Finite Group Databasebeta

← #30649122

Subgroups — #30649122

Order 16256.

424 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 121 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#22550425C211yesmaximal, characteristic
2C4⋊Q8×D127C211yesmaximal, characteristic, complemented
3#22550238C211yesmaximal, characteristic, complemented
4#22550236C211yesmaximal, characteristic, complemented
5#22550235C211yesmaximal, characteristic, complemented
6#22550225C212yesmaximal, complemented
7#22550221C212yesmaximal, complemented
8#22549876C212yesmaximal, complemented
9#22549874C212yesmaximal, complemented
10#22549828C211yesmaximal, characteristic, complemented
11#489×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2211yescharacteristic
13C2×#14523247C2211yescharacteristic
14C2×Q8×D127—21no—
15C2×#14523231C2212yes—
16#16861209C2211yescharacteristic
17#16861208—21no—
18#16861208C2212yes—
19#16861207—21no—
20#16861207C2211yescharacteristic
21#16861205C2211yescharacteristic
22#16861197C2211yescharacteristic
23#16861182—21no—
24#16861182—21no—
25(C22⋊D127)⋊C4—21no—
26(C22⋊D127)⋊C4—21no—
27C4⋊C4×D127—21no—
28C4⋊C4×D127C2212yes—
29C42×D127C2211yescharacteristic
30#16861134C2212yes—
31#16861133C2212yes—
32#16861132C2212yes—
33#16861131C2212yes—
34#16861130C2212yes—
35#16861129C2212yes—
36#16861120C2211yescharacteristic
37C8×Dic127C2212yes—
38C4⋊Q8×C127C2211yescharacteristic
39#124×C127C2211yescharacteristic
40#103×C127C2212yes—
41#102×C127C2212yes—
42#97×C127C2211yescharacteristic
43#14523249—24no—
44#14523247—24no—
45C2×Dic254—22no—
46C2×Dic254—21no—
47C2×Dic254C2311yescharacteristic
48Q8×D127—44no—
49C2×D508—21no—
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—21no—
52C2×C22⋊D127—21no—
53C2×C22⋊D127—21no—
54C22×Dic127—21no—
55C2×C4×D127—22no—
56C2×C4×D127—21no—
57C2×C4×D127D411yescharacteristic
58C2×C4×D127D411yescharacteristic
59C2×C4×D127C2311yescharacteristic
60#14523231—24no—
61C2×C127⋊C8C2312yes—
62C4⋊Dic127—21no—
63C4⋊Dic127C2312yes—
64Dic127⋊C4—22no—
65Dic127⋊C4—22no—
66Dic127⋊C4—21no—
67Dic127⋊C4—21no—
68(C2×D127)⋊C4—21no—
69(C2×D127)⋊C4—21no—
70C4×Dic127D411yescharacteristic
71C4×Dic127D411yescharacteristic
72C4×Dic127C2311yescharacteristic
73C2×Q8×C127—21no—
74C2×Q8×C127C2311yescharacteristic
75C2×D4×C127C2311yescharacteristic
76C2×C8×C127C2312yes—
77C4⋊C4×C127—21no—
78C4⋊C4×C127C2312yes—
79C22⋊C4×C127—21no—
80C22⋊C4×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127—21no—
83C22×D127C2×D411yescharacteristic
84Dic254—22no—
85Dic254—42no—
86Dic254—44no—
87D508—22no—
88D508—42no—
89C22⋊D127—42no—
90C22⋊D127—42no—
91C22⋊D127—42no—
92C2×Dic127—21no—
93C2×Dic127—22no—
94C2×Dic127—42no—
95C2×Dic127C2×D411yescharacteristic
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C4×D127—22no—
99C4×D127—22no—
100C4×D127—44no—
101C4×D127—42no—
102C4×D127#4114yes—
103C127⋊C8—22no—
104C23×C127—21no—
105Q8×C127—22no—
106Q8×C127—42no—
107D4×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C8×C127—22no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127C22×D411yescharacteristic
118C2×D127—21no—
119C2×D127—21no—
120C2×D127—41no—
121C2×D127—41no—
122C2×D127—41no—
123C2×D127#12412yes—
124C4×C127—21no—
125C4×C127—21no—
126C4×C127—41no—
127C4×C127—42no—
128C4×C127C2×#4112yes—
129Dic127—41no—
130Dic127—42no—
131Dic127#12412yes—
132Dic127AGL(1,ℤ/8)12yes—
133Dic127#13712yes—
134C2×C127—81no—
135C2×C127C2×#12411yescharacteristic
