Finite Group Databasebeta

← #30647966

Subgroups — #30647966

Order 16256.

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

normal subgroups only

This group has 738 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×Q8×D127C211yesmaximal, characteristic, complemented
2C2×#16861242C211yesmaximal, characteristic, complemented
3C2×#16861240C211yesmaximal, characteristic, complemented
4#22550388C211yesmaximal, characteristic
5#22550033C214yesmaximal, complemented
6#22550016C214yesmaximal, complemented
7#22549884C211yesmaximal, characteristic, complemented
8#22549839C211yesmaximal, characteristic, complemented
9#419×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×Dic254—21no—
12C22×Dic254C2211yescharacteristic
13C2×Q8×D127—24no—
14C2×Q8×D127—22no—
15C2×Q8×D127—24no—
16C2×Q8×D127C2214yes—
17C22×C4×D127—21no—
18C22×C4×D127C2211yescharacteristic
19#16861242—24no—
20#16861240—24no—
21C2×Dic508C2212yescomplemented
22C2×#14523238C2212yescomplemented
23C2×#14523237C2212yescomplemented
24C2×#14523235C2212yescomplemented
25C2×#14523232C2211yescharacteristic
26#16861193—21no—
27#16861189C2212yes—
28#16861184C2212yes—
29#16861179—21no—
30C4×D508—22no—
31#16861132C2214yescomplemented
32(C2×D127)⋊C8C2214yes—
33C22×Q8×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic
35#117×C127C2211yescharacteristic
36#103×C127C2214yescomplemented
37C23×D127D411yescharacteristic
38#14523248—22no—
39#14523248—22no—
40C2×Dic254—28no—
41C2×Dic254—24no—
42C2×Dic254—21no—
43C2×Dic254—22no—
44C2×Dic254D414yescomplemented
45C2×Dic254C2311yescharacteristic
46C2×Dic254C2312yes—
47Q8×D127—216no—
48Q8×D127—28no—
49Q8×D127—28no—
50C2×D508—22no—
51C2×D508C2311yescharacteristic
52C2×C22⋊D127—21no—
53C22×Dic127—21no—
54C22×Dic127D411yescharacteristic
55C2×C4×D127—28no—
56C2×C4×D127—24no—
57C2×C4×D127—22no—
58C2×C4×D127—22no—
59C2×C4×D127—21no—
60C2×C4×D127D412yes—
61C2×C4×D127C2312yes—
62Dic508—24no—
63#14523238—24no—
64#14523237—24no—
65#14523235—24no—
66#14523232—22no—
67C2×C127⋊C8C2312yes—
68C4⋊Dic127—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71C2×Q8×C127—22no—
72C2×Q8×C127—21no—
73C2×Q8×C127D414yescomplemented
74C2×Q8×C127C2312yes—
75C22×C4×C127—21no—
76C22×C4×C127C2311yescharacteristic
77M16×C127—22no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81C42×C127—21no—
82C22×D127—22no—
83C22×D127—24no—
84C22×D127—21no—
85C22×D127C2×D412yes—
86Dic254—216no—
87Dic254—24no—
88Dic254—22no—
89Dic254—21no—
90Dic254—21no—
91Dic254C2×D414yes—
92D508—21no—
93D508—21no—
94D508—44no—
95C22⋊D127—42no—
96C2×Dic127—24no—
97C2×Dic127—22no—
98C2×Dic127—22no—
99C2×Dic127—21no—
100C2×Dic127C2×D412yes—
101C4×D127—216no—
102C4×D127—24no—
103C4×D127—22no—
104C4×D127—22no—
105C4×D127—42no—
106C127⋊C8—22no—
107C23×C127C2×D411yescharacteristic
108Q8×C127—24no—
109Q8×C127—22no—
110Q8×C127C2×D414yes—
111C2×C4×C127—24no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2411yescharacteristic
118C8×C127—22no—
119C22×C127—22no—
120C22×C127#12012yes—
121C22×C127C22×D411yescharacteristic
122C2×D127—28no—
123C2×D127—22no—
124C2×D127—22no—
125C2×D127—41no—
126C2×D127—42no—
127C2×D127#13714yescomplemented
128C4×C127—24no—
129C4×C127—42no—
130C4×C127#12012yes—
131C4×C127C22×D411yescharacteristic
132C4×C127C22×D411yescharacteristic
133Dic127—24no—
134Dic127—22no—
135Dic127—41no—
136C2×C127—22no—
137C2×C127C2×#12011yescharacteristic
138C2×C127C2×#13712yes—
139D127—24no—
140D127—81no—
141#3004—1271nomaximal, Sylow
142C127#300411yescharacteristic, complemented, Sylow
