Finite Group Databasebeta

← #16856176

Subgroups — #16856176

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×#11758492C212yesmaximal, complemented
2C2×#11758488C212yesmaximal, complemented
3#14521597C211yesmaximal, characteristic, complemented
4#124×D31C211yesmaximal, characteristic, complemented
5#14521444C211yesmaximal, characteristic, complemented
6#14521443C211yesmaximal, characteristic, complemented
7#14521436C211yesmaximal, characteristic, complemented
8#14521432C212yesmaximal, complemented
9#14521430C212yesmaximal, complemented
10#14521001C211yesmaximal, characteristic, complemented
11C31×#496C211yesmaximal, characteristic, complemented
12C2×Q8×D31C2211yescharacteristic
13C2×#92497C2212yes—
14C2×D4×D31C2211yescharacteristic
15C2×SD16×C31C2212yescomplemented
16C2×D8×C31C2212yes—
17C31×#127C2211yescharacteristic, complemented
18C31×#124C2211yescharacteristic
19#11758524C2211yescharacteristic
20C2×#92482C2212yescomplemented
21#11758515—22no—
22#11758513C2211yescharacteristic
23#11758513C2212yescomplemented
24D4×Dic31—22no—
25C2×#92480C2212yes—
26C2×#92480C2212yescomplemented
27C2×#92479C2212yes—
28#11758492—28no—
29#11758488—28no—
30C2×Dic124C2212yescomplemented
31C2×#92470C2212yes—
32C2×#92469C2212yes—
33#11758462—22no—
34C22⋊C4×D31—22no—
35#11758452C2211yescharacteristic
36C42×D31C2211yescharacteristic
37#11758446C2211yescharacteristic, complemented
38C31×#97C2211yescharacteristic, complemented
39C8×Dic31C2212yescomplemented
40#11758391C2211yescharacteristic, complemented
41C23×D31—21no—
42C2×Q8×C31C2311yescharacteristic
43C2×D4×C31—22no—
44C2×D4×C31C2312yes—
45C2×D4×C31C2311yescharacteristic
46C2×C22⋊D31—22no—
47C2×C22⋊D31—21no—
48C22×Dic31—22no—
49Q8×D31—24no—
50#92497—24no—
51#92497—24no—
52D4×D31—24no—
53C2×D124C2311yescharacteristic
54C2×C4×D31—21no—
55C2×C4×D31D412yes—
56C2×C4×D31C2311yescharacteristic
57C2×Dic62—21no—
58C2×Dic62C2312yes—
59C2×Dic62C2311yescharacteristic
60SD16×C31—24no—
61D8×C31—24no—
62C2×C8×C31C2312yes—
63C22⋊C4×C31—22no—
64C42×C31C2311yescharacteristic
65C22⋊Dic31—22no—
66C22⋊Dic31—24no—
67#92482—24no—
68#92480—24no—
69#92480—24no—
70#92479—24no—
71(C2×D31)⋊C4—22no—
72(C2×D31)⋊C4—22no—
73C4⋊Dic31—22no—
74C4×Dic31D412yes—
75C4×Dic31C2311yescharacteristic
76C2×C31⋊C8C2312yes—
77Dic124—24no—
78#92470—24no—
79#92469D418yescomplemented
80C23×C31—21no—
81C23×C31—22no—
82C22×D31—21no—
83C22×D31—44no—
84C22×D31—42no—
85C22×D31C2×D411yescharacteristic
86Q8×C31—22no—
87D4×C31—22no—
88D4×C31—22no—
89D4×C31—22no—
90D4×C31—44no—
91C2×C4×C31—21no—
92C2×C4×C31C2×D412yes—
93C2×C4×C31C2411yescharacteristic
94C22⋊D31—44no—
95C22⋊D31—42no—
96C2×Dic31—22no—
97C2×Dic31—21no—
98C2×Dic31—44no—
99C2×Dic31C2×D412yes—
100C2×Dic31C2×D411yescharacteristic
101D124—22no—
102C4×D31—42no—
103C4×D31—42no—
104C4×D31C2×D414yes—
105Dic62—22no—
106Dic62—22no—
107Dic62—22no—
108Dic62—42no—
109C8×C31C2×D414yes—
110C31⋊C8C2×D414yes—
111#3153—311nomaximal, Sylow
112C22×C31—42no—
113C22×C31—41no—
114C22×C31—44no—
115C22×C31—42no—
116C22×C31C22×D411yescharacteristic
117C2×D31—22no—
118C2×D31—41no—
119C2×D31—42no—
120C2×D31—84no—
121C2×D31#12712yes—
122C4×C31—22no—
123C4×C31—41no—
124C4×C31C22×D412yes—
125Dic31—22no—
126Dic31—41no—
