Finite Group Databasebeta

← #30647992

Subgroups — #30647992

Order 16256.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#14523248C211yesmaximal, characteristic
2C2×#16861242C212yesmaximal, complemented
3C2×#16861188C211yesmaximal, characteristic, complemented
4#22550040C214yesmaximal, complemented
5#22550034C212yesmaximal, complemented
6#22550029C212yesmaximal, complemented
7#22549878C212yesmaximal, complemented
8#407×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—28no—
10C2×#14523248—21no—
11C2×#14523248—21no—
12C2×#14523248C2212yes—
13C2×#14523248C2212yes—
14C22×Dic254C2211yescharacteristic
15C22×Dic254C2211yescharacteristic, complemented
16C22×D508C2211yescharacteristic
17C22×C22⋊D127—21no—
18C22×C4×D127—21no—
19#16861242—28no—
20C2×Dic508C2214yescomplemented
21C2×#14523237C2214yescomplemented
22#16861188—21no—
23#16861188—21no—
24#16861188C2212yes—
25#16861188C2212yescomplemented
26C2×C4⋊Dic127C2211yescharacteristic, complemented
27C2×Dic127⋊C4—21no—
28C2×C22⋊Dic127—21no—
29#16861134C2214yescomplemented
30#16861131C2214yescomplemented
31C23×C4×C127C2211yescharacteristic
32C2×M16×C127C2212yescomplemented
33C22⋊C8×C127C2212yes—
34C22⋊C8×C127C2212yescomplemented
35C23×D127—21no—
36#14523248—28no—
37#14523248—28no—
38#14523248—24no—
39#14523248—24no—
40#14523248—24no—
41#14523248D418yes—
42C2×Dic254—24no—
43C2×Dic254—21no—
44C2×Dic254—21no—
45C2×Dic254—24no—
46C2×Dic254C2312yes—
47C2×Dic254C2312yes—
48C2×D508—24no—
49C2×D508—21no—
50C2×D508C2312yes—
51C2×C22⋊D127—28no—
52C2×C22⋊D127—22no—
53C2×C22⋊D127—22no—
54C22×Dic127—21no—
55C22×Dic127—21no—
56C22×Dic127—21no—
57C2×C4×D127—28no—
58C2×C4×D127—22no—
59C2×C4×D127—22no—
60Dic508—28no—
61#14523237—28no—
62C4⋊Dic127—21no—
63C4⋊Dic127C2312yes—
64Dic127⋊C4—22no—
65Dic127⋊C4—22no—
66C22⋊Dic127—22no—
67C22⋊Dic127—22no—
68C24×C127D411yescharacteristic
69C22×C4×C127—24no—
70C22×C4×C127D411yescharacteristic
71C22×C4×C127D411yescharacteristic
72C22×C4×C127D411yescharacteristic
73C22×C4×C127C2311yescharacteristic
74C22×C4×C127C2311yescharacteristic
75C22×C4×C127C2311yescharacteristic
76M16×C127—24no—
77C2×C8×C127C2314yes—
78C22×D127—24no—
79C22×D127—22no—
80C22×D127—42no—
81C22×D127—42no—
82Dic254—24no—
83Dic254—22no—
84Dic254—24no—
85Dic254—42no—
86Dic254C2×D414yes—
87D508—24no—
88D508—22no—
89D508C2×D414yes—
90C22⋊D127—28no—
91C22⋊D127—28no—
92C2×Dic127—24no—
93C2×Dic127—22no—
94C2×Dic127—22no—
95C2×Dic127—22no—
96C2×Dic127—42no—
97C2×Dic127—42no—
98C4×D127—28no—
99C4×D127—28no—
100C23×C127—22no—
101C23×C127—22no—
102C23×C127—41no—
103C23×C127C2×D411yescharacteristic
104C23×C127C2×D411yescharacteristic
105C23×C127C2×D411yescharacteristic
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127C2×D414yes—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2411yescharacteristic
116C8×C127—24no—
117C22×C127—21no—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—21no—
121C22×C127—21no—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—44no—
125C22×C127#12012yes—
126C22×C127#13712yescomplemented
127C22×C127C22×D411yescharacteristic
128C2×D127—24no—
129C2×D127—48no—
130C2×D127—42no—
131C2×D127—42no—
132C4×C127—21no—
133C4×C127—21no—
134C4×C127#12012yes—
135C4×C127C22×D412yes—
136Dic127—24no—
137Dic127—42no—
138Dic127—42no—
139C2×C127—22no—
140C2×C127—22no—
141C2×C127—41no—
142C2×C127C2×#12011yescharacteristic
143C2×C127C2×#13711yescharacteristic
144C2×C127#57511yescharacteristic
145D127—44no—
146#3007—1271nomaximal, Sylow
147C127#300711yescharacteristic, complemented, Sylow
148C22×#41—1271no—
149C2×#137—1272no—
