Finite Group Databasebeta

← #30650126

Subgroups — #30650126

Order 16256.

354 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
1#22550410C211yesmaximal, characteristic, complemented
2#22550183C211yesmaximal, characteristic, complemented
3#22550180C211yesmaximal, characteristic, complemented
4#22550176C211yesmaximal, characteristic, complemented
5#22550173C211yesmaximal, characteristic, complemented
6#22550132C211yesmaximal, characteristic, complemented
7#22550127C211yesmaximal, characteristic
8#22550124C211yesmaximal, characteristic, complemented
9#22550120C211yesmaximal, characteristic, complemented
10#22550098C211yesmaximal, characteristic, complemented
11#22550095C211yesmaximal, characteristic, complemented
12#22550091C211yesmaximal, characteristic, complemented
13#22550088C211yesmaximal, characteristic, complemented
14#22549900C211yesmaximal, characteristic, complemented
15#529×C127C211yesmaximal, characteristic, complemented
16C2×#14523238C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523235C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523232C2211yescharacteristic
21C2×#14523232C2211yescharacteristic
22#16861203—21no—
23#16861201C2211yescharacteristic
24#16861197C2211yescharacteristic
25#16861192—21no—
26#16861188C2211yescharacteristic
27#16861183C2211yescharacteristic
28C4×Dic254C2211yescharacteristic
29C4×D508C2211yescharacteristic
30C4×C22⋊D127—21no—
31#16861166—21no—
32#16861142C2211yescharacteristic
33#16861142C2211yescharacteristic
34#16861140C2211yescharacteristic
35#16861140C2211yescharacteristic
36C4⋊(C127⋊C8)C2211yescharacteristic
37C4⋊(C127⋊C8)C2211yescharacteristic
38C4⋊(C127⋊C8)C2211yescharacteristic
39C4⋊(C127⋊C8)C2211yescharacteristic
40#16861133C2211yescharacteristic
41#16861133C2211yescharacteristic
42#16861132C2211yescharacteristic
43#16861132C2211yescharacteristic
44#16861130C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48#126×C127—21no—
49#125×C127C2211yescharacteristic
50#124×C127C2211yescharacteristic
51#123×C127—21no—
52C22⋊Q8×C127C2211yescharacteristic
53#121×C127C2211yescharacteristic
54C4×Q8×C127C2211yescharacteristic
55C4×D4×C127—21no—
56C4×D4×C127C2211yescharacteristic
57#117×C127—21no—
58C2×C42×C127C2211yescharacteristic
59C2×Dic254C2311yescharacteristic
60C2×D508C2311yescharacteristic
61C2×C22⋊D127—21no—
62C2×C4×D127—21no—
63#14523238—41no—
64#14523236—41no—
65#14523235—41no—
66#14523233—41no—
67#14523232—41no—
68#14523232—41no—
69C2×C127⋊C8C2311yescharacteristic
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C2×C127⋊C8C2311yescharacteristic
73C4⋊Dic127C2311yescharacteristic
74C4⋊Dic127C2311yescharacteristic
75Dic127⋊C4—21no—
76Dic127⋊C4—21no—
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79C22⋊Dic127—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83(C2×D127)⋊C4—21no—
84C4×Dic127—21no—
85C2×Q8×C127C2311yescharacteristic
86C2×D4×C127—21no—
87C2×D4×C127C2311yescharacteristic
88C22×C4×C127—21no—
89C22×C4×C127D411yescharacteristic
90C22×C4×C127D411yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C4⋊C4×C127—21no—
93C4⋊C4×C127—21no—
94C4⋊C4×C127—21no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127C2311yescharacteristic
97C4⋊C4×C127C2311yescharacteristic
98C22⋊C4×C127—21no—
99C22⋊C4×C127—21no—
100C22⋊C4×C127—21no—
101C22⋊C4×C127—21no—
102C22⋊C4×C127—21no—
103C42×C127—21no—
104C42×C127D411yescharacteristic
105C42×C127D411yescharacteristic
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109Dic254—21no—
110Dic254—21no—
111D508—21no—
112D508—21no—
113C22⋊D127—42no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C4×D127—42no—
118C127⋊C8—21no—
119C127⋊C8—21no—
120C127⋊C8—21no—
121C127⋊C8—21no—
122C23×C127—21no—
123C23×C127C2×D411yescharacteristic
124Q8×C127—21no—
125Q8×C127—21no—
126D4×C127—21no—
127D4×C127—21no—
128D4×C127—42no—
129C2×C4×C127—21no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—42no—
137C2×C4×C127C2×D411yescharacteristic
138C2×C4×C127C2×D411yescharacteristic
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C22×C127—21no—
144C22×C127—21no—
145C22×C127—21no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127C22×D411yescharacteristic
150C2×D127—41no—
151C2×D127—41no—
152C2×D127—41no—
153C4×C127—21no—
154C4×C127—21no—
155C4×C127—21no—
