Finite Group Databasebeta

← #30647608

Subgroups — #30647608

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861230C211yesmaximal, characteristic
2#22550358C211yesmaximal, characteristic, complemented
3#22550355C211yesmaximal, characteristic, complemented
4#22549986C211yesmaximal, characteristic, complemented
5#22549985C211yesmaximal, characteristic, complemented
6#22549981C211yesmaximal, characteristic, complemented
7#22549980C211yesmaximal, characteristic, complemented
8C4×#14523237C211yesmaximal, characteristic, complemented
9SD16×Dic127C211yesmaximal, characteristic, complemented
10#22549963C211yesmaximal, characteristic, complemented
11#106×D127C211yesmaximal, characteristic, complemented
12#22549876C211yesmaximal, characteristic, complemented
13#22549874C211yesmaximal, characteristic, complemented
14#22549830C211yesmaximal, characteristic, complemented
15C4×SD16×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18#16861230C418yescomplemented
19C2×#14523238C2211yescharacteristic
20C2×#14523237C2211yescharacteristic
21C2×#14523233C2211yescharacteristic
22C2×#14523231C2211yescharacteristic
23C2×C8×D127C2211yescharacteristic
24Q8×Dic127C2211yescharacteristic
25C4×Dic254C2211yescharacteristic
26#16861174C2211yescharacteristic
27C4×D508—21no—
28C4×D508C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30#16861171—21no—
31#16861171C2211yescharacteristic
32C4×C22⋊D127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861166—21no—
35#16861166C2211yescharacteristic
36#16861165—21no—
37C2×Dic127⋊C4—21no—
38C4⋊C4×D127—21no—
39C4⋊C4×D127C2211yescharacteristic
40C4⋊C4×D127C2211yescharacteristic
41#16861141C2211yescharacteristic
42#16861140C2211yescharacteristic
43#16861140C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861133C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861129C2211yescharacteristic
50#16861120C2211yescharacteristic
51#16861119C2211yescharacteristic
52C8×Dic127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C4×Q8×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56#106×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C4×C8×C127C2211yescharacteristic
60#14523249—22no—
61#14523249C2×C414yescomplemented
62#14523247—22no—
63#14523247C2×C414yescomplemented
64C2×Dic254C2311yescharacteristic
65C2×D508—21no—
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C22×Dic127—21no—
69C22×Dic127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127C2311yescharacteristic
76#14523238C2×C414yescomplemented
77#14523237C2×C414yescomplemented
78#14523233C2×C414yescomplemented
79#14523231—22no—
80C2×C127⋊C8C2311yescharacteristic
81C2×C127⋊C8C2311yescharacteristic
82C8×D127C2×C414yescomplemented
83C4⋊Dic127—21no—
84C4⋊Dic127C2311yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90Dic127⋊C4D411yescharacteristic
91Dic127⋊C4D411yescharacteristic
92C22⋊Dic127—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96(C2×D127)⋊C4D411yescharacteristic
97C4×Dic127—21no—
98C4×Dic127—21no—
99C4×Dic127—21no—
100C4×Dic127C2311yescharacteristic
101C2×Q8×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104SD16×C127C2×C414yescomplemented
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127—21no—
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C42×C127—21no—
112C42×C127C2311yescharacteristic
113C22×D127—21no—
114C22×D127C2×D411yescharacteristic
115Dic254—21no—
116Dic254C22×C412yes—
117D508—24no—
118D508—21no—
119D508C22×C412yes—
120C22⋊D127—24no—
121C2×Dic127—21no—
122C2×Dic127—24no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—81no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—24no—
130C4×D127—81no—
131C4×D127—81no—
132C4×D127—81no—
133C4×D127—81no—
134C4×D127C22×C412yes—
135C4×D127#4112yes—
136C127⋊C8—21no—
137C127⋊C8C22×C412yes—
138C23×C127—21no—
139Q8×C127—21no—
140Q8×C127C22×C412yes—
141D4×C127—21no—
142D4×C127C22×C412yes—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—81no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127C22×C412yes—
152C22×C127—22no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—22no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127C4×D412yes—
162C4×C127—22no—
163C4×C127—21no—
164C4×C127—21no—
165C4×C127—41no—
166C4×C127—41no—
167C4×C127C23×C411yescharacteristic
168C4×C127C2×#4111yescharacteristic
169Dic127—22no—
170Dic127—21no—
171Dic127—21no—
172Dic127—41no—
173Dic127—41no—
174Dic127C4×D412yes—
175C2×C127—42no—
176C2×C127C2×C4×D411yescharacteristic
177C2×C127#57711yescharacteristic
178C2×C127#57711yescharacteristic
179D127—22no—
