Finite Group Databasebeta

← #30650109

Subgroups — #30650109

Order 16256.

430 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550424C211yesmaximal, characteristic, complemented
2#22550423C211yesmaximal, characteristic, complemented
3#22550167C211yesmaximal, characteristic, complemented
4#22550166C211yesmaximal, characteristic, complemented
5#22550164C212yesmaximal, complemented
6#22550128C212yesmaximal, complemented
7#22550126C212yesmaximal, complemented
8#22550069C212yesmaximal, complemented
9#22550068C211yesmaximal, characteristic, complemented
10#22549898C211yesmaximal, characteristic, complemented
11#459×C127C211yesmaximal, characteristic, complemented
12C2×#14523248—21no—
13C2×#14523247—21no—
14C2×#14523233C2212yes—
15C2×#14523231C2212yes—
16#16861209C2211yescharacteristic
17#16861208C2211yescharacteristic
18#16861205C2211yescharacteristic
19#16861196—21no—
20#16861195—21no—
21#16861184C2211yescharacteristic
22#16861182—21no—
23#16861180—21no—
24#16861180C2211yescharacteristic
25#16861180C2212yes—
26C4×D508C2211yescharacteristic
27D4×Dic127—21no—
28D4×Dic127C2212yes—
29C4×C22⋊D127—21no—
30C42×D127C2211yescharacteristic
31#16861141C2212yes—
32#16861140C2212yes—
33C4⋊(C127⋊C8)C2211yescharacteristic
34Dic127⋊C8C2212yes—
35#16861131C2212yes—
36#16861130C2212yes—
37#16861130C2212yes—
38C2×D8×C127C2212yes—
39#127×C127C2211yescharacteristic
40C4×D4×C127C2211yescharacteristic
41C4⋊C8×C127C2211yescharacteristic
42#102×C127C2212yes—
43#14523248—44no—
44#14523247—44no—
45C2×Dic254—21no—
46C2×Dic254—21no—
47C2×D508—21no—
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—22no—
50C2×C22⋊D127—21no—
51C2×C22⋊D127—21no—
52C2×C22⋊D127—21no—
53C22×Dic127—22no—
54C2×C4×D127—21no—
55C2×C4×D127D411yescharacteristic
56C2×C4×D127D411yescharacteristic
57C2×C4×D127C2311yescharacteristic
58#14523233—42no—
59#14523231—42no—
60C2×C127⋊C8C2312yes—
61C4⋊Dic127—21no—
62C4⋊Dic127C2311yescharacteristic
63C4⋊Dic127C2312yes—
64Dic127⋊C4—21no—
65C22⋊Dic127—21no—
66C22⋊Dic127—22no—
67C22⋊Dic127—22no—
68(C2×D127)⋊C4—21no—
69(C2×D127)⋊C4—21no—
70C4×Dic127D411yescharacteristic
71C4×Dic127D411yescharacteristic
72C4×Dic127C2311yescharacteristic
73C2×D4×C127—21no—
74C2×D4×C127—21no—
75C2×D4×C127C2311yescharacteristic
76C2×D4×C127C2312yes—
77C22×C4×C127—21no—
78D8×C127—42no—
79C2×C8×C127C2312yes—
80C4⋊C4×C127C2311yescharacteristic
81C22⋊C4×C127—21no—
82C42×C127C2311yescharacteristic
83C22×D127—21no—
84C22×D127C2×D411yescharacteristic
85Dic254—42no—
86Dic254—42no—
87D508—21no—
88D508—21no—
89D508—42no—
90C22⋊D127—42no—
91C22⋊D127—44no—
92C22⋊D127—42no—
93C22⋊D127—42no—
94C2×Dic127—21no—
95C2×Dic127—22no—
96C2×Dic127—44no—
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C2×Dic127C2×D411yescharacteristic
100C4×D127—22no—
101C4×D127—21no—
102C4×D127—21no—
103C4×D127—22no—
104C4×D127—42no—
105C127⋊C8—22no—
106C23×C127—21no—
107C23×C127—22no—
108D4×C127—21no—
109D4×C127—22no—
110D4×C127—21no—
111D4×C127—22no—
112D4×C127—42no—
113D4×C127—42no—
114C2×C4×C127—21no—
115C2×C4×C127—42no—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2411yescharacteristic
119C8×C127—22no—
120C22×C127—41no—
121C22×C127—42no—
122C22×C127—41no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127—41no—
126C22×C127C22×D411yescharacteristic
127C2×D127—21no—
128C2×D127—21no—
129C2×D127—21no—
130C2×D127—41no—
131C2×D127—41no—
132C2×D127—41no—
133C4×C127—21no—
134C4×C127—41no—
135C4×C127AGL(1,ℤ/8)12yes—
136C4×C127H(5,𝔽2)11yescharacteristic
137C4×C1272−1+411yescharacteristic
138Dic127—21no—
139Dic127—21no—
140Dic127—41no—
141Dic127—42no—
142Dic127AGL(1,ℤ/8)12yes—
143C2×C127—81no—
