Finite Group Databasebeta

← #30650194

Subgroups — #30650194

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#22550499C211yesmaximal, characteristic, complemented
2#22550493C211yesmaximal, characteristic, complemented
3#22550165C211yesmaximal, characteristic, complemented
4#22550164C211yesmaximal, characteristic, complemented
5#22550160C211yesmaximal, characteristic, complemented
6#22550159C211yesmaximal, characteristic, complemented
7#22550132C211yesmaximal, characteristic, complemented
8#22550127C211yesmaximal, characteristic
9#22550105C211yesmaximal, characteristic, complemented
10#22550101C211yesmaximal, characteristic, complemented
11#22550096C211yesmaximal, characteristic, complemented
12#22550093C211yesmaximal, characteristic, complemented
13#22550090C211yesmaximal, characteristic, complemented
14#22549902C211yesmaximal, characteristic, complemented
15#465×C127C211yesmaximal, characteristic, complemented
16C2×#14523238C2211yescharacteristic
17C2×#14523237C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20#16861208C2211yescharacteristic
21#16861208C2211yescharacteristic
22#16861204—21no—
23#16861202—21no—
24#16861198C2211yescharacteristic
25#16861196—21no—
26#16861196C2211yescharacteristic
27#16861195—21no—
28#16861191—21no—
29#16861190—21no—
30#16861189—21no—
31#16861188C2211yescharacteristic
32#16861186—21no—
33#16861183C2211yescharacteristic
34Q8×Dic127C2211yescharacteristic
35#16861174C2211yescharacteristic
36D4×Dic127C2211yescharacteristic
37#16861171C2211yescharacteristic
38#16861168C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41#16861140C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43Dic127⋊C8C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45Dic127⋊C8C2211yescharacteristic
46#16861134C2211yescharacteristic
47#16861133C2211yescharacteristic
48#16861132C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861130C2211yescharacteristic
51#16861129C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#106×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254—21no—
61C2×Dic254C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C22×Dic127—21no—
65C2×C4×D127—21no—
66#14523238—41no—
67#14523237—41no—
68#14523233—41no—
69#14523232—41no—
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C4⋊Dic127—21no—
73C4⋊Dic127D411yescharacteristic
74C4⋊Dic127D411yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81C22⋊Dic127—21no—
82C22⋊Dic127—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127D411yescharacteristic
86C22⋊Dic127D411yescharacteristic
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91C4×Dic127—21no—
92C4×Dic127—21no—
93C4×Dic127C2311yescharacteristic
94C4×Dic127C2311yescharacteristic
95C2×Q8×C127C2311yescharacteristic
96C2×D4×C127—21no—
97C2×D4×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99SD16×C127—41no—
100C2×C8×C127—41no—
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C4⋊C4×C127—21no—
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C22⋊C4×C127—21no—
108C22×D127—21no—
109Dic254—21no—
110Dic254—21no—
111Dic254—81no—
112Dic254—81no—
113D508—21no—
114D508—21no—
115C22⋊D127—81no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—81no—
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C4×D127—81no—
125C127⋊C8—21no—
126C127⋊C8—21no—
127C23×C127—21no—
128C23×C127C2×D411yescharacteristic
129Q8×C127—21no—
130Q8×C127—21no—
131D4×C127—21no—
132D4×C127—21no—
133D4×C127—81no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2411yescharacteristic
141C8×C127—21no—
142C8×C127—21no—
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—41no—
155C4×C127—41no—
156C4×C127—41no—
157C4×C127H(5,𝔽2)11yescharacteristic
