Finite Group Databasebeta

← #30648343

Subgroups — #30648343

Order 16256.

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

normal subgroups only

This group has 548 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
1C2×M16×D127C211yesmaximal, characteristic, complemented
2#22550383C211yesmaximal, characteristic, complemented
3C2×#16861180C211yesmaximal, characteristic, complemented
4#22550126C212yesmaximal, complemented
5#22550119C212yesmaximal, complemented
6#22550109C212yesmaximal, complemented
7#22550102C212yesmaximal, complemented
8#22549918C211yesmaximal, characteristic, complemented
9#22549881C211yesmaximal, characteristic, complemented
10#22549877C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127—21no—
13C22×C4×D127C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C2×#14523231C2212yescomplemented
16M16×D127—24no—
17C2×C8×D127C2212yescomplemented
18#16861190—21no—
19#16861185C2211yescharacteristic, complemented
20#16861185C2211yescharacteristic, complemented
21#16861180—21no—
22#16861180—21no—
23#16861180C2212yes—
24#16861180C2212yes—
25#16861180C2211yescharacteristic, complemented
26#16861180C2211yescharacteristic, complemented
27#16861177—21no—
28C4×C22⋊D127—21no—
29C4×C22⋊D127—21no—
30#16861168C2211yescharacteristic
31C2×C4⋊Dic127C2211yescharacteristic
32C2×C22⋊Dic127—21no—
33#16861143C2212yescomplemented
34#16861141C2212yescomplemented
35#16861133C2212yescomplemented
36#16861130C2212yescomplemented
37#16861130C2212yescomplemented
38#16861130C2212yescomplemented
39C2×#41×C127C2211yescharacteristic
40C22×D4×C127C2211yescharacteristic
41AGL(1,ℤ/8)×C127—24no—
42C2×SD16×C127C2212yescomplemented
43C2×D8×C127C2212yescomplemented
44C2×M16×C127C2211yescharacteristic
45C23×D127D411yescharacteristic
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—21no—
49C2×C22⋊D127—21no—
50C2×C22⋊D127—44no—
51C22×Dic127—21no—
52C22×Dic127D411yescharacteristic
53C2×C4×D127—22no—
54C2×C4×D127—22no—
55C2×C4×D127D411yescharacteristic
56C2×C4×D127D411yescharacteristic
57C2×C4×D127C2311yescharacteristic
58C2×C4×D127C2311yescharacteristic
59#14523232—22no—
60#14523231—24no—
61C2×C127⋊C8C2312yes—
62C8×D127—24no—
63C4⋊Dic127—21no—
64C4⋊Dic127C2311yescharacteristic
65C4⋊Dic127C2311yescharacteristic
66C4⋊Dic127C2312yes—
67Dic127⋊C4—21no—
68Dic127⋊C4—21no—
69C22⋊Dic127—21no—
70C22⋊Dic127—21no—
71C22⋊Dic127—22no—
72C22⋊Dic127—22no—
73C22⋊Dic127—21no—
74(C2×D127)⋊C4—21no—
75(C2×D127)⋊C4—21no—
76C4×Dic127—21no—
77C24×C127—21no—
78#41×C127—22no—
79#41×C127—22no—
80C2×Q8×C127C2311yescharacteristic
81C2×D4×C127—21no—
82C2×D4×C127—22no—
83C2×D4×C127—21no—
84C2×D4×C127—22no—
85C2×D4×C127C2311yescharacteristic
86C2×D4×C127C2312yes—
87C22×C4×C127—21no—
88C22×C4×C127C2311yescharacteristic
89SD16×C127—24no—
90D8×C127—24no—
91M16×C127D414yescomplemented
92C2×C8×C127C2312yes—
93C22×D127—22no—
94C22×D127—22no—
95C22×D127—41no—
96C22×D127C2×D411yescharacteristic