144C2×C127—82no—
145C2×C127#53711yescharacteristic
146C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
148D127—41no—
149D127—81no—
150#3253—1271nomaximal, Sylow
151C127#325311yescharacteristic, complemented, Sylow
152#532—1271no—
153#532—1271no—
154#476—1271no—
155#476—1271no—
156#476—1272no—
157#469—1272no—
158#469—1272no—
159#459—1272no—
160#459—1271no—
161#459D12711yescharacteristic, complemented
162#423—1271no—
163C2×#41—2541no—
164C2×#41—2541no—
165C2×D8—1272no—
166C2×D8C2×D12712yes—
167C2×M16—1272no—
168C4⋊Q8—1271no—
169C4⋊Q8—1271no—
170#127—1271no—
171#127C2×D12711yescharacteristic
172#124—2541no—
173#124—2541no—
174#121—1271no—
175#121—1272no—
176#121—1271no—
177#121—2541no—
178#121—2541no—
179C4×D4—1272no—
180C4×D4—1271no—
181C4×D4—2541no—
182C4×D4—2541no—
183C4×D4C2×D12711yescharacteristic
184C2×C42—1271no—
185#106—1272no—
186#106—1272no—
187C4⋊C8—1272no—
188C4⋊C8—1271no—
189C4⋊C8C2×D12711yescharacteristic
190#102—1272no—
191#102—1272no—
192#102—1272no—
193#102C2×D12712yes—
194#41—5084no—
195#41—5084no—
196C2×Q8—2541no—
197C2×Q8—2541no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—1271no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4—2541no—
204C2×D4—2541no—
205C2×D4—2542no—
206C2×D4C22×D12711yescharacteristic
207C2×D4C22×D12712yes—
208C22×C4—21no—
209C22×C4—1271no—
210C22×C4—1271no—
211C22×C4—1271no—
212C22×C4—2542no—
213C22×C4—2541no—
214D8—42no—
215D8—5082no—
216M16—5082no—
217C2×C8—1272no—
218C2×C8C22×D12712yes—
219C4⋊C4—1271no—
220C4⋊C4—1272no—
221C4⋊C4—2541no—
222C4⋊C4—2541no—
223C4⋊C4C22×D12711yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—2541no—
226C22⋊C4—2542no—
227C22⋊C4—2541no—
228C22⋊C4—2542no—
229C22⋊C4—2541no—
230C42—1271no—
231C42—1271no—
232C42—1271no—
233C42C22×D12711yescharacteristic
234C23—21no—
235C23—22no—
236C23—1271no—
237C23—2541no—
238Q8—5082no—
239Q8—5082no—
240D4—21no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—42no—
245D4—42no—
246D4—5082no—
247D4—5082no—
248D4—5084no—
249D4—5082no—
250D4—5082no—
251D4—2541no—
252D4—2541no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—1271no—
256C2×C4—1271no—
257C2×C4—1271no—
258C2×C4—5084no—
259C2×C4—5082no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4—2541no—
263C2×C4—2542no—
264C2×C4—2541no—
265C2×C4—2541no—
266C2×C4D4×D12711yescharacteristic
267C2×C4D4×D12711yescharacteristic
268C2×C4C23×D12711yescharacteristic
269C8—22no—
270C8—2542no—
271C22—42no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—42no—
277C22—5081no—
278C22—5081no—
279C22—5081no—
280C22—2541no—
281C22—2541no—
282C22—2541no—
283C22C2×D4×D12711yescharacteristic
284C4—21no—
285C4—41no—
286C4—1272no—
287C4—5082no—
288C4—5081no—
289C4—2541no—
290C4—2541no—
291C4#1686123312yes—
292C4#1686125411yescharacteristic
293C4#1686125611yescharacteristic
294C2—82no—
295C2—81no—
296C2—10161no—
297C2—5081no—
298C2#2255045511yescharacteristic
299C2AGL(1,ℤ/8)×D12711yescharacteristic
300C2C2×#1686123311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3253
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22550424C2semidirect
2. #22550423C2semidirect
3. #22550167C2semidirect
4. #22550166C2semidirect
5. #22550164C2semidirect
6. #22550128C2semidirect
7. #22550126C2semidirect
8. #22550069C2semidirect
9. #22550068C2semidirect
10. #22549898C2semidirect
11. #459×C127C2semidirect
151. C127#3253semidirect
161. #459D127semidirect

Maximal subgroups

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

subgroupindex
#225504242
#225504232
#225501672
#225501662
#225501642
#225501282
#225501262
#225500692
#225500682
#225498982
#459×C1272
#3253127