158C4×C1272−1+411yescharacteristic
159Dic127—21no—
160Dic127—21no—
161Dic127—21no—
162Dic127—21no—
163Dic127—41no—
164Dic127—41no—
165Dic127—41no—
166C2×C127—41no—
167C2×C127—81no—
168C2×C127#53711yescharacteristic
169C2×C127#57711yescharacteristic
170C2×C127#57711yescharacteristic
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#465D12711yescharacteristic, complemented
184#464—1271no—
185#464—1271no—
186#463—1271no—
187#463—1271no—
188#424—1271no—
189C2×Q16—1271no—
190C2×SD16—1271no—
191C2×SD16C2×D12711yescharacteristic
192C2×D8—1271no—
193C2×M16—1271no—
194C22×C8C2×D12711yescharacteristic
195C4⋊Q8—1271no—
196C4⋊Q8—1271no—
197#126—2541no—
198#126—2541no—
199#124—1271no—
200#124—1271no—
201#124—2541no—
202#124—2541no—
203#123—2541no—
204#123—2541no—
205C22⋊Q8—1271no—
206C22⋊Q8—2541no—
207C22⋊Q8—2541no—
208C22⋊Q8C2×D12711yescharacteristic
209#121—1271no—
210#121C2×D12711yescharacteristic
211C4×Q8—1271no—
212C4×Q8—1271no—
213C4×D4—1271no—
214C4×D4—1271no—
215#117—1271no—
216#106—1271no—
217#106—1271no—
218#106—1271no—
219#106C2×D12711yescharacteristic
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222C4⋊C8—1271no—
223C4⋊C8—1271no—
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—1271no—
233C2×Q8—2541no—
234C2×Q8—2541no—
235C2×Q8C22×D12711yescharacteristic
236C2×D4—21no—
237C2×D4—1271no—
238C2×D4—2541no—
239C2×D4C22×D12711yescharacteristic
240C22×C4—2541no—
241C22×C4—2541no—
242C22×C4C22×D12711yescharacteristic
243Q16—5081no—
244SD16—41no—
245SD16—5081no—
246D8—5081no—
247M16—5081no—
248C2×C8—41no—
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—1271no—
258C4⋊C4—2541no—
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—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—1271no—
278C42—1271no—
279C42—2541no—
280C42—2541no—
281C23—21no—
282C23—2541no—
283C23D4×D12711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—10161no—
287Q8—10161no—
288Q8—2541no—
289Q8—2541no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—10161no—
294D4—2541no—
295D4—2541no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—1271no—
302C2×C4—1271no—
303C2×C4—1271no—
304C2×C4—1271no—
305C2×C4—10161no—
306C2×C4—10161no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4—2541no—
310C2×C4D4×D12711yescharacteristic
311C2×C4C23×D12711yescharacteristic
312C8—21no—
313C8—21no—
314C8—2541no—
315C8—2541no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—5081no—
323C22—5081no—
324C22—5081no—
325C22C2×D4×D12711yescharacteristic
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—5081no—
331C4—5081no—
332C4—5081no—
333C4—2541no—
334C4—2541no—
335C4—2541no—
336C4—2541no—
337C4#1686125411yescharacteristic
338C4#1686125611yescharacteristic
339C2—41no—
340C2—81no—
341C2—10161no—
342C2#2255045311yescharacteristic
343C2#2255068211yescharacteristic
344C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#465×C127C2
socleC22×C127C22×D4
prime radical#22550127C2
hypercenter#465D127
hypocenterC127#3263
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #22550499C2semidirect
2. #22550493C2semidirect
3. #22550165C2semidirect
4. #22550164C2semidirect
5. #22550160C2semidirect
6. #22550159C2semidirect
7. #22550132C2semidirect
9. #22550105C2semidirect
10. #22550101C2semidirect
11. #22550096C2semidirect
12. #22550093C2semidirect
13. #22550090C2semidirect
14. #22549902C2semidirect
15. #465×C127C2semidirect
173. C127#3263semidirect
183. #465D127semidirect

Maximal subgroups

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

subgroupindex
#225504992
#225504932
#225501652
#225501642
#225501602
#225501592
#225501322
#225501272
#225501052
#225501012
#225500962
#225500932
#225500902
#225499022
#465×C1272
#3263127