180D127—42no—
181#2952—1271nomaximal, Sylow
182C127#295211yescharacteristic, complemented, Sylow
183C2×#135—1271no—
184#520—1271no—
185#520—1271no—
186#442—1271no—
187#442—1271no—
188#441—1271no—
189#441—1271no—
190C4×SD16—1271no—
191C4×SD16—1271no—
192C4×SD16—1271no—
193C4×SD16D12711yescharacteristic, complemented
194C2×#106—1271no—
195#417—1271no—
196#417—1271no—
197#405—1271no—
198C2×#41—1271no—
199C2×#41—1271no—
200#135—1278no—
201C2×Q16—1271no—
202C2×SD16—1271no—
203C2×SD16C2×D12711yescharacteristic
204C2×D8—1271no—
205C2×M16—1271no—
206C22×C8—1271no—
207C4×Q8—1271no—
208C4×Q8—1271no—
209C4×Q8—1271no—
210C4×Q8C2×D12711yescharacteristic
211C4×D4—1271no—
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—2541no—
215C4×D4—2541no—
216C4×D4—2541no—
217C4×D4—2541no—
218C4×D4C2×D12711yescharacteristic
219#117—1271no—
220#117—2541no—
221#117—2541no—
222C2×C4⋊C4—1271no—
223C2×C4⋊C4—1271no—
224C2×C4⋊C4—2541no—
225C2×C4⋊C4—2541no—
226#106—1271no—
227#106—1271no—
228#106—1271no—
229#106C2×D12711yescharacteristic
230#103—1271no—
231#103—1271no—
232#103—1271no—
233#103C2×D12711yescharacteristic
234#102—1271no—
235#102—1271no—
236#102—1271no—
237#102C2×D12711yescharacteristic
238#97—1271no—
239#97—1271no—
240C4×C8—1271no—
241C4×C8C2×D12711yescharacteristic
242#41—1274no—
243#41—1274no—
244#41—2542no—
245#41—2542no—
246C2×Q8—1271no—
247C2×Q8C22×D12711yescharacteristic
248C2×D4—1271no—
249C2×D4—2541no—
250C2×D4—2541no—
251C2×D4C22×D12711yescharacteristic
252C22×C4—21no—
253C22×C4—1271no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4—2541no—
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4—2541no—
261Q16—1274no—
262SD16—1274no—
263SD16C4×D12714yescomplemented
264D8—1274no—
265M16—2542no—
266C2×C8—1271no—
267C2×C8—1274no—
268C2×C8—1271no—
269C2×C8C22×D12711yescharacteristic
270C2×C8C22×D12711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—1271no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4C22×D12711yescharacteristic
282C4⋊C4C22×D12711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—1271no—
285C22⋊C4—1271no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C42—21no—
290C42—1271no—
291C42—2541no—
292C42—2541no—
293C42—2541no—
294C42C22×D12711yescharacteristic
295C23—21no—
296C23—1271no—
297C23—2541no—
298Q8—21no—
299Q8—1272no—
300Q8—2541no—
301Q8C2×C4×D12712yes—
302D4—21no—
303D4—1272no—
304D4—2544no—
305D4—2544no—
306D4—2541no—
307D4C2×C4×D12712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—1271no—
313C2×C4—1272no—
314C2×C4—1272no—
315C2×C4—1271no—
316C2×C4—1271no—
317C2×C4—10161no—
318C2×C4—10161no—
319C2×C4—10161no—
320C2×C4—10161no—
321C2×C4—10161no—
322C2×C4—2544no—
323C2×C4—2544no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4D4×D12711yescharacteristic
328C2×C4D4×D12711yescharacteristic
329C2×C4C23×D12711yescharacteristic
330C8—21no—
331C8—1272no—
332C8—2541no—
333C8C2×C4×D12712yes—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—1272no—
338C22—5081no—
339C22—5081no—
340C22—2542no—
341C22—2541no—
342C22—2541no—
343C22C2×D4×D12711yescharacteristic
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—1272no—
350C4—5081no—
351C4—5081no—
352C4—2542no—
353C4—2541no—
354C4—2541no—
355C4C22×C4×D12711yescharacteristic
356C4#41×D12711yescharacteristic
357C2—42no—
358C2—5082no—
359C2—2542no—
360C2C4×D4×D12711yescharacteristic
361C2#2255068211yescharacteristic
362C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
FittingC4×SD16×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861230C2
cosocleC2×C4×C127C24
hypercenterC4×SD16D127
hypocenterC127#2952
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #22550358C2semidirect
3. #22550355C2semidirect
4. #22549986C2semidirect
5. #22549985C2semidirect
6. #22549981C2semidirect
7. #22549980C2semidirect
8. C4×#14523237C2semidirect
9. SD16×Dic127C2semidirect
10. #22549963C2semidirect
11. #106×D127C2semidirect
12. #22549876C2semidirect
13. #22549874C2semidirect
14. #22549830C2semidirect
15. C4×SD16×C127C2semidirect
18. #16861230C4semidirect
61. #14523249C2×C4semidirect
63. #14523247C2×C4semidirect
76. #14523238C2×C4semidirect
77. #14523237C2×C4semidirect
78. #14523233C2×C4semidirect
82. C8×D127C2×C4semidirect
104. SD16×C127C2×C4semidirect
182. C127#2952semidirect
193. C4×SD16D127semidirect
263. SD16C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612302
#225503582
#225503552
#225499862
#225499852
#225499812
#225499802
C4×#145232372
SD16×Dic1272
#225499632
#106×D1272
#225498762
#225498742
#225498302
C4×SD16×C1272
#2952127