Finite Group Databasebeta

← #30648236

Subgroups — #30648236

Order 16256.

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

normal subgroups only

This group has 526 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×#16861234C211yesmaximal, characteristic, complemented
2C22×D1016C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550350C211yesmaximal, characteristic, complemented
5#22550094C214yesmaximal, complemented
6#22550070C214yesmaximal, complemented
7#22549902C211yesmaximal, characteristic, complemented
8#22549880C211yesmaximal, characteristic, complemented
9#416×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×D508C2211yescharacteristic
12#16861234—24no—
13C2×#14523235C2212yescomplemented
14C2×D1016—24no—
15C2×D1016C2214yes—
16C2×#14523233C2212yescomplemented
17C2×#14523232C2211yescharacteristic
18#16861205C2212yescomplemented
19#16861198C2212yescomplemented
20#16861183—22no—
21#16861177—22no—
22C4×D508—22no—
23#16861171C2214yescomplemented
24#16861168C2211yescharacteristic
25C22⋊C4×D127—22no—
26Dic127⋊C8C2214yescomplemented
27#16861129C2214yescomplemented
28C2×#41×C127C2211yescharacteristic
29C22×C8×C127C2211yescharacteristic
30#117×C127C2211yescharacteristic
31#103×C127C2212yescomplemented
32#102×C127C2212yescomplemented
33C23×D127—21no—
34C23×D127—21no—
35C2×D508—21no—
36C2×D508—22no—
37C2×D508—21no—
38C2×D508—22no—
39C2×D508—22no—
40C2×D508D414yescomplemented
41C2×D508C2312yes—
42C2×D508C2312yes—
43C2×C22⋊D127—24no—
44C2×C4×D127—24no—
45#14523235—24no—
46D1016—28no—
47#14523233—24no—
48#14523232—22no—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127D412yes—
51C22⋊Dic127D412yes—
52(C2×D127)⋊C4—24no—
53(C2×D127)⋊C4—24no—
54C4×Dic127C2312yes—
55#41×C127—22no—
56#41×C127—22no—
57C2×Q8×C127C2311yescharacteristic
58C2×D4×C127—21no—
59C2×D4×C127C2311yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62C2×C8×C127—22no—
63C2×C8×C127C2312yes—
64C4⋊C4×C127C2312yes—
65C22⋊C4×C127—21no—
66C42×C127—21no—
67C22×D127—22no—
68C22×D127—24no—
69C22×D127—22no—
70C22×D127—42no—
71C22×D127—82no—
72C22×D127—82no—
73D508—24no—
74D508—22no—
75D508—22no—
76D508—22no—
77D508—42no—
78D508C2×D414yes—
79C22⋊D127—84no—
80C2×Dic127C2×D414yes—
81C4×D127—84no—
82C127⋊C8—22no—
83C23×C127—21no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—21no—
86Q8×C127—21no—
87D4×C127—21no—
88D4×C127—21no—
89D4×C127—42no—
90C2×C4×C127—22no—
91C2×C4×C127—21no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127#4112yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C22×C127—22no—
99C22×C127—41no—
100C22×C127—42no—
101C22×C127#12112yes—
102C22×C127C22×D411yescharacteristic
103C2×D127—24no—
104C2×D127—44no—
105C2×D127—42no—
106C2×D127—44no—
107C2×D127—84no—
108C2×D127—84no—
109C4×C127—41no—
110C4×C127—42no—
111C4×C127#12112yes—
112C4×C127C22×D411yescharacteristic
113C4×C127C2×#4111yescharacteristic
114Dic127—24no—
115C2×C127—22no—
116C2×C127—81no—
117C2×C127C2×#12111yescharacteristic
118C2×C127#57612yes—
119D127—44no—
120D127—82no—
121#3041—1271nomaximal, Sylow
122C127#304111yescharacteristic, complemented, Sylow