97C22×D127C2×D411yescharacteristic
98C22⋊D127—42no—
99C22⋊D127—44no—
100C22⋊D127—42no—
101C22⋊D127—44no—
102C2×Dic127—22no—
103C2×Dic127—21no—
104C2×Dic127—21no—
105C2×Dic127—22no—
106C2×Dic127—42no—
107C2×Dic127C2×D411yescharacteristic
108C2×Dic127C2×D411yescharacteristic
109C4×D127—22no—
110C4×D127—22no—
111C4×D127—21no—
112C4×D127—21no—
113C4×D127—21no—
114C4×D127—21no—
115C127⋊C8—22no—
116C23×C127—21no—
117C23×C127—22no—
118C23×C127—42no—
119C23×C127—42no—
120C23×C127—42no—
121C23×C127C2×D411yescharacteristic
122Q8×C127—21no—
123Q8×C127—21no—
124D4×C127—21no—
125D4×C127—22no—
126D4×C127—22no—
127D4×C127—21no—
128D4×C127—42no—
129D4×C127—42no—
130C2×C4×C127—21no—
131C2×C4×C127—42no—
132C2×C4×C127C2×D411yescharacteristic
133C2×C4×C127C2×D412yes—
134C2×C4×C127#4112yes—
135C2×C4×C127C2411yescharacteristic
136C8×C127C2×D414yescomplemented
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—41no—
140C22×C127—42no—
141C22×C127—42no—
142C22×C127—41no—
143C22×C127—42no—
144C22×C127—41no—
145C22×C127—84no—
146C22×C127#12112yes—
147C22×C127C22×D411yescharacteristic
148C2×D127—21no—
149C2×D127—21no—
150C2×D127—21no—
151C2×D127—21no—
152C2×D127—21no—
153C2×D127—44no—
154C2×D127AGL(1,ℤ/8)12yescomplemented
155C4×C127—41no—
156C4×C127#12112yes—
157C4×C127C22×D411yescharacteristic
158C4×C127C2×#4111yescharacteristic
159Dic127—21no—
160Dic127—21no—
161Dic127—42no—
162Dic127—41no—
163Dic127—41no—
164C2×C127—22no—
165C2×C127—81no—
166C2×C127—82no—
167C2×C127C2×#12111yescharacteristic
168C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C127#57511yescharacteristic
170D127—22no—
171D127—41no—
172#3055—1271nomaximal, Sylow
173C127#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
175C22×M16—1271no—
176#525—1271no—
177C2×#121—1271no—
178#469—1272no—
179#468—1272no—
180#466—1272no—
181#465—1272no—
182#428—1271no—
183#418—1271no—
184#417—1271no—
185C2×#41C2×D12711yescharacteristic
186C22×D4—2541no—
187C22×D4C2×D12711yescharacteristic
188C23×C4—1271no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D12712yescomplemented
191C2×D8C2×D12712yescomplemented
192C2×M16—1271no—
193C2×M16—1272no—
194C2×M16—2544no—
195C2×M16C2×D12711yescharacteristic
196C22×C8—1272no—
197#123—2541no—
198C22⋊Q8—1271no—
199C22⋊Q8—1271no—
200#121—1272no—
201#121—1272no—
202#121—1271no—
203#121—1271no—
204#121—2541no—
205#121—2541no—
206#120—2541no—
207C4×D4—2541no—
208C4×D4—2541no—
209#117—1271no—
210C2×C4⋊C4—1271no—
211C2×C22⋊C4—2541no—
212#107—1272no—
213#106—1272no—
214#103—1272no—
215#102—1272no—
216#102—1272no—
217#102—1272no—
218C24—21no—
219C24—1271no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D12711yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—5084no—
228C2×D4—2541no—
229C2×D4—2541no—
230C2×D4—2542no—
231C2×D4—2542no—
232C2×D4C22×D12711yescharacteristic
233C2×D4C22×D12712yes—
234C22×C4—21no—
235C22×C4—1271no—
236C22×C4—1271no—
