Finite Group Databasebeta

← #16858857

Subgroups — #16858857

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#14521863C211yesmaximal, characteristic, complemented
2#14521850C211yesmaximal, characteristic, complemented
3#14521492C211yesmaximal, characteristic, complemented
4#14521488C212yesmaximal, complemented
5#14521483C212yesmaximal, complemented
6#14521479C211yesmaximal, characteristic, complemented
7#14521478C212yesmaximal, complemented
8C2×#11758415C211yesmaximal, characteristic, complemented
9SD32×D31C212yesmaximal, complemented
10#14521467C211yesmaximal, characteristic, complemented
11C31×#509C211yesmaximal, characteristic, complemented
12#11758567—21no—
13#11758561—21no—
14C31×#135C2211yescharacteristic, complemented
15C2×D8×C31C2211yescharacteristic
16#11758528—21no—
17#11758509—21no—
18Q16×D31C2212yescomplemented
19#11758492—22no—
20#11758489C2212yes—
21#11758489C2212yescomplemented
22#11758488—22no—
23D8×D31C2212yes—
24C2×Dic124C2211yescharacteristic, complemented
25#11758481C2211yescharacteristic
26#11758478C2211yescharacteristic
27C31×SD32C2212yescomplemented
28C31×D16C2212yescomplemented
29C31×M32C2211yescharacteristic, complemented
30#11758417C2211yescharacteristic, complemented
31#11758416C2211yescharacteristic, complemented
32#11758415C2212yescomplemented
33#11758415C2212yescomplemented
34#11758415C2211yescharacteristic, complemented
35#11758414C2211yescharacteristic, complemented
36C2×C31⋊C16C2211yescharacteristic, complemented
37Dic248C2212yescomplemented
38#11758388C2212yescomplemented
39#11758386C2212yescomplemented
40C16×D31C2212yescomplemented
41#41×C31—21no—
42C2×D4×C31—21no—
43C2×C22⋊D31—42no—
44Q8×D31—22no—
45#92497—22no—
46#92497—22no—
47#92497—42no—
48#92497—41no—
49D4×D31—22no—
50D4×D31—41no—
51#92495—21no—
52#92495—41no—
53#92495D411yescharacteristic
54C2×Dic62—21no—
55C2×Dic62—41no—
56Q16×C31C2311yescharacteristic
57SD16×C31—21no—
58D8×C31—21no—
59D8×C31C2311yescharacteristic
60D8×C31C2312yes—
61C2×C8×C31C2311yescharacteristic
62#92482—22no—
63#92480—22no—
64#92480—22no—
65#92479—22no—
66#92474D411yescharacteristic
67Dic124—21no—
68Dic124C2312yes—
69Dic124C2311yescharacteristic
70D248C2311yescharacteristic
71#92470—21no—
72#92469D412yes—
73C8×D31C2312yes—
74C16×C31C2312yes—
75C31⋊C16C2311yescharacteristic
76C31⋊C16C2311yescharacteristic
77C23×C31—41no—
78C22×D31—42no—
79Q8×C31—21no—
80D4×C31—22no—
81D4×C31—21no—
82D4×C31—41no—
83D4×C31—41no—
84C2×C4×C31—41no—
85C2×C4×C31C2×D411yescharacteristic
86C22⋊D31—41no—
87C22⋊D31—42no—
88C22⋊D31—42no—
89C22⋊D31—42no—
90C22⋊D31D812yes—
91C2×Dic31—41no—
92C2×Dic31—42no—
93C2×Dic31—42no—
94D124—21no—
95D124D811yescharacteristic
96C4×D31—42no—
97C4×D31—41no—
98C4×D31C2×D412yes—
99Dic62—21no—
100Dic62—22no—
101Dic62—41no—
102Dic62—42no—
103Dic62D811yescharacteristic
104C8×C31C2×D411yescharacteristic
105C8×C31C2411yescharacteristic
106C31⋊C8C2×D412yes—
107#3416—311nomaximal, Sylow
108C22×C31—41no—
109C22×C31—42no—
110C22×C31—161no—
111C22×C31C2×D811yescharacteristic
112C2×D31—41no—
113C2×D31—162no—
114C2×D31C2×D812yes—
115C4×C31—41no—
116C4×C31C2×D811yescharacteristic
117C4×C31C22×D411yescharacteristic
118Dic31—42no—
119Dic31—41no—
120Dic31C2×D812yes—
121#578—311no—
122#576—311no—
123#510—312no—
