Finite Group Databasebeta

← #16858856

Subgroups — #16858856

Order 3968.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#14521862C211yesmaximal, characteristic, complemented
2#14521851C211yesmaximal, characteristic, complemented
3#14521491C212yesmaximal, complemented
4#14521487C211yesmaximal, characteristic, complemented
5#14521486C212yesmaximal, complemented
6#14521482C212yesmaximal, complemented
7#14521479C211yesmaximal, characteristic, complemented
8C2×#11758416C211yesmaximal, characteristic, complemented
9SD32×D31C212yesmaximal, complemented
10#14521467C211yesmaximal, characteristic, complemented
11C31×#510C211yesmaximal, characteristic, complemented
12#11758566—21no—
13#11758564—21no—
14C31×#135C2211yescharacteristic
15C2×Q16×C31C2211yescharacteristic, complemented
16#11758526—21no—
17#11758519—21no—
18#11758496C2212yes—
19#11758496C2212yescomplemented
20#11758495—22no—
21Q16×D31C2212yescomplemented
22#11758491—22no—
23D8×D31C2212yes—
24#11758481C2211yescharacteristic, complemented
25C2×D248C2211yescharacteristic
26#11758478C2211yescharacteristic
27C31×Q32C2212yescomplemented
28C31×SD32C2212yescomplemented
29C31×M32C2211yescharacteristic, complemented
30#11758417C2211yescharacteristic, complemented
31#11758416C2212yescomplemented
32#11758416C2212yescomplemented
33#11758416C2211yescharacteristic, complemented
34#11758415C2211yescharacteristic, complemented
35#11758414C2211yescharacteristic, complemented
36C2×C31⋊C16C2211yescharacteristic, complemented
37#11758388C2212yescomplemented
38D496C2212yescomplemented
39#11758386C2212yescomplemented
40C16×D31C2212yescomplemented
41#41×C31—21no—
42C2×Q8×C31—21no—
43#92499—22no—
44#92499—22no—
45#92499—41no—
46Q8×D31—22no—
47Q8×D31—41no—
48D4×D31—22no—
49D4×D31—42no—
50#92495—21no—
51#92495—42no—
52#92495—41no—
53#92495D411yescharacteristic
54C2×D124—21no—
55C2×D124—41no—
56Q16×C31—21no—
57Q16×C31C2312yes—
58Q16×C31C2311yescharacteristic
59SD16×C31—21no—
60D8×C31C2311yescharacteristic
61C2×C8×C31C2311yescharacteristic
62#92482—22no—
63#92481—22no—
64#92481—22no—
65#92479—22no—
66#92474D411yescharacteristic
67Dic124C2311yescharacteristic
68D248—21no—
69D248C2311yescharacteristic
70D248C2312yes—
71#92470—21no—
72#92469D412yes—
73C8×D31C2312yes—
74C16×C31C2312yes—
75C31⋊C16C2311yescharacteristic
76C31⋊C16C2311yescharacteristic
77C22×D31—41no—
78C22×D31—42no—
79Q8×C31—21no—
80Q8×C31—22no—
81Q8×C31—41no—
82D4×C31—21no—
83D4×C31—41no—
84C2×C4×C31—41no—
85C2×C4×C31—41no—
86C2×C4×C31C2×D411yescharacteristic
87C22⋊D31—41no—
88C22⋊D31—42no—
89C22⋊D31D812yes—
90D124—21no—
91D124—22no—
92D124—42no—
93D124—42no—
94D124—41no—
95D124D811yescharacteristic
96C4×D31—42no—
97C4×D31—41no—
98C4×D31—42no—
99C4×D31—42no—
100C4×D31C2×D412yes—
101Dic62—21no—
102Dic62—42no—
103Dic62D811yescharacteristic
104C8×C31C2×D411yescharacteristic
105C8×C31C2411yescharacteristic
106C31⋊C8C2×D412yes—
107#3416—311nomaximal, Sylow
108C22×C31—41no—
109C22×C31C2×D811yescharacteristic
110C2×D31—41no—
111C2×D31—42no—
112C2×D31—162no—
113C2×D31—161no—
114C2×D31C2×D812yes—
115C4×C31—42no—
116C4×C31—41no—
117C4×C31C2×D811yescharacteristic
118C4×C31C22×D411yescharacteristic
119Dic31—41no—
120Dic31C2×D812yes—
121#578—311no—
122#576—311no—
123#510—312no—
124#510D3111yescharacteristic, complemented
125#509—312no—
126#509—311no—
127#508—311no—
128#508—312no—
129C2×SD32—311no—
130C2×SD32—312no—
131#504—311no—
132C2×C31—21no—
133C2×C31—81no—
134C2×C31C22×D811yescharacteristic
