Finite Group Databasebeta

← #30648090

Subgroups — #30648090

Order 16256.

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

normal subgroups only

This group has 762 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×#16861254C211yesmaximal, characteristic, complemented
2#22550676C212yesmaximal, complemented
3#22550443C211yesmaximal, characteristic, complemented
4#22550056C214yesmaximal, complemented
5#22550046C214yesmaximal, complemented
6#22549892C212yesmaximal, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8#16861254—24no—
9#16861254—22no—
10#16861254C2214yes—
11C2×#14523248C2212yes—
12C2×#14523248C2211yescharacteristic, complemented
13C2×#14523247—22no—
14C22×C22⋊D127—22no—
15C2×D4×D127—22no—
16#16861242—22no—
17#16861242C2214yescomplemented
18#16861237—22no—
19#16861237C2214yescomplemented
20#16861231—24no—
21#16861205C2212yescomplemented
22#16861196C2212yescomplemented
23#16861180—22no—
24#16861178—22no—
25#16861168C2211yescharacteristic, complemented
26#16861135C2218yescomplemented
27C22×D4×C127C2211yescharacteristic
28C2×M16×C127C2212yescomplemented
29#100×C127C2214yescomplemented
30C23×D127—22no—
31#14523248—22no—
32#14523248—22no—
33#14523248—22no—
34#14523248—22no—
35#14523248D414yescomplemented
36#14523248C2314yes—
37#14523247—28no—
38#14523247—28no—
39C2×Dic254D412yescomplemented
40C2×Dic254C2311yescharacteristic, complemented
41C2×D508D412yescomplemented
42C2×D508C2311yescharacteristic, complemented
43C2×C22⋊D127—24no—
44C2×C22⋊D127—28no—
45C2×C22⋊D127—22no—
46C2×C22⋊D127—28no—
47C2×C22⋊D127—21no—
48C2×C22⋊D127—24no—
49D4×D127—28no—
50D4×D127—28no—
51C22×Dic127—22no—
52C2×C4×D127—22no—
53C2×C4×D127—21no—
54Dic508—24no—
55#14523237—24no—
56#14523237—24no—
57D1016—24no—
58C4⋊Dic127—21no—
59C22⋊Dic127—24no—
60C22⋊Dic127—21no—
61C4×Dic127C2312yescomplemented
62C24×C127D412yes—
63C2×D4×C127—22no—
64C2×D4×C127—22no—
65C2×D4×C127D412yes—
66C2×D4×C127C2312yes—
67C22×C4×C127C2311yescharacteristic
68M16×C127—22no—
69M16×C127C2314yescomplemented
70C2×C8×C127—22no—
71C22×D127—22no—
72C22×D127—28no—
73C22×D127—21no—
74C22×D127—84no—
75Dic254—22no—
76Dic254—22no—
77Dic254C2×D414yes—
78D508—22no—
79D508—22no—
80D508C2×D414yes—
81C22⋊D127—216no—
82C22⋊D127—24no—
83C22⋊D127—24no—
84C22⋊D127—41no—
85C22⋊D127—41no—
86C22⋊D127—84no—
87C2×Dic127—28no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C4×D127—24no—
92C4×D127—24no—
93C4×D127—42no—
94C23×C127—22no—
95C23×C127—22no—
96C23×C127—82no—
97C23×C127C2×D414yes—
98C23×C127C2×D411yescharacteristic
99D4×C127—24no—
100D4×C127—24no—
101C2×C4×C127C2×D412yes—
102C2×C4×C127C2×D412yes—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2411yescharacteristic
105C8×C127—24no—
106C22×C127—24no—
107C22×C127—41no—
108C22×C127—44no—
109C22×C127—44no—
110C22×C127—82no—
111C22×C127#12011yescharacteristic
112C22×C127#12011yescharacteristic
113C22×C127C22×D411yescharacteristic
114C2×D127—24no—
115C2×D127—44no—
116C2×D127—42no—
117C2×D127—41no—
118C2×D127—88no—
119C4×C127#12012yes—
120C4×C127C22×D412yes—
121Dic127—24no—
122Dic127—42no—
123Dic127—41no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—44no—
128C2×C127C2×#12011yescharacteristic
129D127—44no—
130D127—81no—
131#3019—1271nomaximal, Sylow
132C127#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)—1271no—
134#575—1272no—
