Finite Group Databasebeta

← #30648649

Subgroups — #30648649

Order 16256.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#22550386C211yesmaximal, characteristic
2C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
3#22550213C212yesmaximal, complemented
4#22550207C211yesmaximal, characteristic, complemented
5#22550206C211yesmaximal, characteristic, complemented
6#22550193C212yesmaximal, complemented
7#22550187C211yesmaximal, characteristic, complemented
8#22549917C212yesmaximal, complemented
9#22549874C212yesmaximal, complemented
10#22549838C211yesmaximal, characteristic, complemented
11#480×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2211yescharacteristic
13C22×C4×D127—21no—
14C22×C4×D127C2211yescharacteristic
15C2×#14523231C2212yes—
16#16861191—21no—
17#16861189C2211yescharacteristic
18#16861188C2211yescharacteristic
19#16861180C2211yescharacteristic
20#16861177—21no—
21D4×Dic127—21no—
22(C22⋊D127)⋊C4—21no—
23#16861167C2211yescharacteristic
24C2×C4⋊Dic127C2211yescharacteristic
25C2×Dic127⋊C4—21no—
26C4⋊C4×D127—21no—
27C4⋊C4×D127—21no—
28C4⋊C4×D127C2212yes—
29C4⋊C4×D127C2212yes—
30#16861142C2212yescomplemented
31#16861141C2212yescomplemented
32#16861140C2212yescomplemented
33#16861134C2212yescomplemented
34#16861132C2212yescomplemented
35#16861130C2212yescomplemented
36C22⋊(C127⋊C8)C2211yescharacteristic
37(C2×D127)⋊C8C2212yes—
38#121×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#107×C127C2212yescomplemented
41#102×C127C2212yescomplemented
42C22⋊C8×C127C2211yescharacteristic
43C23×D127D411yescharacteristic
44#14523247—24no—
45C2×Dic254C2311yescharacteristic
46C2×C22⋊D127—21no—
47C2×C22⋊D127—21no—
48C22×Dic127—21no—
49C22×Dic127—21no—
50C22×Dic127D411yescharacteristic
51C2×C4×D127—24no—
52C2×C4×D127—22no—
53C2×C4×D127—22no—
54C2×C4×D127—44no—
55C2×C4×D127D411yescharacteristic
56C2×C4×D127D411yescharacteristic
57C2×C4×D127C2311yescharacteristic
58C2×C4×D127C2311yescharacteristic
59#14523231—24no—
60C2×C127⋊C8C2312yes—
61C4⋊Dic127—21no—
62C4⋊Dic127—21no—
63C4⋊Dic127C2311yescharacteristic
64C4⋊Dic127C2312yes—
65Dic127⋊C4—22no—
66Dic127⋊C4—22no—
67Dic127⋊C4—21no—
68C22⋊Dic127—21no—
69C22⋊Dic127—21no—
70(C2×D127)⋊C4—21no—
71(C2×D127)⋊C4—21no—
72C4×Dic127—21no—
73C2×D4×C127—21no—
74C2×D4×C127C2311yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127C2311yescharacteristic
77C2×C8×C127C2312yes—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127—22no—
85C22×D127C2×D411yescharacteristic
86C22×D127C2×D411yescharacteristic
87Dic254—22no—
88C22⋊D127—42no—
89C22⋊D127—42no—
90C2×Dic127—22no—
91C2×Dic127—21no—
92C2×Dic127—21no—
93C2×Dic127—22no—
94C2×Dic127—42no—
95C2×Dic127—42no—
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C4×D127—22no—
99C4×D127—22no—
100C4×D127—22no—
101C4×D127—44no—
102C4×D127—44no—
103C4×D127#4114yes—
104C127⋊C8—22no—
105C23×C127—21no—
106C23×C127C2×D411yescharacteristic
107D4×C127—22no—
108D4×C127—42no—
109C2×C4×C127—22no—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—42no—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2411yescharacteristic
115C8×C127—22no—
116C22×C127—21no—
117C22×C127—21no—
118C22×C127—41no—
119C22×C127—41no—
120C22×C127—41no—
121C22×C127AGL(1,ℤ/8)12yes—
122C22×C127C22×D411yescharacteristic
123C2×D127—28no—
124C2×D127—21no—
125C2×D127—21no—
126C2×D127—21no—
127C2×D127—21no—
128C2×D127#12312yes—
129C2×D127#13712yes—
130C4×C127—21no—
131C4×C127—42no—
132C4×C127—41no—
133C4×C127C2×#4112yes—
134Dic127—21no—
135Dic127—41no—
136Dic127—42no—
137Dic127—41no—
138Dic127#12312yes—
139C2×C127—22no—
140C2×C127—81no—
141C2×C127C2×#12311yescharacteristic
142C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C127C2×#13711yescharacteristic