135D31—22no—
136D31—81no—
137D31—82no—
1382−1+4—621no—
139H(5,𝔽2)—621no—
140#137—622no—
141#137—621no—
142AGL(1,ℤ/8)—622no—
143AGL(1,ℤ/8)—621no—
144#135—312no—
145#135—312no—
146#135—311no—
147#135C2×D3111yescharacteristic
148C2×Q16—312no—
149C2×Q16C2×D3111yescharacteristic, complemented
150C2×D8—312no—
151C2×D8—311no—
152#131—311no—
153Q32—311no—
154Q32C2×D3112yescomplemented
155SD32—312no—
156SD32—312no—
157SD32—311no—
158SD32—311no—
159SD32—312no—
160SD32C2×D3112yescomplemented
161D16—311no—
162D16—312no—
163M32—312no—
164M32C2×D3111yescharacteristic, complemented
165C2×C16—311no—
166C2×C16—312no—
167C31#341611yescharacteristic, complemented, Sylow
168#41—21no—
169#41—311no—
170#41—622no—
171#41—622no—
172#41—621no—
173#41—1242no—
174#41—1241no—
175#41—1241no—
176C2×Q8—21no—
177C2×Q8—622no—
178C2×Q8—1241no—
179C2×D4—622no—
180C2×D4—621no—
181C2×D4—1241no—
182C2×D4—1242no—
183Q16—21no—
184Q16—311no—
185Q16—622no—
186Q16C22×D3112yes—
187Q16C22×D3111yescharacteristic
188SD16—21no—
189SD16—622no—
190SD16—622no—
191SD16—621no—
192D8—311no—
193D8—312no—
194D8—622no—
195D8—621no—
196D8C22×D3111yescharacteristic
197M16—311no—
198M16—312no—
199C2×C8—312no—
200C2×C8C22×D3111yescharacteristic
201C16—311no—
202C16—311no—
203C16C22×D3112yes—
204C23—1242no—
205C23—1241no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—311no—
210Q8—621no—
211Q8—1242no—
212D4—21no—
213D4—41no—
214D4—312no—
215D4—311no—
216D4—622no—
217D4—621no—
218D4—1242no—
219D4—1242no—
220D4—1242no—
221D4—1241no—
222D4—1241no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—312no—
226C2×C4—1242no—
227C2×C4—1242no—
228C2×C4—1242no—
229C2×C4—1241no—
230C2×C4D4×D3111yescharacteristic
231C8—312no—
232C8D4×D3111yescharacteristic
233C8C23×D3111yescharacteristic
234C22—41no—
235C22—312no—
236C22—1241no—
237C22—1242no—
238C22—4961no—
239C22—4962no—
240C22D8×D3111yescharacteristic
241C4—41no—
242C4—42no—
243C4—312no—
244C4—1241no—
245C4D8×D3111yescharacteristic
246C4C2×D4×D3111yescharacteristic
247C2—21no—
248C2—81no—
249C2—622no—
250C2—2482no—
251C2—2481no—
252C2C2×D8×D3111yescharacteristic
subgroupG/H
centerC2C2×D8×D31
commutator, cosocleC8×C31C24
FrattiniC8C23×D31
FittingC31×#510C2
socleC2×C31C22×D8
hypercenter#510D31
hypocenterC31#3416
LCSG ▷ C8×C31 ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #14521862C2semidirect
2. #14521851C2semidirect
3. #14521491C2semidirect
4. #14521487C2semidirect
5. #14521486C2semidirect
6. #14521482C2semidirect
7. #14521479C2semidirect
8. C2×#11758416C2semidirect
9. SD32×D31C2semidirect
10. #14521467C2semidirect
11. C31×#510C2semidirect
15. C2×Q16×C31C22semidirect
19. #11758496C22semidirect
21. Q16×D31C22semidirect
24. #11758481C22semidirect
27. C31×Q32C22semidirect
28. C31×SD32C22semidirect
29. C31×M32C22semidirect
30. #11758417C22semidirect
31. #11758416C22semidirect
32. #11758416C22semidirect
33. #11758416C22semidirect
34. #11758415C22semidirect
35. #11758414C22semidirect
36. C2×C31⋊C16C22semidirect
37. #11758388C22semidirect
38. D496C22semidirect
39. #11758386C22semidirect
40. C16×D31C22semidirect
124. #510D31semidirect
149. C2×Q16C2×D31semidirect
154. Q32C2×D31semidirect
160. SD32C2×D31semidirect
164. M32C2×D31semidirect
167. C31#3416semidirect

Maximal subgroups

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

subgroupindex
#145218622
#145218512
#145214912
#145214872
#145214862
#145214822
#145214792
C2×#117584162
SD32×D312
#145214672
C31×#5102
#341631