Finite Group Databasebeta

← #16857492

Subgroups — #16857492

Order 3968.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#14521866C211yesmaximal, characteristic, complemented
2#14521859C212yesmaximal, complemented
3#14521850C211yesmaximal, characteristic, complemented
4#14521432C211yesmaximal, characteristic, complemented
5#14521325C211yesmaximal, characteristic, complemented
6#14521311C212yesmaximal, complemented
7#14521229C212yesmaximal, complemented
8#14521220C212yesmaximal, complemented
9#14521165C211yesmaximal, characteristic, complemented
10#14521164C211yesmaximal, characteristic, complemented
11C31×#472C211yesmaximal, characteristic, complemented
12#11758567C2212yescomplemented
13#11758561C2212yes—
14C31×AGL(1,ℤ/8)C2212yescomplemented
15C2×D8×C31C2211yescharacteristic
16C31×#131C2211yescharacteristic, complemented
17#11758527C2211yescharacteristic
18#11758513C2212yescomplemented
19C2×#92480C2212yescomplemented
20#11758509C2212yes—
21#11758492—22no—
22SD16×D31—22no—
23#11758489—22no—
24#11758489—21no—
25#11758488—22no—
26#11758488—22no—
27D8×D31—21no—
28#11758486—22no—
29#11758486C2212yes—
30#11758486C2212yescomplemented
31#11758484C2212yescomplemented
32C2×Dic124—21no—
33C2×Dic124C2211yescharacteristic, complemented
34#11758481—21no—
35#11758481C2211yescharacteristic
36#11758478C2211yescharacteristic
37C31×#108C2211yescharacteristic, complemented
38C31×#100C2212yescomplemented
39#11758425C2211yescharacteristic, complemented
40#11758413C2211yescharacteristic
41#11758413C2212yescomplemented
42#11758412C2212yescomplemented
43#11758410C2211yescharacteristic, complemented
44C8×Dic31C2211yescharacteristic, complemented
45#41×C31C2311yescharacteristic
46C2×D4×C31C2312yes—
47C2×C22⋊D31—42no—
48Q8×D31—22no—
49#92497—22no—
50#92497—22no—
51#92497—22no—
52#92497—42no—
53D4×D31—22no—
54D4×D31—22no—
55#92495—22no—
56#92495C2311yescharacteristic
57#92495C2312yes—
58C2×Dic62—22no—
59C2×Dic62C2312yes—
60SD16×C31—22no—
61D8×C31—22no—
62D8×C31—21no—
63D8×C31—21no—
64M16×C31—21no—
65M16×C31C2311yescharacteristic
66M16×C31C2312yes—
67C2×C8×C31—21no—
68C2×C8×C31C2311yescharacteristic
69C22⋊Dic31—42no—
70#92482D411yescharacteristic
71#92481D411yescharacteristic
72#92480—22no—
73#92480—22no—
74#92480—22no—
75#92480D411yescharacteristic
76#92479—22no—
77#92479D411yescharacteristic
78C4×Dic31C2311yescharacteristic
79#92474C2311yescharacteristic
80C2×C31⋊C8C2311yescharacteristic
81Dic124—22no—
82Dic124—22no—
83Dic124—21no—
84Dic124—21no—
85Dic124D411yescharacteristic
86D248D411yescharacteristic
87#92470—22no—
88#92470—22no—
89#92470D412yes—
90#92469—21no—
91#92469—22no—
92C8×D31—21no—
93C8×D31—22no—
94C23×C31—22no—
95C22×D31—42no—
96Q8×C31C2×D411yescharacteristic
97D4×C31—21no—
98D4×C31—22no—
99D4×C31—22no—
100D4×C31C2×D411yescharacteristic
101C2×C4×C31—21no—
102C2×C4×C31C2411yescharacteristic
103C22⋊D31—22no—
104C22⋊D31—21no—
105C22⋊D31—42no—
106C22⋊D31—42no—
107C22⋊D31—42no—
108C2×Dic31—21no—
109C2×Dic31—22no—
110C2×Dic31—42no—
111C2×Dic31—42no—
112D124C2×D411yescharacteristic
113D124C2×D412yes—
114C4×D31—21no—
115C4×D31—22no—
116C4×D31—42no—
117Dic62—22no—
118Dic62—22no—
119Dic62—42no—
120Dic62C2×D412yes—
121Dic62C2×D411yescharacteristic
122C8×C31—22no—
123C8×C31—21no—
124C8×C31C2×D411yescharacteristic
125C8×C31C2×D411yescharacteristic
126C31⋊C8—21no—
127C31⋊C8C2×D411yescharacteristic
128C31⋊C8C2×D411yescharacteristic
129#3290—311nomaximal, Sylow
130C22×C31—21no—
131C22×C31—42no—
132C22×C31—82no—
133C22×C31H(5,𝔽2)11yescharacteristic
134C2×D31—22no—
135C2×D31—21no—
136C2×D31—162no—
137C4×C31—21no—
138C4×C31C22×D411yescharacteristic
139C4×C31C22×D411yescharacteristic
140Dic31—22no—
141Dic31—21no—
142Dic31—41no—
143Dic31—42no—
144#578—311no—
145#577—312no—
146#576—311no—
147#495—311no—
148#473—311no—
149#472D3111yescharacteristic, complemented
150#471—312no—
151#455—312no—