156C4×C127—21no—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127H(5,𝔽2)11yescharacteristic
162C4×C1272−1+411yescharacteristic
163Dic127—41no—
164Dic127—41no—
165Dic127—41no—
166C2×C127—41no—
167C2×C127—81no—
168C2×C127#53711yescharacteristic
169C2×C127#57511yescharacteristic
170C2×C127#57511yescharacteristic
171D127—81no—
172#3256—1271nomaximal, Sylow
173C127#325611yescharacteristic, complemented, Sylow
174#529—1271no—
175#529D12711yescharacteristic, complemented
176#479—1271no—
177#479—1271no—
178#478—1271no—
179#478—1271no—
180#470—1271no—
181#469—1271no—
182#468—1271no—
183#468—1271no—
184#464—1271no—
185#464—1271no—
186#463—1271no—
187#463—1271no—
188#423—1271no—
189C2×Q16—1271no—
190C2×SD16—1271no—
191C2×SD16—1271no—
192C2×D8—1271no—
193C2×M16—1271no—
194C2×M16—1271no—
195#126—21no—
196#126—2541no—
197#125—1271no—
198#125C2×D12711yescharacteristic
199#124—1271no—
200#124C2×D12711yescharacteristic
201#123—21no—
202#123—2541no—
203C22⋊Q8—1271no—
204C22⋊Q8C2×D12711yescharacteristic
205#121—1271no—
206#121C2×D12711yescharacteristic
207C4×Q8—1271no—
208C4×Q8C2×D12711yescharacteristic
209C4×D4—21no—
210C4×D4—1271no—
211C4×D4—2541no—
212C4×D4C2×D12711yescharacteristic
213#117—21no—
214#117—2541no—
215C2×C42C2×D12711yescharacteristic
216#107—1271no—
217#107—1271no—
218#106—1271no—
219#106—1271no—
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222C4⋊C8—1271no—
223C4⋊C8—1271no—
224#103—1271no—
225#103—1271no—
226#103—1271no—
227#103—1271no—
228#102—1271no—
229#102—1271no—
230#102—1271no—
231#102—1271no—
232C2×Q8—1271no—
233C2×Q8C22×D12711yescharacteristic
234C2×D4—21no—
235C2×D4—1271no—
236C2×D4—2541no—
237C2×D4C22×D12711yescharacteristic
238C22×C4—21no—
239C22×C4—2541no—
240C22×C4C22⋊D12711yescharacteristic
241C22×C4C22⋊D12711yescharacteristic
242C22×C4C22×D12711yescharacteristic
243Q16—5081no—
244SD16—5081no—
245SD16—5081no—
246D8—5081no—
247M16—5081no—
248M16—5081no—
249C2×C8—1271no—
250C2×C8—1271no—
251C2×C8—1271no—
252C2×C8—1271no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—1271no—
258C4⋊C4—1271no—
259C4⋊C4—2541no—
260C4⋊C4—2541no—
261C4⋊C4—2541no—
262C4⋊C4—2541no—
263C4⋊C4C22×D12711yescharacteristic
264C4⋊C4C22×D12711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—2541no—
271C22⋊C4—2541no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C42—21no—
276C42—2541no—
277C42C22⋊D12711yescharacteristic
278C42C22⋊D12711yescharacteristic
279C42C22×D12711yescharacteristic
280C42C22×D12711yescharacteristic
281C23—21no—
282C23—2541no—
283C23C2×C22⋊D12711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—2541no—
287Q8—2541no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—5082no—
292D4—2541no—
293D4—2541no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—5082no—
303C2×C4—2541no—
304C2×C4—2541no—
305C2×C4—2541no—
306C2×C4C2×C22⋊D12711yescharacteristic
307C2×C4C2×C22⋊D12711yescharacteristic
308C2×C4C2×C22⋊D12711yescharacteristic
309C2×C4C2×C22⋊D12711yescharacteristic
310C2×C4C2×C22⋊D12711yescharacteristic
311C2×C4C23×D12711yescharacteristic
312C8—2541no—
313C8—2541no—
314C8—2541no—
315C8—2541no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—5081no—
323C22—5081no—
324C22—5081no—
325C22C22×C22⋊D12711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—5081no—
335C4—5081no—
336C4—5081no—
337C4#1686125511yescharacteristic
338C4#1686125611yescharacteristic
339C2—41no—
340C2—81no—
341C2—10161no—
342C2#2255045411yescharacteristic
343C2#2255067411yescharacteristic
344C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#529×C127C2
socleC22×C127C22×D4
prime radical#22550127C2
hypercenter#529D127
hypocenterC127#3256
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #22550410C2semidirect
2. #22550183C2semidirect
3. #22550180C2semidirect
4. #22550176C2semidirect
5. #22550173C2semidirect
6. #22550132C2semidirect
8. #22550124C2semidirect
9. #22550120C2semidirect
10. #22550098C2semidirect
11. #22550095C2semidirect
12. #22550091C2semidirect
13. #22550088C2semidirect
14. #22549900C2semidirect
15. #529×C127C2semidirect
173. C127#3256semidirect
175. #529D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
#225501832
#225501802
#225501762
#225501732
#225501322
#225501272
#225501242
#225501202
#225500982
#225500952
#225500912
#225500882
#225499002
#529×C1272
#3256127