Finite Group Databasebeta

← #30648220

Subgroups — #30648220

Order 16256.

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

normal subgroups only

This group has 500 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×#16861240C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic
3C4×#14523248C211yesmaximal, characteristic, complemented
4#22550095C214yesmaximal, complemented
5#22550088C214yesmaximal, complemented
6#22549900C211yesmaximal, characteristic, complemented
7#22549885C211yesmaximal, characteristic, complemented
8#22549881C211yesmaximal, characteristic, complemented
9C2×#124×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11#16861240—24no—
12#16861235—24no—
13C2×#14523238C2212yescomplemented
14C2×#14523236C2212yes—
15C2×#14523235C2212yescomplemented
16C2×#14523233C2212yes—
17C2×#14523232C2211yescharacteristic
18C2×#14523232C2211yescharacteristic
19C4×Dic254C2212yescomplemented
20C4×D508C2212yescomplemented
21C4×C22⋊D127—22no—
22#16861168C2211yescharacteristic
23#16861166—21no—
24C42×D127—21no—
25C4⋊(C127⋊C8)C2214yescomplemented
26#16861133C2214yescomplemented
27#16861130C2214yes—
28C22×Q8×C127C2211yescharacteristic
29C22×D4×C127C2211yescharacteristic
30#124×C127—21no—
31#124×C127—21no—
32#124×C127C2214yes—
33C2×C22⋊C4×C127—22no—
34C2×C42×C127C2211yescharacteristic
35#14523248—22no—
36#14523248D414yes—
37C2×Dic254C2311yescharacteristic
38C2×D508C2311yescharacteristic
39C2×C22⋊D127—21no—
40C2×C4×D127—21no—
41C2×C4×D127—22no—
42#14523238—24no—
43#14523236—24no—
44#14523235—24no—
45#14523233—24no—
46#14523232—22no—
47#14523232—22no—
48C2×C127⋊C8C2312yes—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127C2312yes—
51Dic127⋊C4—22no—
52C22⋊Dic127—21no—
53(C2×D127)⋊C4—22no—
54C4×Dic127—21no—
55C4×Dic127—22no—
56C24×C127—21no—
57C2×Q8×C127—21no—
58C2×Q8×C127—22no—
59C2×Q8×C127—22no—
60C2×Q8×C127C2312yes—
61C2×D4×C127—22no—
62C2×D4×C127—21no—
63C2×D4×C127—22no—
64C2×D4×C127C2312yes—
65C22×C4×C127—21no—
66C22×C4×C127D412yes—
67C22×C4×C127C2311yescharacteristic
68C22⋊C4×C127—24no—
69C22⋊C4×C127—24no—
70C42×C127D411yescharacteristic
71C42×C127D411yescharacteristic
72C42×C127C2312yes—
73C22×D127—21no—
74Dic254—21no—
75Dic254C2×D412yes—
76D508—21no—
77D508C2×D412yes—
78C22⋊D127—24no—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C4×D127—24no—
82C4×D127—44no—
83C127⋊C8—22no—
84C127⋊C8—22no—
85C23×C127—22no—
86C23×C127—44no—
87C23×C127—42no—
88C23×C127C2×D411yescharacteristic
89Q8×C127—22no—
90Q8×C127—22no—
91Q8×C127—42no—
92D4×C127—22no—
93D4×C127—22no—
94D4×C127—42no—
95C2×C4×C127—22no—
96C2×C4×C127—42no—
97C2×C4×C127—42no—
98C2×C4×C127C2×D412yes—
99C2×C4×C127C2×D412yes—
100C2×C4×C127C2×D412yes—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127#4112yes—
103C2×C4×C127C2411yescharacteristic
104C22×C127—22no—
105C22×C127—42no—
106C22×C127—44no—
107C22×C127—84no—
108C22×C127#12112yes—
109C22×C127C22×D411yescharacteristic
110C2×D127—22no—
111C2×D127—42no—
112C4×C127—22no—
113C4×C127—22no—
114C4×C127—42no—
115C4×C127#12112yes—
116C4×C127C22×D411yescharacteristic
117C4×C127C2×#4111yescharacteristic
118Dic127—22no—
119Dic127—42no—
120C2×C127—22no—
121C2×C127—82no—
122C2×C127C2×#12111yescharacteristic
