Finite Group Databasebeta

← #16855082

Subgroups — #16855082

Order 3968.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#14521885C211yesmaximal, characteristic, complemented
2C2×#11758509C212yesmaximal, complemented
3#14521221C214yesmaximal, complemented
4#14521216C214yesmaximal, complemented
5#14521067C212yesmaximal, complemented
6#14521033C211yesmaximal, characteristic, complemented
7C31×#535C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9C22×D4×C31C2211yescharacteristic, complemented
10#11758567—22no—
11#11758566—22no—
12#41×D31—24no—
13#11758564C2212yescomplemented
14#11758561—22no—
15#11758561C2212yescomplemented
16C2×#92495—22no—
17C2×#92495C2212yes—
18C31×#127C2212yescomplemented
19C31×#124C2212yescomplemented
20C31×#121—22no—
21C31×#120—22no—
22C31×#117C2211yescharacteristic, complemented
23C2×#92480—22no—
24#11758509—24no—
25#11758509C2218yescomplemented
26C2×#92479—22no—
27C2×#92474C2212yescomplemented
28#11758424C2212yescomplemented
29#11758421C2212yescomplemented
30#11758393C2218yescomplemented
31C24×C31—21no—
32#41×C31—22no—
33#41×C31—22no—
34C2×Q8×C31C2311yescharacteristic
35C2×D4×C31—24no—
36C2×D4×C31—22no—
37C2×D4×C31—22no—
38C2×D4×C31—21no—
39C2×D4×C31D412yes—
40C2×D4×C31C2312yes—
41C2×D4×C31C2311yescharacteristic
42C22×C4×C31D412yes—
43C22×C4×C31C2311yescharacteristic
44C2×C22⋊D31—24no—
45C2×C22⋊D31—22no—
46#92499—24no—
47Q8×D31—24no—
48#92497—28no—
49#92497—24no—
50D4×D31—28no—
51D4×D31—24no—
52#92495—24no—
53#92495—24no—
54#92495—22no—
55#92495—24no—
56#92495—22no—
57#92495D414yescomplemented
58#92495C2314yes—
59C2×D124—22no—
60C2×D124D412yescomplemented
61C2×C4×D31—24no—
62C2×C4×D31—22no—
63C2×Dic62—22no—
64C2×Dic62D412yescomplemented
65C4⋊C4×C31—21no—
66C22⋊C4×C31—21no—
67C22⋊C4×C31—24no—
68C42×C31C2312yes—
69#92480—28no—
70#92479—28no—
71#92474—22no—
72#92474C2314yes—
73C2×C31⋊C8—22no—
74C23×C31—22no—
75C23×C31—42no—
76C23×C31—82no—
77C23×C31C2×D412yes—
78C23×C31C2×D411yescharacteristic
79C22×D31—24no—
80C22×D31—22no—
81Q8×C31—22no—
82D4×C31—24no—
83D4×C31—22no—
84D4×C31—24no—
85D4×C31—41no—
86D4×C31—41no—
87D4×C31—84no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—22no—
91C2×C4×C31C2×D412yes—
92C2×C4×C31C2×D412yes—
93C2×C4×C31C2×D412yes—
94C2×C4×C31C2×D411yescharacteristic
95C2×C4×C31C2411yescharacteristic
96C22⋊D31—28no—
97C22⋊D31—24no—
98C22⋊D31—24no—
99C2×Dic31—24no—
100C2×Dic31—22no—
101D124—24no—
102D124—22no—
103D124C2×D414yes—
104C4×D31—28no—
105C4×D31—24no—
106C4×D31—24no—
107Dic62—24no—
108Dic62—22no—
109Dic62C2×D414yes—
110C31⋊C8—24no—
111#3021—311nomaximal, Sylow
112C22×C31—22no—
113C22×C31—41no—
114C22×C31—42no—
115C22×C31—44no—
116C22×C31—42no—
117C22×C31—84no—
118C22×C31#12012yes—
119C22×C31C22×D411yescharacteristic
120C2×D31—24no—
121C2×D31—82no—
122C2×D31—84no—
123C4×C31—22no—
124C4×C31—42no—
125C4×C31#12011yescharacteristic
126C4×C31#12011yescharacteristic
127C4×C31C22×D412yes—
128Dic31—24no—
129#585—311no—
130C2×AGL(1,ℤ/8)—312no—
131#535D3111yescharacteristic, complemented
132#454—314no—
133#453—314no—
134#421—312no—
135#413—311no—
136C2×C31—21no—
137C2×C31—22no—
138C2×C31—42no—
139C2×C31—82no—
140C2×C31C2×#12011yescharacteristic
141D31—44no—
