Finite Group Databasebeta

← #30650421

Subgroups — #30650421

Order 16256.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×SD16×D127C212yesmaximal, complemented
2Q8×D508C211yesmaximal, characteristic, complemented
3#22550511C211yesmaximal, characteristic
4#22550248C211yesmaximal, characteristic, complemented
5#22550104C212yesmaximal, complemented
6#22550082C211yesmaximal, characteristic, complemented
7#22550028C212yesmaximal, complemented
8#22550013C212yesmaximal, complemented
9C4×#14523235C211yesmaximal, characteristic, complemented
10#22549941C211yesmaximal, characteristic, complemented
11#460×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2212yescomplemented
13C2×D4×D127—21no—
14C2×D4×D127C2212yes—
15C2×#14523237C2212yes—
16C2×#14523236C2212yescomplemented
17C2×#14523236C2212yescomplemented
18C2×#14523235C2211yescharacteristic
19SD16×D127—28no—
20C2×C8×D127C2212yescomplemented
21#16861209—21no—
22#16861209C2211yescharacteristic, complemented
23#16861189—22no—
24#16861189C2212yescomplemented
25#16861180—21no—
26#16861180C2212yes—
27#16861177—22no—
28C4×D508—21no—
29C4×D508C2211yescharacteristic
30C4×D508C2211yescharacteristic
31#16861140C2211yescharacteristic, complemented
32#16861134C2212yescomplemented
33#16861133C2211yescharacteristic, complemented
34#16861129C2211yescharacteristic
35(C2×D127)⋊C8C2212yescomplemented
36C4×C127⋊C8C2211yescharacteristic, complemented
37C2×SD16×C127C2212yescomplemented
38#127×C127C2211yescharacteristic
39C4×Q8×C127C2211yescharacteristic, complemented
40C4⋊C8×C127C2211yescharacteristic, complemented
41#102×C127C2212yescomplemented
42C23×D127—22no—
43C2×Dic254—22no—
44C2×Dic254C2312yes—
45Q8×D127—24no—
46Q8×D127—42no—
47C2×D508D411yescharacteristic
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—22no—
50C2×C22⋊D127—22no—
51D4×D127—24no—
52D4×D127—42no—
53D4×D127—44no—
54C2×C4×D127—22no—
55C2×C4×D127—21no—
56C2×C4×D127C2312yes—
57#14523237—24no—
58#14523236—24no—
59#14523236—42no—
60#14523235D414yescomplemented
61C2×C127⋊C8C2311yescharacteristic
62C2×C127⋊C8C2311yescharacteristic
63C8×D127—24no—
64C4⋊Dic127—22no—
65C4⋊Dic127—21no—
66C4⋊Dic127—21no—
67C4⋊Dic127D411yescharacteristic
68C4⋊Dic127C2311yescharacteristic
69C22⋊Dic127—22no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—21no—
72(C2×D127)⋊C4D412yes—
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127C2312yes—
77SD16×C127—24no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127—21no—
80C4⋊C4×C127C2311yescharacteristic
81C42×C127—21no—
82C42×C127C2311yescharacteristic
83C22×D127—21no—
84C22×D127—44no—
85C22×D127—44no—
86C22×D127—82no—
87C22×D127C2×D412yes—
88Dic254—22no—
89Dic254—22no—
90Dic254—44no—
91D508—21no—
92D508SD1614yescomplemented
93D508C2×D412yes—
94C22⋊D127—44no—
95C22⋊D127—44no—
96C2×Dic127—21no—
97C2×Dic127—22no—
98C2×Dic127C2×D412yes—
99C4×D127—22no—
100C4×D127—22no—
101C4×D127—44no—
102C4×D127—42no—
103C127⋊C8—21no—
104C127⋊C8C2×D412yescomplemented
105C23×C127—22no—
106Q8×C127—21no—
107Q8×C127C2×D412yescomplemented
108D4×C127—22no—
109D4×C127—22no—
110D4×C127—42no—
111D4×C127—81no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2411yescharacteristic
117C8×C127—22no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127C22×D411yescharacteristic
122C2×D127—22no—
123C2×D127—22no—
124C2×D127—22no—
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—88no—
128C2×D127C2×SD1614yescomplemented
129C4×C127—22no—
130C4×C127—21no—
131C4×C127—41no—
132C4×C127C2×SD1612yescomplemented
133C4×C127C22×D411yescharacteristic
134C4×C127H(5,𝔽2)11yescharacteristic
135Dic127—22no—
136Dic127—42no—
137Dic127—41no—
138C2×C127—82no—
139C2×C127D4211yescharacteristic
140C2×C127C22×SD1611yescharacteristic
141C2×C127#57711yescharacteristic
142D127—24no—
143D127—42no—
144D4×SD16—1271nomaximal, Sylow
145C127D4×SD1611yescharacteristic, complemented, Sylow
146C22×SD16—1272no—
147D4×Q8—1271no—
148D42—1271no—
149#492—1271no—
150#465—1272no—
151#461—1271no—
