Finite Group Databasebeta

← #16856191

Subgroups — #16856191

Order 3968.

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

normal subgroups only

This group has 596 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×#11758488C212yesmaximal, complemented
2C2×#11758489C211yesmaximal, characteristic, complemented
3C2×D8×D31C211yesmaximal, characteristic, complemented
4#14521612C212yesmaximal, complemented
5#14521437C212yesmaximal, complemented
6#14521436C212yesmaximal, complemented
7#14521433C211yesmaximal, characteristic, complemented
8#14521432C211yesmaximal, characteristic, complemented
9#14521424C211yesmaximal, characteristic, complemented
10#14521003C211yesmaximal, characteristic, complemented
11C31×#493C211yesmaximal, characteristic, complemented
12C2×#92497C2212yes—
13C2×D4×D31C2212yes—
14C2×D8×C31C2211yescharacteristic
15C2×D8×C31C2211yescharacteristic, complemented
16C2×D8×C31C2212yescomplemented
17C31×#127C2212yescomplemented
18#11758517C2212yescomplemented
19#11758516—24no—
20#11758514—24no—
21#11758513C2212yescomplemented
22C2×#92480C2212yescomplemented
23C2×#92480C2212yescomplemented
24C2×#92479C2212yes—
25C2×#92479C2212yescomplemented
26#11758489—24no—
27#11758488—28no—
28D8×D31—24no—
29C2×Dic124C2211yescharacteristic, complemented
30C2×D248C2211yescharacteristic
31C2×#92470C2212yescomplemented
32C2×#92469C2211yescharacteristic
33C2×C8×D31C2211yescharacteristic
34#11758452C2212yescomplemented
35#11758449C2211yescharacteristic
36C4×C8×C31C2211yescharacteristic, complemented
37#11758403C2211yescharacteristic, complemented
38C8×Dic31C2211yescharacteristic, complemented
39#11758391C2211yescharacteristic, complemented
40C23×D31—22no—
41C2×D4×C31—24no—
42C2×D4×C31C2312yes—
43C2×D4×C31C2312yes—
44C2×C22⋊D31—22no—
45C2×C22⋊D31—22no—
46C2×C22⋊D31—24no—
47C22×Dic31—22no—
48#92497—24no—
49#92497—24no—
50D4×D31—24no—
51D4×D31—24no—
52C2×D124C2312yes—
53C2×C4×D31C2311yescharacteristic
54C2×Dic62C2312yes—
55D8×C31—22no—
56D8×C31—24no—
57D8×C31—22no—
58C2×C8×C31C2311yescharacteristic
59C2×C8×C31C2311yescharacteristic
60C42×C31C2311yescharacteristic
61C22⋊Dic31—24no—
62#92480—24no—
63#92480—24no—
64#92479—24no—
65#92479—24no—
66(C2×D31)⋊C4—24no—
67(C2×D31)⋊C4D412yes—
68Dic31⋊C4D412yes—
69C4×Dic31C2311yescharacteristic
70C2×C31⋊C8C2311yescharacteristic
71C2×C31⋊C8C2311yescharacteristic
72Dic124—22no—
73D248—22no—
74#92470—24no—
75#92469D414yescomplemented
76C8×D31D414yescomplemented
77C23×C31—22no—
78C23×C31—22no—
79C22×D31—22no—
80C22×D31—44no—
81C22×D31—84no—
82C22×D31C2×D411yescharacteristic
83D4×C31—22no—
84D4×C31—22no—
85D4×C31—22no—
86D4×C31—22no—
87D4×C31—84no—
88C2×C4×C31C2×D412yes—
89C2×C4×C31C2411yescharacteristic
90C22⋊D31—44no—
91C22⋊D31—44no—
92C22⋊D31—84no—
93C2×Dic31—22no—
94C2×Dic31—44no—
95C2×Dic31C2×D412yes—
96C2×Dic31C2×D411yescharacteristic
97D124—22no—
98D124—22no—
99C4×D31C2×D412yes—
100C4×D31C2×D412yes—
101Dic62—22no—
102Dic62—22no—
103C8×C31C2×D412yes—
104C8×C31C2×D412yes—
105C31⋊C8C2×D412yes—
106C31⋊C8C2×D412yes—
107#3155—311nomaximal, Sylow
108C22×C31—42no—
109C22×C31—42no—
110C22×C31—44no—
111C22×C31—44no—
112C22×C31C22×D411yescharacteristic
113C2×D31—22no—
114C2×D31—42no—
115C2×D31—44no—
116C2×D31—88no—
117C2×D31#12712yes—
118C4×C31—22no—
119C4×C31C22×D411yescharacteristic
120C4×C31C22×D411yescharacteristic
121Dic31—22no—
122Dic31—42no—
123Dic31#12712yes—
124C2×AGL(1,ℤ/8)—312no—
125C2×#135—311no—
126C22×D8—311no—
127#535—312no—
128#496—312no—
129#496—312no—
130#495—311no—
131#495—311no—
132#493—311no—
