Finite Group Databasebeta

← #483×D127 #30648604

Subgroups — #483×D127 #30648604

Order 16256.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
2C2×C4⋊C4×D127C211yesmaximal, characteristic
3#22550208C212yesmaximal, complemented
4#22550207C211yesmaximal, characteristic, complemented
5#22550206C211yesmaximal, characteristic, complemented
6#22550205C212yesmaximal, complemented
7#22550204C211yesmaximal, characteristic, complemented
8#106×D127C212yesmaximal, complemented
9#103×D127C212yesmaximal, complemented
10C22⋊C8×D127C211yesmaximal, characteristic
11#483×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2211yescharacteristic
13C22×C4×D127—21no—
14C22×C4×D127C2211yescharacteristic
15C2×C8×D127C2212yes—
16#16861189C2211yescharacteristic
17#16861188C2211yescharacteristic
18#16861187—21no—
19#16861186—21no—
20#16861185C2211yescharacteristic
21C2×C4⋊Dic127C2211yescharacteristic
22C2×Dic127⋊C4—21no—
23C4⋊C4×D127—21no—
24C4⋊C4×D127—21no—
25C4⋊C4×D127—21no—
26C4⋊C4×D127C2212yes—
27C4⋊C4×D127C2212yes—
28C4⋊C4×D127C2211yescharacteristic
29C22⋊C4×D127—21no—
30#16861141C2212yescomplemented
31#16861140C2212yescomplemented
32#16861140C2212yescomplemented
33#16861134C2212yescomplemented
34#16861133C2212yescomplemented
35#16861132C2212yescomplemented
36C22⋊(C127⋊C8)C2211yescharacteristic
37(C2×D127)⋊C8C2212yes—
38C22⋊Q8×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#106×C127C2212yescomplemented
41#103×C127C2212yescomplemented
42C22⋊C8×C127C2211yescharacteristic
43C23×D127D411yescharacteristic
44C2×Dic254—21no—
45C2×Dic254C2311yescharacteristic
46Q8×D127—24no—
47C22×Dic127—21no—
48C22×Dic127D411yescharacteristic
49C2×C4×D127—24no—
50C2×C4×D127—22no—
51C2×C4×D127—22no—
52C2×C4×D127—21no—
53C2×C4×D127—21no—
54C2×C4×D127—44no—
55C2×C4×D127D411yescharacteristic
56C2×C4×D127D411yescharacteristic
57C2×C4×D127C2311yescharacteristic
58C2×C4×D127C2311yescharacteristic
59C2×C127⋊C8C2312yes—
60C8×D127—24no—
61C4⋊Dic127—21no—
62C4⋊Dic127—21no—
63C4⋊Dic127C2311yescharacteristic
64C4⋊Dic127C2312yes—
65Dic127⋊C4—21no—
66Dic127⋊C4—21no—
67Dic127⋊C4—22no—
68Dic127⋊C4—22no—
69Dic127⋊C4—21no—
70C22⋊Dic127—21no—
71(C2×D127)⋊C4—21no—
72(C2×D127)⋊C4—21no—
73C2×Q8×C127C2311yescharacteristic
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76C2×C8×C127C2312yes—
77C4⋊C4×C127—21no—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127SD1614yes—
85C22×D127C2×D411yescharacteristic
86C22×D127C2×D411yescharacteristic
87Dic254—22no—
88Dic254—42no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127—22no—
93C2×Dic127—42no—
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—22no—
97C4×D127—22no—
98C4×D127—22no—
99C4×D127—42no—
100C4×D127—44no—
101C4×D127—44no—
102C4×D127—42no—
103C4×D127#4114yes—
104C127⋊C8—22no—
105C23×C127C2×D411yescharacteristic
106Q8×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—42no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2411yescharacteristic
114C8×C127—22no—
115C22×C127—21no—
116C22×C127—21no—
117C22×C127C2×SD1612yes—
118C22×C127C22×D411yescharacteristic
119C2×D127—21no—
120C2×D127—21no—
121C2×D127—28no—
122C2×D127#12312yes—
123C2×D127C2×SD1612yes—
124C2×D127C2×SD1612yes—
125C2×D127#13712yes—
126C4×C127—21no—
127C4×C127—41no—
128C4×C127—42no—
129C4×C127—41no—
130C4×C127C2×#4112yes—
131Dic127—21no—
132Dic127—41no—
133Dic127—42no—
134Dic127—41no—
135Dic127#12312yes—
136C2×C127—22no—
137C2×C127C2×#12311yescharacteristic
138C2×C127C22×SD1611yescharacteristic
139C2×C127C2×#13711yescharacteristic
140D127—22no—
141D127#48314yesdirect product factor