237C22×C4—1271no—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—2541no—
241C22×C4—2542no—
242C22×C4—2542no—
243C22×C4C22×D12711yescharacteristic
244SD16—24no—
245D8—24no—
246M16—2542no—
247M16—2544no—
248M16C22⋊D12714yescomplemented
249C2×C8—1272no—
250C2×C8—2544no—
251C2×C8C22×D12712yes—
252C4⋊C4—1271no—
253C4⋊C4—1271no—
254C4⋊C4—1272no—
255C4⋊C4—2541no—
256C4⋊C4—2541no—
257C4⋊C4—2541no—
258C22⋊C4—2541no—
259C22⋊C4—2541no—
260C22⋊C4—2542no—
261C22⋊C4—2542no—
262C22⋊C4—2541no—
263C22⋊C4—2541no—
264C22⋊C4—2541no—
265C42—2541no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—1271no—
272C23—1271no—
273C23—5081no—
274C23—2542no—
275C23—2542no—
276C23D4×D12711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—5084no—
286D4—5082no—
287D4—5084no—
288D4—5082no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—1271no—
292C2×C4—1271no—
293C2×C4—5082no—
294C2×C4—2541no—
295C2×C4—2541no—
296C2×C4—2542no—
297C2×C4—2542no—
298C2×C4—2541no—
299C2×C4—2541no—
300C2×C4—2541no—
301C2×C4—2541no—
302C2×C4—2542no—
303C2×C4—2542no—
304C2×C4D4×D12711yescharacteristic
305C2×C4C2×C22⋊D12712yes—
306C2×C4#1452324712yes—
307C2×C4C23×D12711yescharacteristic
308C8—2542no—
309C8C2×C22⋊D12714yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—1272no—
320C22—5084no—
321C22—2541no—
322C22—2541no—
323C22—2541no—
324C22—2541no—
325C22—2541no—
326C22#1686118012yes—
327C22C2×D4×D12711yescharacteristic
328C4—41no—
329C4—5082no—
330C4—5081no—
331C4—5081no—
332C4—2541no—
333C4—2541no—
334C4#1686118012yes—
335C4C22×C22⋊D12711yescharacteristic
336C4C2×#1452324711yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—5081no—
341C2—2542no—
342C2C2×#1686118011yescharacteristic
343C2AGL(1,ℤ/8)×D12711yescharacteristic
344C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3055
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×M16×D127C2semidirect
2. #22550383C2semidirect
3. C2×#16861180C2semidirect
4. #22550126C2semidirect
5. #22550119C2semidirect
6. #22550109C2semidirect
7. #22550102C2semidirect
8. #22549918C2semidirect
9. #22549881C2semidirect
10. #22549877C2semidirect
11. C2×AGL(1,ℤ/8)×C127C2semidirect
15. C2×#14523231C22semidirect
17. C2×C8×D127C22semidirect
19. #16861185C22semidirect
20. #16861185C22semidirect
25. #16861180C22semidirect
26. #16861180C22semidirect
33. #16861143C22semidirect
34. #16861141C22semidirect
35. #16861133C22semidirect
36. #16861130C22semidirect
37. #16861130C22semidirect
38. #16861130C22semidirect
42. C2×SD16×C127C22semidirect
43. C2×D8×C127C22semidirect
91. M16×C127D4semidirect
136. C8×C127C2×D4semidirect
154. C2×D127AGL(1,ℤ/8)semidirect
173. C127#3055semidirect
174. C2×AGL(1,ℤ/8)D127semidirect
190. C2×SD16C2×D127semidirect
191. C2×D8C2×D127semidirect
248. M16C22⋊D127semidirect
309. C8C2×C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1272
#225503832
C2×#168611802
#225501262
#225501192
#225501092
#225501022
#225499182
#225498812
#225498772
C2×AGL(1,ℤ/8)×C1272
#3055127