Finite Group Databasebeta

← #30639742

Subgroups — #30639742

Order 16256.

478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#22550440C211yesmaximal, characteristic, complemented
2C2×#16861205C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C22⋊D127—22no—
10C2×#14523232C2211yescharacteristic
11#16861205—22no—
12#16861205—22no—
13#16861205—21no—
14#16861198—21no—
15#16861183—22no—
16#16861177—22no—
17#16861168C2211yescharacteristic
18C2×C4×Dic127C2211yescharacteristic
19#16861136—22no—
20#16861130—22no—
21#16861125—22no—
22C2×#41×C127C2211yescharacteristic
23C22×D4×C127C2211yescharacteristic
24AGL(1,ℤ/8)×C127—22no—
25AGL(1,ℤ/8)×C127—22no—
26C2×SD16×C127—21no—
27C2×D8×C127—21no—
28C2×M16×C127C2211yescharacteristic
29C23×D127D412yes—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×C22⋊D127—28no—
37C2×C22⋊D127—24no—
38C2×C22⋊D127—48no—
39C22×Dic127—21no—
40#14523232—21no—
41#14523232—21no—
42C2×C127⋊C8—21no—
43C4⋊Dic127D412yescomplemented
44C22⋊Dic127—21no—
45(C2×D127)⋊C4—24no—
46C4×Dic127—21no—
47C4×Dic127—21no—
48C4×Dic127D412yescomplemented
49C24×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127—21no—
53#41×C127—21no—
54C2×Q8×C127D411yescharacteristic, complemented
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127D411yescharacteristic, complemented
62C2×D4×C127D412yescomplemented
63C22×C4×C127—21no—
64C22×C4×C127C2311yescharacteristic
65SD16×C127—24no—
66D8×C127—24no—
67M16×C127—21no—
68M16×C127D412yescomplemented
69C2×C8×C127—21no—
70C22×D127—22no—
71C22×D127—22no—
72C22×D127—22no—
73C22×D127—44no—
74D508—41no—
75D508—41no—
76D508—41no—
77D508—41no—
78C22⋊D127—416no—
79C22⋊D127—84no—
80C2×Dic127—24no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C127⋊C8—41no—
84C23×C127—21no—
85C23×C127—22no—
86C23×C127—44no—
87C23×C127—41no—
88C23×C127—41no—
89C23×C127C2×D411yescharacteristic
90Q8×C127—21no—
91Q8×C127—21no—
92D4×C127—21no—
93D4×C127—22no—
94D4×C127—22no—
95D4×C127—21no—
96D4×C127—41no—
97D4×C127—41no—
98D4×C127—41no—
99D4×C127—41no—
100C2×C4×C127—21no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127#4111yescharacteristic
109C8×C127—22no—
110C22×C127—22no—
111C22×C127—44no—
112C22×C127—41no—
113C22×C127—42no—
114C22×C127—42no—
115C22×C127—82no—
116C22×C127—82no—
117C22×C127#12011yescharacteristic
118C22×C127#12011yescharacteristic
119C22×C127#12111yescharacteristic
120C2×D127—44no—
121C2×D127—42no—
122C2×D127—44no—
123C2×D127—82no—
124C4×C127—41no—
125C4×C127#12011yescharacteristic
126C4×C127#12111yescharacteristic
127C4×C127#12111yescharacteristic
128C4×C127#12711yescharacteristic
129Dic127—24no—
130Dic127—42no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127—81no—
134C2×C127—82no—
135C2×C127#39211yescharacteristic
136C2×C127#45312yes—
137D127—82no—
138#2002—1271nomaximal, Sylow
139C127#200211yescharacteristic, complemented, Sylow
140C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
141#535—1271no—
142C2×#127—1271no—
143C2×#104—1271no—
144#418—1271no—
145C2×#100—1271no—
146#339—1271no—
147C2×#41C2×D12711yescharacteristic
148C22×D4—1271no—
149C22×D4—2542no—
150C22×D4C2×D12711yescharacteristic
