Finite Group Databasebeta

← #30650371

Subgroups — #30650371

Order 16256.

472 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#16861241C212yesmaximal, complemented
2Q8×D508C211yesmaximal, characteristic, complemented
3#22550529C211yesmaximal, characteristic
4#22550255C211yesmaximal, characteristic, complemented
5#22550133C212yesmaximal, complemented
6#22550082C211yesmaximal, characteristic, complemented
7#22550041C212yesmaximal, complemented
8#22550030C212yesmaximal, complemented
9C4×#14523235C211yesmaximal, characteristic, complemented
10#22549953C211yesmaximal, characteristic, complemented
11#462×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13C2×Q8×D127C2212yes—
14#16861241—28no—
15C2×#14523238C2212yes—
16C2×#14523238C2212yes—
17C2×#14523237C2212yes—
18C2×#14523235C2211yescharacteristic
19C2×#14523231C2212yes—
20#16861209—21no—
21#16861209C2211yescharacteristic
22#16861194—22no—
23#16861189—22no—
24#16861189C2212yes—
25#16861185C2212yes—
26#16861184—21no—
27C4×D508—21no—
28C4×D508C2211yescharacteristic
29C4×D508C2211yescharacteristic
30C4⋊C4×D127—21no—
31#16861140C2211yescharacteristic
32#16861134C2212yes—
33#16861133C2211yescharacteristic
34#16861129C2211yescharacteristic
35(C2×D127)⋊C8C2212yes—
36C4×C127⋊C8C2211yescharacteristic
37C2×Q16×C127C2212yes—
38C4⋊Q8×C127C2211yescharacteristic
39C4×Q8×C127C2211yescharacteristic
40C4⋊C8×C127C2211yescharacteristic
41#103×C127C2212yes—
42#14523249—24no—
43#14523249—42no—
44C2×Dic254—22no—
45C2×Dic254C2312yes—
46Q8×D127—24no—
47Q8×D127—42no—
48C2×D508—22no—
49C2×D508D411yescharacteristic
50C2×D508C2311yescharacteristic
51C2×C4×D127—22no—
52C2×C4×D127—22no—
53C2×C4×D127—21no—
54C2×C4×D127C2312yes—
55#14523238—24no—
56#14523238—42no—
57#14523237—24no—
58#14523235D414yescomplemented
59#14523231—24no—
60C2×C127⋊C8C2311yescharacteristic
61C2×C127⋊C8C2311yescharacteristic
62C4⋊Dic127—22no—
63C4⋊Dic127—21no—
64C4⋊Dic127—21no—
65C4⋊Dic127D411yescharacteristic
66C4⋊Dic127C2311yescharacteristic
67Dic127⋊C4—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—21no—
71(C2×D127)⋊C4D412yes—
72C2×Q8×C127C2312yes—
73C2×Q8×C127C2311yescharacteristic
74Q16×C127—24no—
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—21no—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C42×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127—21no—
83C22×D127C2×D412yes—
84Dic254—22no—
85Dic254—22no—
86Dic254—44no—
87D508—22no—
88D508—21no—
89D508—44no—
90D508C2×D412yes—
91C2×Dic127—21no—
92C2×Dic127—22no—
93C2×Dic127C2×D412yes—
94C4×D127—22no—
95C4×D127—22no—
96C4×D127—44no—
97C4×D127—44no—
98C4×D127—42no—
99C127⋊C8—21no—
100C127⋊C8C2×D412yes—
101Q8×C127—21no—
102Q8×C127—22no—
103Q8×C127—22no—
104Q8×C127C2×D412yes—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2411yescharacteristic
111C8×C127—22no—
112C22×C127C22×D411yescharacteristic
113C2×D127—22no—
114C2×D127—22no—
115C2×D127—22no—
116C2×D127—22no—
117C2×D127—41no—
118C2×D127—41no—
119C4×C127—22no—
120C4×C127—21no—
121C4×C127—42no—
122C4×C127—41no—
123C4×C127#13712yescomplemented
124C4×C127C22×D411yescharacteristic
125C4×C127H(5,𝔽2)11yescharacteristic
126Dic127—22no—
127Dic127—42no—
128Dic127—41no—
129C2×C127D4211yescharacteristic
130C2×C127C2×#13711yescharacteristic
131C2×C127#57811yescharacteristic
132D127—42no—
133D127—42no—