123C2×AGL(1,ℤ/8)—1271no—
124C22×D8—1271no—
125#535—1271no—
126#518—1271no—
127#463—1274no—
128#459—1274no—
129#424—1271no—
130#418—1271no—
131#416D12711yescharacteristic, complemented
132C2×#41C2×D12711yescharacteristic
133C22×D4—1271no—
134C22×D4—1271no—
135AGL(1,ℤ/8)—2544no—
136C2×SD16—1272no—
137C2×D8—1272no—
138C2×D8—1274no—
139C2×D8—2544no—
140C2×M16—1271no—
141C22×C8C2×D12711yescharacteristic
142#127—1272no—
143#124—1272no—
144#121—2542no—
145#120—2542no—
146C4×D4—1274no—
147C4×D4—2542no—
148#117—1271no—
149#117C2×D12711yescharacteristic
150C2×C22⋊C4—2542no—
151C4⋊C8—1274no—
152#103C2×D12712yescomplemented
153#102—1274no—
154#102C2×D12712yescomplemented
155C24—2541no—
156C24—2541no—
157#41—22no—
158#41—22no—
159C2×Q8C22×D12711yescharacteristic
160C2×D4—21no—
161C2×D4—1274no—
162C2×D4—1272no—
163C2×D4—1272no—
164C2×D4—2544no—
165C2×D4—2541no—
166C2×D4—2542no—
167C2×D4—2541no—
168C2×D4—2542no—
169C2×D4—2542no—
170C2×D4C22×D12711yescharacteristic
171C22×C4—21no—
172C22×C4—2544no—
173C22×C4C22×D12711yescharacteristic
174SD16—2544no—
175D8—2548no—
176D8—2544no—
177M16—2542no—
178C2×C8—22no—
179C2×C8—1272no—
180C2×C8C22×D12712yes—
181C4⋊C4—1272no—
182C4⋊C4C22×D12712yes—
183C22⋊C4—21no—
184C22⋊C4—1272no—
185C22⋊C4—2544no—
186C22⋊C4—2544no—
187C42—21no—
188C42—1272no—
189C23—21no—
190C23—10162no—
191C23—10162no—
192C23—5082no—
193C23—2544no—
194C23—2542no—
195C23—2542no—
196C23D4×D12711yescharacteristic
197Q8—21no—
198Q8—21no—
199D4—21no—
200D4—21no—
201D4—42no—
202D4—1274no—
203D4—10164no—
204D4—5082no—
205D4—2542no—
206D4—2542no—
207D4—2542no—
208D4—2544no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—1274no—
213C2×C4—10164no—
214C2×C4D4×D12712yes—
215C2×C4D4×D12711yescharacteristic
216C2×C4#1452324812yes—
217C2×C4C23×D12711yescharacteristic
218C8—22no—
219C8—2542no—
220C22—22no—
221C22—41no—
222C22—42no—
223C22—10164no—
224C22—10164no—
225C22—5082no—
226C22—5084no—
227C22—5084no—
228C22—2544no—
229C22#1686118212yes—
230C22C2×D4×D12711yescharacteristic
231C4—41no—
232C4—42no—
233C4—2544no—
234C4#1686118212yes—
235C4C2×D4×D12711yescharacteristic
236C4C2×#1452324811yescharacteristic
237C2—22no—
238C2—81no—
239C2—10162no—
240C2—5084no—
241C2C2×#1686118211yescharacteristic
242C2#2255067812yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
hypercenter#416D127
hypocenterC127#3041
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C22×D1016C2semidirect
3. #22550440C2semidirect
4. #22550350C2semidirect
5. #22550094C2semidirect
6. #22550070C2semidirect
7. #22549902C2semidirect
8. #22549880C2semidirect
9. #416×C127C2semidirect
13. C2×#14523235C22semidirect
16. C2×#14523233C22semidirect
18. #16861205C22semidirect
19. #16861198C22semidirect
23. #16861171C22semidirect
26. Dic127⋊C8C22semidirect
27. #16861129C22semidirect
31. #103×C127C22semidirect
32. #102×C127C22semidirect
40. C2×D508D4semidirect
122. C127#3041semidirect
131. #416D127semidirect
152. #103C2×D127semidirect
154. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C22×D10162
#225504402
#225503502
#225500942
#225500702
#225499022
#225498802
#416×C1272
#3041127