Finite Group Databasebeta

← #30648524

Subgroups — #30648524

Order 16256.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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×D4×Dic127C211yesmaximal, characteristic
3#22550178C212yesmaximal, complemented
4#22550174C212yesmaximal, complemented
5#22550164C212yesmaximal, complemented
6#22550154C212yesmaximal, complemented
7#22549919C211yesmaximal, characteristic, complemented
8#22549918C211yesmaximal, characteristic, complemented
9#22549899C211yesmaximal, characteristic, complemented
10#22549881C211yesmaximal, characteristic
11#418×C127C211yesmaximal, characteristic
12C23×Dic127—21no—
13C2×#14523232C2211yescharacteristic
14#16861208C2211yescharacteristic, complemented
15#16861208C2211yescharacteristic, complemented
16#16861200C2211yescharacteristic, complemented
17#16861200C2211yescharacteristic, complemented
18#16861186—21no—
19#16861186—21no—
20C4×Dic254—21no—
21C4×Dic254—21no—
22D4×Dic127—21no—
23D4×Dic127—21no—
24D4×Dic127C2212yes—
25D4×Dic127C2212yes—
26#16861168C2211yescharacteristic
27C2×C4⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861143C2212yescomplemented
31#16861142C2212yescomplemented
32#16861141C2212yescomplemented
33#16861140C2212yescomplemented
34Dic127⋊C8C2212yescomplemented
35Dic127⋊C8C2212yescomplemented
36#16861130C2212yes—
37#16861130C2212yes—
38C22×D4×C127C2211yescharacteristic
39C2×M16×C127C2211yescharacteristic
40#117×C127C2211yescharacteristic
41#102×C127C2212yes—
42#102×C127C2212yes—
43C2×Dic254—21no—
44C2×Dic254—21no—
45C22×Dic127—22no—
46C22×Dic127—22no—
47C22×Dic127—21no—
48C22×Dic127—44no—
49C22×Dic127D411yescharacteristic
50C22×Dic127D411yescharacteristic
51#14523232—22no—
52C2×C127⋊C8C2312yes—
53C4⋊Dic127—21no—
54C4⋊Dic127—21no—
55C4⋊Dic127—21no—
56C4⋊Dic127C2311yescharacteristic
57C4⋊Dic127C2311yescharacteristic
58C4⋊Dic127C2312yes—
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63C22⋊Dic127—22no—
64C22⋊Dic127—22no—
65C22⋊Dic127—21no—
66C4×Dic127—21no—
67C4×Dic127D411yescharacteristic
68C4×Dic127D411yescharacteristic
69C4×Dic127C2311yescharacteristic
70C4×Dic127C2311yescharacteristic
71C24×C127—21no—
72C2×D4×C127—21no—
73C2×D4×C127—22no—
74C2×D4×C127Q814yescomplemented
75C2×D4×C127C2312yes—
76C22×C4×C127C2311yescharacteristic
77M16×C127—22no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2311yescharacteristic
81C22⋊C4×C127—21no—
82C42×C127—21no—
83Dic254—42no—
84Dic254—42no—
85C2×Dic127—22no—
86C2×Dic127—21no—
87C2×Dic127—21no—
88C2×Dic127—22no—
89C2×Dic127—41no—
90C2×Dic127—44no—
91C2×Dic127—42no—
92C2×Dic127—44no—
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C2×Dic127C2×D411yescharacteristic
97C127⋊C8—22no—
98C23×C127—24no—
99C23×C127—22no—
100C23×C127—42no—
101C23×C127—42no—
102C23×C127C2×D411yescharacteristic
103D4×C127—24no—
104D4×C127—22no—
105D4×C127C2×Q814yes—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
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×C127—24no—
116C22×C127—48no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127C22⋊Q812yes—
120C22×C127C22×D411yescharacteristic
121C4×C127—41no—
122C4×C127—41no—
123C4×C127C22⋊Q812yes—
124C4×C127C22×Q811yescharacteristic
125C4×C127C2×#4111yescharacteristic
126Dic127—21no—
127Dic127—21no—
