Finite Group Databasebeta

← #30648345

Subgroups — #30648345

Order 16256.

548 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861235C211yesmaximal, characteristic, complemented
2C22×#14523232C211yesmaximal, characteristic, complemented
3#22550387C211yesmaximal, characteristic, complemented
4#22550127C212yesmaximal, complemented
5#22550120C212yesmaximal, complemented
6#22550110C212yesmaximal, complemented
7#22550103C212yesmaximal, complemented
8#22549916C211yesmaximal, characteristic, complemented
9#22549881C211yesmaximal, characteristic, complemented
10#22549875C211yesmaximal, characteristic, complemented
11C2×#121×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861235—24no—
14C2×#14523236C2212yescomplemented
15C2×#14523233C2212yescomplemented
16C2×#14523232—24no—
17C2×#14523232C2211yescharacteristic
18C2×#14523232C2211yescharacteristic
19C2×#14523232C2212yescomplemented
20C22×C127⋊C8C2212yescomplemented
21#16861192—21no—
22#16861188C2211yescharacteristic, complemented
23#16861188C2211yescharacteristic, complemented
24#16861183C2211yescharacteristic, complemented
25#16861183C2211yescharacteristic, complemented
26#16861178—21no—
27C4×C22⋊D127—21no—
28C4×C22⋊D127—21no—
29#16861168C2211yescharacteristic
30#16861142C2212yescomplemented
31#16861140C2212yescomplemented
32#16861132C2212yescomplemented
33#16861130C2212yescomplemented
34#16861130C2212yescomplemented
35#16861129C2212yescomplemented
36C22×D4×C127—21no—
37C22×D4×C127C2211yescharacteristic
38C23×C4×C127C2211yescharacteristic
39#121×C127—21no—
40#121×C127—21no—
41#121×C127C2212yes—
42#121×C127C2212yes—
43C2×C4⋊C4×C127C2211yescharacteristic
44C2×C22⋊C4×C127—21no—
45#14523248—22no—
46#14523248—22no—
47C2×Dic254C2311yescharacteristic
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—21no—
50C2×C22⋊D127—21no—
51C2×C22⋊D127—21no—
52C2×C4×D127—21no—
53#14523236—24no—
54#14523233—24no—
55#14523232—24no—
56#14523232—22no—
57#14523232D414yescomplemented
58C2×C127⋊C8—24no—
59C2×C127⋊C8C2312yes—
60C2×C127⋊C8C2312yes—
61C4⋊Dic127C2311yescharacteristic
62C4⋊Dic127C2311yescharacteristic
63Dic127⋊C4—21no—
64Dic127⋊C4—21no—
65C22⋊Dic127—21no—
66C22⋊Dic127—21no—
67C22⋊Dic127—21no—
68(C2×D127)⋊C4—21no—
69(C2×D127)⋊C4—21no—
70C4×Dic127—21no—
71C24×C127—21no—
72C24×C127D411yescharacteristic
73C2×D4×C127—22no—
74C2×D4×C127—22no—
75C2×D4×C127—22no—
76C2×D4×C127—21no—
77C2×D4×C127—22no—
78C2×D4×C127—44no—
79C2×D4×C127C2312yes—
80C22×C4×C127—22no—
81C22×C4×C127—22no—
82C22×C4×C127—21no—
83C22×C4×C127D411yescharacteristic
84C22×C4×C127D411yescharacteristic
85C22×C4×C127D411yescharacteristic
86C22×C4×C127C2311yescharacteristic
87C22×C4×C127C2311yescharacteristic
88C22×C4×C127C2311yescharacteristic
89C4⋊C4×C127—21no—
90C4⋊C4×C127C2312yes—
91C22⋊C4×C127—22no—
92C22⋊C4×C127—22no—
93C22×D127—21no—
94Dic254—21no—
95Dic254—21no—
96D508—21no—
97D508—21no—
98C22⋊D127—42no—
99C22⋊D127—42no—
100C22⋊D127—42no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C4×D127—42no—
105C127⋊C8—22no—
106C127⋊C8C2×D414yescomplemented
107C23×C127—22no—
108C23×C127—22no—
109C23×C127—22no—
110C23×C127—41no—
111C23×C127—42no—
112C23×C127—42no—
113C23×C127—42no—
114C23×C127C2×D411yescharacteristic
115C23×C127C2×D411yescharacteristic
116C23×C127C2×D411yescharacteristic
117D4×C127—22no—
118D4×C127—22no—
119D4×C127—44no—
120D4×C127—44no—
121D4×C127—42no—
122C2×C4×C127—22no—
123C2×C4×C127—22no—
124C2×C4×C127—22no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—22no—
130C2×C4×C127—42no—
131C2×C4×C127C2×D412yes—
132C2×C4×C127C2×D411yescharacteristic
133C2×C4×C127C2×D411yescharacteristic
134C2×C4×C127C2×D411yescharacteristic
135C2×C4×C127#4112yes—
136C2×C4×C127C2411yescharacteristic
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—44no—
145C22×C127—42no—
146C22×C127—42no—
147C22×C127—42no—
148C22×C127—84no—
149C22×C127#12112yes—
150C22×C127AGL(1,ℤ/8)12yescomplemented
151C22×C127C22×D411yescharacteristic
152C2×D127—41no—
153C2×D127—41no—
154C2×D127—41no—
155C4×C127—21no—
156C4×C127—21no—
157C4×C127—42no—
158C4×C127#12112yes—
159C4×C127C22×D411yescharacteristic
160C4×C127C2×#4111yescharacteristic
161Dic127—41no—
162Dic127—41no—
163Dic127—41no—
164C2×C127—22no—
165C2×C127—22no—
166C2×C127—41no—
167C2×C127—82no—
168C2×C127C2×#12111yescharacteristic
169C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C127#57511yescharacteristic
