Finite Group Databasebeta

← #30648102

Subgroups — #30648102

Order 16256.

736 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 161 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×#16861234C212yesmaximal, complemented
2#22550440C211yesmaximal, characteristic, complemented
3#22550049C214yesmaximal, complemented
4#22550044C214yesmaximal, complemented
5#22549893C212yesmaximal, complemented
6#22549860C211yesmaximal, characteristic, complemented
7#585×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic, complemented
9#16861234—24no—
10#16861234C2218yescomplemented
11C2×#14523235—22no—
12C2×#14523233—22no—
13C2×#14523232C2212yescomplemented
14#16861205C2212yescomplemented
15#16861198C2212yescomplemented
16#16861183—22no—
17#16861177—22no—
18#16861168C2211yescharacteristic, complemented
19#16861136C2218yescomplemented
20#16861128C2212yescomplemented
21#16861126C2212yescomplemented
222−1+4×C127—22no—
232−1+4×C127C2212yescomplemented
24H(5,𝔽2)×C127—22no—
25H(5,𝔽2)×C127—22no—
26H(5,𝔽2)×C127C2212yescomplemented
27C2×#41×C127—24no—
28C2×#41×C127—22no—
29C2×#41×C127C2212yes—
30C2×#41×C127C2211yescharacteristic
31C23×D127—21no—
32C2×D508—22no—
33C2×D508—22no—
34C2×D508—21no—
35C2×D508C2312yes—
36C2×C22⋊D127—24no—
37#14523235—28no—
38#14523233—28no—
39#14523232—22no—
40#14523232C2314yes—
41C2×C127⋊C8—22no—
42C4⋊Dic127—21no—
43C22⋊Dic127—21no—
44(C2×D127)⋊C4—24no—
45C4×Dic127C2312yes—
46#41×C127—24no—
47#41×C127—28no—
48#41×C127—24no—
49#41×C127—24no—
50#41×C127—24no—
51#41×C127—22no—
52#41×C127—22no—
53#41×C127—24no—
54#41×C127—22no—
55#41×C127—22no—
56#41×C127D414yescomplemented
57#41×C127C2314yes—
58C2×Q8×C127—22no—
59C2×Q8×C127—24no—
60C2×Q8×C127D412yescomplemented
61C2×Q8×C127C2311yescharacteristic
62C2×D4×C127—28no—
63C2×D4×C127—22no—
64C2×D4×C127—24no—
65C2×D4×C127—22no—
66C2×D4×C127—24no—
67C2×D4×C127D412yes—
68C2×D4×C127D412yescomplemented
69C2×D4×C127C2311yescharacteristic
70C22×C4×C127—24no—
71C22×C4×C127—22no—
72C22×C4×C127D412yes—
73C22×C4×C127C2311yescharacteristic
74C22×D127—22no—
75C22×D127—42no—
76C22×D127—82no—
77D508—24no—
78D508—41no—
79D508—41no—
80C22⋊D127—84no—
81C2×Dic127—22no—
82C127⋊C8—24no—
83C23×C127—24no—
84C23×C127—22no—
85C23×C127C2×D412yes—
86C23×C127C2×D411yescharacteristic
87Q8×C127—24no—
88Q8×C127—22no—
89Q8×C127—22no—
90Q8×C127C2×D414yes—
91D4×C127—28no—
92D4×C127—24no—
93D4×C127—24no—
94D4×C127—24no—
95D4×C127—24no—
96D4×C127—22no—
97D4×C127—22no—
98D4×C127C2×D414yes—
99C2×C4×C127—24no—
100C2×C4×C127—28no—
101C2×C4×C127—22no—
102C2×C4×C127—24no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—24no—
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D412yes—
109C2×C4×C127C2×D412yes—
110C2×C4×C127C2411yescharacteristic
111C22×C127—22no—
112C22×C127—24no—
113C22×C127—41no—
114C22×C127—42no—
115C22×C127—82no—
116C22×C127—84no—
117C22×C127#12012yes—
118C22×C127C22×D411yescharacteristic
119C2×D127—44no—
120C2×D127—42no—
121C2×D127—84no—
122C4×C127—24no—
123C4×C127—22no—
124C4×C127#12011yescharacteristic
125C4×C127#12011yescharacteristic
126C4×C127C22×D412yes—
127Dic127—42no—
128C2×C127—21no—
129C2×C127—22no—
130C2×C127—42no—
131C2×C127—44no—
132C2×C127C2×#12011yescharacteristic
133D127—82no—
134#3021—1271nomaximal, Sylow
