Finite Group Databasebeta

← #30651295

Subgroups — #30651295

Order 16256.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550576C211yesmaximal, characteristic
2#22550567C211yesmaximal, characteristic, complemented
3#22550278C211yesmaximal, characteristic, complemented
4#22550215C211yesmaximal, characteristic, complemented
5#22550205C211yesmaximal, characteristic, complemented
6#22550199C211yesmaximal, characteristic, complemented
7#22550193C211yesmaximal, characteristic, complemented
8#22550183C211yesmaximal, characteristic, complemented
9#22550176C211yesmaximal, characteristic, complemented
10#22550119C211yesmaximal, characteristic, complemented
11#22550119C211yesmaximal, characteristic, complemented
12#22549955C211yesmaximal, characteristic, complemented
13#22549915C211yesmaximal, characteristic, complemented
14#22549915C211yesmaximal, characteristic, complemented
15C4×SD16×C127C211yesmaximal, characteristic, complemented
16C2×#14523231C2211yescharacteristic
17C2×#14523231C2211yescharacteristic
18#16861203—21no—
19#16861202—21no—
20#16861201C2211yescharacteristic
21#16861201C2211yescharacteristic
22#16861199—21no—
23#16861194—21no—
24#16861192—21no—
25#16861185C2211yescharacteristic
26#16861185C2211yescharacteristic
27#16861182—21no—
28#16861180C2211yescharacteristic
29#16861180C2211yescharacteristic
30C4×D508C2211yescharacteristic
31C4×C22⋊D127—21no—
32#16861167C2211yescharacteristic
33#16861167C2211yescharacteristic
34#16861166—21no—
35C4⋊C4×D127C2211yescharacteristic
36C4⋊C4×D127C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861142C2211yescharacteristic
41#16861141C2211yescharacteristic
42#16861141C2211yescharacteristic
43#16861140C2211yescharacteristic
44#16861140C2211yescharacteristic
45C4⋊(C127⋊C8)C2211yescharacteristic
46#16861133C2211yescharacteristic
47#16861133C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C4×Q8×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C4×C8×C127C2211yescharacteristic
59C2×D508D411yescharacteristic
60C2×C22⋊D127—21no—
61C2×C22⋊D127—21no—
62C2×C4×D127—21no—
63C2×C4×D127—21no—
64C2×C4×D127C2311yescharacteristic
65C2×C4×D127C2311yescharacteristic
66#14523231—22no—
67#14523231—22no—
68C2×C127⋊C8C2311yescharacteristic
69C2×C127⋊C8C2311yescharacteristic
70C4⋊Dic127D411yescharacteristic
71C4⋊Dic127C2311yescharacteristic
72C4⋊Dic127C2311yescharacteristic
73C4⋊Dic127C2311yescharacteristic
74C4⋊Dic127C2311yescharacteristic
75Dic127⋊C4—21no—
76Dic127⋊C4—21no—
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4D411yescharacteristic
90(C2×D127)⋊C4D411yescharacteristic
91C4×Dic127—21no—
92C4×Dic127—21no—
93C2×Q8×C127C2311yescharacteristic
94C2×D4×C127C2311yescharacteristic
95C22×C4×C127—21no—
96SD16×C127—22no—
97C2×C8×C127C2311yescharacteristic
98C2×C8×C127C2311yescharacteristic
99C4⋊C4×C127—21no—
100C4⋊C4×C127C2311yescharacteristic
101C4⋊C4×C127C2311yescharacteristic
102C22⋊C4×C127—21no—
103C42×C127—21no—
104C42×C127C2311yescharacteristic
105C22×D127C2×D411yescharacteristic
106C22×D127C2×D411yescharacteristic
107D508—41no—
108C22⋊D127—42no—
109C22⋊D127—81no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127C2×D411yescharacteristic
115C2×Dic127C2×D411yescharacteristic
116C4×D127—21no—
117C4×D127—21no—
118C4×D127—21no—
119C4×D127—21no—
120C4×D127—42no—
121C4×D127—81no—
122C127⋊C8—21no—
123C127⋊C8—21no—
124C23×C127—21no—
125Q8×C127—21no—
126Q8×C127—21no—
127D4×C127—21no—
128D4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—81no—
133C2×C4×C127C2×D411yescharacteristic
134C2×C4×C127C2×D411yescharacteristic
135C2×C4×C127C2411yescharacteristic
136C8×C127—21no—
137C8×C127#4112yes—
138C22×C127—41no—
139C22×C127—41no—
140C22×C127—41no—
141C22×C127C22×D411yescharacteristic
142C2×D127—21no—
143C2×D127—21no—
144C2×D127—21no—
145C2×D127—21no—
146C2×D127—21no—
147C2×D127—21no—
148C4×C127—21no—
149C4×C127—21no—
150C4×C127—41no—
151C4×C127—41no—
152C4×C127—41no—
153C4×C127C2×#4111yescharacteristic
154C4×C1272−1+411yescharacteristic
155Dic127—21no—
156Dic127—21no—
157Dic127—41no—
158Dic127—41no—
159Dic127—41no—
160Dic127—41no—
161C2×C127—81no—
162C2×C127#54711yescharacteristic