136C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C127C2×#13711yescharacteristic
138D127—22no—
139D127—81no—
140#3141—1271nomaximal, Sylow
141C127#314111yescharacteristic, complemented, Sylow
142#532—1271no—
143C2×C4⋊Q8—1271no—
144#489—1271no—
145#489—1271no—
146#489—1271no—
147#489D12711yescharacteristic, complemented
148#487—1272no—
149#486—1272no—
150#417—1272no—
151#417—1272no—
152C4×M16—1271no—
153C2×#41—1271no—
154C2×#41—1271no—
155C22×Q8—2541no—
156C2×M16—1272no—
157C4⋊Q8—1272no—
158C4⋊Q8—1271no—
159C4⋊Q8—1271no—
160C4⋊Q8—2541no—
161C4⋊Q8—2541no—
162C4⋊Q8C2×D12711yescharacteristic
163#127—1271no—
164#124—1271no—
165#124C2×D12711yescharacteristic
166#121—2541no—
167#121—2541no—
168C4×D4—2541no—
169C4×D4—2541no—
170C2×C4⋊C4—1272no—
171C2×C4⋊C4—2541no—
172C2×C42—1271no—
173#103—1272no—
174#103—1272no—
175#103—1272no—
176#103C2×D12712yes—
177#102—1272no—
178#102—1272no—
179#102—1272no—
180#102C2×D12712yes—
181#97—1271no—
182#97C2×D12711yescharacteristic
183C4×C8—1272no—
184#41—2544no—
185#41—2544no—
186C2×Q8—21no—
187C2×Q8—1271no—
188C2×Q8—5084no—
189C2×Q8—2542no—
190C2×Q8—2541no—
191C2×Q8C22×D12711yescharacteristic
192C2×D4—1271no—
193C2×D4—2541no—
194C2×D4—2541no—
195C2×D4—2541no—
196C2×D4—2541no—
197C2×D4C22×D12711yescharacteristic
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—1271no—
201C22×C4—2541no—
202C22×C4—2542no—
203C22×C4—2541no—
204M16—2544no—
205C2×C8—1272no—
206C2×C8C22×D12712yes—
207C4⋊C4—21no—
208C4⋊C4—1272no—
209C4⋊C4—2542no—
210C4⋊C4—2542no—
211C4⋊C4—2541no—
212C4⋊C4—2541no—
213C4⋊C4—2541no—
214C4⋊C4C22×D12712yes—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—2541no—
218C22⋊C4—2541no—
219C42—1271no—
220C42—1271no—
221C42—1271no—
222C42C22×D12711yescharacteristic
223C23—21no—
224C23—1271no—
225C23—2541no—
226Q8—22no—
227Q8—42no—
228Q8—5082no—
229Q8—5084no—
230Q8—2542no—
231D4—22no—
232D4—5082no—
233D4—5082no—
234D4—5082no—
235D4—5082no—
236D4—2542no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—1271no—
240C2×C4—1271no—
241C2×C4—1271no—
242C2×C4—1274no—
243C2×C4—5082no—
244C2×C4—5084no—
245C2×C4—5082no—
246C2×C4—2542no—
247C2×C4—2542no—
248C2×C4—2541no—
249C2×C4—2542no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—22no—
254C8—2542no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—1272no—
259C22—5081no—
260C22—5081no—
261C22—5081no—
262C22—2541no—
263C22—2541no—
264C22C2×D4×D12711yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—42no—
269C4—1272no—
270C4—1272no—
271C4—1272no—
272C4—5081no—
273C4—5082no—
274C4#41×D12712yes—
275C2—81no—
276C2—10161no—
277C2—2542no—
278C2#124×D12711yescharacteristic
279C2AGL(1,ℤ/8)×D12711yescharacteristic
280C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#489×C127C2
socleC22×C127C22×D4
prime radical#22550425C2
hypercenter#489D127
hypocenterC127#3141
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D127C2semidirect
3. #22550238C2semidirect
4. #22550236C2semidirect
5. #22550235C2semidirect
6. #22550225C2semidirect
7. #22550221C2semidirect
8. #22549876C2semidirect
9. #22549874C2semidirect
10. #22549828C2semidirect
11. #489×C127C2semidirect
141. C127#3141semidirect
147. #489D127semidirect

Maximal subgroups

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

subgroupindex
#225504252
C4⋊Q8×D1272
#225502382
#225502362
#225502352
#225502252
#225502212
#225498762
#225498742
#225498282
#489×C1272
#3141127