133#493D3111yescharacteristic, complemented
134#405—311no—
135C2×C31—82no—
136C2×C31—82no—
137C2×C31C2×#12711yescharacteristic
138C2×C31#57612yes—
139D31—22no—
140D31—82no—
141C2×#41—312no—
142C22×D4—312no—
143AGL(1,ℤ/8)—628no—
144#135—624no—
145C2×Q16—311no—
146C2×SD16—312no—
147C2×SD16—312no—
148C2×SD16—312no—
149C2×D8—312no—
150C2×D8—312no—
151C2×D8—311no—
152C2×D8—624no—
153C2×D8C2×D3111yescharacteristic
154C2×D8C2×D3111yescharacteristic, complemented
155C2×D8C2×D3112yescomplemented
156C2×M16—311no—
157C22×C8—311no—
158#127—312no—
159#127C2×D3112yescomplemented
160#124—312no—
161#124—312no—
162#121—624no—
163#120—624no—
164#117—311no—
165#97—311no—
166#97—311no—
167C4×C8—311no—
168C4×C8C2×D3111yescharacteristic, complemented
169C31#315511yescharacteristic, complemented, Sylow
170C24—622no—
171#41—624no—
172#41—624no—
173C2×Q8—312no—
174C2×D4—24no—
175C2×D4—312no—
176C2×D4—624no—
177C2×D4—624no—
178C2×D4—622no—
179C2×D4—624no—
180C2×D4—622no—
181C2×D4C22×D3112yes—
182C2×D4C22×D3112yes—
183C22×C4—311no—
184C22×C4—622no—
185Q16—622no—
186SD16—624no—
187SD16—624no—
188SD16—624no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—624no—
193D8—624no—
194D8—622no—
195M16—314no—
196C2×C8—311no—
197C2×C8—311no—
198C2×C8—314no—
199C2×C8C22×D3111yescharacteristic
200C2×C8C22×D3111yescharacteristic
201C4⋊C4—312no—
202C22⋊C4—312no—
203C22⋊C4—624no—
204C22⋊C4—624no—
205C42—311no—
206C42C22×D3111yescharacteristic
207C23—22no—
208C23—22no—
209C23—311no—
210C23—622no—
211C23—1244no—
212C23—2484no—
213Q8—622no—
214Q8—622no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—622no—
221D4—622no—
222D4—1244no—
223D4—1244no—
224D4—2484no—
225C2×C4—311no—
226C2×C4—312no—
227C2×C4—312no—
228C2×C4—312no—
229C2×C4—622no—
230C2×C4—1244no—
231C2×C4D4×D3112yes—
232C2×C4C23×D3111yescharacteristic
233C8—312no—
234C8—312no—
235C8D4×D3112yes—
236C8D4×D3112yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—312no—
242C22—622no—
243C22—1242no—
244C22—1244no—
245C22—2488no—
246C22C2×D4×D3111yescharacteristic
247C4—22no—
248C4—312no—
249C4—622no—
250C4—1242no—
251C4C2×D4×D3111yescharacteristic
252C4C2×D4×D3111yescharacteristic
253C2—82no—
254C2—82no—
255C2—622no—
256C2—2482no—
257C2#127×D3111yescharacteristic
258C2#1452185012yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#493C2
socleC22×C31C22×D4
hypercenter#493D31
hypocenterC31#3155
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#11758488C2semidirect
2. C2×#11758489C2semidirect
3. C2×D8×D31C2semidirect
4. #14521612C2semidirect
5. #14521437C2semidirect
6. #14521436C2semidirect
7. #14521433C2semidirect
8. #14521432C2semidirect
9. #14521424C2semidirect
10. #14521003C2semidirect
11. C31×#493C2semidirect
15. C2×D8×C31C22semidirect
16. C2×D8×C31C22semidirect
17. C31×#127C22semidirect
18. #11758517C22semidirect
21. #11758513C22semidirect
22. C2×#92480C22semidirect
23. C2×#92480C22semidirect
25. C2×#92479C22semidirect
29. C2×Dic124C22semidirect
31. C2×#92470C22semidirect
34. #11758452C22semidirect
36. C4×C8×C31C22semidirect
37. #11758403C22semidirect
38. C8×Dic31C22semidirect
39. #11758391C22semidirect
75. #92469D4semidirect
76. C8×D31D4semidirect
133. #493D31semidirect
154. C2×D8C2×D31semidirect
155. C2×D8C2×D31semidirect
159. #127C2×D31semidirect
168. C4×C8C2×D31semidirect
169. C31#3155semidirect

Maximal subgroups

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

subgroupindex
C2×#117584882
C2×#117584892
C2×D8×D312
#145216122
#145214372
#145214362
#145214332
#145214322
#145214242
#145210032
C31×#4932
#315531