Finite Group Databasebeta

← #30648224

Subgroups — #30648224

Order 16256.

500 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861239C211yesmaximal, characteristic, complemented
2C2×#16861234C211yesmaximal, characteristic
3C2×#16861197C211yesmaximal, characteristic, complemented
4#22550098C214yesmaximal, complemented
5#22550091C214yesmaximal, complemented
6#22549900C211yesmaximal, characteristic, complemented
7#22549884C211yesmaximal, characteristic, complemented
8#22549880C211yesmaximal, characteristic, complemented
9C4×#41×C127C211yesmaximal, characteristic, complemented
10C22×Dic254C2211yescharacteristic
11C22×D508C2211yescharacteristic
12#16861239—24no—
13#16861234—24no—
14C2×#14523238C2212yescomplemented
15C2×#14523236C2212yescomplemented
16C2×#14523235C2212yes—
17C2×#14523233C2212yes—
18C2×#14523232C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20#16861197—21no—
21#16861197—21no—
22#16861197C2214yes—
23C2×(C2×D127)⋊C4—22no—
24C4⋊(C127⋊C8)C2214yescomplemented
25#16861132C2214yescomplemented
26#16861129C2214yes—
27C2×#41×C127C2211yescharacteristic
28C4×Q8×C127C2212yescomplemented
29C4×D4×C127—22no—
30C4×D4×C127C2212yescomplemented
31#117×C127—21no—
32#117×C127C2211yescharacteristic
33C2×C42×C127—21no—
34C2×C42×C127C2211yescharacteristic
35C23×D127—21no—
36C2×Dic254—21no—
37C2×Dic254—22no—
38C2×Dic254—22no—
39C2×Dic254C2312yes—
40C2×D508—22no—
41C2×D508—21no—
42C2×D508—22no—
43C2×D508C2312yes—
44C22×Dic127—21no—
45#14523238—24no—
46#14523236—24no—
47#14523235—24no—
48#14523233—24no—
49#14523232—22no—
50#14523232—22no—
51C2×C127⋊C8C2312yes—
52C2×C127⋊C8C2312yes—
53(C2×D127)⋊C4—24no—
54(C2×D127)⋊C4—24no—
55#41×C127—22no—
56#41×C127D414yes—
57C2×Q8×C127C2311yescharacteristic
58C2×D4×C127—21no—
59C2×D4×C127C2311yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127—22no—
62C22×C4×C127D412yes—
63C22×C4×C127C2311yescharacteristic
64C4⋊C4×C127—22no—
65C4⋊C4×C127C2312yes—
66C22⋊C4×C127—21no—
67C22⋊C4×C127—22no—
68C42×C127—21no—
69C42×C127—22no—
70C42×C127D411yescharacteristic
71C42×C127D411yescharacteristic
72C42×C127C2312yes—
73C22×D127—22no—
74C22×D127—44no—
75C22×D127—42no—
76Dic254—22no—
77Dic254—22no—
78Dic254—42no—
79D508—22no—
80D508—22no—
81D508—42no—
82C2×Dic127—22no—
83C2×Dic127—42no—
84C127⋊C8—22no—
85C127⋊C8—22no—
86C23×C127—21no—
87C23×C127C2×D411yescharacteristic
88Q8×C127—21no—
89Q8×C127C2×D412yes—
90D4×C127—24no—
91D4×C127—21no—
92D4×C127C2×D412yes—
93C2×C4×C127—21no—
94C2×C4×C127—24no—
95C2×C4×C127—22no—
96C2×C4×C127—42no—
97C2×C4×C127—44no—
98C2×C4×C127C2×D412yes—
99C2×C4×C127C2×D412yes—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D412yes—
102C2×C4×C127#4112yes—
103C2×C4×C127C2411yescharacteristic
104C22×C127—22no—
105C22×C127—22no—
106C22×C127—42no—
107C22×C127#12112yes—
108C22×C127C22×D411yescharacteristic
109C2×D127—42no—
110C2×D127—44no—
111C2×D127—84no—
112C4×C127—22no—
113C4×C127—22no—
114C4×C127—22no—
115C4×C127—42no—
116C4×C127#12112yes—
117C4×C127C22×D411yescharacteristic
118C4×C127C2×#4111yescharacteristic
119Dic127—42no—
120C2×C127—22no—
121C2×C127—42no—
