Finite Group Databasebeta

← #30648227

Subgroups — #30648227

Order 16256.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16861237C211yesmaximal, characteristic
3C2×#16861198C211yesmaximal, characteristic, complemented
4#22550344C211yesmaximal, characteristic, complemented
5#22550101C214yesmaximal, complemented
6#22550094C214yesmaximal, complemented
7#22549899C211yesmaximal, characteristic, complemented
8#22549880C211yesmaximal, characteristic, complemented
9#419×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×D508C2211yescharacteristic
12#16861240—24no—
13#16861237—24no—
14C2×#14523238C2212yescomplemented
15C2×#14523237C2212yes—
16C2×#14523235C2212yescomplemented
17C2×D1016C2212yes—
18C2×#14523232C2211yescharacteristic
19#16861198—21no—
20#16861198—21no—
21#16861198C2214yes—
22#16861174C2212yescomplemented
23#16861171C2212yescomplemented
24(C22⋊D127)⋊C4—22no—
25#16861166—21no—
26C2×(C2×D127)⋊C4—22no—
27C2×C4×Dic127C2211yescharacteristic
28C42×D127—21no—
29Dic127⋊C8C2214yescomplemented
30#16861129C2214yes—
31C22×Q8×C127C2211yescharacteristic
32C2×M16×C127C2211yescharacteristic
33#117×C127C2211yescharacteristic
34#103×C127C2214yescomplemented
35C23×D127—21no—
36#14523248—22no—
37#14523248D414yes—
38C2×Dic254C2311yescharacteristic
39C2×D508—22no—
40C2×D508—21no—
41C2×D508—22no—
42C2×D508C2312yes—
43C2×D508C2311yescharacteristic
44C2×C22⋊D127—21no—
45C22×Dic127D412yes—
46C2×C4×D127—22no—
47C2×C4×D127—21no—
48#14523238—24no—
49#14523237—24no—
50#14523235—24no—
51D1016—24no—
52#14523232—22no—
53C2×C127⋊C8C2312yes—
54Dic127⋊C4—22no—
55(C2×D127)⋊C4—24no—
56(C2×D127)⋊C4—24no—
57(C2×D127)⋊C4—22no—
58C4×Dic127—22no—
59C4×Dic127D411yescharacteristic
60C4×Dic127D411yescharacteristic
61C4×Dic127C2312yes—
62C2×Q8×C127—21no—
63C2×Q8×C127—22no—
64C2×Q8×C127—22no—
65C2×Q8×C127C2312yes—
66C22×C4×C127—21no—
67C22×C4×C127C2311yescharacteristic
68M16×C127—22no—
69C2×C8×C127C2312yes—
70C4⋊C4×C127C2312yes—
71C22⋊C4×C127—21no—
72C42×C127—21no—
73C22×D127—21no—
74C22×D127—22no—
75C22×D127—44no—
76C22×D127—42no—
77Dic254—21no—
78Dic254C2×D412yes—
79D508—22no—
80D508—21no—
81D508—22no—
82D508—42no—
83D508C2×D412yes—
84C22⋊D127—24no—
85C2×Dic127—21no—
86C2×Dic127—42no—
87C2×Dic127C2×D412yes—
88C2×Dic127C2×D412yes—
89C4×D127—24no—
90C4×D127—44no—
91C127⋊C8—22no—
92C23×C127C2×D411yescharacteristic
93Q8×C127—22no—
94Q8×C127—22no—
95Q8×C127—42no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—42no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D412yes—
101C2×C4×C127#4112yes—
102C2×C4×C127C2411yescharacteristic
103C8×C127—22no—
104C22×C127—22no—
105C22×C127#12112yes—
106C22×C127C22×D411yescharacteristic
107C2×D127—22no—
108C2×D127—42no—
109C2×D127—42no—
110C2×D127—44no—
111C2×D127—84no—
112C4×C127—42no—
113C4×C127—42no—
114C4×C127#12112yes—
115C4×C127C22×D411yescharacteristic
116C4×C127C2×#4111yescharacteristic
117Dic127—22no—
118Dic127—22no—
119Dic127—22no—
120C2×C127—22no—
121C2×C127C2×#12111yescharacteristic
122C2×C127#57512yes—
123D127—42no—
