×C8C2×D3111yescharacteristic
173#127—621no—
174#124—621no—
175#121—314no—
176#120—622no—
177#117—311no—
178#103—314no—
179#103—312no—
180#102—312no—
181C22⋊C8—314no—
182C31#301011yescharacteristic, complemented, Sylow
183C24—21no—
184#41—314no—
185#41—6216no—
186#41—624no—
187#41—628no—
188#41—624no—
189#41—622no—
190#41—622no—
191#41—622no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—311no—
195C2×Q8—311no—
196C2×Q8—622no—
197C2×Q8—624no—
198C2×Q8—628no—
199C2×Q8—624no—
200C2×Q8C22⋊D3114yescomplemented
201C2×Q8C22×D3112yes—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—312no—
206C2×D4—311no—
207C2×D4—311no—
208C2×D4—622no—
209C2×D4—621no—
210C2×D4—624no—
211C2×D4C22×D3112yes—
212C22×C4—21no—
213C22×C4—312no—
214C22×C4—622no—
215C22×C4—622no—
216C22×C4—621no—
217C22×C4C22×D3111yescharacteristic
218SD16—28no—
219SD16—624no—
220D8—624no—
221M16—622no—
222C2×C8—22no—
223C2×C8—312no—
224C2×C8C22×D3112yes—
225C4⋊C4—312no—
226C22⋊C4—624no—
227C22⋊C4—621no—
228C42—621no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—312no—
233C23—621no—
234C23D4×D3111yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—312no—
238Q8—628no—
239Q8—622no—
240Q8—621no—
241Q8C2×C22⋊D3114yes—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—312no—
246D4—628no—
247D4—624no—
248D4—624no—
249D4—622no—
250D4—621no—
251D4—2484no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—312no—
255C2×C4—628no—
256C2×C4—628no—
257C2×C4—621no—
258C2×C4—624no—
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—42no—
270C22—44no—
271C22—84no—
272C22—622no—
273C22—622no—
274C22—624no—
275C22—1242no—
276C22#1175851412yes—
277C22C2×D4×D3111yescharacteristic
278C4—24no—
279C4—622no—
280C4—622no—
281C4—1242no—
282C4#1175851412yes—
283C4C2×D4×D3111yescharacteristic
284C4C22×C22⋊D3111yescharacteristic
285C2—22no—
286C2—82no—
287C2—1242no—
288C2—1242no—
289C2C2×#1175851411yescharacteristic
290C2#1452185512yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC22×SD16×C31C2
socleC22×C31C22×D4
hypercenterC22×SD16D31
hypocenterC31#3010
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#11758564C2semidirect
2. C2×#11758509C2semidirect
3. #14521616C2semidirect
4. #14521193C2semidirect
5. #14521186C2semidirect
6. #14521058C2semidirect
7. #14521046C2semidirect
8. #14521015C2semidirect
9. C22×SD16×C31C2semidirect
15. C2×#92499C22semidirect
17. C2×Q8×D31C22semidirect
33. #11758423C22semidirect
34. #11758409C22semidirect
35. (C2×D31)⋊C8C22semidirect
36. #11758404C22semidirect
40. C2×Q8×C31D4semidirect
61. #92495D4semidirect
142. C22×SD16D31semidirect
182. C31#3010semidirect
200. C2×Q8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585642
C2×#117585092
#145216162
#145211932
#145211862
#145210582
#145210462
#145210152
C22×SD16×C312
#301031