Finite Group Databasebeta

← #30650683

Subgroups — #30650683

Order 16256.

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

normal subgroups only

This group has 408 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#22550591C211yesmaximal, characteristic, complemented
2#22550546C211yesmaximal, characteristic, complemented
3#22550222C211yesmaximal, characteristic, complemented
4#22550198C212yesmaximal, complemented
5#22550172C211yesmaximal, characteristic, complemented
6#22550109C212yesmaximal, complemented
7#22550018C212yesmaximal, complemented
8C4×#14523236C211yesmaximal, characteristic, complemented
9#22549941C211yesmaximal, characteristic, complemented
10#22549869C212yesmaximal, complemented
11#459×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yes—
13C2×#14523236C2211yescharacteristic
14C2×C8×D127C2212yes—
15#16861209C2211yescharacteristic
16#16861205—21no—
17#16861195—22no—
18#16861183—21no—
19#16861181—22no—
20#16861180C2212yes—
21#16861180C2212yes—
22C4×Dic254C2211yescharacteristic
23C4×D508C2211yescharacteristic
24D4×Dic127—21no—
25#16861168—21no—
26#16861167C2212yes—
27#16861143C2212yes—
28#16861140C2211yescharacteristic
29#16861134C2212yes—
30#16861132C2211yescharacteristic
31#16861130C2212yes—
32#16861130C2211yescharacteristic
33#16861130C2212yes—
34(C2×D127)⋊C8C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C2×D8×C127C2212yes—
37#127×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41#14523247—24no—
42#14523247—42no—
43C2×Dic254—21no—
44C2×Dic254C2311yescharacteristic
45C2×D508D411yescharacteristic
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—22no—
49C22×Dic127—22no—
50C2×C4×D127C2312yes—
51#14523236—22no—
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2311yescharacteristic
54C8×D127—42no—
55C4⋊Dic127—21no—
56C4⋊Dic127D411yescharacteristic
57C4⋊Dic127C2312yes—
58C4⋊Dic127C2311yescharacteristic
59Dic127⋊C4—21no—
60C22⋊Dic127—22no—
61C22⋊Dic127—22no—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4D412yes—
64C4×Dic127—22no—
65C4×Dic127—21no—
66C2×D4×C127—21no—
67C2×D4×C127—21no—
68C2×D4×C127C2311yescharacteristic
69C2×D4×C127C2312yes—
70C22×C4×C127—21no—
71D8×C127—42no—
72C2×C8×C127C2312yes—
73C4⋊C4×C127C2311yescharacteristic
74C22⋊C4×C127—21no—
75C42×C127C2311yescharacteristic
76C22×D127C2×D412yes—
77Dic254—21no—
78Dic254—81no—
79Dic254#4112yes—
80D508—22no—
81C22⋊D127—44no—
82C22⋊D127—82no—
83C22⋊D127—82no—
84C2×Dic127—21no—
85C2×Dic127—22no—
86C2×Dic127—21no—
87C2×Dic127—44no—
88C2×Dic127C2×D412yes—
89C4×D127—22no—
90C4×D127—22no—
91C127⋊C8—21no—
92C127⋊C8—21no—
93C23×C127—21no—
94C23×C127—22no—
95D4×C127—21no—
96D4×C127—22no—
97D4×C127—21no—
98D4×C127—22no—
99D4×C127—42no—
100D4×C127—81no—
101C2×C4×C127—21no—
102C2×C4×C127—42no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2411yescharacteristic
106C8×C127—22no—
107C22×C127—41no—
108C22×C127—42no—
109C22×C127—41no—
110C22×C127—42no—
111C22×C127—42no—
112C22×C127—41no—
113C22×C127C22×D411yescharacteristic
114C2×D127—22no—
115C2×D127—22no—
116C2×D127—22no—
117C4×C127—21no—
118C4×C127—41no—
119C4×C127#13512yes—
120C4×C127C2×#4111yescharacteristic
121C4×C127H(5,𝔽2)11yescharacteristic
122Dic127—22no—
123Dic127—22no—
124Dic127—41no—
125Dic127—42no—
126C2×C127—81no—
