Finite Group Databasebeta

← #30647988

Subgroups — #30647988

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×#16861240C212yesmaximal, complemented
3#22550036C214yesmaximal, complemented
4#22550031C212yesmaximal, complemented
5#22550026C212yesmaximal, complemented
6#22549875C212yesmaximal, complemented
7#22549840C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—28no—
10C2×#14523248—21no—
11C2×#14523248—21no—
12C2×#14523248C2212yes—
13C2×#14523248C2212yes—
14C22×Dic254C2211yescharacteristic
15C22×D508C2211yescharacteristic
16C22×C22⋊D127—21no—
17C22×C4×D127—21no—
18#16861240—28no—
19C2×#14523238C2214yescomplemented
20C2×#14523235C2214yescomplemented
21C2×#14523232C2212yescomplemented
22#16861132C2214yescomplemented
23#16861129C2214yescomplemented
24C22⋊(C127⋊C8)C2212yes—
25C22⋊(C127⋊C8)C2212yescomplemented
26C22×Q8×C127C2211yescharacteristic, complemented
27C23×C4×C127C2211yescharacteristic
28C22⋊Q8×C127—21no—
29C22⋊Q8×C127—21no—
30C22⋊Q8×C127C2212yes—
31C22⋊Q8×C127C2212yescomplemented
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127C2211yescharacteristic, complemented
34C2×C22⋊C4×C127—21no—
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×Dic254C2312yes—
45C2×D508—24no—
46C2×D508—21no—
47C2×D508C2312yes—
48C2×C22⋊D127—28no—
49C2×C22⋊D127—22no—
50C2×C22⋊D127—22no—
51C22×Dic127—21no—
52C2×C4×D127—28no—
53C2×C4×D127—22no—
54C2×C4×D127—22no—
55#14523238—28no—
56#14523235—28no—
57#14523232—24no—
58C2×C127⋊C8C2314yes—
59C24×C127D411yescharacteristic
60C2×Q8×C127—21no—
61C2×Q8×C127—24no—
62C2×Q8×C127C2312yes—
63C22×C4×C127—24no—
64C22×C4×C127—21no—
65C22×C4×C127—21no—
66C22×C4×C127D411yescharacteristic
67C22×C4×C127D411yescharacteristic
68C22×C4×C127D411yescharacteristic
69C22×C4×C127C2311yescharacteristic
70C22×C4×C127C2311yescharacteristic
71C22×C4×C127C2311yescharacteristic
72C4⋊C4×C127—22no—
73C4⋊C4×C127—22no—
74C4⋊C4×C127—21no—
75C4⋊C4×C127C2312yes—
76C22⋊C4×C127—22no—
77C22⋊C4×C127—22no—
78C22×D127—24no—
79C22×D127—22no—
80C22×D127—42no—
81C22×D127—42no—
82Dic254—24no—
83Dic254—22no—
84Dic254C2×D414yes—
85D508—24no—
86D508—22no—
87D508C2×D414yes—
88C22⋊D127—28no—
89C22⋊D127—28no—
90C2×Dic127—24no—
91C2×Dic127—22no—
92C4×D127—28no—
93C4×D127—28no—
94C127⋊C8—24no—
95C23×C127—22no—
96C23×C127—22no—
97C23×C127—41no—
98C23×C127C2×D411yescharacteristic
99C23×C127C2×D411yescharacteristic
100C23×C127C2×D411yescharacteristic
101Q8×C127—24no—
102Q8×C127—42no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—42no—
111C2×C4×C127—42no—
112C2×C4×C127C2×D414yes—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2411yescharacteristic
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—42no—
135C4×C127—42no—
136C4×C127#12012yes—
137C4×C127C22×D412yes—
138Dic127—24no—
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⋊Q8D12711yescharacteristic, complemented
151#452—1274no—
152#451—1272no—
153#450—1272no—
154#417—1272no—
155#407—1271no—
156C2×#41—1272no—
157C2×#41—1272no—
158C2×#41—2548no—
159C2×#41—2541no—
160C2×#41—2541no—
161C22×Q8—1271no—
162C22×Q8C2×D12711yescharacteristic, complemented
163C22×D4—1271no—
164C22×D4—2541no—
165C23×C4—2541no—