135#535—1271no—
136#455—1274no—
137#453—1274no—
138#421—1272no—
139C2×#100D12711yescharacteristic, complemented
140H(5,𝔽2)—1274no—
141H(5,𝔽2)—2544no—
142H(5,𝔽2)—2542no—
143C2×#41—1272no—
144C2×#41—1271no—
145C2×#41—2542no—
146C22×D4—2542no—
147C22×D4—2542no—
148C22×D4C2×D12711yescharacteristic
149#137—1274no—
150#137—2542no—
151AGL(1,ℤ/8)—1274no—
152AGL(1,ℤ/8)—2542no—
153#135—2544no—
154C2×M16C2×D12712yescomplemented
155#127—1272no—
156#124—1272no—
157#121—2542no—
158#120—2542no—
159#117—1271no—
160#104—1278no—
161#100C2×D12714yescomplemented
162C24—2542no—
163C24D50812yes—
164#41—1274no—
165#41—1274no—
166#41—2542no—
167#41—2542no—
168#41—2548no—
169#41—2548no—
170#41—2542no—
171#41—2542no—
172C2×Q8—1272no—
173C2×Q8—1271no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—1272no—
177C2×D4—1271no—
178C2×D4—2548no—
179C2×D4—2548no—
180C2×D4—2544no—
181C2×D4—2541no—
182C2×D4—2544no—
183C2×D4—2548no—
184C2×D4—2542no—
185C2×D4—2548no—
186C2×D4D50812yes—
187C2×D4C22×D12712yes—
188C22×C4—2541no—
189C22×C4—2542no—
190C22×C4—2542no—
191C22×C4C22×D12711yescharacteristic
192Q16—2544no—
193SD16—2544no—
194SD16—2544no—
195D8—2544no—
196M16—22no—
197M16C22×D12714yescomplemented
198C2×C8—22no—
199C4⋊C4—2541no—
200C22⋊C4—2541no—
201C22⋊C4—2544no—
202C42—1272no—
203C23—22no—
204C23—22no—
205C23—82no—
206C23—10164no—
207C23—2548no—
208C23—2541no—
209C23—2542no—
210C23C2×D50814yes—
211C23C2×D50811yescharacteristic
212Q8—1274no—
213Q8—2542no—
214Q8—2542no—
215D4—24no—
216D4—24no—
217D4—1274no—
218D4—10164no—
219D4—5081no—
220D4—5081no—
221D4—25416no—
222D4—2544no—
223D4—2544no—
224D4—2542no—
225D4—2542no—
226C2×C4—5082no—
227C2×C4—2542no—
228C2×C4—2541no—
229C2×C4—2548no—
230C2×C4—2542no—
231C2×C4—2544no—
232C2×C4—2544no—
233C2×C4D4×D12712yes—
234C2×C4D4×D12712yes—
235C2×C4C2×D50811yescharacteristic
236C2×C4C23×D12711yescharacteristic
237C8—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—82no—
243C22—10168no—
244C22—5082no—
245C22—5084no—
246C22—5081no—
247C22—2544no—
248C22#1686117911yescharacteristic
249C22#1686117911yescharacteristic
250C22C22×D50811yescharacteristic
251C4—5082no—
252C4—5081no—
253C4—2544no—
254C4#1686117912yes—
255C4C2×D4×D12712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—10161no—
261C2—5084no—
262C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#100×C127C2
socleC2×C127C2×#120
hypercenterC2×#100D127
hypocenterC127#3019
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#16861254C2semidirect
2. #22550676C2semidirect
3. #22550443C2semidirect
4. #22550056C2semidirect
5. #22550046C2semidirect
6. #22549892C2semidirect
7. C2×#100×C127C2semidirect
12. C2×#14523248C22semidirect
17. #16861242C22semidirect
19. #16861237C22semidirect
21. #16861205C22semidirect
22. #16861196C22semidirect
25. #16861168C22semidirect
26. #16861135C22semidirect
28. C2×M16×C127C22semidirect
29. #100×C127C22semidirect
35. #14523248D4semidirect
39. C2×Dic254D4semidirect
40. C2×Dic254C23semidirect
41. C2×D508D4semidirect
42. C2×D508C23semidirect
61. C4×Dic127C23semidirect
69. M16×C127C23semidirect
132. C127#3019semidirect
139. C2×#100D127semidirect
154. C2×M16C2×D127semidirect
161. #100C2×D127semidirect
197. M16C22×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
#225506762
#225504432
#225500562
#225500462
#225498922
C2×#100×C1272
#3019127