127C2×C127—82no—
128C2×C127#54611yescharacteristic
129C2×C127C2×#13511yescharacteristic
130C2×C127#57611yescharacteristic
131D127—42no—
132#3305—1271nomaximal, Sylow
133C127#330511yescharacteristic, complemented, Sylow
134#555—1271no—
135#550—1271no—
136#486—1271no—
137#482—1272no—
138#477—1271no—
139#466—1272no—
140#459D12711yescharacteristic, complemented
141#449—1272no—
142C4×SD16—1271no—
143C8×D4—1271no—
144#416—1272no—
145C2×#41—1272no—
146C2×SD16—1271no—
147C2×D8C2×D12712yes—
148C22×C8—1272no—
149C4⋊Q8—1271no—
150#127—2541no—
151#127C2×D12711yescharacteristic
152#124—2542no—
153#121—1272no—
154#121—1272no—
155#121—2542no—
156#121—2541no—
157C4×Q8—1271no—
158C4×D4—1271no—
159C4×D4—2541no—
160C4×D4C2×D12711yescharacteristic
161#117—1272no—
162#117—2541no—
163#107—1272no—
164#106—1271no—
165C4⋊C8C2×D12711yescharacteristic
166#103—1272no—
167#103—1271no—
168#102—1272no—
169#102—1271no—
170#102—1272no—
171#102C2×D12712yes—
172C22⋊C8—1272no—
173C4×C8—1271no—
174#41—5082no—
175#41—2544no—
176C2×Q8—1271no—
177C2×Q8—2541no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—1271no—
181C2×D4—2542no—
182C2×D4—2542no—
183C2×D4—2542no—
184C2×D4C22×D12712yes—
185C2×D4C22×D12711yescharacteristic
186C22×C4—21no—
187C22×C4—1272no—
188C22×C4—2542no—
189SD16—2542no—
190D8—42no—
191C2×C8—1271no—
192C2×C8—1271no—
193C2×C8—5082no—
194C2×C8C22×D12712yes—
195C4⋊C4—1271no—
196C4⋊C4—1272no—
197C4⋊C4—1271no—
198C4⋊C4—2541no—
199C4⋊C4—2541no—
200C4⋊C4C22×D12711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—1272no—
203C22⋊C4—2542no—
204C22⋊C4—2542no—
205C22⋊C4—2542no—
206C42—2542no—
207C42—2541no—
208C42C22×D12711yescharacteristic
209C23—21no—
210C23—22no—
211C23—1272no—
212Q8—1272no—
213Q8—10161no—
214Q8—2541no—
215D4—21no—
216D4—22no—
217D4—21no—
218D4—22no—
219D4—42no—
220D4—81no—
221D4—10162no—
222D4—10162no—
223D4—5084no—
224D4—2542no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—1272no—
228C2×C4—5084no—
229C2×C4—2541no—
230C2×C4—2542no—
231C2×C4—2541no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4D4×D12711yescharacteristic
235C2×C4D4×D12711yescharacteristic
236C2×C4C23×D12711yescharacteristic
237C8—22no—
238C8—2541no—
239C8—2541no—
240C22—41no—
241C22—42no—
242C22—42no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—2542no—
247C22—2542no—
248C22—2542no—
249C22C2×D4×D12711yescharacteristic
250C4—21no—
251C4—41no—
252C4—5081no—
253C4—5082no—
254C4—2542no—
255C4—2542no—
256C4#1686122812yes—
257C4#41×D12711yescharacteristic
258C4#1686125411yescharacteristic
259C2—81no—
260C2—82no—
261C2—5082no—
262C2#2255052311yescharacteristic
263C2C2×#1686122811yescharacteristic
264C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3305
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. #22550591C2semidirect
2. #22550546C2semidirect
3. #22550222C2semidirect
4. #22550198C2semidirect
5. #22550172C2semidirect
6. #22550109C2semidirect
7. #22550018C2semidirect
8. C4×#14523236C2semidirect
9. #22549941C2semidirect
10. #22549869C2semidirect
11. #459×C127C2semidirect
133. C127#3305semidirect
140. #459D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225505462
#225502222
#225501982
#225501722
#225501092
#225500182
C4×#145232362
#225499412
#225498692
#459×C1272
#3305127