152#460D12711yescharacteristic, complemented
153#450—1272no—
154#448—1272no—
155C4×SD16—1271no—
156C8×D4—1271no—
157C22×Q8—1272no—
158C22×D4—1272no—
159C22×D4—2541no—
160C2×SD16—1271no—
161C2×SD16—1272no—
162C2×SD16—1272no—
163C2×SD16—1272no—
164C2×SD16—2548no—
165C2×SD16C2×D12712yescomplemented
166C22×C8—1272no—
167C4⋊Q8—1271no—
168C4⋊Q8—2541no—
169#127C2×D12711yescharacteristic
170C22⋊Q8—1272no—
171C22⋊Q8—2542no—
172#121—1272no—
173#121—2541no—
174#120—2542no—
175C4×Q8C2×D12711yescharacteristic, complemented
176C4×D4—1271no—
177C4×D4—1271no—
178C4×D4—2541no—
179#106—1271no—
180C4⋊C8C2×D12711yescharacteristic, complemented
181#103—1271no—
182#103—1272no—
183#102—1271no—
184#102C2×D12712yescomplemented
185C22⋊C8—1272no—
186C4×C8—1271no—
187C24—2542no—
188C2×Q8—1272no—
189C2×Q8—5082no—
190C2×Q8—2542no—
191C2×Q8—2544no—
192C2×Q8C22×D12711yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—1271no—
196C2×D4—1271no—
197C2×D4—5084no—
198C2×D4—5082no—
199C2×D4—2542no—
200C2×D4—2544no—
201C2×D4—2542no—
202C2×D4C22×D12712yes—
203C22×C4—1272no—
204C22×C4—2542no—
205C22×C4—2541no—
206SD16—24no—
207SD16—1274no—
208SD16—5082no—
209SD16—2544no—
210SD16—2544no—
211C2×C8—1271no—
212C2×C8—1271no—
213C2×C8—2544no—
214C2×C8C22×D12712yes—
215C4⋊C4—21no—
216C4⋊C4—1271no—
217C4⋊C4—1271no—
218C4⋊C4—2542no—
219C4⋊C4—2541no—
220C4⋊C4—2541no—
221C4⋊C4C22×D12711yescharacteristic
222C22⋊C4—1272no—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C22⋊C4—2541no—
226C42—21no—
227C42C22×D12711yescharacteristic
228C23—22no—
229C23—1272no—
230C23—10162no—
231C23—5084no—
232C23—5084no—
233C23—2541no—
234Q8—21no—
235Q8—5084no—
236Q8—2542no—
237Q8—2542no—
238Q8D4×D12712yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—1274no—
244D4—1272no—
245D4—5084no—
246D4—5084no—
247D4—2541no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—1272no—
251C2×C4—5084no—
252C2×C4—5082no—
253C2×C4—2541no—
254C2×C4—2542no—
255C2×C4—2542no—
256C2×C4—2542no—
257C2×C4D4×D12711yescharacteristic
258C2×C4D4×D12711yescharacteristic
259C2×C4C23×D12711yescharacteristic
260C8—22no—
261C8—1272no—
262C8—2541no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—1274no—
267C22—10168no—
268C22—5081no—
269C22—5081no—
270C22—2542no—
271C22—2542no—
272C22—2542no—
273C22C2×D4×D12711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—5081no—
278C4—5082no—
279C4—2542no—
280C4SD16×D12712yescomplemented
281C4C2×D4×D12711yescharacteristic
282C4#1686125511yescharacteristic
283C2—82no—
284C2—5082no—
285C2—2544no—
286C2#2255051011yescharacteristic
287C2C2×SD16×D12711yescharacteristic
288C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
prime radical#22550511C2
hypercenter#460D127
hypocenterC127D4×SD16
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D127C2semidirect
2. Q8×D508C2semidirect
4. #22550248C2semidirect
5. #22550104C2semidirect
6. #22550082C2semidirect
7. #22550028C2semidirect
8. #22550013C2semidirect
9. C4×#14523235C2semidirect
10. #22549941C2semidirect
11. #460×C127C2semidirect
12. C2×Q8×D127C22semidirect
16. C2×#14523236C22semidirect
17. C2×#14523236C22semidirect
20. C2×C8×D127C22semidirect
22. #16861209C22semidirect
24. #16861189C22semidirect
31. #16861140C22semidirect
32. #16861134C22semidirect
33. #16861133C22semidirect
35. (C2×D127)⋊C8C22semidirect
36. C4×C127⋊C8C22semidirect
37. C2×SD16×C127C22semidirect
39. C4×Q8×C127C22semidirect
40. C4⋊C8×C127C22semidirect
41. #102×C127C22semidirect
60. #14523235D4semidirect
92. D508SD16semidirect
104. C127⋊C8C2×D4semidirect
107. Q8×C127C2×D4semidirect
128. C2×D127C2×SD16semidirect
132. C4×C127C2×SD16semidirect
145. C127D4×SD16semidirect
152. #460D127semidirect
165. C2×SD16C2×D127semidirect
175. C4×Q8C2×D127semidirect
180. C4⋊C8C2×D127semidirect
184. #102C2×D127semidirect
238. Q8D4×D127semidirect
280. C4SD16×D127semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1272
Q8×D5082
#225505112
#225502482
#225501042
#225500822
#225500282
#225500132
C4×#145232352
#225499412
#460×C1272
D4×SD16127