Finite Group Databasebeta

← #30648012

Subgroups — #30648012

Order 16256.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861257C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3#22550443C211yesmaximal, characteristic, complemented
4#22550020C214yesmaximal, complemented
5#22550013C214yesmaximal, complemented
6#22549885C211yesmaximal, characteristic, complemented
7#22549873C211yesmaximal, characteristic, complemented
8#22549842C211yesmaximal, characteristic, complemented
9C22×SD16×C127C211yesmaximal, characteristic, complemented
10#16861257—24no—
11#16861257—24no—
12C2×#14523249—21no—
13C2×#14523249C2212yescomplemented
14C2×#14523248—22no—
15C2×#14523248C2211yescharacteristic
16C2×#14523248C2211yescharacteristic
17C2×Q8×D127—21no—
18C2×Q8×D127C2212yescomplemented
19#16861235—24no—
20C2×#14523236C2212yes—
21C2×#14523233C2212yes—
22C2×#14523232C2211yescharacteristic
23#16861205—21no—
24#16861196—21no—
25#16861180C2214yes—
26#16861178—22no—
27#16861168C2211yescharacteristic
28#16861134C2212yescomplemented
29#16861133C2214yescomplemented
30#16861131C2212yescomplemented
31(C2×D127)⋊C8C2214yescomplemented
32C22×Q8×C127C2211yescharacteristic
33C22×D4×C127C2211yescharacteristic
34C2×SD16×C127—24no—
35C2×SD16×C127C2214yes—
36C22×C8×C127C2211yescharacteristic
37#14523249—28no—
38#14523249—24no—
39#14523249—24no—
40#14523248—216no—
41#14523248—22no—
42#14523248—22no—
43#14523248—22no—
44#14523248D414yescomplemented
45C2×Dic254—22no—
46C2×Dic254D411yescharacteristic
47C2×Dic254C2311yescharacteristic
48Q8×D127—28no—
49Q8×D127—24no—
50Q8×D127—24no—
51C2×D508—22no—
52C2×D508D411yescharacteristic
53C2×D508C2311yescharacteristic
54C2×C22⋊D127—24no—
55C2×C22⋊D127—21no—
56C2×C22⋊D127D412yes—
57C2×C4×D127—22no—
58C2×C4×D127—22no—
59C2×C4×D127—21no—
60C2×C4×D127C2312yes—
61#14523236—24no—
62#14523233—24no—
63#14523232—22no—
64C2×C127⋊C8C2312yes—
65C4⋊Dic127C2312yes—
66C22⋊Dic127—24no—
67C22⋊Dic127—21no—
68C4×Dic127—21no—
69C24×C127—21no—
70C2×Q8×C127—21no—
71C2×Q8×C127—22no—
72C2×Q8×C127D414yescomplemented
73C2×Q8×C127C2312yes—
74C2×D4×C127—22no—
75C2×D4×C127—22no—
76C2×D4×C127—21no—
77C2×D4×C127C2312yes—
78C22×C4×C127—21no—
79C22×C4×C127C2311yescharacteristic
80SD16×C127—28no—
81C2×C8×C127—22no—
82C2×C8×C127C2312yes—
83C22×D127—21no—
84C22×D127C2×D412yes—
85Dic254—28no—
86Dic254—22no—
87Dic254—21no—
88Dic254C2×D412yes—
89D508—28no—
90D508—22no—
91D508—21no—
92D508C2×D412yes—
93C22⋊D127—24no—
94C22⋊D127—24no—
95C22⋊D127—84no—
96C2×Dic127—21no—
97C2×Dic127—22no—
98C2×Dic127C2×D412yes—
99C4×D127—28no—
100C4×D127—28no—
101C4×D127—24no—
102C4×D127—22no—
103C4×D127—22no—
104C127⋊C8—22no—
105C23×C127—22no—
106C23×C127—42no—
107C23×C127—82no—
108C23×C127C2×D411yescharacteristic
109Q8×C127—24no—
110Q8×C127—22no—
111Q8×C127C2×D414yes—
112D4×C127—22no—
113D4×C127—22no—
114D4×C127—42no—
115C2×C4×C127—24no—
116C2×C4×C127—22no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2411yescharacteristic
121C8×C127—22no—
122C22×C127—22no—
123C22×C127—42no—
124C22×C127—44no—
125C22×C127—84no—
126C22×C127#12012yes—
127C22×C127C22×D411yescharacteristic
128C2×D127—22no—
129C2×D127—22no—
130C2×D127—24no—
131C2×D127—42no—
132C4×C127—24no—
133C4×C127#12012yes—
134C4×C127C22×D411yescharacteristic
135C4×C127C22×D411yescharacteristic
136Dic127—22no—
137Dic127—22no—
138Dic127—42no—
139C2×C127—22no—
140C2×C127—82no—
141C2×C127C2×#12011yescharacteristic
142C2×C127#57712yes—
143D127—42no—
144D127—42no—
145#3010—1271nomaximal, Sylow