127Dic31—42no—
128Dic31#12712yes—
129C2×#137—312no—
130C2×AGL(1,ℤ/8)—312no—
131#532—311no—
132C2×#124—311no—
133#497—311no—
134#497—311no—
135#496—311no—
136#496D3111yescharacteristic, complemented
137#495—312no—
138#495—312no—
139C4×M16—311no—
140C2×C31—81no—
141C2×C31—82no—
142C2×C31C2×#12711yescharacteristic
143C2×C31#57512yes—
144D31—22no—
145D31—81no—
146C2×#41—312no—
147C22×Q8—311no—
148C22×D4—311no—
149#137—628no—
150AGL(1,ℤ/8)—628no—
151C2×Q16—312no—
152C2×Q16—312no—
153C2×SD16—312no—
154C2×SD16—312no—
155C2×SD16—312no—
156C2×SD16C2×D3112yescomplemented
157C2×D8—312no—
158C2×D8C2×D3112yes—
159C2×M16—312no—
160C4⋊Q8—311no—
161#127C2×D3111yescharacteristic, complemented
162#124—311no—
163#124—311no—
164#124—312no—
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—312no—
181C2×Q8—621no—
182C2×Q8—624no—
183C2×Q8C22×D3111yescharacteristic
184C2×D4—22no—
185C2×D4—311no—
186C2×D4—624no—
187C2×D4—621no—
188C2×D4—622no—
189C2×D4C22×D3111yescharacteristic
190C2×D4C22×D3112yes—
191C22×C4—311no—
192C22×C4—312no—
193C22×C4—621no—
194C22×C4—622no—
195Q16—624no—
196Q16—624no—
197SD16—24no—
198SD16—624no—
199SD16—624no—
200SD16—624no—
201D8—24no—
202D8—624no—
203M16—318no—
204C2×C8—312no—
205C2×C8C22×D3112yes—
206C4⋊C4—622no—
207C22⋊C4—22no—
208C22⋊C4—622no—
209C22⋊C4—622no—
210C22⋊C4—624no—
211C22⋊C4—622no—
212C42—312no—
213C42—311no—
214C42C22×D3111yescharacteristic
215C23—21no—
216C23—22no—
217C23—311no—
218C23—621no—
219C23—1242no—
220C23—1244no—
221Q8—22no—
222Q8—622no—
223Q8—622no—
224Q8—622no—
225Q8—1242no—
226D4—22no—
227D4—22no—
228D4—22no—
229D4—44no—
230D4—622no—
231D4—1242no—
232D4—1244no—
233C2×C4—21no—
234C2×C4—311no—
235C2×C4—314no—
236C2×C4—312no—
237C2×C4—621no—
238C2×C4—622no—
239C2×C4—1242no—
240C2×C4—1242no—
241C2×C4—1244no—
242C2×C4D4×D3112yes—
243C2×C4C23×D3111yescharacteristic
244C8—314no—
245C8D4×D3114yes—
246C22—42no—
247C22—41no—
248C22—44no—
249C22—42no—
250C22—312no—
251C22—622no—
252C22—1241no—
253C22—1242no—
254C22—2484no—
255C22C2×D4×D3111yescharacteristic
256C4—22no—
257C4—41no—
258C4—312no—
259C4—622no—
260C4—1242no—
261C4—1241no—
262C4C2×D4×D3112yes—
263C2—81no—
264C2—82no—
265C2—622no—
266C2—2481no—
267C2#127×D3111yescharacteristic
268C2#1452184512yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#496C2
socleC22×C31C22×D4
hypercenter#496D31
hypocenterC31#3153
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×#11758492C2semidirect
2. C2×#11758488C2semidirect
3. #14521597C2semidirect
4. #124×D31C2semidirect
5. #14521444C2semidirect
6. #14521443C2semidirect
7. #14521436C2semidirect
8. #14521432C2semidirect
9. #14521430C2semidirect
10. #14521001C2semidirect
11. C31×#496C2semidirect
15. C2×SD16×C31C22semidirect
17. C31×#127C22semidirect
20. C2×#92482C22semidirect
23. #11758513C22semidirect
26. C2×#92480C22semidirect
30. C2×Dic124C22semidirect
37. #11758446C22semidirect
38. C31×#97C22semidirect
39. C8×Dic31C22semidirect
40. #11758391C22semidirect
79. #92469D4semidirect
136. #496D31semidirect
156. C2×SD16C2×D31semidirect
161. #127C2×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×#117584922
C2×#117584882
#145215972
#124×D312
#145214442
#145214432
#145214362
#145214322
#145214302
#145210012
C31×#4962
#315331