Finite Group Databasebeta

← #30650115

Subgroups — #30650115

Order 16256.

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

normal subgroups only

This group has 386 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#22550430C211yesmaximal, characteristic, complemented
2#22550424C211yesmaximal, characteristic
3#22550172C211yesmaximal, characteristic, complemented
4#22550171C211yesmaximal, characteristic, complemented
5#22550154C212yesmaximal, complemented
6#22550121C212yesmaximal, complemented
7#22550119C212yesmaximal, complemented
8#22550079C212yesmaximal, complemented
9#22550073C211yesmaximal, characteristic, complemented
10#22549898C211yesmaximal, characteristic, complemented
11#460×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C2×#14523236C2212yes—
14C2×#14523231C2212yes—
15#16861209C2211yescharacteristic
16#16861208C2211yescharacteristic
17#16861207C2211yescharacteristic
18#16861199—21no—
19#16861196—21no—
20#16861189C2211yescharacteristic
21#16861187—21no—
22#16861185C2211yescharacteristic
23#16861180—21no—
24#16861180C2212yes—
25C4×Dic254—21no—
26C4×Dic254C2211yescharacteristic
27D4×Dic127—21no—
28D4×Dic127C2212yes—
29#16861166—21no—
30C42×D127C2211yescharacteristic
31#16861143C2212yes—
32#16861142C2212yes—
33C4⋊(C127⋊C8)C2211yescharacteristic
34Dic127⋊C8C2212yes—
35#16861134C2212yes—
36#16861133C2212yes—
37#16861130C2212yes—
38C2×SD16×C127C2212yes—
39#127×C127C2211yescharacteristic
40C4×Q8×C127C2211yescharacteristic
41C4⋊C8×C127C2211yescharacteristic
42#102×C127C2212yes—
43#14523247—44no—
44C2×Dic254—21no—
45C2×Dic254—21no—
46C2×Dic254C2311yescharacteristic
47C2×C22⋊D127—22no—
48C22×Dic127—22no—
49C2×C4×D127D411yescharacteristic
50C2×C4×D127D411yescharacteristic
51C2×C4×D127C2311yescharacteristic
52#14523236—42no—
53#14523231—42no—
54C2×C127⋊C8C2312yes—
55C4⋊Dic127—21no—
56C4⋊Dic127—21no—
57C4⋊Dic127C2311yescharacteristic
58C4⋊Dic127C2312yes—
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64C22⋊Dic127—22no—
65C22⋊Dic127—22no—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—21no—
68C4×Dic127—21no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127C2311yescharacteristic
73C2×D4×C127—21no—
74C2×D4×C127—21no—
75C2×D4×C127C2312yes—
76SD16×C127—42no—
77C2×C8×C127C2312yes—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C42×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127C2×D411yescharacteristic
83Dic254—21no—
84Dic254—21no—
85Dic254—42no—
86Dic254—42no—
87C22⋊D127—44no—
88C2×Dic127—21no—
89C2×Dic127—21no—
90C2×Dic127—22no—
91C2×Dic127—44no—
92C2×Dic127C2×D411yescharacteristic
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C4×D127—22no—
96C4×D127—21no—
97C4×D127—21no—
98C4×D127—22no—
99C127⋊C8—22no—
100C23×C127—22no—
101Q8×C127—21no—
102Q8×C127—21no—
103D4×C127—22no—
104D4×C127—22no—
105D4×C127—42no—
106D4×C127—42no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—42no—
114C22×C127—42no—
115C22×C127—42no—
116C22×C127C22×D411yescharacteristic
117C2×D127—21no—
118C2×D127—21no—
119C2×D127—21no—
120C4×C127—21no—
121C4×C127—41no—
122C4×C127—41no—
123C4×C127#13712yes—
124C4×C127H(5,𝔽2)11yescharacteristic
125C4×C1272−1+411yescharacteristic
126Dic127—21no—
127Dic127—21no—
128Dic127—41no—
129Dic127—42no—
130Dic127—41no—
131Dic127AGL(1,ℤ/8)12yes—
132C2×C127—82no—
133C2×C127#53711yescharacteristic