146C127#301011yescharacteristic, complemented, Sylow
147C2×2−1+4—1271no—
148C2×AGL(1,ℤ/8)—1271no—
149C22×SD16D12711yescharacteristic, complemented
150#535—1271no—
151#449—1274no—
152#448—1274no—
153#419—1271no—
154#416—1271no—
155#408—1271no—
1562−1+4—2544no—
1572−1+4—2544no—
158C2×#41—1271no—
159C2×#41—1272no—
160C2×#41—1271no—
161C2×#41—2542no—
162C2×#41—2541no—
163C22×Q8—1272no—
164C22×Q8—2541no—
165C22×Q8C2×D12711yescharacteristic
166C22×D4C2×D12711yescharacteristic
167AGL(1,ℤ/8)—2544no—
168C2×SD16—24no—
169C2×SD16—1272no—
170C2×SD16C2×D12714yes—
171C2×D8—1272no—
172C2×M16—1271no—
173C22×C8C2×D12711yescharacteristic
174#127—2541no—
175#124—2541no—
176#121—1274no—
177#120—2542no—
178#117—1271no—
179#103—1274no—
180#103—1272no—
181#102—1272no—
182C22⋊C8—1274no—
183C24—21no—
184#41—1274no—
185#41—25416no—
186#41—2544no—
187#41—2548no—
188#41—2544no—
189#41—2542no—
190#41—2542no—
191#41—2542no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—1271no—
195C2×Q8—1271no—
196C2×Q8—2542no—
197C2×Q8—2544no—
198C2×Q8—2548no—
199C2×Q8—2544no—
200C2×Q8C22⋊D12714yescomplemented
201C2×Q8C22×D12712yes—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—1272no—
206C2×D4—1271no—
207C2×D4—1271no—
208C2×D4—2542no—
209C2×D4—2544no—
210C2×D4—2541no—
211C2×D4C22×D12712yes—
212C22×C4—21no—
213C22×C4—1272no—
214C22×C4—2542no—
215C22×C4—2542no—
216C22×C4—2541no—
217C22×C4C22×D12711yescharacteristic
218SD16—28no—
219SD16—2544no—
220D8—2544no—
221M16—2542no—
222C2×C8—22no—
223C2×C8—1272no—
224C2×C8C22×D12712yes—
225C4⋊C4—1272no—
226C22⋊C4—2544no—
227C22⋊C4—2541no—
228C42—2541no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—1272no—
233C23—2541no—
234C23D4×D12711yescharacteristic
235Q8—22no—
236Q8—24no—
237Q8—1272no—
238Q8—2548no—
239Q8—2541no—
240Q8—2542no—
241Q8C2×C22⋊D12714yes—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—1272no—
246D4—10164no—
247D4—2548no—
248D4—2544no—
249D4—2544no—
250D4—2541no—
251D4—2542no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—1272no—
255C2×C4—2548no—
256C2×C4—2548no—
257C2×C4—2542no—
258C2×C4—2541no—
259C2×C4—2542no—
260C2×C4—2542no—
261C2×C4—2544no—
262C2×C4D4×D12712yes—
263C2×C4D4×D12711yescharacteristic
264C2×C4C2×C22⋊D12712yes—
265C2×C4C23×D12711yescharacteristic
266C8—22no—
267C8—2542no—
268C22—22no—
269C22—42no—
270C22—44no—
271C22—84no—
272C22—5082no—
273C22—2542no—
274C22—2542no—
275C22—2544no—
276C22#1686117712yes—
277C22C2×D4×D12711yescharacteristic
278C4—24no—
279C4—5082no—
280C4—2542no—
281C4—2542no—
282C4#1686117712yes—
283C4C2×D4×D12711yescharacteristic
284C4C22×C22⋊D12711yescharacteristic
285C2—22no—
286C2—82no—
287C2—5082no—
288C2—5082no—
289C2C2×#1686117711yescharacteristic
290C2#2255068212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC22×C127C22×D4
hypercenterC22×SD16D127
hypocenterC127#3010
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. C2×#16861235C2semidirect
3. #22550443C2semidirect
4. #22550020C2semidirect
5. #22550013C2semidirect
6. #22549885C2semidirect
7. #22549873C2semidirect
8. #22549842C2semidirect
9. C22×SD16×C127C2semidirect
13. C2×#14523249C22semidirect
18. C2×Q8×D127C22semidirect
28. #16861134C22semidirect
29. #16861133C22semidirect
30. #16861131C22semidirect
31. (C2×D127)⋊C8C22semidirect
44. #14523248D4semidirect
72. C2×Q8×C127D4semidirect
146. C127#3010semidirect
149. C22×SD16D127semidirect
200. C2×Q8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
C2×#168612352
#225504432
#225500202
#225500132
#225498852
#225498732
#225498422
C22×SD16×C1272
#3010127