144D127—22no—
145D127—22no—
146#3092—1271nomaximal, Sylow
147C127#309211yescharacteristic, complemented, Sylow
148#525—1271no—
149C22×C4⋊C4—1271no—
150#484—1272no—
151#483—1271no—
152#483—1271no—
153#481—1272no—
154#480—1271no—
155#480D12711yescharacteristic, complemented
156#428—1272no—
157#417—1272no—
158#407—1271no—
159C2×#41—1271no—
160C23×C4—1271no—
161C23×C4—2541no—
162C2×M16—1272no—
163#123—2541no—
164C22⋊Q8—1271no—
165C22⋊Q8—1271no—
166#121—1271no—
167#121C2×D12711yescharacteristic
168#120—2541no—
169C4×D4—2541no—
170C4×D4—2541no—
171#117—1271no—
172C2×C4⋊C4—1271no—
173C2×C4⋊C4—1272no—
174C2×C4⋊C4—1272no—
175C2×C4⋊C4—2541no—
176C2×C4⋊C4—2541no—
177C2×C4⋊C4—2541no—
178C2×C4⋊C4C2×D12711yescharacteristic
179#107—1272no—
180#107C2×D12712yescomplemented
181#106—1272no—
182#106—1272no—
183#103—1272no—
184#103—1272no—
185#102—1272no—
186#102C2×D12712yescomplemented
187C22⋊C8—1271no—
188C22⋊C8—1272no—
189C22⋊C8C2×D12711yescharacteristic
190C24—1271no—
191#41—2544no—
192C2×Q8—1271no—
193C2×D4—21no—
194C2×D4—2541no—
195C2×D4—2541no—
196C2×D4C22×D12711yescharacteristic
197C22×C4—21no—
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—1271no—
201C22×C4—1271no—
202C22×C4—1271no—
203C22×C4—5084no—
204C22×C4—2544no—
205C22×C4—2542no—
206C22×C4—2542no—
207C22×C4—2541no—
208C22×C4—2541no—
209C22×C4C22×D12711yescharacteristic
210M16—2544no—
211C2×C8—1272no—
212C2×C8C22×D12712yes—
213C4⋊C4—21no—
214C4⋊C4—1272no—
215C4⋊C4—1271no—
216C4⋊C4—2542no—
217C4⋊C4—2542no—
218C4⋊C4—2541no—
219C4⋊C4—2541no—
220C4⋊C4—2541no—
221C4⋊C4C22×D12712yes—
222C4⋊C4C22×D12711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2541no—
225C22⋊C4—2541no—
226C22⋊C4—2541no—
227C22⋊C4—2541no—
228C42—2541no—
229C23—21no—
230C23—1271no—
231C23—1271no—
232C23—2542no—
233C23—2542no—
234C23—2542no—
235C23D4×D12711yescharacteristic
236Q8—2542no—
237D4—22no—
238D4—42no—
239D4—5082no—
240D4—5082no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—1271no—
246C2×C4—1274no—
247C2×C4—1271no—
248C2×C4—5084no—
249C2×C4—5084no—
250C2×C4—5082no—
251C2×C4—5082no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4—2541no—
256C2×C4—2541no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4D4×D12711yescharacteristic
260C2×C4C23×D12711yescharacteristic
261C8—22no—
262C8—2542no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—1272no—
269C22—1272no—
270C22—2548no—
271C22—2541no—
272C22—2541no—
273C22—2541no—
274C22—2541no—
275C22#1686123312yes—
276C22C2×D4×D12711yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—1272no—
281C4—5081no—
282C4—5082no—
283C4—5081no—
284C4—2541no—
285C4#41×D12712yes—
286C2—22no—
287C2—81no—
288C2—2542no—
289C2—2542no—
290C2#123×D12711yescharacteristic
291C2C2×#1686123311yescharacteristic
292C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#480×C127C2
socleC22×C127C22×D4
prime radical#22550386C2
hypercenter#480D127
hypocenterC127#3092
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D127C2semidirect
3. #22550213C2semidirect
4. #22550207C2semidirect
5. #22550206C2semidirect
6. #22550193C2semidirect
7. #22550187C2semidirect
8. #22549917C2semidirect
9. #22549874C2semidirect
10. #22549838C2semidirect
11. #480×C127C2semidirect
30. #16861142C22semidirect
31. #16861141C22semidirect
32. #16861140C22semidirect
33. #16861134C22semidirect
34. #16861132C22semidirect
35. #16861130C22semidirect
40. #107×C127C22semidirect
41. #102×C127C22semidirect
147. C127#3092semidirect
155. #480D127semidirect
180. #107C2×D127semidirect
186. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503862
C2×C4⋊C4×D1272
#225502132
#225502072
#225502062
#225501932
#225501872
#225499172
#225498742
#225498382
#480×C1272
#3092127