128Dic127—21no—
129Dic127—42no—
130Dic127—41no—
131Dic127—41no—
132Dic127AGL(1,ℤ/8)12yes—
133C2×C127—22no—
134C2×C127—44no—
135C2×C127C2×C22⋊Q811yescharacteristic
136C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C127#57511yescharacteristic
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#428—1271no—
148#423—1271no—
149#418—1271no—
150#418D12711yescharacteristic
151C22×D4C2×D12711yescharacteristic
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⋊C4—1271no—
170C2×C22⋊C4—2541no—
171C2×C42—1271no—
172#107—1272no—
173#107—1272no—
174#106—1272no—
175#106—1272no—
176C4⋊C8—1272no—
177C4⋊C8—1272no—
178#102—1272no—
179#102—1272no—
180#102C2×D12712yes—
181#102C2×D12712yes—
182C24—21no—
183C2×Q8—2541no—
184C2×Q8—2541no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic25414yescomplemented
188C2×D4C22×D12712yes—
189C22×C4—1271no—
190C22×C4—1271no—
191C22×C4—5084no—
192C22×C4—2542no—
193C22×C4—2542no—
194C22×C4—2541no—
195C22×C4C22×D12711yescharacteristic
196M16—22no—
197M16—2542no—
198C2×C8—1272no—
199C2×C8C22×D12712yes—
200C4⋊C4—1271no—
201C4⋊C4—1271no—
202C4⋊C4—1272no—
203C4⋊C4—2541no—
204C4⋊C4—2541no—
205C4⋊C4—2541no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4—2541no—
210C4⋊C4C22×D12711yescharacteristic
211C4⋊C4C22×D12711yescharacteristic
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—24no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23D4×D12711yescharacteristic
227Q8—5082no—
228Q8—5082no—
229D4—24no—
230D4—22no—
231D4C2×Dic25414yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—1271no—
235C2×C4—1271no—
236C2×C4—1271no—
237C2×C4—1271no—
238C2×C4—5084no—
239C2×C4—5082no—
240C2×C4—5084no—
241C2×C4—5081no—
242C2×C4—2542no—
243C2×C4—2541no—
244C2×C4—2541no—
245C2×C4—2542no—
246C2×C4D4×D12711yescharacteristic
247C2×C4C2×Dic25412yes—
248C2×C4#1452324712yes—
249C2×C4C23×D12711yescharacteristic
250C8—22no—
251C8—2542no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#1686118612yes—
259C22C2×D4×D12711yescharacteristic
260C4—41no—
261C4—41no—
262C4—1272no—
263C4—5082no—
264C4—5081no—
265C4—5081no—
266C4—2541no—
267C4—2541no—
268C4—2541no—
269C4#1686118612yes—
270C4C22×Dic25411yescharacteristic
271C4C2×#1452324711yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#1686118611yescharacteristic
275C2AGL(1,ℤ/8)×D12711yescharacteristic
276C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×C127C22
hypercenter#418D127
hypocenterC127#3077
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #22550429C2semidirect
3. #22550178C2semidirect
4. #22550174C2semidirect
5. #22550164C2semidirect
6. #22550154C2semidirect
7. #22549919C2semidirect
8. #22549918C2semidirect
9. #22549899C2semidirect
14. #16861208C22semidirect
15. #16861208C22semidirect
16. #16861200C22semidirect
17. #16861200C22semidirect
30. #16861143C22semidirect
31. #16861142C22semidirect
32. #16861141C22semidirect
33. #16861140C22semidirect
34. Dic127⋊C8C22semidirect
35. Dic127⋊C8C22semidirect
74. C2×D4×C127Q8semidirect
139. C127#3077semidirect
187. C2×D4Dic254semidirect

Maximal subgroups

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

subgroupindex
#225504292
C2×D4×Dic1272
#225501782
#225501742
#225501642
#225501542
#225499192
#225499182
#225498992
#225498812
#418×C1272
#3077127