Finite Group Databasebeta

← #16856174

Subgroups — #16856174

Order 3968.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#11758495C212yesmaximal, complemented
2C2×#11758491C212yesmaximal, complemented
3#14521600C211yesmaximal, characteristic, complemented
4#124×D31C211yesmaximal, characteristic, complemented
5#14521441C211yesmaximal, characteristic, complemented
6#14521439C211yesmaximal, characteristic, complemented
7#14521438C211yesmaximal, characteristic, complemented
8#14521434C212yesmaximal, complemented
9#14521430C212yesmaximal, complemented
10#14521001C211yesmaximal, characteristic, complemented
11C31×#497C211yesmaximal, characteristic, complemented
12C2×#92499C2212yes—
13C2×Q8×D31C2211yescharacteristic
14C2×D4×D31C2211yescharacteristic
15C2×Q16×C31C2212yescomplemented
16C2×SD16×C31C2212yes—
17C31×C4⋊Q8C2211yescharacteristic, complemented
18C31×#124C2211yescharacteristic
19#11758524C2211yescharacteristic
20#11758524C2212yescomplemented
21C2×#92482C2212yescomplemented
22C2×#92481C2212yes—
23C2×#92481C2212yescomplemented
24#11758513C2211yescharacteristic
25C2×#92479C2212yes—
26#11758495—28no—
27#11758491—28no—
28C2×D248C2212yes—
29C2×#92470C2212yescomplemented
30C2×#92469C2212yes—
31#11758472—22no—
32#11758470—22no—
33#11758462—22no—
34C22⋊C4×D31—22no—
35#11758452C2211yescharacteristic
36#11758451C2211yescharacteristic, complemented
37C42×D31C2211yescharacteristic
38C31×#97C2211yescharacteristic, complemented
39C8×Dic31C2212yescomplemented
40#11758391C2211yescharacteristic, complemented
41C23×D31—21no—
42C2×Q8×C31C2312yes—
43C2×Q8×C31C2311yescharacteristic
44C2×D4×C31C2311yescharacteristic
45C2×C22⋊D31—21no—
46#92499—24no—
47#92499—24no—
48Q8×D31—24no—
49D4×D31—24no—
50C2×D124—22no—
51C2×D124—22no—
52C2×D124C2312yes—
53C2×D124C2311yescharacteristic
54C2×C4×D31—21no—
55C2×C4×D31—22no—
56C2×C4×D31D412yes—
57C2×C4×D31C2311yescharacteristic
58C2×Dic62—21no—
59C2×Dic62C2311yescharacteristic
60Q16×C31—24no—
61SD16×C31—24no—
62C2×C8×C31C2312yes—
63C4⋊C4×C31—22no—
64C22⋊C4×C31—22no—
65C42×C31C2311yescharacteristic
66C22⋊Dic31—22no—
67#92482—24no—
68#92481—24no—
69#92481—24no—
70#92479—24no—
71(C2×D31)⋊C4—22no—
72(C2×D31)⋊C4—22no—
73(C2×D31)⋊C4—24no—
74C4×Dic31D412yes—
75C4×Dic31C2311yescharacteristic
76C2×C31⋊C8C2312yes—
77D248—24no—
78#92470—24no—
79#92469D418yescomplemented
80C23×C31—21no—
81C22×D31—21no—
82C22×D31—22no—
83C22×D31—44no—
84C22×D31—42no—
85C22×D31C2×D411yescharacteristic
86Q8×C31—22no—
87Q8×C31—22no—
88Q8×C31—22no—
89D4×C31—22no—
90C2×C4×C31—21no—
91C2×C4×C31—22no—
92C2×C4×C31C2×D412yes—
93C2×C4×C31C2411yescharacteristic
94C22⋊D31—42no—
95C2×Dic31—21no—
96C2×Dic31C2×D412yes—
97C2×Dic31C2×D411yescharacteristic
98D124—22no—
99D124—22no—
100D124—22no—
101D124—44no—
102D124—44no—
103C4×D31—42no—
104C4×D31—42no—
105C4×D31—44no—
106C4×D31C2×D414yes—
107Dic62—22no—
108Dic62—42no—
109C8×C31C2×D414yes—
110C31⋊C8C2×D414yes—
111#3153—311nomaximal, Sylow
112C22×C31—41no—
113C22×C31—42no—
114C22×C31C22×D411yescharacteristic
115C2×D31—22no—
116C2×D31—42no—
117C2×D31—41no—
118C2×D31—44no—
119C2×D31—42no—
120C2×D31—84no—
121C2×D31#12712yes—
122C4×C31—22no—
123C4×C31—41no—
124C4×C31—42no—
125C4×C31C22×D412yes—
126Dic31—22no—
127Dic31—41no—
128Dic31#12712yes—
129C2×#137—312no—
130C2×AGL(1,ℤ/8)—312no—
131#532—311no—
132C2×#124—311no—
133#497—311no—
134#497D3111yescharacteristic, complemented
135#496—311no—
136#496—311no—
137#495—312no—