135C127#302111yescharacteristic, complemented, Sylow
136#585D12711yescharacteristic, complemented
137C2×AGL(1,ℤ/8)—1272no—
138#535—1271no—
139#454—1274no—
140#453—1274no—
141#421—1272no—
142#413—1271no—
1432−1+4—22no—
1442−1+4C2×D12712yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C2×D12712yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C2×D12712yes—
151C2×#41C2×D12711yescharacteristic
152C22×D4—1271no—
153AGL(1,ℤ/8)—1278no—
154AGL(1,ℤ/8)—2544no—
155C2×SD16—2542no—
156C2×D8—2542no—
157C2×M16—1272no—
158#127—1272no—
159#124—1272no—
160#121—2542no—
161#120—2542no—
162#117—1271no—
163#104—1278no—
164#101—1272no—
165#100—1272no—
166C24—2541no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41C22⋊D12714yescomplemented
178#41C22×D12714yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D12712yescomplemented
182C2×Q8C22×D12711yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—1272no—
189C2×D4—2544no—
190C2×D4—2541no—
191C2×D4—2542no—
192C2×D4—2542no—
193C2×D4C22⋊D12712yes—
194C2×D4C22⋊D12712yescomplemented
195C2×D4C22×D12711yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D12712yes—
199C22×C4C22×D12711yescharacteristic
200SD16—2548no—
201D8—2548no—
202M16—1274no—
203M16—2542no—
204C2×C8—2542no—
205C4⋊C4—2541no—
206C22⋊C4—2541no—
207C22⋊C4—2544no—
208C42—1272no—
209C23—24no—
210C23—22no—
211C23—10162no—
212C23—5082no—
213C23—2542no—
214C23C2×C22⋊D12712yes—
215C23C2×C22⋊D12711yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D12714yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—10164no—
228D4—5081no—
229D4—5081no—
230D4—2544no—
231D4C2×C22⋊D12714yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—2542no—
240C2×C4C2×C22⋊D12712yes—
241C2×C4C2×C22⋊D12712yes—
242C2×C4C2×C22⋊D12712yes—
243C2×C4C2×C22⋊D12711yescharacteristic
244C2×C4C23×D12711yescharacteristic
245C8—2544no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—10164no—
253C22—5084no—
254C22—5082no—
255C22#1686117812yes—
256C22C22×C22⋊D12711yescharacteristic
257C4—24no—
258C4—22no—
259C4—5082no—
260C4#1686117811yescharacteristic
261C4#1686117811yescharacteristic
262C4C22×C22⋊D12712yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—10162no—
268C2C2×#1686117811yescharacteristic
subgroupG/H
centerC2C2×#16861178
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#585×C127C2
socleC2×C127C2×#120
hypercenter#585D127
hypocenterC127#3021
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. #22550440C2semidirect
3. #22550049C2semidirect
4. #22550044C2semidirect
5. #22549893C2semidirect
6. #22549860C2semidirect
7. #585×C127C2semidirect
8. C22×D508C22semidirect
10. #16861234C22semidirect
13. C2×#14523232C22semidirect
14. #16861205C22semidirect
15. #16861198C22semidirect
18. #16861168C22semidirect
19. #16861136C22semidirect
20. #16861128C22semidirect
21. #16861126C22semidirect
23. 2−1+4×C127C22semidirect
26. H(5,𝔽2)×C127C22semidirect
56. #41×C127D4semidirect
60. C2×Q8×C127D4semidirect
68. C2×D4×C127D4semidirect
135. C127#3021semidirect
136. #585D127semidirect
144. 2−1+4C2×D127semidirect
147. H(5,𝔽2)C2×D127semidirect
177. #41C22⋊D127semidirect
181. C2×Q8C22⋊D127semidirect
194. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
#225504402
#225500492
#225500442
#225498932
#225498602
#585×C1272
#3021127