124#510—311no—
125#509—312no—
126#509D3111yescharacteristic, complemented
127#508—311no—
128#508—312no—
129C2×SD32—311no—
130C2×SD32—312no—
131#504—311no—
132C2×C31—21no—
133C2×C31—82no—
134C2×C31—81no—
135C2×C31C22×D811yescharacteristic
136D31—22no—
137D31—81no—
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, complemented
148C2×Q16—312no—
149C2×Q16—311no—
150C2×D8—312no—
151C2×D8C2×D3111yescharacteristic
152#131—311no—
153Q32—311no—
154Q32—312no—
155SD32—311no—
156SD32—312no—
157SD32—312no—
158SD32—311no—
159SD32—312no—
160SD32C2×D3112yescomplemented
161D16—311no—
162D16C2×D3112yescomplemented
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—1241no—
174#41—1242no—
175#41—1241no—
176C2×Q8—622no—
177C2×Q8—621no—
178C2×Q8—1241no—
179C2×D4—21no—
180C2×D4—622no—
181C2×D4—1241no—
182C2×D4—1242no—
183Q16—312no—
184Q16—311no—
185Q16—622no—
186Q16—621no—
187Q16C22×D3111yescharacteristic
188SD16—21no—
189SD16—622no—
190SD16—622no—
191SD16—621no—
192D8—21no—
193D8—311no—
194D8—622no—
195D8C22×D3111yescharacteristic
196D8C22×D3112yes—
197M16—311no—
198M16—312no—
199C2×C8—312no—
200C2×C8C22×D3111yescharacteristic
201C16—311no—
202C16—311no—
203C16C22×D3112yes—
204C23—41no—
205C23—1242no—
206Q8—21no—
207Q8—311no—
208Q8—621no—
209Q8—622no—
210Q8—1242no—
211Q8—1241no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—312no—
217D4—311no—
218D4—621no—
219D4—1242no—
220D4—1242no—
221D4—1242no—
222D4—1241no—
223C2×C4—41no—
224C2×C4—312no—
225C2×C4—1242no—
226C2×C4—1242no—
227C2×C4—1242no—
228C2×C4—1241no—
229C2×C4—1241no—
230C2×C4D4×D3111yescharacteristic
231C8—312no—
232C8D4×D3111yescharacteristic
233C8C23×D3111yescharacteristic
234C22—41no—
235C22—42no—
236C22—161no—
237C22—312no—
238C22—1241no—
239C22—4962no—
240C22D8×D3111yescharacteristic
241C4—41no—
242C4—312no—
243C4—1241no—
244C4—1242no—
245C4D8×D3111yescharacteristic
246C4C2×D4×D3111yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—622no—
251C2—2481no—
252C2C2×D8×D3111yescharacteristic
subgroupG/H
centerC2C2×D8×D31
commutator, cosocleC8×C31C24
FrattiniC8C23×D31
FittingC31×#509C2
socleC2×C31C22×D8
hypercenter#509D31
hypocenterC31#3416
LCSG ▷ C8×C31 ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #14521863C2semidirect
2. #14521850C2semidirect
3. #14521492C2semidirect
4. #14521488C2semidirect
5. #14521483C2semidirect
6. #14521479C2semidirect
7. #14521478C2semidirect
8. C2×#11758415C2semidirect
9. SD32×D31C2semidirect
10. #14521467C2semidirect
11. C31×#509C2semidirect
14. C31×#135C22semidirect
18. Q16×D31C22semidirect
21. #11758489C22semidirect
24. C2×Dic124C22semidirect
27. C31×SD32C22semidirect
28. C31×D16C22semidirect
29. C31×M32C22semidirect
30. #11758417C22semidirect
31. #11758416C22semidirect
32. #11758415C22semidirect
33. #11758415C22semidirect
34. #11758415C22semidirect
35. #11758414C22semidirect
36. C2×C31⋊C16C22semidirect
37. Dic248C22semidirect
38. #11758388C22semidirect
39. #11758386C22semidirect
40. C16×D31C22semidirect
126. #509D31semidirect
147. #135C2×D31semidirect
160. SD32C2×D31semidirect
162. D16C2×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
#145218632
#145218502
#145214922
#145214882
#145214832
#145214792
#145214782
C2×#117584152
SD32×D312
#145214672
C31×#5092
#341631