171D127—81no—
172#3055—1271nomaximal, Sylow
173C127#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—1271no—
175C22×M16—1271no—
176#525—1271no—
177C2×#121D12711yescharacteristic, complemented
178#469—1272no—
179#468—1272no—
180#466—1272no—
181#465—1272no—
182#428—1271no—
183#418—1271no—
184#417—1271no—
185C2×#41—1271no—
186C22×D4—21no—
187C22×D4C2×D12711yescharacteristic
188C23×C4C2×D12711yescharacteristic
189AGL(1,ℤ/8)—2544no—
190C2×SD16—1272no—
191C2×D8—1272no—
192C2×M16—1271no—
193C2×M16—1271no—
194C2×M16—1272no—
195C2×M16—2544no—
196C22×C8—1272no—
197#123—2541no—
198C22⋊Q8—1271no—
199C22⋊Q8—1271no—
200#121—21no—
201#121—21no—
202#121—1271no—
203#121—1271no—
204#121C2×D12712yes—
205#121C2×D12712yes—
206#120—2541no—
207C4×D4—2541no—
208C4×D4—2541no—
209#117—1271no—
210C2×C4⋊C4C2×D12711yescharacteristic
211C2×C22⋊C4—21no—
212#107—1272no—
213#106—1272no—
214#103—1272no—
215#102—1272no—
216#102—1272no—
217#102—1272no—
218C24—21no—
219C24C22⋊D12711yescharacteristic
220#41—2542no—
221#41—2542no—
222C2×Q8—1271no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—21no—
227C2×D4—22no—
228C2×D4—44no—
229C2×D4—1271no—
230C2×D4—2541no—
231C2×D4—2541no—
232C2×D4—2541no—
233C2×D4C22×D12712yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—2541no—
238C22×C4C22⋊D12711yescharacteristic
239C22×C4C22⋊D12711yescharacteristic
240C22×C4C22⋊D12711yescharacteristic
241C22×C4C22×D12711yescharacteristic
242C22×C4C22×D12711yescharacteristic
243C22×C4C22×D12711yescharacteristic
244SD16—2544no—
245D8—2544no—
246M16—1274no—
247M16—2544no—
248M16—2542no—
249C2×C8—1272no—
250C2×C8—1272no—
251C2×C8—2544no—
252C4⋊C4—21no—
253C4⋊C4—1271no—
254C4⋊C4—1271no—
255C4⋊C4—2541no—
256C4⋊C4—2541no—
257C4⋊C4C22×D12712yes—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—2541no—
261C22⋊C4—2541no—
262C22⋊C4—2541no—
263C22⋊C4—2541no—
264C22⋊C4—2541no—
265C42—2541no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—2541no—
274C23C2×C22⋊D12711yescharacteristic
275C23C2×C22⋊D12711yescharacteristic
276C23C2×C22⋊D12711yescharacteristic
277Q8—2541no—
278Q8—2541no—
279D4—22no—
280D4—22no—
281D4—44no—
282D4—44no—
283D4—42no—
284D4—5082no—
285D4—5082no—
286D4—5082no—
287D4—2541no—
288D4—2541no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—42no—
298C2×C4—5082no—
299C2×C4—2541no—
300C2×C4—2541no—
301C2×C4—2541no—
302C2×C4D4×D12712yes—
303C2×C4C2×C22⋊D12711yescharacteristic
304C2×C4C2×C22⋊D12711yescharacteristic
305C2×C4C2×C22⋊D12711yescharacteristic
306C2×C4#1452324712yes—
307C2×C4C23×D12711yescharacteristic
308C8—1274no—
309C8—2542no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—44no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—84no—
322C22—5081no—
323C22—5081no—
324C22—5081no—
325C22#1686118112yes—
326C22#1686123512yescomplemented
327C22C22×C22⋊D12711yescharacteristic
328C4—21no—
329C4—21no—
330C4—42no—
331C4—5081no—
332C4—5081no—
333C4—5081no—
334C4#1686118112yes—
335C4C2×D4×D12711yescharacteristic
336C4C2×#1452324711yescharacteristic
337C2—22no—
338C2—22no—
339C2—41no—
340C2—82no—
341C2—10161no—
342C2C2×#1686118111yescharacteristic
343C2C2×#1686123511yescharacteristic
344C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#121×C127C2
socleC22×C127C22×D4
hypercenterC2×#121D127
hypocenterC127#3055
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. C22×#14523232C2semidirect
3. #22550387C2semidirect
4. #22550127C2semidirect
5. #22550120C2semidirect
6. #22550110C2semidirect
7. #22550103C2semidirect
8. #22549916C2semidirect
9. #22549881C2semidirect
10. #22549875C2semidirect
11. C2×#121×C127C2semidirect
14. C2×#14523236C22semidirect
15. C2×#14523233C22semidirect
19. C2×#14523232C22semidirect
20. C22×C127⋊C8C22semidirect
22. #16861188C22semidirect
23. #16861188C22semidirect
24. #16861183C22semidirect
25. #16861183C22semidirect
30. #16861142C22semidirect
31. #16861140C22semidirect
32. #16861132C22semidirect
33. #16861130C22semidirect
34. #16861130C22semidirect
35. #16861129C22semidirect
57. #14523232D4semidirect
106. C127⋊C8C2×D4semidirect
150. C22×C127AGL(1,ℤ/8)semidirect
173. C127#3055semidirect
177. C2×#121D127semidirect
326. C22#16861235semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C22×#145232322
#225503872
#225501272
#225501202
#225501102
#225501032
#225499162
#225498812
#225498752
C2×#121×C1272
#3055127