Finite Group Databasebeta

← #16856160

Subgroups — #16856160

Order 3968.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11758491C212yesmaximal, complemented
2C2×#11758488C212yesmaximal, complemented
3#14521598C211yesmaximal, characteristic, complemented
4#127×D31C211yesmaximal, characteristic, complemented
5#14521439C211yesmaximal, characteristic, complemented
6#14521438C211yesmaximal, characteristic, complemented
7#14521436C211yesmaximal, characteristic, complemented
8#14521424C212yesmaximal, complemented
9#14521420C212yesmaximal, complemented
10#14521001C211yesmaximal, characteristic, complemented
11C31×#496C211yesmaximal, characteristic, complemented
12C2×#92499C2211yescharacteristic
13C2×#92497C2211yescharacteristic
14C2×D4×D31—22no—
15C2×D4×D31C2212yes—
16C2×SD16×C31C2212yescomplemented
17C2×D8×C31C2212yescomplemented
18C31×#127C2211yescharacteristic
19C31×#124C2211yescharacteristic, complemented
20#11758521C2211yescharacteristic, complemented
21C2×#92481C2212yescomplemented
22#11758517—22no—
23#11758517C2212yes—
24#11758517C2211yescharacteristic, complemented
25C2×#92480C2212yescomplemented
26C2×#92479C2212yescomplemented
27C2×#92479C2212yescomplemented
28#11758491—28no—
29#11758488—28no—
30C2×D248C2212yescomplemented
31C2×#92470C2212yescomplemented
32C2×#92469C2212yes—
33#11758461—22no—
34(C22⋊D31)⋊C4—22no—
35#11758452C2211yescharacteristic, complemented
36#11758451C2211yescharacteristic
37C42×D31C2211yescharacteristic
38C31×#97C2211yescharacteristic, complemented
39C8×Dic31C2212yescomplemented
40#11758391C2211yescharacteristic, complemented
41C23×D31—22no—
42C2×Q8×C31C2311yescharacteristic
43C2×D4×C31—22no—
44C2×D4×C31C2311yescharacteristic
45C2×D4×C31C2312yes—
46C2×C22⋊D31—21no—
47C2×C22⋊D31—22no—
48C2×C22⋊D31—22no—
49C2×C22⋊D31—24no—
50C22×Dic31—21no—
51#92499—24no—
52#92497—24no—
53D4×D31—24no—
54D4×D31—24no—
55D4×D31—48no—
56C2×D124—21no—
57C2×D124—22no—
58C2×D124C2312yes—
59C2×D124C2311yescharacteristic
60C2×C4×D31—21no—
61C2×C4×D31D412yes—
62C2×C4×D31C2311yescharacteristic
63C2×Dic62C2311yescharacteristic
64SD16×C31—24no—
65D8×C31—24no—
66C2×C8×C31C2312yes—
67C22⋊C4×C31—22no—
68C42×C31C2311yescharacteristic
69#92481—24no—
70#92480—24no—
71#92479—24no—
72#92479—24no—
73(C2×D31)⋊C4—22no—
74Dic31⋊C4—22no—
75C4×Dic31D412yes—
76C4×Dic31C2311yescharacteristic
77C2×C31⋊C8C2312yes—
78D248—24no—
79#92470—24no—
80#92469D418yescomplemented
81C23×C31—21no—
82C23×C31—22no—
83C22×D31—21no—
84C22×D31—22no—
85C22×D31—48no—
86C22×D31—44no—
87C22×D31C2×D411yescharacteristic
88Q8×C31—22no—
89D4×C31—22no—
90D4×C31—22no—
91D4×C31—22no—
92D4×C31—44no—
93C2×C4×C31—21no—
94C2×C4×C31C2×D412yes—
95C2×C4×C31C2411yescharacteristic
96C22⋊D31—42no—
97C22⋊D31—44no—
98C22⋊D31—48no—
99C22⋊D31—44no—
100C2×Dic31—21no—
101C2×Dic31—42no—
102C2×Dic31C2×D411yescharacteristic
103C2×Dic31C2×D412yes—
104D124—22no—
105D124—22no—
106D124—22no—
107D124—44no—
108D124—42no—
109C4×D31—24no—
110C4×D31—42no—
111C4×D31C2×D414yes—
112Dic62—22no—
113C8×C31C2×D414yescomplemented
114C31⋊C8C2×D414yescomplemented
115#3151—311nomaximal, Sylow
116C22×C31—41no—
117C22×C31—42no—
118C22×C31—44no—
119C22×C31—42no—
120C22×C31C22×D411yescharacteristic
121C2×D31—22no—
122C2×D31—41no—
123C2×D31—42no—
124C2×D31—44no—
125C2×D31—42no—
126C2×D31—88no—
127C2×D31#12712yes—
128C4×C31—22no—
129C4×C31—41no—
130C4×C31C22×D412yes—
131Dic31—41no—
132Dic31#12712yes—
133Dic31AGL(1,ℤ/8)14yescomplemented
134C2×AGL(1,ℤ/8)—312no—
135C2×AGL(1,ℤ/8)—312no—
136#532—311no—
137C2×#127—311no—
138#496—311no—
139#496—311no—
140#496—311no—
141#496D3111yescharacteristic, complemented
142#493—312no—
143#492—312no—
144C4×M16—311no—
145C2×C31—82no—