142C2×#483—1271nomaximal, Sylow
143C127C2×#48311yescharacteristic, complemented, Sylow
144C2×C22⋊Q8—1271no—
145C22×C4⋊C4—1271no—
146#483—1271no—
147#483—1271no—
148#483—1272no—
149#483—1272no—
150#483—1271no—
151#483D12711yescharacteristic, direct product factor
152C2×#106—1272no—
153C2×#103—1272no—
154C2×C22⋊C8—1271no—
155C22×Q8—1271no—
156C23×C4—1271no—
157C23×C4—2541no—
158C22×C8—1272no—
159C22⋊Q8—1271no—
160C22⋊Q8—1271no—
161C22⋊Q8—1271no—
162C22⋊Q8—2541no—
163C22⋊Q8—2541no—
164C22⋊Q8C2×D12711yescharacteristic
165C2×C4⋊C4—1271no—
166C2×C4⋊C4—1272no—
167C2×C4⋊C4—1272no—
168C2×C4⋊C4—1271no—
169C2×C4⋊C4—2541no—
170C2×C4⋊C4—2541no—
171C2×C4⋊C4—2541no—
172C2×C4⋊C4—2541no—
173C2×C4⋊C4C2×D12711yescharacteristic
174C2×C22⋊C4—2541no—
175#106—1272no—
176#106—1272no—
177#106—1272no—
178#106C2×D12712yescomplemented
179#103—1272no—
180#103—1272no—
181#103—1272no—
182#103C2×D12712yescomplemented
183C22⋊C8—1271no—
184C22⋊C8—1272no—
185C22⋊C8C2×D12711yescharacteristic
186C24—1271no—
187C2×Q8—1271no—
188C2×Q8—2541no—
189C2×Q8—2544no—
190C2×Q8C22×D12711yescharacteristic
191C22×C4—21no—
192C22×C4—1271no—
193C22×C4—1271no—
194C22×C4—1271no—
195C22×C4—1271no—
196C22×C4—1271no—
197C22×C4—5084no—
198C22×C4—2544no—
199C22×C4—2541no—
200C22×C4—2541no—
201C22×C4—2542no—
202C22×C4—2542no—
203C22×C4—2541no—
204C22×C4C22×D12711yescharacteristic
205C2×C8—1272no—
206C2×C8—2544no—
207C2×C8C22×D12712yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—1272no—
211C4⋊C4—1271no—
212C4⋊C4—2542no—
213C4⋊C4—2542no—
214C4⋊C4—2541no—
215C4⋊C4—2541no—
216C4⋊C4—2541no—
217C4⋊C4—2541no—
218C4⋊C4—2541no—
219C4⋊C4C22×D12712yes—
220C4⋊C4C22×D12711yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—2541no—
223C22⋊C4—2541no—
224C22⋊C4—2541no—
225C23—1271no—
226C23—1274no—
227C23—1271no—
228C23—2542no—
229C23—2542no—
230C23D4×D12711yescharacteristic
231Q8—22no—
232Q8—5082no—
233Q8—2542no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—1271no—
240C2×C4—1274no—
241C2×C4—1271no—
242C2×C4—5084no—
243C2×C4—5084no—
244C2×C4—5082no—
245C2×C4—5082no—
246C2×C4—5082no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4D4×D12711yescharacteristic
255C2×C4C23×D12711yescharacteristic
256C8—22no—
257C8—2542no—
258C22—21no—
259C22—21no—
260C22—1272no—
261C22—1272no—
262C22—1272no—
263C22—1272no—
264C22—2548no—
265C22—2541no—
266C22—2541no—
267C22SD16×D12712yes—
268C22C2×D4×D12711yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—1272no—
274C4—5081no—
275C4—5082no—
276C4—5081no—
277C4—2541no—
278C4#41×D12712yes—
279C2—22no—
280C2—1274no—
281C2—2542no—
282C2#123×D12711yescharacteristic
283C2C2×SD16×D12711yescharacteristic
284C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#483×C127C2
socleC22×C127C22×D4
prime radicalC23×D127D4
hypercenter#483D127
hypocenterC127C2×#483
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #22550208C2semidirect
4. #22550207C2semidirect
5. #22550206C2semidirect
6. #22550205C2semidirect
7. #22550204C2semidirect
8. #106×D127C2semidirect
9. #103×D127C2semidirect
11. #483×C127C2semidirect
30. #16861141C22semidirect
31. #16861140C22semidirect
32. #16861140C22semidirect
33. #16861134C22semidirect
34. #16861133C22semidirect
35. #16861132C22semidirect
40. #106×C127C22semidirect
41. #103×C127C22semidirect
141. D127#483direct (N × H)
143. C127C2×#483semidirect
151. #483D127direct (N × H)
178. #106C2×D127semidirect
182. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1272
C2×C4⋊C4×D1272
#225502082
#225502072
#225502062
#225502052
#225502042
#106×D1272
#103×D1272
C22⋊C8×D1272
#483×C1272
C2×#483127