Finite Group Databasebeta

← #30650198

Subgroups — #30650198

Order 16256.

346 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#22550498C211yesmaximal, characteristic, complemented
2#22550495C211yesmaximal, characteristic, complemented
3#22550167C211yesmaximal, characteristic, complemented
4#22550164C211yesmaximal, characteristic, complemented
5#22550161C211yesmaximal, characteristic, complemented
6#22550159C211yesmaximal, characteristic, complemented
7#22550131C211yesmaximal, characteristic, complemented
8#22550128C211yesmaximal, characteristic
9#22550108C211yesmaximal, characteristic, complemented
10#22550102C211yesmaximal, characteristic, complemented
11#22550100C211yesmaximal, characteristic, complemented
12#22550093C211yesmaximal, characteristic, complemented
13#22550088C211yesmaximal, characteristic, complemented
14#22549901C211yesmaximal, characteristic, complemented
15#464×C127C211yesmaximal, characteristic, complemented
16C2×#14523235C2211yescharacteristic
17C2×#14523233C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20#16861209C2211yescharacteristic
21#16861208C2211yescharacteristic
22#16861203—21no—
23#16861202—21no—
24#16861198C2211yescharacteristic
25#16861197—21no—
26#16861195—21no—
27#16861194—21no—
28#16861193—21no—
29#16861187—21no—
30#16861186—21no—
31#16861185C2211yescharacteristic
32#16861185C2211yescharacteristic
33#16861184C2211yescharacteristic
34#16861180C2211yescharacteristic
35Q8×Dic127C2211yescharacteristic
36C4×D508C2211yescharacteristic
37D4×Dic127C2211yescharacteristic
38#16861166C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861141C2211yescharacteristic
41#16861140C2211yescharacteristic
42#16861140C2211yescharacteristic
43C4⋊(C127⋊C8)C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45Dic127⋊C8C2211yescharacteristic
46#16861133C2211yescharacteristic
47#16861133C2211yescharacteristic
48#16861133C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861130C2211yescharacteristic
52C2×Q16×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54#124×C127C2211yescharacteristic
55C4×Q8×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254—21no—
61C2×D508—21no—
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C22×Dic127—21no—
65C2×C4×D127—21no—
66C2×C4×D127C2311yescharacteristic
67#14523235—41no—
68#14523233—41no—
69#14523231—41no—
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C8×D127—41no—
73C4⋊Dic127—21no—
74C4⋊Dic127—21no—
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4D411yescharacteristic
82Dic127⋊C4D411yescharacteristic
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)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4D411yescharacteristic
92(C2×D127)⋊C4D411yescharacteristic
93C4×Dic127—21no—
94C4×Dic127C2311yescharacteristic
95C2×Q8×C127C2311yescharacteristic
96C2×Q8×C127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98Q16×C127—41no—
99SD16×C127—41no—
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127—21no—
103C4⋊C4×C127C2311yescharacteristic
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C42×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127C2×D411yescharacteristic
110Dic254—81no—
111Dic254—81no—
112D508—21no—
113D508—21no—
114D508—81no—
115C22⋊D127—81no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—81no—
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C4×D127—21no—
124C4×D127—21no—
125C4×D127—81no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C23×C127—21no—
129Q8×C127—21no—
130Q8×C127—21no—
131Q8×C127—21no—
132Q8×C127—21no—
133D4×C127—21no—
134D4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127C2×D411yescharacteristic
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2411yescharacteristic
141C8×C127—21no—
142C8×C127—21no—
143C22×C127—41no—
144C22×C127—41no—
145C22×C127—41no—
146C22×C127C22×D411yescharacteristic
147C2×D127—21no—
148C2×D127—21no—
149C2×D127—21no—
150C2×D127—41no—
151C2×D127—41no—
152C2×D127—41no—
153C4×C127—21no—
154C4×C127—21no—
155C4×C127—41no—
156C4×C127—41no—
157C4×C127—41no—
158C4×C127H(5,𝔽2)11yescharacteristic
159C4×C1272−1+411yescharacteristic
160Dic127—21no—
161Dic127—21no—
162Dic127—21no—
163Dic127—41no—
164Dic127—41no—
