Finite Group Databasebeta

← #30648208

Subgroups — #30648208

Order 16256.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16861234C211yesmaximal, characteristic, complemented
2C2×#16861205C211yesmaximal, characteristic, complemented
3#41×Dic127C211yesmaximal, characteristic, complemented
4#22550074C214yesmaximal, complemented
5#22550069C214yesmaximal, complemented
6#22549899C211yesmaximal, characteristic, complemented
7#22549882C211yesmaximal, characteristic, complemented
8#22549881C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×C22⋊D127—22no—
12#16861234—24no—
13C2×#14523235C2212yescomplemented
14C2×#14523233C2212yescomplemented
15C2×#14523232C2211yescharacteristic
16#16861205—21no—
17#16861205—21no—
18#16861205C2214yes—
19Q8×Dic127C2212yescomplemented
20D4×Dic127—22no—
21D4×Dic127C2212yescomplemented
22#16861168—21no—
23#16861168C2211yescharacteristic
24C2×C4×Dic127—21no—
25C2×C4×Dic127C2211yescharacteristic
26Dic127⋊C8C2214yescomplemented
27#16861131C2214yescomplemented
28#16861130C2214yescomplemented
29C2×#41×C127C2211yescharacteristic
30C22×D4×C127C2211yescharacteristic
31AGL(1,ℤ/8)×C127—24no—
32C2×SD16×C127C2212yescomplemented
33C2×D8×C127C2212yescomplemented
34C2×M16×C127C2211yescharacteristic
35C23×D127—21no—
36C2×D508—22no—
37C2×D508—21no—
38C2×D508—22no—
39C2×D508C2312yes—
40C2×C22⋊D127—24no—
41C2×C22⋊D127—24no—
42C2×C22⋊D127—48no—
43C22×Dic127—22no—
44C22×Dic127D412yes—
45#14523235—24no—
46#14523233—24no—
47#14523232—22no—
48C2×C127⋊C8C2312yes—
49C4⋊Dic127—22no—
50C4⋊Dic127C2312yes—
51C22⋊Dic127—22no—
52C22⋊Dic127—21no—
53C4×Dic127—22no—
54C4×Dic127—21no—
55C4×Dic127D411yescharacteristic
56C4×Dic127D411yescharacteristic
57C4×Dic127C2312yes—
58C24×C127—21no—
59#41×C127—22no—
60#41×C127D414yescomplemented
61C2×Q8×C127C2311yescharacteristic
62C2×D4×C127—21no—
63C2×D4×C127—22no—
64C2×D4×C127—21no—
65C2×D4×C127—22no—
66C2×D4×C127C2312yes—
67C2×D4×C127C2311yescharacteristic
68C22×C4×C127—21no—
69C22×C4×C127C2311yescharacteristic
70SD16×C127—24no—
71D8×C127—24no—
72M16×C127—22no—
73C2×C8×C127C2312yes—
74C22×D127—22no—
75C22×D127—44no—
76C22×D127—42no—
77D508—22no—
78D508—22no—
79D508—42no—
80C22⋊D127—48no—
81C22⋊D127—48no—
82C2×Dic127—24no—
83C2×Dic127—22no—
84C2×Dic127—44no—
85C2×Dic127C2×D412yes—
86C2×Dic127C2×D412yes—
87C127⋊C8—22no—
88C23×C127—21no—
89C23×C127—22no—
90C23×C127—44no—
91C23×C127—42no—
92C23×C127C2×D411yescharacteristic
93Q8×C127—21no—
94Q8×C127C2×D412yescomplemented
95D4×C127—24no—
96D4×C127—22no—
97D4×C127—21no—
98D4×C127—22no—
99D4×C127—42no—
100D4×C127C2×D412yescomplemented
101C2×C4×C127—21no—
102C2×C4×C127—24no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D412yes—
105C2×C4×C127#4112yes—
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127—22no—
109C22×C127—22no—
110C22×C127—42no—
111C22×C127—42no—
112C22×C127—44no—
113C22×C127—84no—
114C22×C127#12112yes—
115C22×C127C22×D411yescharacteristic
116C2×D127—42no—
117C2×D127—44no—
118C2×D127—84no—
119C4×C127—22no—
120C4×C127#12112yes—
121C4×C127C22×D411yescharacteristic
122C4×C127C2×#4111yescharacteristic
123Dic127—22no—
124Dic127—42no—
125Dic127AGL(1,ℤ/8)14yescomplemented
126C2×C127—22no—
127C2×C127—42no—
128C2×C127—82no—
129C2×C127C2×#12111yescharacteristic
130C2×C127C2×AGL(1,ℤ/8)12yes—
131D127—82no—
132#3038—1271nomaximal, Sylow
133C127#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1271no—
