Finite Group Databasebeta

← #30648337

Subgroups — #30648337

Order 16256.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#16861228C211yesmaximal, characteristic, complemented
2C2×M16×D127C211yesmaximal, characteristic, complemented
3#22550386C212yesmaximal, complemented
4#22550135C211yesmaximal, characteristic, complemented
5#22550134C211yesmaximal, characteristic, complemented
6#22550126C211yesmaximal, characteristic, complemented
7#22550121C212yesmaximal, complemented
8#22550120C212yesmaximal, complemented
9#22549911C211yesmaximal, characteristic, complemented
10#22549869C212yesmaximal, complemented
11#469×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yes—
13C22×C4×D127C2211yescharacteristic
14#16861228—24no—
15C2×Dic508C2211yescharacteristic, complemented
16C2×#14523236C2212yescomplemented
17C2×#14523232C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×#14523231C2211yescharacteristic
20M16×D127—24no—
21C2×C8×D127C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861191—22no—
24#16861189C2212yes—
25#16861188C2212yes—
26#16861180C2212yes—
27#16861177—22no—
28D4×Dic127—22no—
29(C22⋊D127)⋊C4—22no—
30#16861167C2212yes—
31#16861142C2212yescomplemented
32#16861141C2211yescharacteristic, complemented
33#16861134C2212yescomplemented
34#16861132C2212yescomplemented
35#16861130C2212yescomplemented
36C2×D8×C127C2211yescharacteristic, complemented
37C2×M16×C127C2211yescharacteristic
38#121×C127C2212yes—
39#106×C127C2211yescharacteristic, complemented
40#102×C127C2212yescomplemented
41C23×D127D411yescharacteristic
42#14523247—24no—
43#14523247—24no—
44C2×Dic254C2312yes—
45C2×C22⋊D127—22no—
46C2×C22⋊D127—22no—
47C22×Dic127—22no—
48C22×Dic127D411yescharacteristic
49C2×C4×D127—22no—
50C2×C4×D127—22no—
51C2×C4×D127D411yescharacteristic
52C2×C4×D127D411yescharacteristic
53C2×C4×D127C2311yescharacteristic
54C2×C4×D127C2311yescharacteristic
55Dic508—22no—
56#14523236—24no—
57#14523232—22no—
58#14523231—22no—
59#14523231—22no—
60C2×C127⋊C8C2311yescharacteristic
61C2×C127⋊C8C2311yescharacteristic
62C8×D127—22no—
63C8×D127D414yescomplemented
64C4⋊Dic127—22no—
65C4⋊Dic127C2312yes—
66Dic127⋊C4—22no—
67C22⋊Dic127—22no—
68C22⋊Dic127—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71C4×Dic127—22no—
72C2×D4×C127—22no—
73C2×D4×C127C2312yes—
74C22×C4×C127C2311yescharacteristic
75D8×C127—22no—
76M16×C127—22no—
77C2×C8×C127C2311yescharacteristic
78C2×C8×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—22no—
81C22×D127—22no—
82C22×D127—42no—
83C22×D127C2×D411yescharacteristic
84C22×D127C2×D411yescharacteristic
85Dic254—22no—
86Dic254—22no—
87C22⋊D127—44no—
88C22⋊D127—44no—
89C2×Dic127—22no—
90C2×Dic127—22no—
91C2×Dic127—22no—
92C2×Dic127—44no—
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C4×D127—22no—
96C4×D127—21no—
97C4×D127—21no—
98C4×D127—22no—
99C4×D127C2×D412yes—
100C4×D127#4112yes—
101C127⋊C8—21no—
102C127⋊C8C2×D412yes—
103C23×C127—22no—
104C23×C127C2×D411yescharacteristic
105D4×C127—22no—
106D4×C127—22no—
107D4×C127—44no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C8×C127—21no—
114C8×C127C2×D412yes—
115C22×C127—21no—
116C22×C127—22no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127C22×D411yescharacteristic
121C2×D127—22no—
122C2×D127—21no—
123C2×D127—22no—
124C2×D127—44no—
125C2×D127#12112yes—
126C4×C127—21no—
127C4×C127—42no—
128C4×C127C22×D411yescharacteristic
129C4×C127C2×#4111yescharacteristic
130Dic127—21no—
131Dic127—42no—
132Dic127—42no—
133Dic127#12112yes—
134C2×C127—41no—
135C2×C127—82no—
136C2×C127C2×#12111yescharacteristic
137C2×C127#57512yes—
138D127—22no—
139D127—41no—
140#3054—1271nomaximal, Sylow