124D127—82no—
125#3040—1271nomaximal, Sylow
126C127#304011yescharacteristic, complemented, Sylow
127C2×#137—1271no—
128C2×AGL(1,ℤ/8)—1271no—
129C2×#124—1271no—
130C4×#41—1271no—
131#464—1274no—
132#463—1274no—
133#423—1271no—
134#419D12711yescharacteristic, complemented
135#418—1271no—
136C2×#41—1271no—
137C22×Q8C2×D12711yescharacteristic
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×M16C2×D12711yescharacteristic
147#124—1274no—
148#124—2541no—
149#124—2541no—
150C4×Q8—1272no—
151C4×D4—1272no—
152C4×D4—2542no—
153#117—2541no—
154#117C2×D12711yescharacteristic
155C2×C22⋊C4—2542no—
156C2×C42—1271no—
157C2×C42—2541no—
158C4⋊C8—1274no—
159#103C2×D12714yescomplemented
160#102—1274no—
161C24—2541no—
162#41—1274no—
163#41—2542no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—1271no—
168C2×Q8C22×D12712yes—
169C2×D4—1271no—
170C2×D4—1272no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4—2542no—
174C2×D4—2542no—
175C22×C4—21no—
176C22×C4—1272no—
177C22×C4—2542no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180Q16—2544no—
181SD16—2544no—
182SD16—2544no—
183D8—2544no—
184M16—22no—
185M16—2542no—
186C2×C8—1272no—
187C2×C8C22×D12712yes—
188C4⋊C4—2542no—
189C4⋊C4C22×D12712yes—
190C22⋊C4—21no—
191C22⋊C4—2544no—
192C22⋊C4—2544no—
193C22⋊C4—2542no—
194C42—21no—
195C42—1271no—
196C42—1271no—
197C42—1272no—
198C42—2542no—
199C23—5084no—
200C23—5082no—
201C23—2542no—
202C23—2541no—
203C23D4×D12711yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—1272no—
208Q8—2541no—
209D4—1272no—
210D4—5082no—
211D4—2544no—
212D4—2542no—
213D4—2542no—
214D4—2541no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—1272no—
219C2×C4—1272no—
220C2×C4—5084no—
221C2×C4—5082no—
222C2×C4—2541no—
223C2×C4—2544no—
224C2×C4D4×D12712yes—
225C2×C4D4×D12711yescharacteristic
226C2×C4#1452324812yes—
227C2×C4C23×D12711yescharacteristic
228C8—22no—
229C8—2542no—
230C22—22no—
231C22—10164no—
232C22—5082no—
233C22—5084no—
234C22—5082no—
235C22—2542no—
236C22#1686118212yes—
237C22C2×D4×D12711yescharacteristic
238C4—42no—
239C4—42no—
240C4—2542no—
241C4—2542no—
242C4—2542no—
243C4#1686118212yes—
244C4C2×D4×D12711yescharacteristic
245C4C2×#1452324811yescharacteristic
246C2—22no—
247C2—10162no—
248C2—5082no—
249C2C2×#1686118211yescharacteristic
250C2#2255067512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861237C2
hypercenter#419D127
hypocenterC127#3040
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. C2×#16861198C2semidirect
4. #22550344C2semidirect
5. #22550101C2semidirect
6. #22550094C2semidirect
7. #22549899C2semidirect
8. #22549880C2semidirect
9. #419×C127C2semidirect
14. C2×#14523238C22semidirect
16. C2×#14523235C22semidirect
22. #16861174C22semidirect
23. #16861171C22semidirect
29. Dic127⋊C8C22semidirect
34. #103×C127C22semidirect
126. C127#3040semidirect
134. #419D127semidirect
159. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612372
C2×#168611982
#225503442
#225501012
#225500942
#225498992
#225498802
#419×C1272
#3040127