134C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C127C2×#13711yescharacteristic
136D127—41no—
137#3254—1271nomaximal, Sylow
138C127#325411yescharacteristic, complemented, Sylow
139#533—1271no—
140#532—1271no—
141#477—1271no—
142#477—1271no—
143#474—1272no—
144#468—1272no—
145#468—1272no—
146#461—1272no—
147#460—1271no—
148#460D12711yescharacteristic, complemented
149#423—1271no—
150C2×#41—2541no—
151C2×SD16—1272no—
152C2×SD16C2×D12712yes—
153C2×M16—1272no—
154C4⋊Q8—1271no—
155C4⋊Q8—1271no—
156C4⋊Q8—1271no—
157#127C2×D12711yescharacteristic
158#125—2541no—
159#124—2541no—
160C22⋊Q8—1271no—
161C22⋊Q8—1271no—
162C22⋊Q8—2541no—
163#121—1272no—
164#121—2541no—
165C4×Q8—1271no—
166C4×Q8—2541no—
167C4×Q8C2×D12711yescharacteristic
168C4×D4—1272no—
169C4×D4—2541no—
170#117—2541no—
171C2×C42—1271no—
172#107—1272no—
173#107—1272no—
174C4⋊C8—1272no—
175C4⋊C8—1271no—
176C4⋊C8C2×D12711yescharacteristic
177#103—1272no—
178#103—1272no—
179#102—1272no—
180#102C2×D12712yes—
181#41—5084no—
182C2×Q8—1271no—
183C2×Q8—2541no—
184C2×Q8—2541no—
185C2×Q8C22×D12711yescharacteristic
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—2542no—
189C2×D4C22×D12712yes—
190C22×C4—1271no—
191C22×C4—1271no—
192C22×C4—1271no—
193C22×C4—2542no—
194SD16—42no—
195SD16—5082no—
196M16—5082no—
197C2×C8—1272no—
198C2×C8C22×D12712yes—
199C4⋊C4—21no—
200C4⋊C4—1271no—
201C4⋊C4—1272no—
202C4⋊C4—2541no—
203C4⋊C4—2541no—
204C4⋊C4—2541no—
205C4⋊C4—2541no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4C22×D12711yescharacteristic
210C22⋊C4—2541no—
211C22⋊C4—2541no—
212C22⋊C4—2542no—
213C22⋊C4—2542no—
214C42—21no—
215C42—1271no—
216C42—1271no—
217C42—1271no—
218C42—2541no—
219C42C22×D12711yescharacteristic
220C23—22no—
221C23—1271no—
222Q8—21no—
223Q8—21no—
224Q8—5082no—
225Q8—5082no—
226Q8—2541no—
227Q8—2541no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—42no—
232D4—5084no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—1271no—
236C2×C4—1271no—
237C2×C4—1271no—
238C2×C4—5084no—
239C2×C4—2542no—
240C2×C4—2541no—
241C2×C4—2541no—
242C2×C4—2542no—
243C2×C4—2542no—
244C2×C4—2541no—
245C2×C4—2541no—
246C2×C4D4×D12711yescharacteristic
247C2×C4D4×D12711yescharacteristic
248C2×C4C23×D12711yescharacteristic
249C8—22no—
250C8—2542no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—2541no—
255C22—2541no—
256C22—2541no—
257C22C2×D4×D12711yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—1272no—
262C4—5082no—
263C4—5081no—
264C4—5081no—
265C4—2541no—
266C4—2541no—
267C4#1686123812yes—
268C4#1686125411yescharacteristic
269C4#1686125611yescharacteristic
270C2—82no—
271C2—5081no—
272C2#2255045511yescharacteristic
273C2AGL(1,ℤ/8)×D12711yescharacteristic
274C2C2×#1686123811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
prime radical#22550424C2
hypercenter#460D127
hypocenterC127#3254
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #22550430C2semidirect
3. #22550172C2semidirect
4. #22550171C2semidirect
5. #22550154C2semidirect
6. #22550121C2semidirect
7. #22550119C2semidirect
8. #22550079C2semidirect
9. #22550073C2semidirect
10. #22549898C2semidirect
11. #460×C127C2semidirect
138. C127#3254semidirect
148. #460D127semidirect

Maximal subgroups

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

subgroupindex
#225504302
#225504242
#225501722
#225501712
#225501542
#225501212
#225501192
#225500792
#225500732
#225498982
#460×C1272
#3254127