1422−1+4—312no—
1432−1+4—622no—
144H(5,𝔽2)—312no—
145H(5,𝔽2)—622no—
146H(5,𝔽2)—622no—
147C2×#41—312no—
148C2×#41—624no—
149C2×#41—622no—
150C2×#41C2×D3111yescharacteristic
151C22×D4C2×D3111yescharacteristic, complemented
152AGL(1,ℤ/8)—318no—
153AGL(1,ℤ/8)—624no—
154C2×SD16—622no—
155C2×D8—622no—
156C2×M16—312no—
157#127C2×D3112yescomplemented
158#124C2×D3112yescomplemented
159#121—22no—
160#120—22no—
161#117C2×D3111yescharacteristic, complemented
162#104—318no—
163#101—312no—
164#100—312no—
165C31#302111yescharacteristic, complemented, Sylow
166C24—21no—
167#41—22no—
168#41—22no—
169#41—314no—
170#41—314no—
171#41—624no—
172#41—624no—
173#41—624no—
174#41—628no—
175#41—624no—
176#41—622no—
177#41—622no—
178#41—624no—
179C2×Q8—312no—
180C2×Q8—622no—
181C2×Q8—624no—
182C2×Q8C22×D3111yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—312no—
188C2×D4—622no—
189C2×D4—624no—
190C2×D4—628no—
191C2×D4—622no—
192C2×D4—624no—
193C2×D4C22⋊D3112yes—
194C2×D4C22×D3112yes—
195C2×D4C22×D3111yescharacteristic
196C22×C4—624no—
197C22×C4—622no—
198C22×C4C22⋊D3112yes—
199C22×C4C22×D3111yescharacteristic
200SD16—628no—
201D8—628no—
202M16—314no—
203M16—622no—
204C2×C8—622no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D3112yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—624no—
213C23—622no—
214C23C2×C22⋊D3112yes—
215C23C2×C22⋊D3111yescharacteristic
216Q8—22no—
217Q8—314no—
218Q8—624no—
219Q8—622no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—314no—
227D4—624no—
228D4—628no—
229D4—624no—
230D4—624no—
231D4—622no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—628no—
236C2×C4—624no—
237C2×C4—622no—
238C2×C4—624no—
239C2×C4—624no—
240C2×C4D4×D3112yes—
241C2×C4D4×D3112yes—
242C2×C4C2×C22⋊D3112yes—
243C2×C4C2×C22⋊D3111yescharacteristic
244C2×C4C23×D3111yescharacteristic
245C8—624no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—624no—
253C22—2482no—
254C22—2484no—
255C22#1175851412yes—
256C22C22×C22⋊D3111yescharacteristic
257C4—22no—
258C4—42no—
259C4—624no—
260C4#1175851411yescharacteristic
261C4#1175851411yescharacteristic
262C4C2×D4×D3112yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—1244no—
268C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#535C2
socleC2×C31C2×#120
hypercenter#535D31
hypocenterC31#3021
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. C2×#11758509C2semidirect
3. #14521221C2semidirect
4. #14521216C2semidirect
5. #14521067C2semidirect
6. #14521033C2semidirect
7. C31×#535C2semidirect
9. C22×D4×C31C22semidirect
13. #11758564C22semidirect
15. #11758561C22semidirect
18. C31×#127C22semidirect
19. C31×#124C22semidirect
22. C31×#117C22semidirect
25. #11758509C22semidirect
27. C2×#92474C22semidirect
28. #11758424C22semidirect
29. #11758421C22semidirect
30. #11758393C22semidirect
57. #92495D4semidirect
60. C2×D124D4semidirect
64. C2×Dic62D4semidirect
131. #535D31semidirect
151. C22×D4C2×D31semidirect
157. #127C2×D31semidirect
158. #124C2×D31semidirect
161. #117C2×D31semidirect
165. C31#3021semidirect

Maximal subgroups

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

subgroupindex
#145218852
C2×#117585092
#145212212
#145212162
#145210672
#145210332
C31×#5352
#302131