123C2×C127#57512yes—
124D127—42no—
125#3040—1271nomaximal, Sylow
126C127#304011yescharacteristic, complemented, Sylow
127C2×#137—1271no—
128C2×AGL(1,ℤ/8)—1271no—
129C2×#124D12711yescharacteristic, complemented
130C4×#41—1271no—
131#464—1274no—
132#463—1274no—
133#423—1271no—
134#419—1271no—
135#418—1271no—
136C2×#41—1271no—
137C22×Q8C2×D12711yescharacteristic
138C22×D4C2×D12711yescharacteristic
139#137—2544no—
140AGL(1,ℤ/8)—2544no—
141C2×Q16—1272no—
142C2×SD16—1272no—
143C2×SD16—1272no—
144C2×D8—1272no—
145C2×M16—1271no—
146C2×M16—1271no—
147#124—21no—
148#124—21no—
149#124C2×D12714yes—
150C4×Q8—1272no—
151C4×D4—1272no—
152C4×D4—2542no—
153#117—1271no—
154#117—2541no—
155C2×C22⋊C4—22no—
156C2×C42—2541no—
157C2×C42C2×D12711yescharacteristic
158C4⋊C8—1274no—
159#103—1274no—
160#102—1274no—
161C24—21no—
162#41—1274no—
163#41—2542no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—1271no—
168C2×Q8C22×D12712yes—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—1271no—
173C2×D4—2541no—
174C2×D4C22×D12712yes—
175C22×C4—21no—
176C22×C4—2542no—
177C22×C4—2541no—
178C22×C4C22⋊D12712yes—
179C22×C4C22×D12711yescharacteristic
180Q16—2544no—
181SD16—2544no—
182SD16—2544no—
183D8—2544no—
184M16—2542no—
185M16—2542no—
186C2×C8—1272no—
187C2×C8—1272no—
188C4⋊C4—1272no—
189C4⋊C4—2542no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—2541no—
193C22⋊C4—2542no—
194C42—2541no—
195C42—2542no—
196C42C22⋊D12711yescharacteristic
197C42C22⋊D12711yescharacteristic
198C42C22×D12712yes—
199C23—22no—
200C23—44no—
201C23—42no—
202C23—2541no—
203C23C2×C22⋊D12711yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—1272no—
208Q8—2541no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—1272no—
213D4—2544no—
214D4—2541no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—5084no—
219C2×C4—2542no—
220C2×C4—2541no—
221C2×C4—2544no—
222C2×C4D4×D12712yes—
223C2×C4C2×C22⋊D12712yes—
224C2×C4C2×C22⋊D12712yes—
225C2×C4C2×C22⋊D12711yescharacteristic
226C2×C4#1452324712yes—
227C2×C4C23×D12711yescharacteristic
228C8—2542no—
229C8—2542no—
230C22—22no—
231C22—42no—
232C22—44no—
233C22—84no—
234C22—5082no—
235C22—2542no—
236C22#1686118012yes—
237C22C22×C22⋊D12711yescharacteristic
238C4—22no—
239C4—22no—
240C4—42no—
241C4—5082no—
242C4—2542no—
243C4#1686118012yes—
244C4C2×D4×D12711yescharacteristic
245C4C2×#1452324711yescharacteristic
246C2—22no—
247C2—82no—
248C2—5082no—
249C2C2×#1686118011yescharacteristic
250C2#2255067412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#124×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861235C2
hypercenterC2×#124D127
hypocenterC127#3040
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. C4×#14523248C2semidirect
4. #22550095C2semidirect
5. #22550088C2semidirect
6. #22549900C2semidirect
7. #22549885C2semidirect
8. #22549881C2semidirect
9. C2×#124×C127C2semidirect
13. C2×#14523238C22semidirect
15. C2×#14523235C22semidirect
19. C4×Dic254C22semidirect
20. C4×D508C22semidirect
25. C4⋊(C127⋊C8)C22semidirect
26. #16861133C22semidirect
126. C127#3040semidirect
129. C2×#124D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612352
C4×#145232482
#225500952
#225500882
#225499002
#225498852
#225498812
C2×#124×C1272
#3040127