163C2×C127#57511yescharacteristic
164C2×C127#57711yescharacteristic
165D127—41no—
166D127—41no—
167#3351—1271nomaximal, Sylow
168C127#335111yescharacteristic, complemented, Sylow
169#553—1271no—
170#552—1271no—
171#499—1271no—
172#485—1271no—
173#483—1271no—
174#482—1271no—
175#481—1271no—
176#479—1271no—
177#478—1271no—
178#468—1271no—
179#468—1271no—
180C4×SD16D12711yescharacteristic, complemented
181#436—1271no—
182#428—1271no—
183#428—1271no—
184C2×SD16C2×D12711yescharacteristic
185C2×M16—1271no—
186C2×M16—1271no—
187#126—2541no—
188#126—2541no—
189#125—1271no—
190#125—1271no—
191#125—2541no—
192#123—2541no—
193#123—2541no—
194C22⋊Q8—1271no—
195C22⋊Q8—1271no—
196#121—1271no—
197#121—1271no—
198#121—2541no—
199C4×Q8C2×D12711yescharacteristic
200C4×D4—1271no—
201C4×D4—2541no—
202C4×D4C2×D12711yescharacteristic
203#117—1271no—
204#117—1271no—
205#117—2541no—
206C2×C4⋊C4—1271no—
207C2×C4⋊C4—1271no—
208#107—1271no—
209#107—1271no—
210#107—1271no—
211#107—1271no—
212#106—1271no—
213#106—1271no—
214#106—1271no—
215#106—1271no—
216#106C2×D12711yescharacteristic
217C4⋊C8—1271no—
218#103—1271no—
219#103—1271no—
220#103C2×D12711yescharacteristic
221#102—1271no—
222#102—1271no—
223#102C2×D12711yescharacteristic
224C22⋊C8—1271no—
225C22⋊C8—1271no—
226C4×C8C2×D12711yescharacteristic
227C2×Q8C22×D12711yescharacteristic
228C2×D4—1271no—
229C2×D4—2541no—
230C2×D4—2541no—
231C2×D4C22×D12711yescharacteristic
232C22×C4—21no—
233C22×C4—1271no—
234C22×C4—1271no—
235C22×C4—2541no—
236C22×C4—2541no—
237SD16—22no—
238M16—2542no—
239M16—2542no—
240C2×C8—1271no—
241C2×C8—1271no—
242C2×C8C22×D12711yescharacteristic
243C2×C8C22×D12711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—1271no—
246C4⋊C4—1271no—
247C4⋊C4—1271no—
248C4⋊C4—1271no—
249C4⋊C4—1271no—
250C4⋊C4—2541no—
251C4⋊C4—2541no—
252C4⋊C4—2541no—
253C4⋊C4—2541no—
254C4⋊C4—2541no—
255C4⋊C4—2541no—
256C4⋊C4—2541no—
257C4⋊C4—2541no—
258C4⋊C4C22×D12711yescharacteristic
259C4⋊C4C22×D12711yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—1271no—
262C22⋊C4—1271no—
263C22⋊C4—2541no—
264C22⋊C4—2541no—
265C22⋊C4—2541no—
266C22⋊C4—2541no—
267C22⋊C4—2541no—
268C22⋊C4—2541no—
269C42—21no—
270C42—2541no—
271C42—2541no—
272C42C22×D12711yescharacteristic
273C23—21no—
274C23—1271no—
275C23—1271no—
276Q8—21no—
277Q8—21no—
278D4—21no—
279D4—21no—
280D4—10161no—
281D4—5081no—
282D4—5082no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—81no—
287C2×C4—1271no—
288C2×C4—1271no—
289C2×C4—10161no—
290C2×C4—5082no—
291C2×C4—2541no—
292C2×C4—2541no—
293C2×C4—2541no—
294C2×C4—2541no—
295C2×C4—2541no—
296C2×C4—2541no—
297C2×C4—2541no—
298C2×C4—2541no—
299C2×C4D4×D12711yescharacteristic
300C2×C4D4×D12711yescharacteristic
301C2×C4C23×D12711yescharacteristic
302C8—21no—
303C8—2541no—
304C8—2541no—
305C8#1452324812yes—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—2541no—
310C22—2541no—
311C22—2541no—
312C22—2541no—
313C22—2541no—
314C22—2541no—
315C22C2×D4×D12711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—5081no—
322C4—5081no—
323C4—5081no—
324C4—5081no—
325C4—2541no—
326C4—2541no—
327C4C2×#1452324811yescharacteristic
328C4#1686125611yescharacteristic
329C2—81no—
330C2—5081no—
331C2—5081no—
332C2#2255053111yescharacteristic
333C2#2255067311yescharacteristic
334C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×SD16×C127C2
socleC22×C127C22×D4
prime radical#22550576C2
hypercenterC4×SD16D127
hypocenterC127#3351
LCSG ▷ C2×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. #22550567C2semidirect
3. #22550278C2semidirect
4. #22550215C2semidirect
5. #22550205C2semidirect
6. #22550199C2semidirect
7. #22550193C2semidirect
8. #22550183C2semidirect
9. #22550176C2semidirect
10. #22550119C2semidirect
11. #22550119C2semidirect
12. #22549955C2semidirect
13. #22549915C2semidirect
14. #22549915C2semidirect
15. C4×SD16×C127C2semidirect
168. C127#3351semidirect
180. C4×SD16D127semidirect

Maximal subgroups

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

subgroupindex
#225505762
#225505672
#225502782
#225502152
#225502052
#225501992
#225501932
#225501832
#225501762
#225501192
#225499552
#225499152
C4×SD16×C1272
#3351127