141C127#305411yescharacteristic, complemented, Sylow
142C2×#135—1271no—
143C22×M16—1271no—
144#525—1272no—
145#470—1271no—
146#470—1271no—
147#469—1271no—
148#469D12711yescharacteristic, complemented
149#468—1272no—
150#468—1272no—
151#427—1271no—
152#416—1272no—
153C2×#41—1272no—
154C23×C4—1271no—
155#135—2544no—
156C2×Q16—1271no—
157C2×SD16—1272no—
158C2×D8C2×D12711yescharacteristic, complemented
159C2×M16—1271no—
160C2×M16—1271no—
161C2×M16—1271no—
162C2×M16—2544no—
163C2×M16C2×D12711yescharacteristic
164C22×C8—1271no—
165C22×C8—1271no—
166#123—2542no—
167C22⋊Q8—1272no—
168C22⋊Q8—1272no—
169#121—1272no—
170#121C2×D12712yes—
171#120—2542no—
172C4×D4—2542no—
173C4×D4—2542no—
174#117—1272no—
175#107—1272no—
176#106—1271no—
177#106C2×D12711yescharacteristic, complemented
178#103—1272no—
179#103—1272no—
180#102—1272no—
181#102C2×D12712yescomplemented
182C24—1271no—
183#41—2544no—
184#41—2544no—
185C2×Q8—1272no—
186C2×D4—22no—
187C2×D4—2542no—
188C2×D4—2542no—
189C2×D4C22×D12712yes—
190C22×C4—1271no—
191C22×C4—1271no—
192C22×C4—1271no—
193C22×C4—1271no—
194C22×C4—1271no—
195C22×C4—2542no—
196C22×C4—2542no—
197C22×C4—2542no—
198C22×C4C22×D12711yescharacteristic
199Q16—2542no—
200SD16—2544no—
201D8—22no—
202M16—22no—
203M16—2542no—
204M16—2542no—
205M16—2542no—
206C2×C8—1271no—
207C2×C8—1274no—
208C2×C8—1271no—
209C2×C8—2542no—
210C2×C8C22×D12711yescharacteristic
211C2×C8C22×D12711yescharacteristic
212C4⋊C4—1272no—
213C4⋊C4—2542no—
214C4⋊C4—2542no—
215C4⋊C4C22×D12712yes—
216C22⋊C4—22no—
217C22⋊C4—2542no—
218C22⋊C4—2542no—
219C22⋊C4—2542no—
220C22⋊C4—2542no—
221C42—2542no—
222C23—22no—
223C23—1271no—
224C23—1271no—
225C23—5082no—
226C23—2542no—
227C23D4×D12711yescharacteristic
228Q8—2542no—
229Q8—2542no—
230D4—22no—
231D4—22no—
232D4—44no—
233D4—5084no—
234D4—5084no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—1271no—
239C2×C4—1272no—
240C2×C4—1272no—
241C2×C4—1271no—
242C2×C4—5084no—
243C2×C4—2542no—
244C2×C4—2542no—
245C2×C4—2542no—
246C2×C4—2542no—
247C2×C4—2542no—
248C2×C4—2541no—
249C2×C4—2541no—
250C2×C4D4×D12711yescharacteristic
251C2×C4C23×D12711yescharacteristic
252C8—21no—
253C8—1272no—
254C8—2541no—
255C8D4×D12712yes—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—1272no—
262C22—5084no—
263C22—2541no—
264C22—2542no—
265C22—2542no—
266C22C2×D4×D12711yescharacteristic
267C4—21no—
268C4—42no—
269C4—1272no—
270C4—5082no—
271C4—5082no—
272C4—2541no—
273C4C2×D4×D12711yescharacteristic
274C4#41×D12711yescharacteristic
275C2—41no—
276C2—82no—
277C2—5081no—
278C2—2542no—
279C2#121×D12711yescharacteristic
280C2#2255067212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#469×C127C2
socleC22×C127C22×D4
hypercenter#469D127
hypocenterC127#3054
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#16861228C2semidirect
2. C2×M16×D127C2semidirect
3. #22550386C2semidirect
4. #22550135C2semidirect
5. #22550134C2semidirect
6. #22550126C2semidirect
7. #22550121C2semidirect
8. #22550120C2semidirect
9. #22549911C2semidirect
10. #22549869C2semidirect
11. #469×C127C2semidirect
15. C2×Dic508C22semidirect
16. C2×#14523236C22semidirect
31. #16861142C22semidirect
32. #16861141C22semidirect
33. #16861134C22semidirect
34. #16861132C22semidirect
35. #16861130C22semidirect
36. C2×D8×C127C22semidirect
39. #106×C127C22semidirect
40. #102×C127C22semidirect
63. C8×D127D4semidirect
141. C127#3054semidirect
148. #469D127semidirect
158. C2×D8C2×D127semidirect
177. #106C2×D127semidirect
181. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612282
C2×M16×D1272
#225503862
#225501352
#225501342
#225501262
#225501212
#225501202
#225499112
#225498692
#469×C1272
#3054127