Finite Group Databasebeta

← #30651332

Subgroups — #30651332

Order 16256.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550574C211yesmaximal, characteristic, complemented
2#22550570C211yesmaximal, characteristic, complemented
3#22550276C211yesmaximal, characteristic, complemented
4#22550202C211yesmaximal, characteristic, complemented
5#22550200C211yesmaximal, characteristic, complemented
6#22550191C211yesmaximal, characteristic, complemented
7#22550189C211yesmaximal, characteristic, complemented
8#22550179C211yesmaximal, characteristic, complemented
9#22550175C211yesmaximal, characteristic, complemented
10#22550130C211yesmaximal, characteristic, complemented
11#22550129C211yesmaximal, characteristic, complemented
12#22549958C211yesmaximal, characteristic, complemented
13#22549952C211yesmaximal, characteristic, complemented
14#22549917C211yesmaximal, characteristic, complemented
15#428×C127C211yesmaximal, characteristic, complemented
16C2×D1016C2211yescharacteristic
17C2×#14523231C2211yescharacteristic
18#16861204—21no—
19#16861203—21no—
20#16861200C2211yescharacteristic
21#16861200C2211yescharacteristic
22#16861194—21no—
23#16861191—21no—
24#16861184C2211yescharacteristic
25#16861184C2211yescharacteristic
26#16861183—21no—
27#16861183C2211yescharacteristic
28#16861183C2211yescharacteristic
29#16861182—21no—
30C4×D508—21no—
31#16861171C2211yescharacteristic
32#16861171C2211yescharacteristic
33C4×C22⋊D127—21no—
34C4×C22⋊D127C2211yescharacteristic
35#16861167C2211yescharacteristic
36C4⋊C4×D127C2211yescharacteristic
37#16861142C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861141C2211yescharacteristic
41#16861140C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861131C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861129C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×M16×C127C2211yescharacteristic
53#117×C127C2211yescharacteristic
54C2×C4⋊C4×C127C2211yescharacteristic
55#107×C127C2211yescharacteristic
56#107×C127C2211yescharacteristic
57#106×C127C2211yescharacteristic
58#106×C127C2211yescharacteristic
59C2×D508—21no—
60C2×D508—21no—
61C2×D508C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127D411yescharacteristic
66C2×C4×D127—21no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127C2311yescharacteristic
70D1016—22no—
71#14523231—22no—
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C4⋊Dic127—21no—
75C4⋊Dic127—21no—
76C4⋊Dic127C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4D411yescharacteristic
82C22⋊Dic127D411yescharacteristic
83(C2×D127)⋊C4—21no—
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4D411yescharacteristic
91C4×Dic127—21no—
92C4×Dic127C2311yescharacteristic
93C22×C4×C127—21no—
94C22×C4×C127C2311yescharacteristic
95M16×C127—22no—
96C2×C8×C127C2311yescharacteristic
97C2×C8×C127C2311yescharacteristic
98C4⋊C4×C127—21no—
99C4⋊C4×C127C2311yescharacteristic
100C4⋊C4×C127C2311yescharacteristic
101C4⋊C4×C127C2311yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C22⋊C4×C127—21no—
104C42×C127—21no—
105C22×D127—21no—
106C22×D127—21no—
107C22×D127C2×D411yescharacteristic
108D508—21no—
109D508—21no—
110D508—21no—
111D508—21no—
112D508—42no—
113D508—81no—
114C22⋊D127—41no—
115C22⋊D127—42no—
116C22⋊D127—81no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127C2×D411yescharacteristic
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C4×D127—21no—
123C4×D127—21no—
124C4×D127—81no—
125C4×D127—81no—
126C4×D127—81no—
127C127⋊C8—21no—
128C127⋊C8—21no—
129C23×C127C2×D411yescharacteristic
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—81no—
137C2×C4×C127C2×D411yescharacteristic
138C2×C4×C127C2411yescharacteristic
139C8×C127—21no—
140C8×C127#4112yes—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127C22×D411yescharacteristic
145C2×D127—21no—
146C2×D127—21no—
147C2×D127—21no—
148C2×D127—41no—
149C2×D127—41no—
150C2×D127—41no—
151C2×D127—41no—
152C2×D127—41no—
153C2×D127—41no—
154C4×C127—21no—
155C4×C127—41no—
156C4×C127—41no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127C2×#4111yescharacteristic
160C4×C1272−1+411yescharacteristic
161Dic127—21no—
162Dic127—21no—
163Dic127—21no—
164Dic127—41no—
165Dic127—41no—
166C2×C127—41no—