166C23×C4C2×D12711yescharacteristic
167#137—2548no—
168C2×Q16—1274no—
169C2×SD16—1274no—
170C2×M16—1272no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D12712yes—
174C22⋊Q8C2×D12712yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D12711yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—1274no—
179#102—1274no—
180C22⋊C8—1272no—
181C22⋊C8—1272no—
182C24—2541no—
183C24C22⋊D12711yescharacteristic
184#41—1278no—
185#41—2544no—
186#41—2544no—
187#41—2548no—
188#41—2548no—
189#41—2544no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—1272no—
193C2×Q8—2544no—
194C2×Q8—2541no—
195C2×Q8C22×D12712yes—
196C2×D4—1272no—
197C2×D4—2544no—
198C2×D4—2548no—
199C2×D4—2542no—
200C2×D4—2542no—
201C2×D4—2541no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—2548no—
206C22×C4—2541no—
207C22×C4—2542no—
208C22×C4—2542no—
209C22×C4C22⋊D12711yescharacteristic
210C22×C4C22⋊D12711yescharacteristic
211C22×C4C22⋊D12711yescharacteristic
212C22×C4C22×D12711yescharacteristic
213C22×C4C22×D12711yescharacteristic
214C22×C4C22×D12711yescharacteristic
215Q16—2548no—
216SD16—2548no—
217M16—2544no—
218C2×C8—1274no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D12712yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—5082no—
229C23—5082no—
230C23—2544no—
231C23—2542no—
232C23C2×C22⋊D12711yescharacteristic
233C23C2×C22⋊D12711yescharacteristic
234C23C2×C22⋊D12711yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—1274no—
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—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—2548no—
255C2×C4—2544no—
256C2×C4—2542no—
257C2×C4—2548no—
258C2×C4D4×D12714yes—
259C2×C4C2×C22⋊D12711yescharacteristic
260C2×C4C2×C22⋊D12711yescharacteristic
261C2×C4C2×C22⋊D12711yescharacteristic
262C2×C4C23×D12711yescharacteristic
263C8—2544no—
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#1686117712yes—
277C22#1686124012yescomplemented
278C22C22×C22⋊D12711yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—2544no—
284C4#1686117712yes—
285C4C2×D4×D12712yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—5084no—
290C2C2×#1686117711yescharacteristic
291C2C2×#1686124011yescharacteristic
292C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C22⋊Q8×C127C2
socleC22×C127C22×D4
prime radicalC22×#14523248C2
hypercenterC2×C22⋊Q8D127
hypocenterC127#3007
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. #22550036C2semidirect
4. #22550031C2semidirect
5. #22550026C2semidirect
6. #22549875C2semidirect
7. #22549840C2semidirect
8. C2×C22⋊Q8×C127C2semidirect
19. C2×#14523238C22semidirect
20. C2×#14523235C22semidirect
21. C2×#14523232C22semidirect
22. #16861132C22semidirect
23. #16861129C22semidirect
25. C22⋊(C127⋊C8)C22semidirect
26. C22×Q8×C127C22semidirect
31. C22⋊Q8×C127C22semidirect
33. C2×C4⋊C4×C127C22semidirect
126. C22×C127#137semidirect
147. C127#3007semidirect
150. C2×C22⋊Q8D127semidirect
162. C22×Q8C2×D127semidirect
174. C22⋊Q8C2×D127semidirect
176. C2×C4⋊C4C2×D127semidirect
277. C22#16861240semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168612402
#225500362
#225500312
#225500262
#225498752
#225498402
C2×C22⋊Q8×C1272
#3007127