Finite Group Databasebeta

← #16855083

Subgroups — #16855083

Order 3968.

736 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 161 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×#11758526C212yesmaximal, complemented
2#14521613C211yesmaximal, characteristic, complemented
3#14521222C214yesmaximal, complemented
4#14521217C214yesmaximal, complemented
5#14521066C212yesmaximal, complemented
6#14521033C211yesmaximal, characteristic, complemented
7C31×#585C211yesmaximal, characteristic, complemented
8C31×2−1+4—22no—
9C31×2−1+4C2212yescomplemented
10C31×H(5,𝔽2)—22no—
11C31×H(5,𝔽2)—22no—
12C31×H(5,𝔽2)C2212yescomplemented
13C2×#41×C31—24no—
14C2×#41×C31—22no—
15C2×#41×C31C2212yes—
16C2×#41×C31C2211yescharacteristic
17C22×D124C2211yescharacteristic, complemented
18#11758526—24no—
19#11758526C2218yescomplemented
20#11758524C2212yescomplemented
21C2×#92481—22no—
22#11758517C2212yescomplemented
23#11758514—22no—
24C2×#92479—22no—
25#11758507—22no—
26#11758503C2211yescharacteristic, complemented
27C2×#92474C2212yescomplemented
28#11758425C2218yescomplemented
29#11758424C2212yescomplemented
30#11758421C2212yescomplemented
31C23×D31—21no—
32#41×C31—24no—
33#41×C31—28no—
34#41×C31—24no—
35#41×C31—24no—
36#41×C31—24no—
37#41×C31—22no—
38#41×C31—22no—
39#41×C31—24no—
40#41×C31—22no—
41#41×C31—22no—
42#41×C31D414yescomplemented
43#41×C31C2314yes—
44C2×Q8×C31—22no—
45C2×Q8×C31—24no—
46C2×Q8×C31D412yescomplemented
47C2×Q8×C31C2311yescharacteristic
48C2×D4×C31—28no—
49C2×D4×C31—22no—
50C2×D4×C31—24no—
51C2×D4×C31—22no—
52C2×D4×C31—24no—
53C2×D4×C31D412yes—
54C2×D4×C31D412yescomplemented
55C2×D4×C31C2311yescharacteristic
56C22×C4×C31—24no—
57C22×C4×C31—22no—
58C22×C4×C31D412yes—
59C22×C4×C31C2311yescharacteristic
60C2×C22⋊D31—24no—
61C2×D124—22no—
62C2×D124—22no—
63C2×D124—21no—
64C2×D124C2312yes—
65C22⋊Dic31—21no—
66#92481—28no—
67#92479—28no—
68(C2×D31)⋊C4—24no—
69C4⋊Dic31—21no—
70C4×Dic31C2312yes—
71#92474—22no—
72#92474C2314yes—
73C2×C31⋊C8—22no—
74C23×C31—24no—
75C23×C31—22no—
76C23×C31C2×D412yes—
77C23×C31C2×D411yescharacteristic
78C22×D31—22no—
79C22×D31—42no—
80C22×D31—82no—
81Q8×C31—24no—
82Q8×C31—22no—
83Q8×C31—22no—
84Q8×C31C2×D414yes—
85D4×C31—28no—
86D4×C31—24no—
87D4×C31—24no—
88D4×C31—24no—
89D4×C31—24no—
90D4×C31—22no—
91D4×C31—22no—
92D4×C31C2×D414yes—
93C2×C4×C31—24no—
94C2×C4×C31—28no—
95C2×C4×C31—22no—
96C2×C4×C31—24no—
97C2×C4×C31—22no—
98C2×C4×C31—22no—
99C2×C4×C31—24no—
100C2×C4×C31C2×D412yes—
101C2×C4×C31C2×D411yescharacteristic
102C2×C4×C31C2×D412yes—
103C2×C4×C31C2×D412yes—
104C2×C4×C31C2411yescharacteristic
105C22⋊D31—84no—
106C2×Dic31—22no—
107D124—24no—
108D124—41no—
109D124—41no—
110C31⋊C8—24no—
111#3021—311nomaximal, Sylow
112C22×C31—22no—
113C22×C31—24no—
114C22×C31—41no—
115C22×C31—42no—
116C22×C31—82no—
117C22×C31—84no—
118C22×C31#12012yes—
119C22×C31C22×D411yescharacteristic
120C2×D31—44no—
121C2×D31—42no—
122C2×D31—84no—
123C4×C31—24no—
124C4×C31—22no—
125C4×C31#12011yescharacteristic
126C4×C31#12011yescharacteristic
127C4×C31C22×D412yes—
128Dic31—42no—
129#585D3111yescharacteristic, complemented
130C2×AGL(1,ℤ/8)—312no—
131#535—311no—
132#454—314no—
133#453—314no—
134#421—312no—
135#413—311no—