152#454—312no—
153#443—311no—
154#443—311no—
155C2×C31—21no—
156C2×C31—41no—
157C2×C31—82no—
158C2×C31D4211yescharacteristic
159D31—42no—
160D31—41no—
1612−1+4—312no—
162H(5,𝔽2)—312no—
163#137—312no—
164#137—312no—
165#137—622no—
166#137—622no—
167AGL(1,ℤ/8)—312no—
168AGL(1,ℤ/8)—622no—
169AGL(1,ℤ/8)—622no—
170AGL(1,ℤ/8)C2×D3112yescomplemented
171#135—311no—
172#135—311no—
173#135—621no—
174#135—622no—
175#135—621no—
176C2×Q16—311no—
177C2×Q16—621no—
178C2×SD16—312no—
179C2×SD16—622no—
180C2×D8—621no—
181C2×D8C2×D3111yescharacteristic
182#131—312no—
183#131—311no—
184#131C2×D3111yescharacteristic, complemented
185#124—312no—
186#108—311no—
187#108C2×D3111yescharacteristic, complemented
188#104—311no—
189#104—311no—
190#104—312no—
191#101—312no—
192#100C2×D3112yescomplemented
193C4×C8—311no—
194C31#329011yescharacteristic, complemented, Sylow
195#41—312no—
196#41—311no—
197#41—622no—
198#41—622no—
199#41—622no—
200#41—622no—
201#41—1242no—
202#41C22×D3111yescharacteristic
203C2×Q8—312no—
204C2×Q8—622no—
205C2×Q8—622no—
206C2×D4—622no—
207C2×D4—622no—
208C2×D4—1242no—
209C2×D4C22×D3112yes—
210Q16—311no—
211Q16—311no—
212Q16—622no—
213Q16—622no—
214Q16—621no—
215Q16—621no—
216SD16—22no—
217SD16—311no—
218SD16—311no—
219SD16—312no—
220SD16—622no—
221SD16—622no—
222SD16—622no—
223SD16—622no—
224SD16—622no—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—311no—
229D8—311no—
230D8—622no—
231M16—21no—
232M16—311no—
233M16—622no—
234M16—621no—
235M16C22×D3111yescharacteristic
236M16C22×D3112yes—
237C2×C8—21no—
238C2×C8—311no—
239C2×C8—622no—
240C2×C8—621no—
241C2×C8C22×D3111yescharacteristic
242C22⋊C4—1242no—
243C42—311no—
244C23—22no—
245C23—1242no—
246Q8—312no—
247Q8—311no—
248Q8—622no—
249Q8—622no—
250Q8—1242no—
251Q8D4×D3111yescharacteristic
252D4—21no—
253D4—22no—
254D4—22no—
255D4—312no—
256D4—311no—
257D4—621no—
258D4—622no—
259D4—1242no—
260D4—1242no—
261D4—1242no—
262D4D4×D3111yescharacteristic
263C2×C4—21no—
264C2×C4—621no—
265C2×C4—622no—
266C2×C4—621no—
267C2×C4—622no—
268C2×C4—1242no—
269C2×C4—1242no—
270C2×C4—1242no—
271C2×C4C23×D3111yescharacteristic
272C8—21no—
273C8—22no—
274C8—311no—
275C8—311no—
276C8—621no—
277C8D4×D3111yescharacteristic
278C8D4×D3111yescharacteristic
279C22—21no—
280C22—42no—
281C22—82no—
282C22—622no—
283C22—621no—
284C22—4962no—
285C22#1175856611yescharacteristic
286C4—21no—
287C4—621no—
288C4—622no—
289C4—1241no—
290C4—1242no—
291C4C2×D4×D3111yescharacteristic
292C4C2×D4×D3111yescharacteristic
293C2—21no—
294C2—41no—
295C2—82no—
296C2—1241no—
297C2—1242no—
298C2#1452168311yescharacteristic
subgroupG/H
centerC2#14521683
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#472C2
socleC2×C31D42
hypercenter#472D31
hypocenterC31#3290
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #14521866C2semidirect
2. #14521859C2semidirect
3. #14521850C2semidirect
4. #14521432C2semidirect
5. #14521325C2semidirect
6. #14521311C2semidirect
7. #14521229C2semidirect
8. #14521220C2semidirect
9. #14521165C2semidirect
10. #14521164C2semidirect
11. C31×#472C2semidirect
12. #11758567C22semidirect
14. C31×AGL(1,ℤ/8)C22semidirect
16. C31×#131C22semidirect
18. #11758513C22semidirect
19. C2×#92480C22semidirect
30. #11758486C22semidirect
31. #11758484C22semidirect
33. C2×Dic124C22semidirect
37. C31×#108C22semidirect
38. C31×#100C22semidirect
39. #11758425C22semidirect
41. #11758413C22semidirect
42. #11758412C22semidirect
43. #11758410C22semidirect
44. C8×Dic31C22semidirect
149. #472D31semidirect
170. AGL(1,ℤ/8)C2×D31semidirect
184. #131C2×D31semidirect
187. #108C2×D31semidirect
192. #100C2×D31semidirect
194. C31#3290semidirect

Maximal subgroups

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

subgroupindex
#145218662
#145218592
#145218502
#145214322
#145213252
#145213112
#145212292
#145212202
#145211652
#145211642
C31×#4722
#329031