150C2×C22⋊Q8—1271no—
151#452—1274no—
152#451—1272no—
153#450—1272no—
154#417—1272no—
155#407D12711yescharacteristic, complemented
156C2×#41—1272no—
157C2×#41—1272no—
158C2×#41—2548no—
159C2×#41—2541no—
160C2×#41—2541no—
161C22×Q8—1271no—
162C22×Q8—1271no—
163C22×D4—1271no—
164C22×D4—2541no—
165C23×C4—2541no—
166C23×C4C2×D12711yescharacteristic
167#137—2548no—
168C2×Q16—1274no—
169C2×SD16—1274no—
170C2×M16C2×D12712yescomplemented
171C22⋊Q8—1272no—
172C22⋊Q8—1272no—
173C22⋊Q8—2541no—
174C22⋊Q8—2541no—
175C2×C4⋊C4—1271no—
176C2×C4⋊C4—2541no—
177C2×C22⋊C4—2541no—
178#103—1274no—
179#102—1274no—
180C22⋊C8C2×D12712yes—
181C22⋊C8C2×D12712yescomplemented
182C24—2541no—
183C24D50811yescharacteristic
184#41—1278no—
185#41—2544no—
186#41—2544no—
187#41—2548no—
188#41—2548no—
189#41—2544no—
190C2×Q8—1272no—
191C2×Q8—1272no—
192C2×Q8—2544no—
193C2×Q8—2544no—
194C2×Q8—2541no—
195C2×Q8—2541no—
196C2×D4—1272no—
197C2×D4—2544no—
198C2×D4—2548no—
199C2×D4—2542no—
200C2×D4—2542no—
201C2×D4—2541no—
202C22×C4—24no—
203C22×C4—2548no—
204C22×C4—2541no—
205C22×C4—2541no—
206C22×C4—2541no—
207C22×C4—2542no—
208C22×C4—2542no—
209C22×C4D50811yescharacteristic
210C22×C4D50811yescharacteristic
211C22×C4D50811yescharacteristic
212C22×C4C22×D12711yescharacteristic
213C22×C4C22×D12711yescharacteristic
214C22×C4C22×D12711yescharacteristic
215Q16—2548no—
216SD16—2548no—
217M16—24no—
218C2×C8C22×D12714yes—
219C4⋊C4—1272no—
220C4⋊C4—2542no—
221C4⋊C4—2542no—
222C4⋊C4—2541no—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—5082no—
229C23—5082no—
230C23—2544no—
231C23—2542no—
232C23C2×D50811yescharacteristic
233C23C2×D50811yescharacteristic
234C23C2×D50811yescharacteristic
235Q8—1274no—
236Q8—5082no—
237Q8—2544no—
238Q8—2544no—
239Q8—2542no—
240D4—1274no—
241D4—2544no—
242D4—2548no—
243D4—2548no—
244D4—2542no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—5082no—
251C2×C4—5082no—
252C2×C4—2548no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4—2544no—
256C2×C4—2542no—
257C2×C4—2548no—
258C2×C4D4×D12714yes—
259C2×C4C2×D50811yescharacteristic
260C2×C4C2×D50811yescharacteristic
261C2×C4C2×D50811yescharacteristic
262C2×C4C23×D12711yescharacteristic
263C8—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—5088no—
273C22—5082no—
274C22—5082no—
275C22—2544no—
276C22#1686117912yes—
277C22#1686124212yescomplemented
278C22C22×D50811yescharacteristic
279C4—21no—
280C4—21no—
281C4—5082no—
282C4—5082no—
283C4—2544no—
284C4#1686117912yes—
285C4C2×D4×D12712yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—5084no—
290C2C2×#1686117911yescharacteristic
291C2C2×#1686124211yescharacteristic
292C2#2255067611yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#407×C127C2
socleC22×C127C22×D4
prime radicalC22×#14523248C2
hypercenter#407D127
hypocenterC127#3007
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16861242C2semidirect
3. C2×#16861188C2semidirect
4. #22550040C2semidirect
5. #22550034C2semidirect
6. #22550029C2semidirect
7. #22549878C2semidirect
8. #407×C127C2semidirect
15. C22×Dic254C22semidirect
20. C2×Dic508C22semidirect
21. C2×#14523237C22semidirect
25. #16861188C22semidirect
26. C2×C4⋊Dic127C22semidirect
29. #16861134C22semidirect
30. #16861131C22semidirect
32. C2×M16×C127C22semidirect
34. C22⋊C8×C127C22semidirect
126. C22×C127#137semidirect
147. C127#3007semidirect
155. #407D127semidirect
170. C2×M16C2×D127semidirect
181. C22⋊C8C2×D127semidirect
277. C22#16861242semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168612422
C2×#168611882
#225500402
#225500342
#225500292
#225498782
#407×C1272
#3007127