151AGL(1,ℤ/8)—22no—
152AGL(1,ℤ/8)—22no—
153C2×SD16—21no—
154C2×D8—21no—
155C2×M16—1271no—
156C2×M16C2×D12711yescharacteristic
157#127—2542no—
158#127—2542no—
159#127—2541no—
160#124—2541no—
161#121—2542no—
162#120—2542no—
163#117—1271no—
164C2×C42—1271no—
165#104—2542no—
166#102—2542no—
167#100—2542no—
168C24—21no—
169C24—1272no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174C2×Q8C22⋊D12711yescharacteristic, complemented
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—5088no—
182C2×D4—2544no—
183C2×D4—2548no—
184C2×D4—2541no—
185C2×D4—2541no—
186C2×D4—2541no—
187C2×D4—2541no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4C22⋊D12711yescharacteristic, complemented
191C2×D4C22⋊D12712yescomplemented
192C22×C4—21no—
193C22×C4—2541no—
194C22×C4C22×D12711yescharacteristic
195SD16—24no—
196D8—24no—
197M16—21no—
198M16—2541no—
199M16—2541no—
200M16C22⋊D12712yescomplemented
201C2×C8—21no—
202C2×C8—2541no—
203C4⋊C4—1272no—
204C22⋊C4—2544no—
205C22⋊C4—2541no—
206C42—1272no—
207C42—2541no—
208C42—2541no—
209C23—22no—
210C23—21no—
211C23—44no—
212C23—41no—
213C23—41no—
214C23—5084no—
215C23—2542no—
216C23—2542no—
217C23—2542no—
218C23D4×D12711yescharacteristic
219Q8—21no—
220Q8—21no—
221D4—21no—
222D4—22no—
223D4—22no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—10164no—
230D4—50816no—
231D4—5081no—
232D4—5081no—
233D4—5081no—
234D4—5081no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—2542no—
239C2×C4—2544no—
240C2×C4—2542no—
241C2×C4D4×D12711yescharacteristic
242C2×C4D4×D12711yescharacteristic
243C2×C4C2×C22⋊D12711yescharacteristic
244C2×C4C2×C22⋊D12711yescharacteristic
245C2×C4C2×C22⋊D12711yescharacteristic
246C2×C4#1452324711yescharacteristic
247C8—22no—
248C8—5081no—
249C22—22no—
250C22—44no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—82no—
255C22—82no—
256C22—10162no—
257C22—5084no—
258C22—5082no—
259C22—5084no—
260C22#1686117711yescharacteristic
261C22#1686117711yescharacteristic
262C22#1686118011yescharacteristic
263C4—41no—
264C4—5082no—
265C4—2544no—
266C4#1686117811yescharacteristic
267C4#1686118111yescharacteristic
268C4#1686118111yescharacteristic
269C4#1686120511yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—81no—
274C2—10162no—
275C2#2254978411yescharacteristic
276C2#2255004212yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#120
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#2002
LCSG ▷ C22×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. #22550440C2semidirect
2. C2×#16861205C2semidirect
3. C2×#16861136C2semidirect
4. #22549881C2semidirect
5. C2×#16861125C2semidirect
6. #22549637C2semidirect
7. C2×AGL(1,ℤ/8)×C127C2semidirect
43. C4⋊Dic127D4semidirect
48. C4×Dic127D4semidirect
54. C2×Q8×C127D4semidirect
61. C2×D4×C127D4semidirect
62. C2×D4×C127D4semidirect
68. M16×C127D4semidirect
139. C127#2002semidirect
140. C2×AGL(1,ℤ/8)D127semidirect
174. C2×Q8C22⋊D127semidirect
190. C2×D4C22⋊D127semidirect
191. C2×D4C22⋊D127semidirect
200. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504402
C2×#168612052
C2×#168611362
#225498812
C2×#168611252
#225496372
C2×AGL(1,ℤ/8)×C1272
#2002127