Finite Group Databasebeta

← #16857506

Subgroups — #16857506

Order 3968.

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

normal subgroups only

This group has 460 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#14521860C212yesmaximal, complemented
2#14521859C212yesmaximal, complemented
3#14521421C211yesmaximal, characteristic, complemented
4#14521311C212yesmaximal, complemented
5#14521310C212yesmaximal, complemented
6#14521233C212yesmaximal, complemented
7#14521216C211yesmaximal, characteristic, complemented
8#14521163C212yesmaximal, complemented
9C31×#453C211yesmaximal, characteristic, complemented
10C31×H(5,𝔽2)C2211yescharacteristic, complemented
11#11758567C2212yes—
12#11758561C2211yescharacteristic, complemented
13C31×AGL(1,ℤ/8)C2212yes—
14C31×#127C2211yescharacteristic
15#11758528—22no—
16#11758528C2212yescomplemented
17#11758527—22no—
18#11758527C2212yes—
19#11758525C2212yescomplemented
20C2×#92480—22no—
21C2×#92480C2212yescomplemented
22C2×#92480C2212yescomplemented
23#11758509—22no—
24#11758509C2212yes—
25#11758492—22no—
26SD16×D31—22no—
27#11758489—22no—
28#11758488—22no—
29#11758486C2212yescomplemented
30#11758484C2212yescomplemented
31#11758446C2211yescharacteristic, complemented
32C31×#104C2212yescomplemented
33C31×#100C2211yescharacteristic, complemented
34#11758421C2211yescharacteristic, complemented
35#11758412C2212yescomplemented
36#11758410C2212yescomplemented
37#11758393C2212yescomplemented
38C4×C31⋊C8C2211yescharacteristic, complemented
39#41×C31—22no—
40#41×C31C2312yes—
41C2×D4×C31—22no—
42C2×D4×C31—41no—
43C2×D4×C31—41no—
44C2×D4×C31C2311yescharacteristic
45C2×D4×C31C2311yescharacteristic
46C2×C22⋊D31—41no—
47Q8×D31—22no—
48#92497—22no—
49#92497—22no—
50#92497—42no—
51D4×D31—22no—
52#92495—22no—
53#92495C2312yes—
54C2×Dic62—22no—
55C2×Dic62C2312yes—
56SD16×C31—22no—
57D8×C31—22no—
58M16×C31C2312yes—
59C42×C31C2311yescharacteristic
60#92482—22no—
61#92482D412yescomplemented
62#92481D412yescomplemented
63#92480—22no—
64#92480—22no—
65#92480—22no—
66#92480—22no—
67#92480—22no—
68#92480—22no—
69#92480D412yescomplemented
70#92479—22no—
71#92479D412yescomplemented
72C4⋊Dic31—41no—
73#92474—22no—
74#92474C2312yes—
75C2×C31⋊C8—22no—
76C2×C31⋊C8C2312yes—
77Dic124—22no—
78#92470—22no—
79#92469—22no—
80C8×D31—22no—
81C23×C31—21no—
82C23×C31—21no—
83C23×C31—41no—
84C22×D31—41no—
85Q8×C31C2×D412yes—
86D4×C31—22no—
87D4×C31—22no—
88D4×C31—22no—
89D4×C31—42no—
90D4×C31—82no—
91D4×C31C2×D412yes—
92C2×C4×C31—21no—
93C2×C4×C31—22no—
94C2×C4×C31—41no—
95C2×C4×C31C2411yescharacteristic
96C22⋊D31—22no—
97C22⋊D31—42no—
98C22⋊D31—42no—
99C2×Dic31—22no—
100C2×Dic31—41no—
101C2×Dic31—42no—
102D124C2×D412yes—
103C4×D31—22no—
104C4×D31—42no—
105Dic62—22no—
106Dic62—22no—
107Dic62—42no—
108Dic62C2×D412yes—
109C8×C31—22no—
110C31⋊C8—22no—
111C31⋊C8C2×D412yes—
112C31⋊C8C2×D412yes—
113#3291—311nomaximal, Sylow
114C22×C31—22no—
115C22×C31—41no—
116C22×C31—41no—
117C22×C31—81no—
118C22×C31—81no—
119C22×C31—161no—
120C22×C31H(5,𝔽2)11yescharacteristic
121C2×D31—22no—
122C2×D31—161no—
123C4×C31—22no—
124C4×C31—22no—
125C4×C31C22×D412yes—
126Dic31—22no—
127Dic31—42no—
128#577—312no—
129#577—312no—
130#492—311no—
131#471—312no—
132#471—312no—
133#456—312no—
134#453—311no—
135#453D3111yescharacteristic, complemented
136#443—312no—
137C2×C31—21no—
138C2×C31—42no—
139C2×C31—81no—
140C2×C31—81no—