143C23×Q8—1271no—
144C2×#137—1271no—
145C2×#137—1271no—
146#525—1271no—
147#451—1274no—
148#448—1274no—
149#419—1271no—
150#419D12711yescharacteristic, complemented
151#407—1271no—
152C2×#41—1271no—
153C22×Q8—1274no—
154C22×Q8—1271no—
155C22×Q8—2541no—
156C22×Q8—2542no—
157C22×Q8—2544no—
158C22×Q8—2544no—
159C22×Q8C2×D12711yescharacteristic
160C23×C4—1271no—
161C23×C4—2541no—
162#137—2544no—
163#137—2544no—
164C2×Q16—1272no—
165C2×Q16—1272no—
166C2×SD16—1272no—
167C2×SD16—1272no—
168C2×M16—1271no—
169C2×M16C2×D12711yescharacteristic
170#123—2541no—
171C22⋊Q8—1272no—
172#121—1272no—
173#120—2541no—
174C4×D4—2542no—
175#117C2×D12711yescharacteristic
176#103—1274no—
177#103C2×D12714yescomplemented
178C22⋊C8—1274no—
179C24—1271no—
180#41—2542no—
181#41—2542no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—1274no—
185C2×Q8—1272no—
186C2×Q8—1271no—
187C2×Q8—2544no—
188C2×Q8—2548no—
189C2×Q8—2548no—
190C2×Q8—25416no—
191C2×Q8—2548no—
192C2×Q8—2541no—
193C2×Q8—2542no—
194C2×Q8D50814yescomplemented
195C2×Q8C22×D12712yes—
196C2×D4—1271no—
197C2×D4—2542no—
198C2×D4—2541no—
199C22×C4—21no—
200C22×C4—1271no—
201C22×C4—1272no—
202C22×C4—1272no—
203C22×C4—2544no—
204C22×C4—2548no—
205C22×C4—2541no—
206C22×C4—2541no—
207C22×C4—2542no—
208C22×C4—2542no—
209C22×C4C22×D12711yescharacteristic
210Q16—2544no—
211Q16—2544no—
212SD16—2544no—
213SD16—2544no—
214M16—22no—
215M16—2542no—
216C2×C8—1272no—
217C2×C8C22×D12712yes—
218C4⋊C4—2542no—
219C4⋊C4C22×D12712yes—
220C22⋊C4—21no—
221C22⋊C4—2542no—
222C22⋊C4—2542no—
223C42—21no—
224C23—1272no—
225C23—2544no—
226C23—2541no—
227C23—2542no—
228C23D4×D12711yescharacteristic
229Q8—22no—
230Q8—24no—
231Q8—1274no—
232Q8—25416no—
233Q8—2541no—
234Q8—2541no—
235Q8—2542no—
236Q8—2544no—
237Q8C2×D50814yes—
238D4—5084no—
239D4—5082no—
240D4—2541no—
241D4—2541no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—1272no—
246C2×C4—5082no—
247C2×C4—25416no—
248C2×C4—2541no—
249C2×C4—2542no—
250C2×C4—2542no—
251C2×C4—2544no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2544no—
255C2×C4D4×D12712yes—
256C2×C4D4×D12711yescharacteristic
257C2×C4C2×D50812yes—
258C2×C4C23×D12711yescharacteristic
259C8—22no—
260C8—2542no—
261C22—22no—
262C22—1274no—
263C22—5081no—
264C22—5082no—
265C22—2548no—
266C22—2542no—
267C22—2542no—
268C22#1686117912yes—
269C22C2×D4×D12711yescharacteristic
270C4—24no—
271C4—42no—
272C4—5081no—
273C4—2542no—
274C4—2544no—
275C4#1686117912yes—
276C4C2×D4×D12711yescharacteristic
277C4C22×D50811yescharacteristic
278C2—22no—
279C2—10161no—
280C2—2544no—
281C2C2×#1686117911yescharacteristic
282C2#137×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radical#22550388C2
hypercenter#419D127
hypocenterC127#3004
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D127C2semidirect
2. C2×#16861242C2semidirect
3. C2×#16861240C2semidirect
5. #22550033C2semidirect
6. #22550016C2semidirect
7. #22549884C2semidirect
8. #22549839C2semidirect
9. #419×C127C2semidirect
21. C2×Dic508C22semidirect
22. C2×#14523238C22semidirect
23. C2×#14523237C22semidirect
24. C2×#14523235C22semidirect
31. #16861132C22semidirect
36. #103×C127C22semidirect
44. C2×Dic254D4semidirect
73. C2×Q8×C127D4semidirect
127. C2×D127#137semidirect
142. C127#3004semidirect
150. #419D127semidirect
177. #103C2×D127semidirect
194. C2×Q8D508semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1272
C2×#168612422
C2×#168612402
#225503882
#225500332
#225500162
#225498842
#225498392
#419×C1272
#3004127