Finite Group Databasebeta

← #30648018

Subgroups — #30648018

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×#16861256C211yesmaximal, characteristic, complemented
2C2×#16861237C211yesmaximal, characteristic, complemented
3C22×#14523237C211yesmaximal, characteristic, complemented
4#22550440C211yesmaximal, characteristic, complemented
5#22550023C214yesmaximal, complemented
6#22550017C214yesmaximal, complemented
7#22549886C211yesmaximal, characteristic, complemented
8#22549873C211yesmaximal, characteristic, complemented
9#408×C127C211yesmaximal, characteristic, complemented
10#16861256—24no—
11#16861256—24no—
12C2×#14523248C2211yescharacteristic
13C2×#14523248C2212yescomplemented
14C2×#14523247—21no—
15C2×#14523247—22no—
16C22×Dic254C2211yescharacteristic
17C22×Dic254C2212yescomplemented
18C2×Q8×D127—21no—
19C22×D508C2211yescharacteristic
20#16861237—24no—
21C2×#14523237—24no—
22C2×#14523237C2212yes—
23C2×#14523237C2214yes—
24C2×D1016C2212yes—
25#16861205—21no—
26#16861198—21no—
27#16861183C2214yes—
28#16861177—22no—
29#16861168C2211yescharacteristic
30#16861134C2212yescomplemented
31#16861134C2214yescomplemented
32#16861131C2212yescomplemented
33C2×#41×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic
35C22×C8×C127C2211yescharacteristic
36C22⋊C8×C127C2214yescomplemented
37C23×D127—21no—
38#14523248—24no—
39#14523248—24no—
40#14523248—22no—
41#14523248D414yescomplemented
42#14523247—216no—
43#14523247—28no—
44C2×Dic254—24no—
45C2×Dic254—24no—
46C2×Dic254—21no—
47C2×Dic254—22no—
48C2×Dic254—22no—
49C2×Dic254D414yescomplemented
50C2×Dic254C2312yes—
51C2×Dic254C2311yescharacteristic
52Q8×D127—28no—
53C2×D508—22no—
54C2×D508—22no—
55C2×D508—21no—
56C2×D508C2312yes—
57C2×D508C2311yescharacteristic
58C2×C22⋊D127—21no—
59C2×C22⋊D127—22no—
60C2×C22⋊D127—24no—
61C22×Dic127—21no—
62C22×Dic127—22no—
63C2×C4×D127—21no—
64C2×C4×D127—22no—
65#14523237—28no—
66#14523237—24no—
67D1016—24no—
68C4⋊Dic127C2312yes—
69C22⋊Dic127—21no—
70(C2×D127)⋊C4—24no—
71C4×Dic127—21no—
72#41×C127—22no—
73#41×C127—22no—
74C2×Q8×C127D411yescharacteristic
75C2×D4×C127D411yescharacteristic
76C2×D4×C127D412yes—
77C22×C4×C127C2312yes—
78C22×C4×C127C2311yescharacteristic
79M16×C127—22no—
80C2×C8×C127—22no—
81C2×C8×C127C2312yes—
82C2×C8×C127C2312yes—
83C22×D127—21no—
84C22×D127—22no—
85C22×D127—42no—
86C22×D127—82no—
87Dic254—28no—
88Dic254—24no—
89Dic254—21no—
90Dic254—22no—
91Dic254C2×D412yes—
92Dic254C2×D414yes—
93D508—21no—
94D508—22no—
95D508—22no—
96D508—42no—
97D508C2×D412yes—
98C22⋊D127—28no—
99C22⋊D127—24no—
100C22⋊D127—84no—
101C2×Dic127—28no—
102C2×Dic127—24no—
103C2×Dic127—21no—
104C2×Dic127—22no—
105C2×Dic127—22no—
106C4×D127—28no—
107C4×D127—24no—
108C23×C127C2×D412yes—
109C23×C127C2×D411yescharacteristic
110Q8×C127—22no—
111D4×C127—24no—
112D4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2411yescharacteristic
120C8×C127—22no—
121C8×C127—22no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—24no—
125C22×C127#12012yes—
126C22×C127C22×D411yescharacteristic
127C2×D127—22no—
128C2×D127—42no—
129C2×D127—42no—
130C2×D127—44no—
131C2×D127—84no—
132C4×C127—22no—
133C4×C127#12012yes—
134C4×C127C22×D411yescharacteristic
135C4×C127C22×D411yescharacteristic
136Dic127—24no—
137Dic127—22no—
138Dic127—42no—
139C2×C127—22no—
140C2×C127—42no—
141C2×C127C2×#12011yescharacteristic
142C2×C127#57712yes—
143D127—42no—
144D127—82no—
145#3010—1271nomaximal, Sylow
146C127#301011yescharacteristic, complemented, Sylow
147C2×2−1+4—1271no—