141C2×C31D4211yescharacteristic
142D31—42no—
1432−1+4—312no—
144H(5,𝔽2)—311no—
145H(5,𝔽2)C2×D3111yescharacteristic, complemented
146#137—312no—
147#137—312no—
148#137—622no—
149#137—622no—
150AGL(1,ℤ/8)—312no—
151AGL(1,ℤ/8)—622no—
152AGL(1,ℤ/8)—622no—
153AGL(1,ℤ/8)C2×D3112yes—
154#135—312no—
155#135—622no—
156#135—622no—
157C2×SD16—312no—
158C2×SD16—312no—
159C2×SD16—622no—
160C2×SD16—622no—
161#131—312no—
162#131—312no—
163C4⋊Q8—311no—
164#127C2×D3111yescharacteristic
165#108—312no—
166#104—312no—
167#104C2×D3112yescomplemented
168#101—312no—
169#100—311no—
170#100C2×D3111yescharacteristic, complemented
171C4×C8—311no—
172C31#329111yescharacteristic, complemented, Sylow
173#41—22no—
174#41—312no—
175#41—622no—
176#41—622no—
177#41—622no—
178#41—1242no—
179#41C22×D3112yes—
180C2×Q8—312no—
181C2×Q8—622no—
182C2×Q8—622no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—622no—
187C2×D4—1241no—
188C2×D4C22×D3111yescharacteristic
189C2×D4C22×D3111yescharacteristic
190Q16—312no—
191Q16—622no—
192Q16—622no—
193SD16—22no—
194SD16—312no—
195SD16—312no—
196SD16—622no—
197SD16—622no—
198SD16—622no—
199SD16—622no—
200SD16—622no—
201SD16—622no—
202SD16—622no—
203D8—22no—
204D8—312no—
205D8—622no—
206M16—312no—
207M16—622no—
208M16—622no—
209M16C22×D3112yes—
210C2×C8—312no—
211C2×C8—622no—
212C2×C8—622no—
213C4⋊C4—1241no—
214C42C22×D3111yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—1241no—
219Q8—312no—
220Q8—622no—
221Q8—622no—
222Q8—1242no—
223Q8D4×D3112yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—312no—
230D4—622no—
231D4—1242no—
232D4—1242no—
233D4D4×D3112yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—622no—
238C2×C4—622no—
239C2×C4—1242no—
240C2×C4—1241no—
241C2×C4—1242no—
242C2×C4C23×D3111yescharacteristic
243C8—22no—
244C8—312no—
245C8—312no—
246C8—622no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—622no—
254C22—4961no—
255C22#1175856111yescharacteristic
256C4—22no—
257C4—22no—
258C4—622no—
259C4—1242no—
260C4C2×D4×D3112yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—1242no—
266C2#1452168111yescharacteristic
subgroupG/H
centerC2#14521681
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#453C2
socleC2×C31D42
hypercenter#453D31
hypocenterC31#3291
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #14521860C2semidirect
2. #14521859C2semidirect
3. #14521421C2semidirect
4. #14521311C2semidirect
5. #14521310C2semidirect
6. #14521233C2semidirect
7. #14521216C2semidirect
8. #14521163C2semidirect
9. C31×#453C2semidirect
10. C31×H(5,𝔽2)C22semidirect
12. #11758561C22semidirect
16. #11758528C22semidirect
19. #11758525C22semidirect
21. C2×#92480C22semidirect
22. C2×#92480C22semidirect
29. #11758486C22semidirect
30. #11758484C22semidirect
31. #11758446C22semidirect
32. C31×#104C22semidirect
33. C31×#100C22semidirect
34. #11758421C22semidirect
35. #11758412C22semidirect
36. #11758410C22semidirect
37. #11758393C22semidirect
38. C4×C31⋊C8C22semidirect
61. #92482D4semidirect
62. #92481D4semidirect
69. #92480D4semidirect
71. #92479D4semidirect
135. #453D31semidirect
145. H(5,𝔽2)C2×D31semidirect
167. #104C2×D31semidirect
170. #100C2×D31semidirect
172. C31#3291semidirect

Maximal subgroups

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

subgroupindex
#145218602
#145218592
#145214212
#145213112
#145213102
#145212332
#145212162
#145211632
C31×#4532
#329131