135C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
136C2×#127—1271no—
137C4×#41—1271no—
138#460—1274no—
139#459—1274no—
140#423—1271no—
141#418—1271no—
142#418—1271no—
143C2×#41C2×D12711yescharacteristic
144C22×D4—1271no—
145C22×D4—2542no—
146C22×D4C2×D12711yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—2544no—
149C2×SD16—1272no—
150C2×SD16C2×D12712yescomplemented
151C2×D8—1272no—
152C2×D8C2×D12712yescomplemented
153C2×M16—1271no—
154C2×M16C2×D12711yescharacteristic
155#127—1274no—
156#127—2541no—
157#127—2541no—
158C4×Q8—1272no—
159C4×D4—1272no—
160C4×D4—2542no—
161#117—1271no—
162#117—2541no—
163C2×C42—1271no—
164C2×C42—2541no—
165C4⋊C8—1274no—
166#102—1274no—
167#102—1274no—
168C24—21no—
169C24—2541no—
170#41—22no—
171#41C22⋊D12714yescomplemented
172C2×Q8C22×D12711yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—1272no—
178C2×D4—5088no—
179C2×D4—2544no—
180C2×D4—2544no—
181C2×D4—2541no—
182C2×D4—2542no—
183C2×D4—2542no—
184C2×D4C22×D12712yes—
185C2×D4C22×D12711yescharacteristic
186C22×C4—21no—
187C22×C4—1272no—
188C22×C4—2542no—
189C22×C4C22×D12711yescharacteristic
190SD16—24no—
191SD16—2544no—
192D8—24no—
193D8—2544no—
194M16—22no—
195M16—2542no—
196C2×C8—1272no—
197C2×C8C22×D12712yes—
198C4⋊C4—1272no—
199C4⋊C4—2542no—
200C22⋊C4—2542no—
201C22⋊C4—2541no—
202C42—1271no—
203C42—1271no—
204C42—1272no—
205C42—2542no—
206C42—2541no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—5084no—
212C23—5082no—
213C23—2542no—
214C23D4×D12711yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D12712yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—5088no—
223D4—5088no—
224D4—5082no—
225D4—2542no—
226D4—2542no—
227D4C2×C22⋊D12712yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—1272no—
231C2×C4—1272no—
232C2×C4—5084no—
233C2×C4—2544no—
234C2×C4—2542no—
235C2×C4D4×D12711yescharacteristic
236C2×C4C2×C22⋊D12712yes—
237C2×C4#1452324712yes—
238C2×C4C23×D12711yescharacteristic
239C8—22no—
240C8—2542no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—10164no—
248C22—5082no—
249C22—5084no—
250C22#1686118112yes—
251C22C2×D4×D12711yescharacteristic
252C4—22no—
253C4—1274no—
254C4—5082no—
255C4—2542no—
256C4#1686118112yes—
257C4C22×C22⋊D12711yescharacteristic
258C4C2×#1452324711yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—10162no—
263C2C2×#1686118111yescharacteristic
264C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3038
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×#16861205C2semidirect
3. #41×Dic127C2semidirect
4. #22550074C2semidirect
5. #22550069C2semidirect
6. #22549899C2semidirect
7. #22549882C2semidirect
8. #22549881C2semidirect
9. C2×AGL(1,ℤ/8)×C127C2semidirect
13. C2×#14523235C22semidirect
14. C2×#14523233C22semidirect
19. Q8×Dic127C22semidirect
21. D4×Dic127C22semidirect
26. Dic127⋊C8C22semidirect
27. #16861131C22semidirect
28. #16861130C22semidirect
32. C2×SD16×C127C22semidirect
33. C2×D8×C127C22semidirect
60. #41×C127D4semidirect
94. Q8×C127C2×D4semidirect
100. D4×C127C2×D4semidirect
125. Dic127AGL(1,ℤ/8)semidirect
133. C127#3038semidirect
135. C2×AGL(1,ℤ/8)D127semidirect
150. C2×SD16C2×D127semidirect
152. C2×D8C2×D127semidirect
171. #41C22⋊D127semidirect
216. Q8C2×C22⋊D127semidirect
227. D4C2×C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168612052
#41×Dic1272
#225500742
#225500692
#225498992
#225498822
#225498812
C2×AGL(1,ℤ/8)×C1272
#3038127