134#3276—1271nomaximal, Sylow
135C127#327611yescharacteristic, complemented, Sylow
136C2×#137—1272no—
137D4×Q8—1271no—
138#547—1271no—
139#494—1271no—
140#470—1272no—
141#462D12711yescharacteristic, complemented
142#461—1271no—
143#452—1272no—
144#451—1272no—
145C4×SD16—1271no—
146#436—1271no—
147C2×#41—1272no—
148C22×Q8—1272no—
149#137—2548no—
150C2×Q16—1272no—
151C2×Q16—1272no—
152C2×Q16C2×D12712yes—
153C2×SD16—1271no—
154C2×SD16—1272no—
155C2×M16—1272no—
156C4⋊Q8—1271no—
157C4⋊Q8—2541no—
158C4⋊Q8C2×D12711yescharacteristic
159#123—2542no—
160C22⋊Q8—1272no—
161C22⋊Q8—1272no—
162C22⋊Q8—2542no—
163#121—2541no—
164C4×Q8C2×D12711yescharacteristic
165C4×D4—1271no—
166C4×D4—1271no—
167C4×D4—2541no—
168C2×C4⋊C4—2541no—
169#106—1271no—
170C4⋊C8C2×D12711yescharacteristic
171#103—1271no—
172#103—1272no—
173#103C2×D12712yes—
174#102—1271no—
175C22⋊C8—1272no—
176C4×C8—1271no—
177#41—5082no—
178#41—2544no—
179C2×Q8—1272no—
180C2×Q8—5082no—
181C2×Q8—2542no—
182C2×Q8—2544no—
183C2×Q8C22×D12712yes—
184C2×Q8C22×D12711yescharacteristic
185C2×D4—1271no—
186C2×D4—1271no—
187C2×D4—2542no—
188C22×C4—1272no—
189C22×C4—2542no—
190C22×C4—2542no—
191C22×C4—2541no—
192Q16—24no—
193Q16—5082no—
194Q16—2544no—
195SD16—1274no—
196SD16—2544no—
197M16—2544no—
198C2×C8—1271no—
199C2×C8—1271no—
200C2×C8C22×D12712yes—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—1271no—
205C4⋊C4—1271no—
206C4⋊C4—2542no—
207C4⋊C4—2542no—
208C4⋊C4—2541no—
209C4⋊C4—2541no—
210C4⋊C4C22×D12711yescharacteristic
211C22⋊C4—1272no—
212C22⋊C4—2542no—
213C22⋊C4—2542no—
214C22⋊C4—2541no—
215C42—21no—
216C42C22×D12711yescharacteristic
217C23—1272no—
218C23—2541no—
219Q8—21no—
220Q8—22no—
221Q8—22no—
222Q8—5084no—
223Q8—2542no—
224Q8—2542no—
225Q8D4×D12712yes—
226D4—1272no—
227D4—5084no—
228D4—2542no—
229D4—2541no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—1272no—
234C2×C4—5084no—
235C2×C4—5084no—
236C2×C4—5082no—
237C2×C4—2541no—
238C2×C4—2542no—
239C2×C4—2542no—
240C2×C4—2542no—
241C2×C4D4×D12711yescharacteristic
242C2×C4D4×D12711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—22no—
245C8—1272no—
246C8—2541no—
247C22—5081no—
248C22—5081no—
249C22—2542no—
250C22—2542no—
251C22—2542no—
252C22—2542no—
253C22C2×D4×D12711yescharacteristic
254C4—21no—
255C4—22no—
256C4—41no—
257C4—42no—
258C4—5081no—
259C4—5082no—
260C4—2542no—
261C4#1686124112yescomplemented
262C4C2×D4×D12711yescharacteristic
263C4#1686125511yescharacteristic
264C2—5082no—
265C2—5082no—
266C2#2255051011yescharacteristic
267C2C2×#1686124111yescharacteristic
268C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#462×C127C2
socleC22×C127C22×D4
prime radical#22550529C2
hypercenter#462D127
hypocenterC127#3276
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#16861241C2semidirect
2. Q8×D508C2semidirect
4. #22550255C2semidirect
5. #22550133C2semidirect
6. #22550082C2semidirect
7. #22550041C2semidirect
8. #22550030C2semidirect
9. C4×#14523235C2semidirect
10. #22549953C2semidirect
11. #462×C127C2semidirect
58. #14523235D4semidirect
123. C4×C127#137semidirect
135. C127#3276semidirect
141. #462D127semidirect
261. C4#16861241semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
Q8×D5082
#225505292
#225502552
#225501332
#225500822
#225500412
#225500302
C4×#145232352
#225499532
#462×C1272
#3276127