138#495—312no—
139C4×M16—311no—
140C2×C31—81no—
141C2×C31C2×#12711yescharacteristic
142C2×C31#57512yes—
143D31—22no—
144D31—82no—
145D31—81no—
146C2×#41—312no—
147C22×Q8—311no—
148C22×D4—311no—
149#137—628no—
150AGL(1,ℤ/8)—628no—
151C2×Q16—312no—
152C2×Q16C2×D3112yescomplemented
153C2×SD16—312no—
154C2×SD16—312no—
155C2×SD16—312no—
156C2×SD16C2×D3112yes—
157C2×D8—312no—
158C2×D8—312no—
159C2×M16—312no—
160C4⋊Q8C2×D3111yescharacteristic, complemented
161#127—311no—
162#124—311no—
163#124—312no—
164#124—311no—
165#124—311no—
166#124—622no—
167#124C2×D3111yescharacteristic
168#121—622no—
169C4×D4—622no—
170C2×C22⋊C4—622no—
171C2×C42—311no—
172#97—311no—
173#97C2×D3111yescharacteristic, complemented
174C4×C8—312no—
175C31#315311yescharacteristic, complemented, Sylow
176C24—621no—
177#41—624no—
178#41—624no—
179C2×Q8—311no—
180C2×Q8—621no—
181C2×Q8—624no—
182C2×Q8C22×D3111yescharacteristic
183C2×Q8C22×D3112yes—
184C2×D4—311no—
185C2×D4—312no—
186C2×D4—622no—
187C2×D4—624no—
188C2×D4—621no—
189C2×D4—622no—
190C2×D4C22×D3111yescharacteristic
191C22×C4—311no—
192C22×C4—312no—
193C22×C4—622no—
194C22×C4—621no—
195Q16—24no—
196Q16—624no—
197SD16—24no—
198SD16—624no—
199SD16—624no—
200SD16—624no—
201D8—624no—
202D8—624no—
203M16—318no—
204C2×C8—312no—
205C2×C8C22×D3112yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—622no—
209C22⋊C4—624no—
210C22⋊C4—622no—
211C22⋊C4—622no—
212C42—312no—
213C42—311no—
214C42C22×D3111yescharacteristic
215C23—21no—
216C23—311no—
217C23—621no—
218C23—622no—
219C23—1242no—
220C23—1244no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—622no—
225Q8—1242no—
226D4—22no—
227D4—622no—
228D4—622no—
229D4—622no—
230D4—1244no—
231D4—1244no—
232D4—1242no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—311no—
236C2×C4—314no—
237C2×C4—312no—
238C2×C4—621no—
239C2×C4—1242no—
240C2×C4—1244no—
241C2×C4—1242no—
242C2×C4D4×D3112yes—
243C2×C4C23×D3111yescharacteristic
244C8—314no—
245C8D4×D3114yes—
246C22—41no—
247C22—42no—
248C22—312no—
249C22—622no—
250C22—1242no—
251C22—1241no—
252C22—1244no—
253C22—1242no—
254C22—2484no—
255C22C2×D4×D3111yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—312no—
260C4—622no—
261C4—1241no—
262C4C2×D4×D3112yes—
263C2—81no—
264C2—622no—
265C2—2481no—
266C2—2482no—
267C2#127×D3111yescharacteristic
268C2#1452184812yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#497C2
socleC22×C31C22×D4
hypercenter#497D31
hypocenterC31#3153
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#11758495C2semidirect
2. C2×#11758491C2semidirect
3. #14521600C2semidirect
4. #124×D31C2semidirect
5. #14521441C2semidirect
6. #14521439C2semidirect
7. #14521438C2semidirect
8. #14521434C2semidirect
9. #14521430C2semidirect
10. #14521001C2semidirect
11. C31×#497C2semidirect
15. C2×Q16×C31C22semidirect
17. C31×C4⋊Q8C22semidirect
20. #11758524C22semidirect
21. C2×#92482C22semidirect
23. C2×#92481C22semidirect
29. C2×#92470C22semidirect
36. #11758451C22semidirect
38. C31×#97C22semidirect
39. C8×Dic31C22semidirect
40. #11758391C22semidirect
79. #92469D4semidirect
134. #497D31semidirect
152. C2×Q16C2×D31semidirect
160. C4⋊Q8C2×D31semidirect
173. #97C2×D31semidirect
175. C31#3153semidirect

Maximal subgroups

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

subgroupindex
C2×#117584952
C2×#117584912
#145216002
#124×D312
#145214412
#145214392
#145214382
#145214342
#145214302
#145210012
C31×#4972
#315331