167C2×C127#54711yescharacteristic
168C2×C127#57511yescharacteristic
169C2×C127#57611yescharacteristic
170D127—41no—
171D127—81no—
172D127—81no—
173#3355—1271nomaximal, Sylow
174C127#335511yescharacteristic, complemented, Sylow
175#553—1271no—
176#553—1271no—
177#499—1271no—
178#482—1271no—
179#482—1271no—
180#480—1271no—
181#480—1271no—
182#478—1271no—
183#478—1271no—
184#469—1271no—
185#469—1271no—
186C4×D8—1271no—
187#436—1271no—
188#428—1271no—
189#428D12711yescharacteristic, complemented
190C2×D8—1271no—
191C2×M16—1271no—
192C2×M16C2×D12711yescharacteristic
193#126—2541no—
194#126—2541no—
195#125—1271no—
196#125—1271no—
197#123—2541no—
198#123—2541no—
199#121—1271no—
200#121—1271no—
201#121—1271no—
202#121—1271no—
203#121—2541no—
204#121—2541no—
205C4×D4—1271no—
206C4×D4—1271no—
207C4×D4—1271no—
208C4×D4—2541no—
209C4×D4—2541no—
210#117—1271no—
211#117C2×D12711yescharacteristic
212C2×C4⋊C4—1271no—
213C2×C4⋊C4C2×D12711yescharacteristic
214#107—1271no—
215#107—1271no—
216#107—1271no—
217#107C2×D12711yescharacteristic
218#107C2×D12711yescharacteristic
219#106—1271no—
220#106—1271no—
221#106C2×D12711yescharacteristic
222#106C2×D12711yescharacteristic
223C4⋊C8—1271no—
224#102—1271no—
225#102—1271no—
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102—1271no—
230C22⋊C8—1271no—
231C22⋊C8—1271no—
232C4×C8—1271no—
233C2×D4—1271no—
234C2×D4—1271no—
235C2×D4—1271no—
236C2×D4—2541no—
237C2×D4—2541no—
238C2×D4—2541no—
239C2×D4—2541no—
240C22×C4—21no—
241C22×C4—1271no—
242C22×C4—2541no—
243C22×C4—2541no—
244C22×C4—2541no—
245C22×C4C22×D12711yescharacteristic
246D8—2542no—
247M16—22no—
248M16—2542no—
249C2×C8—1271no—
250C2×C8—1271no—
251C2×C8C22×D12711yescharacteristic
252C2×C8C22×D12711yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—1271no—
255C4⋊C4—1271no—
256C4⋊C4—1271no—
257C4⋊C4—2541no—
258C4⋊C4—2541no—
259C4⋊C4—2541no—
260C4⋊C4—2541no—
261C4⋊C4—2541no—
262C4⋊C4C22×D12711yescharacteristic
263C4⋊C4C22×D12711yescharacteristic
264C4⋊C4C22×D12711yescharacteristic
265C4⋊C4C22×D12711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—1271no—
268C22⋊C4—1271no—
269C22⋊C4—2541no—
270C22⋊C4—2541no—
271C22⋊C4—2541no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C42—21no—
277C42—1271no—
278C42—2541no—
279C23—1271no—
280C23—2541no—
281C23—2541no—
282C23D4×D12711yescharacteristic
283D4—10161no—
284D4—10161no—
285D4—5082no—
286D4—5082no—
287D4—5081no—
288D4—2541no—
289D4—2541no—
290D4—2541no—
291D4—2541no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—81no—
299C2×C4—1271no—
300C2×C4—1271no—
301C2×C4—1271no—
302C2×C4—10161no—
303C2×C4—10161no—
304C2×C4—10161no—
305C2×C4—2541no—
306C2×C4—2541no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4D4×D12711yescharacteristic
310C2×C4C23×D12711yescharacteristic
311C8—21no—
312C8—2541no—
313C8—2541no—
314C8#1452324812yes—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—5081no—
319C22—5081no—
320C22—5081no—
321C22—5081no—
322C22—5081no—
323C22—5081no—
324C22—2541no—
325C22—2541no—
326C22—2541no—
327C22C2×D4×D12711yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—5081no—
334C4—5081no—
335C4—2541no—
336C4—2541no—
337C4—2541no—
338C4C2×#1452324811yescharacteristic
339C4#1686125711yescharacteristic
340C2—41no—
341C2—10161no—
342C2—10161no—
343C2—5081no—
344C2#2255052711yescharacteristic
345C2#2255067511yescharacteristic
346C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
hypercenter#428D127
hypocenterC127#3355
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #22550574C2semidirect
2. #22550570C2semidirect
3. #22550276C2semidirect
4. #22550202C2semidirect
5. #22550200C2semidirect
6. #22550191C2semidirect
7. #22550189C2semidirect
8. #22550179C2semidirect
9. #22550175C2semidirect
10. #22550130C2semidirect
11. #22550129C2semidirect
12. #22549958C2semidirect
13. #22549952C2semidirect
14. #22549917C2semidirect
15. #428×C127C2semidirect
174. C127#3355semidirect
189. #428D127semidirect

Maximal subgroups

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

subgroupindex
#225505742
#225505702
#225502762
#225502022
#225502002
#225501912
#225501892
#225501792
#225501752
#225501302
#225501292
#225499582
#225499522
#225499172
#428×C1272
#3355127