Finite Group Databasebeta

← #30648526

Subgroups — #30648526

Order 16256.

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

normal subgroups only

This group has 440 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
1#22550429C211yesmaximal, characteristic, complemented
2C2×#16861171C211yesmaximal, characteristic
3#22550179C212yesmaximal, complemented
4#22550175C212yesmaximal, complemented
5#22550165C212yesmaximal, complemented
6#22550155C212yesmaximal, complemented
7#22549916C211yesmaximal, characteristic, complemented
8#22549899C211yesmaximal, characteristic, complemented
9#22549882C211yesmaximal, characteristic
10#22549880C211yesmaximal, characteristic
11#428×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C4×D127—21no—
14C2×#14523232C2211yescharacteristic
15#16861208C2211yescharacteristic, complemented
16#16861208C2211yescharacteristic, complemented
17#16861200C2211yescharacteristic, complemented
18#16861200C2211yescharacteristic, complemented
19#16861186—21no—
20#16861186—21no—
21C4×Dic254—21no—
22C4×Dic254—21no—
23#16861171—21no—
24#16861171—21no—
25#16861171C2212yes—
26#16861171C2212yes—
27#16861168C2211yescharacteristic
28C2×(C2×D127)⋊C4—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861142C2212yescomplemented
31#16861140C2212yescomplemented
32Dic127⋊C8C2212yescomplemented
33Dic127⋊C8C2212yescomplemented
34#16861131C2212yes—
35#16861131C2212yes—
36#16861129C2212yes—
37#16861129C2212yes—
38C2×M16×C127C2211yescharacteristic
39#117×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#107×C127C2212yescomplemented
42#106×C127C2212yescomplemented
43C23×D127—21no—
44C2×Dic254—21no—
45C2×Dic254—21no—
46C2×D508—21no—
47C2×D508—22no—
48C2×D508Q814yescomplemented
49C2×D508C2312yes—
50C22×Dic127D411yescharacteristic
51C22×Dic127D411yescharacteristic
52C2×C4×D127—22no—
53C2×C4×D127—22no—
54C2×C4×D127—44no—
55#14523232—22no—
56C2×C127⋊C8C2312yes—
57C4⋊Dic127—21no—
58C4⋊Dic127—21no—
59C4⋊Dic127C2311yescharacteristic
60C4⋊Dic127C2311yescharacteristic
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64Dic127⋊C4—21no—
65C22⋊Dic127—21no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—22no—
68C4×Dic127—21no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C4×Dic127C2311yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127C2311yescharacteristic
75M16×C127—22no—
76C2×C8×C127C2312yes—
77C4⋊C4×C127—21no—
78C4⋊C4×C127C2311yescharacteristic
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C42×C127—21no—
83C22×D127—24no—
84C22×D127—22no—
85C22×D127—42no—
86C22×D127—42no—
87Dic254—42no—
88Dic254—42no—
89D508—24no—
90D508—22no—
91D508C2×Q814yes—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—21no—
95C2×Dic127—41no—
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C2×Dic127C2×D411yescharacteristic
100C4×D127—44no—
101C4×D127—44no—
102C127⋊C8—22no—
103C23×C127C2×D411yescharacteristic
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—42no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×Q812yes—
110C2×C4×C127#4112yes—
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127C22⋊Q812yes—
116C22×C127C22×D411yescharacteristic
117C2×D127—24no—
118C2×D127—48no—
119C2×D127—42no—
120C2×D127—42no—
121C4×C127—42no—
122C4×C127—41no—
123C4×C127—41no—
124C4×C127C22⋊Q812yes—
125C4×C127C22×Q811yescharacteristic
126C4×C127C2×#4111yescharacteristic
127Dic127—21no—
128Dic127—21no—
129Dic127—21no—
130Dic127—41no—
131Dic127—41no—
132Dic127AGL(1,ℤ/8)12yes—
133C2×C127—22no—
134C2×C127C2×C22⋊Q811yescharacteristic
135C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C127#57511yescharacteristic