146C2×C31—81no—
147C2×C31C2×#12711yescharacteristic
148C2×C31C2×AGL(1,ℤ/8)12yes—
149D31—22no—
150D31—81no—
151D31—82no—
152C2×#41—311no—
153C2×#41—311no—
154C22×D4—312no—
155C22×D4—622no—
156AGL(1,ℤ/8)—628no—
157AGL(1,ℤ/8)—628no—
158C2×SD16—312no—
159C2×SD16—312no—
160C2×SD16—312no—
161C2×SD16C2×D3112yescomplemented
162C2×D8—312no—
163C2×D8—312no—
164C2×D8—312no—
165C2×D8C2×D3112yescomplemented
166C2×M16—312no—
167C4⋊Q8—311no—
168#127—312no—
169#127—311no—
170#127—311no—
171#127—622no—
172#127C2×D3111yescharacteristic
173#124—311no—
174#124C2×D3111yescharacteristic, complemented
175#121—622no—
176C4×D4—622no—
177C2×C42—311no—
178#97—311no—
179#97C2×D3111yescharacteristic, complemented
180C4×C8—312no—
181C31#315111yescharacteristic, complemented, Sylow
182C24—622no—
183#41—624no—
184#41—624no—
185C2×Q8—311no—
186C2×Q8C22×D3111yescharacteristic
187C2×D4—22no—
188C2×D4—311no—
189C2×D4—312no—
190C2×D4—621no—
191C2×D4—624no—
192C2×D4—624no—
193C2×D4—622no—
194C2×D4—624no—
195C2×D4—622no—
196C2×D4—621no—
197C2×D4—622no—
198C2×D4—1248no—
199C2×D4C22×D3111yescharacteristic
200C2×D4C22×D3112yes—
201C22×C4—311no—
202C22×C4—312no—
203C22×C4—621no—
204C22×C4—621no—
205SD16—24no—
206SD16—624no—
207SD16—624no—
208SD16—624no—
209D8—24no—
210D8—624no—
211D8—624no—
212D8—624no—
213M16—318no—
214C2×C8—312no—
215C2×C8C22×D3112yes—
216C4⋊C4—622no—
217C22⋊C4—22no—
218C22⋊C4—622no—
219C42—312no—
220C42—311no—
221C42C22×D3111yescharacteristic
222C23—21no—
223C23—22no—
224C23—311no—
225C23—621no—
226C23—622no—
227C23—1244no—
228C23—1248no—
229Q8—22no—
230Q8—622no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—622no—
236D4—622no—
237D4—622no—
238D4—1244no—
239D4—1242no—
240D4—1244no—
241D4—1244no—
242D4—1248no—
243D4—1242no—
244C2×C4—21no—
245C2×C4—312no—
246C2×C4—311no—
247C2×C4—314no—
248C2×C4—624no—
249C2×C4—621no—
250C2×C4—1242no—
251C2×C4—1242no—
252C2×C4D4×D3112yes—
253C2×C4C23×D3111yescharacteristic
254C8—314no—
255C8D4×D3114yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—312no—
261C22—622no—
262C22—1244no—
263C22—1242no—
264C22—1241no—
265C22—1242no—
266C22—2488no—
267C22C2×D4×D3111yescharacteristic
268C4—22no—
269C4—41no—
270C4—314no—
271C4—312no—
272C4—1241no—
273C4C2×D4×D3112yes—
274C2—82no—
275C2—81no—
276C2—622no—
277C2—2482no—
278C2—2481no—
279C2#127×D3111yescharacteristic
280C2AGL(1,ℤ/8)×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#496C2
socleC22×C31C22×D4
hypercenter#496D31
hypocenterC31#3151
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#11758491C2semidirect
2. C2×#11758488C2semidirect
3. #14521598C2semidirect
4. #127×D31C2semidirect
5. #14521439C2semidirect
6. #14521438C2semidirect
7. #14521436C2semidirect
8. #14521424C2semidirect
9. #14521420C2semidirect
10. #14521001C2semidirect
11. C31×#496C2semidirect
16. C2×SD16×C31C22semidirect
17. C2×D8×C31C22semidirect
19. C31×#124C22semidirect
20. #11758521C22semidirect
21. C2×#92481C22semidirect
24. #11758517C22semidirect
25. C2×#92480C22semidirect
26. C2×#92479C22semidirect
27. C2×#92479C22semidirect
30. C2×D248C22semidirect
31. C2×#92470C22semidirect
35. #11758452C22semidirect
38. C31×#97C22semidirect
39. C8×Dic31C22semidirect
40. #11758391C22semidirect
80. #92469D4semidirect
113. C8×C31C2×D4semidirect
114. C31⋊C8C2×D4semidirect
133. Dic31AGL(1,ℤ/8)semidirect
141. #496D31semidirect
161. C2×SD16C2×D31semidirect
165. C2×D8C2×D31semidirect
174. #124C2×D31semidirect
179. #97C2×D31semidirect
181. C31#3151semidirect
255. C8D4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584912
C2×#117584882
#145215982
#127×D312
#145214392
#145214382
#145214362
#145214242
#145214202
#145210012
C31×#4962
#315131