Finite Group Databasebeta

← #30648371

Subgroups — #30648371

Order 16256.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14523235C211yesmaximal, characteristic, complemented
2C2×#16861216C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic
4#22550124C214yesmaximal, complemented
5#22550107C214yesmaximal, complemented
6#22549912C211yesmaximal, characteristic, complemented
7#22549885C211yesmaximal, characteristic, complemented
8#22549880C211yesmaximal, characteristic, complemented
9#536×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C2×#14523235—24no—
12C2×#14523235C2212yes—
13C2×#14523235C2212yes—
14#16861216—24no—
15C2×#14523232C2211yescharacteristic
16C2×#14523232C2212yescomplemented
17C22×C127⋊C8C2211yescharacteristic
18C22×C127⋊C8C2212yescomplemented
19#16861205—21no—
20#16861198—21no—
21#16861183C2214yes—
22#16861177—22no—
23#16861168C2211yescharacteristic
24#16861142C2212yescomplemented
25#16861140C2212yescomplemented
26#16861133C2214yescomplemented
27#16861129C2214yes—
28C2×#41×C127C2211yescharacteristic
29C22×Q8×C127C2211yescharacteristic
30C4⋊Q8×C127—21no—
31#124×C127—21no—
32#123×C127—22no—
33C22⋊Q8×C127C2214yescomplemented
34#117×C127C2211yescharacteristic
35C23×D127—21no—
36C2×D508—22no—
37C2×D508—21no—
38C2×D508—22no—
39C2×D508C2312yes—
40C2×C22⋊D127—24no—
41#14523235—24no—
42#14523235—24no—
43#14523232—24no—
44#14523232—22no—
45C2×C127⋊C8—24no—
46C2×C127⋊C8D414yescomplemented
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2311yescharacteristic
49C2×C127⋊C8C2311yescharacteristic
50C4⋊Dic127C2312yes—
51C22⋊Dic127—21no—
52(C2×D127)⋊C4—24no—
53C4×Dic127—21no—
54#41×C127—22no—
55#41×C127—22no—
56C2×Q8×C127—21no—
57C2×Q8×C127—22no—
58C2×Q8×C127—22no—
59C2×Q8×C127D411yescharacteristic
60C2×Q8×C127C2312yes—
61C2×D4×C127D411yescharacteristic
62C2×D4×C127D412yes—
63C22×C4×C127—21no—
64C22×C4×C127C2312yes—
65C22×C4×C127C2311yescharacteristic
66C4⋊C4×C127—24no—
67C4⋊C4×C127C2312yes—
68C22⋊C4×C127—21no—
69C22⋊C4×C127—24no—
70C42×C127—21no—
71C22×D127—22no—
72C22×D127—42no—
73C22×D127—82no—
74D508—22no—
75D508—22no—
76D508—42no—
77C22⋊D127—84no—
78C2×Dic127—22no—
79C127⋊C8—22no—
80C127⋊C8C2×D412yes—
81C127⋊C8C2×D412yes—
82C23×C127C2×D412yes—
83C23×C127C2×D411yescharacteristic
84Q8×C127—22no—
85Q8×C127—22no—
86Q8×C127—22no—
87Q8×C127—42no—
88D4×C127—24no—
89D4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—42no—
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2×D412yes—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127#4112yes—
99C2×C4×C127C2411yescharacteristic
100C22×C127—22no—
101C22×C127—22no—
102C22×C127—24no—
103C22×C127#12112yes—
104C22×C127C22×D411yescharacteristic
105C2×D127—42no—
106C2×D127—44no—
107C2×D127—84no—
108C4×C127—22no—
109C4×C127—42no—
110C4×C127—42no—
111C4×C127#12112yes—
112C4×C127C22×D411yescharacteristic
113C4×C127C2×#4111yescharacteristic
114Dic127—42no—
115C2×C127—22no—
116C2×C127—42no—
117C2×C127C2×#12111yescharacteristic
118C2×C127#57712yes—
119D127—82no—
120#3057—1271nomaximal, Sylow