122C2×C127C2×#12111yescharacteristic
123C2×C127#57512yes—
124D127—82no—
125#3040—1271nomaximal, Sylow
126C127#304011yescharacteristic, complemented, Sylow
127C2×#137—1271no—
128C2×AGL(1,ℤ/8)—1271no—
129C2×#124—1271no—
130C4×#41D12711yescharacteristic, complemented
131#464—1274no—
132#463—1274no—
133#423—1271no—
134#419—1271no—
135#418—1271no—
136C2×#41C2×D12711yescharacteristic
137C22×Q8—1271no—
138C22×D4—1271no—
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—1274no—
148#124—2541no—
149#124—2541no—
150C4×Q8C2×D12712yescomplemented
151C4×D4—22no—
152C4×D4C2×D12712yescomplemented
153#117—21no—
154#117C2×D12711yescharacteristic
155C2×C22⋊C4—2542no—
156C2×C42—21no—
157C2×C42C2×D12711yescharacteristic
158C4⋊C8—1274no—
159#103—1274no—
160#102—1274no—
161C24—2541no—
162#41—22no—
163#41D50814yes—
164C2×Q8—1272no—
165C2×Q8—2542no—
166C2×Q8—2542no—
167C2×Q8—2541no—
168C2×Q8C22×D12711yescharacteristic
169C2×D4—21no—
170C2×D4—1272no—
171C2×D4—2541no—
172C2×D4—2542no—
173C2×D4—2542no—
174C2×D4C22×D12711yescharacteristic
175C22×C4—21no—
176C22×C4—22no—
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—22no—
189C4⋊C4C22×D12712yes—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—2544no—
193C22⋊C4—2544no—
194C42—21no—
195C42—22no—
196C42C22⋊D12711yescharacteristic
197C42C22⋊D12711yescharacteristic
198C42C22×D12712yes—
199C23—21no—
200C23—5084no—
201C23—5082no—
202C23—2542no—
203C23C2×C22⋊D12711yescharacteristic
204Q8—21no—
205Q8—5082no—
206Q8—2542no—
207Q8—2542no—
208Q8C2×D50812yes—
209D4—24no—
210D4—21no—
211D4—5082no—
212D4—2542no—
213D4—2542no—
214D4C2×D50812yes—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—5082no—
221C2×C4—2542no—
222C2×C4C2×C22⋊D12712yes—
223C2×C4C2×C22⋊D12712yes—
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C2×D50812yes—
226C2×C4#1452324812yes—
227C2×C4C23×D12711yescharacteristic
228C8—2542no—
229C8—2542no—
230C22—22no—
231C22—22no—
232C22—42no—
233C22—10164no—
234C22—5082no—
235C22—5084no—
236C22#1686118312yes—
237C22C22×C22⋊D12711yescharacteristic
238C4—22no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—5082no—
243C4#1686118312yes—
244C4C22×D50811yescharacteristic
245C4C2×#1452324811yescharacteristic
246C2—22no—
247C2—42no—
248C2—10162no—
249C2C2×#1686118311yescharacteristic
250C2#2255067412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×#41×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861234C2
hypercenterC4×#41D127
hypocenterC127#3040
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
3. C2×#16861197C2semidirect
4. #22550098C2semidirect
5. #22550091C2semidirect
6. #22549900C2semidirect
7. #22549884C2semidirect
8. #22549880C2semidirect
9. C4×#41×C127C2semidirect
14. C2×#14523238C22semidirect
15. C2×#14523236C22semidirect
24. C4⋊(C127⋊C8)C22semidirect
25. #16861132C22semidirect
28. C4×Q8×C127C22semidirect
30. C4×D4×C127C22semidirect
126. C127#3040semidirect
130. C4×#41D127semidirect
150. C4×Q8C2×D127semidirect
152. C4×D4C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168612342
C2×#168611972
#225500982
#225500912
#225499002
#225498842
#225498802
C4×#41×C1272
#3040127