136C2×C31—21no—
137C2×C31—22no—
138C2×C31—42no—
139C2×C31—44no—
140C2×C31C2×#12011yescharacteristic
141D31—82no—
1422−1+4—22no—
1432−1+4C2×D3112yescomplemented
144H(5,𝔽2)—22no—
145H(5,𝔽2)—22no—
146H(5,𝔽2)C2×D3112yescomplemented
147C2×#41—24no—
148C2×#41—22no—
149C2×#41C2×D3112yes—
150C2×#41C2×D3111yescharacteristic
151C22×D4—311no—
152AGL(1,ℤ/8)—318no—
153AGL(1,ℤ/8)—624no—
154C2×SD16—622no—
155C2×D8—622no—
156C2×M16—312no—
157#127—312no—
158#124—312no—
159#121—622no—
160#120—622no—
161#117—311no—
162#104—318no—
163#101—312no—
164#100—312no—
165C31#302111yescharacteristic, complemented, Sylow
166C24—621no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41C22⋊D3114yescomplemented
178#41C22×D3114yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D3112yescomplemented
182C2×Q8C22×D3111yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—312no—
189C2×D4—624no—
190C2×D4—622no—
191C2×D4—622no—
192C2×D4—621no—
193C2×D4C22⋊D3112yes—
194C2×D4C22⋊D3112yescomplemented
195C2×D4C22×D3111yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D3112yes—
199C22×C4C22×D3111yescharacteristic
200SD16—628no—
201D8—628no—
202M16—314no—
203M16—622no—
204C2×C8—622no—
205C4⋊C4—621no—
206C22⋊C4—621no—
207C22⋊C4—624no—
208C42—312no—
209C23—24no—
210C23—22no—
211C23—622no—
212C23—1242no—
213C23—2482no—
214C23C2×C22⋊D3112yes—
215C23C2×C22⋊D3111yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D3114yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—624no—
228D4—1241no—
229D4—1241no—
230D4—2484no—
231D4C2×C22⋊D3114yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—622no—
240C2×C4C2×C22⋊D3112yes—
241C2×C4C2×C22⋊D3111yescharacteristic
242C2×C4C2×C22⋊D3112yes—
243C2×C4C2×C22⋊D3112yes—
244C2×C4C23×D3111yescharacteristic
245C8—624no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—1244no—
253C22—1242no—
254C22—2484no—
255C22#1175853012yes—
256C22C22×C22⋊D3111yescharacteristic
257C4—24no—
258C4—22no—
259C4—1242no—
260C4#1175853011yescharacteristic
261C4#1175853011yescharacteristic
262C4C22×C22⋊D3112yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—2482no—
268C2C2×#1175853011yescharacteristic
subgroupG/H
centerC2C2×#11758530
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#585C2
socleC2×C31C2×#120
hypercenter#585D31
hypocenterC31#3021
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#11758526C2semidirect
2. #14521613C2semidirect
3. #14521222C2semidirect
4. #14521217C2semidirect
5. #14521066C2semidirect
6. #14521033C2semidirect
7. C31×#585C2semidirect
9. C31×2−1+4C22semidirect
12. C31×H(5,𝔽2)C22semidirect
17. C22×D124C22semidirect
19. #11758526C22semidirect
20. #11758524C22semidirect
22. #11758517C22semidirect
26. #11758503C22semidirect
27. C2×#92474C22semidirect
28. #11758425C22semidirect
29. #11758424C22semidirect
30. #11758421C22semidirect
42. #41×C31D4semidirect
46. C2×Q8×C31D4semidirect
54. C2×D4×C31D4semidirect
129. #585D31semidirect
143. 2−1+4C2×D31semidirect
146. H(5,𝔽2)C2×D31semidirect
165. C31#3021semidirect
177. #41C22⋊D31semidirect
181. C2×Q8C22⋊D31semidirect
194. C2×D4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585262
#145216132
#145212222
#145212172
#145210662
#145210332
C31×#5852
#302131