137D127—44no—
138#3077—1271nomaximal, Sylow
139C127#307711yescharacteristic, complemented, Sylow
140#533—1271no—
141C2×C4×D4—1271no—
142#478—1272no—
143#478—1272no—
144#476—1272no—
145#474—1272no—
146#428—1271no—
147#428D12711yescharacteristic, complemented
148#423—1271no—
149#418—1271no—
150#418—1271no—
151C22×D4—1271no—
152C23×C4—2541no—
153C2×M16—1271no—
154C2×M16C2×D12711yescharacteristic
155C4⋊Q8—1271no—
156C4⋊Q8—1271no—
157#125—1271no—
158#125—1271no—
159C22⋊Q8—2541no—
160C22⋊Q8—2541no—
161C4×Q8—2541no—
162C4×Q8—2541no—
163C4×D4—1272no—
164C4×D4—1272no—
165C4×D4—2541no—
166C4×D4—2541no—
167#117—1271no—
168#117C2×D12711yescharacteristic
169C2×C4⋊C4C2×D12711yescharacteristic
170C2×C22⋊C4—2541no—
171C2×C42—1271no—
172#107—1272no—
173#107C2×D12712yescomplemented
174#106—1272no—
175#106C2×D12712yescomplemented
176C4⋊C8—1272no—
177C4⋊C8—1272no—
178#102—1272no—
179#102—1272no—
180#102—1272no—
181#102—1272no—
182C24—2541no—
183C2×Q8—2541no—
184C2×Q8—2541no—
185C2×D4—1274no—
186C2×D4—1272no—
187C2×D4—2541no—
188C2×D4—2542no—
189C22×C4—21no—
190C22×C4—1271no—
191C22×C4—1271no—
192C22×C4—5084no—
193C22×C4—2542no—
194C22×C4—2542no—
195C22×C4C22×D12711yescharacteristic
196M16—22no—
197M16—2542no—
198C2×C8—1272no—
199C2×C8C22×D12712yes—
200C4⋊C4—21no—
201C4⋊C4—1271no—
202C4⋊C4—1271no—
203C4⋊C4—2541no—
204C4⋊C4—2541no—
205C4⋊C4—2541no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4C22×D12711yescharacteristic
210C4⋊C4C22×D12711yescharacteristic
211C4⋊C4C22×D12712yes—
212C22⋊C4—21no—
213C22⋊C4—2542no—
214C22⋊C4—2542no—
215C22⋊C4—2541no—
216C42—21no—
217C42—1271no—
218C42—1271no—
219C42—1271no—
220C42—1271no—
221C42—2541no—
222C23—5082no—
223C23—5082no—
224C23—2544no—
225C23—2542no—
226C23D4×D12711yescharacteristic
227Q8—5082no—
228Q8—5082no—
229D4—1274no—
230D4—2544no—
231D4—2542no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—1271no—
237C2×C4—1271no—
238C2×C4—1271no—
239C2×C4—1271no—
240C2×C4—5084no—
241C2×C4—5084no—
242C2×C4—5081no—
243C2×C4—2542no—
244C2×C4—2541no—
245C2×C4—2541no—
246C2×C4D4×D12711yescharacteristic
247C2×C4Q8×D12712yes—
248C2×C4#1452324812yes—
249C2×C4C23×D12711yescharacteristic
250C8—22no—
251C8—2542no—
252C22—21no—
253C22—21no—
254C22—5088no—
255C22—5082no—
256C22—5082no—
257C22—2544no—
258C22#1686118712yes—
259C22C2×D4×D12711yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—1272no—
264C4—5081no—
265C4—5081no—
266C4—2541no—
267C4—2541no—
268C4—2541no—
269C4#1686118712yes—
270C4C2×Q8×D12711yescharacteristic
271C4C2×#1452324811yescharacteristic
272C2—22no—
273C2—5084no—
274C2C2×#1686118711yescharacteristic
275C2AGL(1,ℤ/8)×D12711yescharacteristic
276C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
prime radicalC22×D508C22
hypercenter#428D127
hypocenterC127#3077
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #22550429C2semidirect
3. #22550179C2semidirect
4. #22550175C2semidirect
5. #22550165C2semidirect
6. #22550155C2semidirect
7. #22549916C2semidirect
8. #22549899C2semidirect
11. #428×C127C2semidirect
15. #16861208C22semidirect
16. #16861208C22semidirect
17. #16861200C22semidirect
18. #16861200C22semidirect
30. #16861142C22semidirect
31. #16861140C22semidirect
32. Dic127⋊C8C22semidirect
33. Dic127⋊C8C22semidirect
41. #107×C127C22semidirect
42. #106×C127C22semidirect
48. C2×D508Q8semidirect
139. C127#3077semidirect
147. #428D127semidirect
173. #107C2×D127semidirect
175. #106C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504292
C2×#168611712
#225501792
#225501752
#225501652
#225501552
#225499162
#225498992
#225498822
#225498802
#428×C1272
#3077127