165Dic127—41no—
166C2×C127—81no—
167C2×C127#53711yescharacteristic
168C2×C127#57711yescharacteristic
169C2×C127#57711yescharacteristic
170D127—41no—
171D127—81no—
172#3263—1271nomaximal, Sylow
173C127#326311yescharacteristic, complemented, Sylow
174#542—1271no—
175#542—1271no—
176#476—1271no—
177#476—1271no—
178#475—1271no—
179#475—1271no—
180#470—1271no—
181#469—1271no—
182#465—1271no—
183#465—1271no—
184#464—1271no—
185#464D12711yescharacteristic, complemented
186#463—1271no—
187#463—1271no—
188#424—1271no—
189C2×Q16C2×D12711yescharacteristic
190C2×SD16—1271no—
191C2×SD16C2×D12711yescharacteristic
192C2×D8—1271no—
193C2×M16—1271no—
194C22×C8—1271no—
195C4⋊Q8—1271no—
196C4⋊Q8—1271no—
197#126—2541no—
198#126—2541no—
199#124—1271no—
200#124—2541no—
201#124—2541no—
202#124C2×D12711yescharacteristic
203#123—2541no—
204#123—2541no—
205C22⋊Q8—1271no—
206C22⋊Q8—1271no—
207C22⋊Q8—2541no—
208C22⋊Q8—2541no—
209#121—1271no—
210#121—1271no—
211C4×Q8—1271no—
212C4×Q8C2×D12711yescharacteristic
213C4×D4—1271no—
214C4×D4—1271no—
215#117—1271no—
216#106—1271no—
217#106—1271no—
218#106—1271no—
219#106—1271no—
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222C4⋊C8—1271no—
223C4⋊C8C2×D12711yescharacteristic
224#103—1271no—
225#103—1271no—
226#103—1271no—
227#103C2×D12711yescharacteristic
228#102—1271no—
229#102—1271no—
230#102—1271no—
231#102C2×D12711yescharacteristic
232C2×Q8—2541no—
233C2×Q8—2541no—
234C2×Q8C22×D12711yescharacteristic
235C2×Q8C22×D12711yescharacteristic
236C2×D4—1271no—
237C2×D4—2541no—
238C2×D4—2541no—
239C2×D4C22×D12711yescharacteristic
240C22×C4—1271no—
241C22×C4—2541no—
242C22×C4—2541no—
243Q16—41no—
244SD16—41no—
245SD16—5081no—
246D8—5081no—
247M16—5081no—
248C2×C8—1271no—
249C2×C8—1271no—
250C2×C8—5081no—
251C2×C8C22×D12711yescharacteristic
252C2×C8C22×D12711yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—1271no—
255C4⋊C4—1271no—
256C4⋊C4—1271no—
257C4⋊C4—1271no—
258C4⋊C4—1271no—
259C4⋊C4—2541no—
260C4⋊C4—2541no—
261C4⋊C4—2541no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4C22×D12711yescharacteristic
265C22⋊C4—21no—
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—
276C22⋊C4—2541no—
277C42—21no—
278C42—1271no—
279C42—2541no—
280C42C22×D12711yescharacteristic
281C23—21no—
282C23—1271no—
283C23—2541no—
284Q8—21no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—10161no—
289Q8—10161no—
290D4—21no—
291D4—21no—
292D4—10161no—
293D4—10161no—
294D4—2541no—
295D4—2541no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—1271no—
300C2×C4—1271no—
301C2×C4—1271no—
302C2×C4—10161no—
303C2×C4—10161no—
304C2×C4—2541no—
305C2×C4—2541no—
306C2×C4—2541no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4D4×D12711yescharacteristic
310C2×C4D4×D12711yescharacteristic
311C2×C4C23×D12711yescharacteristic
312C8—21no—
313C8—21no—
314C8—2541no—
315C8—2541no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—5081no—
320C22—5081no—
321C22—5081no—
322C22—2541no—
323C22—2541no—
324C22—2541no—
325C22C2×D4×D12711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—5081no—
332C4—5081no—
333C4—5081no—
334C4—2541no—
335C4—2541no—
336C4—2541no—
337C4#1686125411yescharacteristic
338C4#1686125611yescharacteristic
339C2—81no—
340C2—10161no—
341C2—5081no—
342C2#2255045511yescharacteristic
343C2#2255068311yescharacteristic
344C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#464×C127C2
socleC22×C127C22×D4
prime radical#22550128C2
hypercenter#464D127
hypocenterC127#3263
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #22550498C2semidirect
2. #22550495C2semidirect
3. #22550167C2semidirect
4. #22550164C2semidirect
5. #22550161C2semidirect
6. #22550159C2semidirect
7. #22550131C2semidirect
9. #22550108C2semidirect
10. #22550102C2semidirect
11. #22550100C2semidirect
12. #22550093C2semidirect
13. #22550088C2semidirect
14. #22549901C2semidirect
15. #464×C127C2semidirect
173. C127#3263semidirect
185. #464D127semidirect

Maximal subgroups

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

subgroupindex
#225504982
#225504952
#225501672
#225501642
#225501612
#225501592
#225501312
#225501282
#225501082
#225501022
#225501002
#225500932
#225500882
#225499012
#464×C1272
#3263127