148C2×AGL(1,ℤ/8)—1271no—
149C22×SD16—1271no—
150#535—1271no—
151#449—1274no—
152#448—1274no—
153#419—1271no—
154#416—1271no—
155#408D12711yescharacteristic, complemented
1562−1+4—2544no—
1572−1+4—2544no—
158C2×#41—1271no—
159C2×#41—1272no—
160C2×#41—2542no—
161C2×#41—2541no—
162C2×#41C2×D12711yescharacteristic
163C22×Q8—1271no—
164C22×Q8—1272no—
165C22×Q8—2541no—
166C22×D4—1271no—
167AGL(1,ℤ/8)—2544no—
168C2×SD16—1274no—
169C2×SD16—1272no—
170C2×SD16—2544no—
171C2×D8—1272no—
172C2×M16C2×D12711yescharacteristic
173C22×C8C2×D12711yescharacteristic
174#127—2541no—
175#124—2541no—
176#121—1274no—
177#120—2542no—
178#117—1271no—
179#103—1272no—
180#103—1274no—
181#102—1272no—
182C22⋊C8C2×D12714yescomplemented
183C24—2541no—
184#41—22no—
185#41—22no—
186#41—1274no—
187#41—2544no—
188#41—2544no—
189#41—25416no—
190#41—2548no—
191#41—2542no—
192C2×Q8—1274no—
193C2×Q8—1272no—
194C2×Q8—1271no—
195C2×Q8—2544no—
196C2×Q8—2544no—
197C2×Q8—2548no—
198C2×Q8—2542no—
199C2×Q8—2542no—
200C2×Q8—2541no—
201C2×Q8D50811yescharacteristic
202C2×D4—1271no—
203C2×D4—1272no—
204C2×D4—2544no—
205C2×D4—2542no—
206C2×D4—2541no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4—2541no—
210C2×D4D50812yes—
211C2×D4D50811yescharacteristic
212C22×C4—2542no—
213C22×C4—2541no—
214C22×C4—2542no—
215C22×C4—2541no—
216C22×C4C22×D12712yes—
217C22×C4C22×D12711yescharacteristic
218SD16—2548no—
219SD16—2544no—
220D8—2544no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D12712yes—
224C2×C8C22×D12712yes—
225C4⋊C4—1272no—
226C22⋊C4—2541no—
227C22⋊C4—2544no—
228C42—2541no—
229C23—10162no—
230C23—5082no—
231C23—2542no—
232C23—2541no—
233C23C2×D50812yes—
234C23C2×D50811yescharacteristic
235Q8—22no—
236Q8—1274no—
237Q8—1272no—
238Q8—2548no—
239Q8—2542no—
240Q8—2541no—
241Q8—2544no—
242D4—24no—
243D4—22no—
244D4—1272no—
245D4—10164no—
246D4—5082no—
247D4—2548no—
248D4—2544no—
249D4—2542no—
250D4—2542no—
251D4—2541no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—2548no—
255C2×C4—2542no—
256C2×C4—2548no—
257C2×C4—2542no—
258C2×C4—2541no—
259C2×C4—2544no—
260C2×C4—2544no—
261C2×C4D4×D12712yes—
262C2×C4D4×D12712yes—
263C2×C4C2×D50812yes—
264C2×C4C2×D50811yescharacteristic
265C2×C4C23×D12711yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—10164no—
272C22—5082no—
273C22—5084no—
274C22—5082no—
275C22—2542no—
276C22#1686117912yes—
277C22C22×D50811yescharacteristic
278C4—22no—
279C4—5082no—
280C4—2544no—
281C4—2542no—
282C4#1686117912yes—
283C4C2×D4×D12711yescharacteristic
284C4C2×D4×D12711yescharacteristic
285C2—22no—
286C2—42no—
287C2—10162no—
288C2—5082no—
289C2C2×#1686117911yescharacteristic
290C2#2255068812yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#408×C127C2
socleC22×C127C22×D4
hypercenter#408D127
hypocenterC127#3010
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#16861256C2semidirect
2. C2×#16861237C2semidirect
3. C22×#14523237C2semidirect
4. #22550440C2semidirect
5. #22550023C2semidirect
6. #22550017C2semidirect
7. #22549886C2semidirect
8. #22549873C2semidirect
9. #408×C127C2semidirect
13. C2×#14523248C22semidirect
17. C22×Dic254C22semidirect
30. #16861134C22semidirect
31. #16861134C22semidirect
32. #16861131C22semidirect
36. C22⋊C8×C127C22semidirect
41. #14523248D4semidirect
49. C2×Dic254D4semidirect
146. C127#3010semidirect
155. #408D127semidirect
182. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
C2×#168612372
C22×#145232372
#225504402
#225500232
#225500172
#225498862
#225498732
#408×C1272
#3010127