121C127#305711yescharacteristic, complemented, Sylow
122C22×SD16—1271no—
123C2×#131—1271no—
124#536D12711yescharacteristic, complemented
125#535—1271no—
126#468—1274no—
127#465—1274no—
128#427—1271no—
129#419—1271no—
130#418—1271no—
131C2×#41C2×D12711yescharacteristic
132C22×Q8C2×D12711yescharacteristic
133C22×D4—1271no—
134C2×SD16—1272no—
135C2×SD16—1272no—
136C2×SD16—2544no—
137#131—2544no—
138C2×M16—1271no—
139C2×M16—1272no—
140C22×C8—1271no—
141C22×C8—1272no—
142C4⋊Q8—21no—
143#127—2541no—
144#124—21no—
145#124—2541no—
146#123—22no—
147C22⋊Q8C2×D12714yescomplemented
148#121—1274no—
149#120—2542no—
150#117—1271no—
151#117C2×D12711yescharacteristic
152#107—1272no—
153#106—1272no—
154#103—1274no—
155#102—1274no—
156C24—2541no—
157#41—22no—
158#41—22no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8C22⋊D12711yescharacteristic
163C2×Q8C22×D12712yes—
164C2×D4—1272no—
165C2×D4—2544no—
166C2×D4—2542no—
167C2×D4—2541no—
168C2×D4—2542no—
169C2×D4C22⋊D12712yes—
170C2×D4C22⋊D12711yescharacteristic
171C22×C4—21no—
172C22×C4C22×D12712yes—
173C22×C4C22×D12711yescharacteristic
174SD16—2544no—
175SD16—2544no—
176M16—2544no—
177M16—2542no—
178C2×C8—1274no—
179C2×C8—1271no—
180C2×C8—1271no—
181C2×C8—1272no—
182C2×C8—2544no—
183C4⋊C4—24no—
184C4⋊C4—1272no—
185C4⋊C4C22×D12712yes—
186C22⋊C4—21no—
187C22⋊C4—24no—
188C22⋊C4—2541no—
189C22⋊C4—2544no—
190C42—21no—
191C42—2541no—
192C23—10162no—
193C23—5082no—
194C23—2542no—
195C23C2×C22⋊D12712yes—
196C23C2×C22⋊D12711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—24no—
202D4—22no—
203D4—10164no—
204D4—5082no—
205D4—2542no—
206D4—2542no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—2542no—
213C2×C4D4×D12712yes—
214C2×C4C2×C22⋊D12712yes—
215C2×C4C2×C22⋊D12711yescharacteristic
216C2×C4#1452324712yes—
217C2×C4C23×D12711yescharacteristic
218C8—1272no—
219C8—1272no—
220C8—2542no—
221C22—22no—
222C22—22no—
223C22—24no—
224C22—10164no—
225C22—5082no—
226C22—5084no—
227C22#1686118112yes—
228C22C22×C22⋊D12711yescharacteristic
229C4—22no—
230C4—42no—
231C4—42no—
232C4—5082no—
233C4#1686118112yes—
234C4C2×D4×D12711yescharacteristic
235C4C2×#1452324711yescharacteristic
236C2—22no—
237C2—42no—
238C2—10162no—
239C2C2×#1686118111yescharacteristic
240C2#2255068412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC22×C127C22×D4
prime radical#22550440C2
hypercenter#536D127
hypocenterC127#3057
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#14523235C2semidirect
2. C2×#16861216C2semidirect
4. #22550124C2semidirect
5. #22550107C2semidirect
6. #22549912C2semidirect
7. #22549885C2semidirect
8. #22549880C2semidirect
9. #536×C127C2semidirect
16. C2×#14523232C22semidirect
18. C22×C127⋊C8C22semidirect
24. #16861142C22semidirect
25. #16861140C22semidirect
26. #16861133C22semidirect
33. C22⋊Q8×C127C22semidirect
46. C2×C127⋊C8D4semidirect
121. C127#3057semidirect
124. #536D127semidirect
147. C22⋊Q8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232352
C2×#168612162
#225504402
#225501242
#225501072
#225499122
#225498852
#225498802
#536×C1272
#3057127