Finite Group Databasebeta

← #30648368

Subgroups — #30648368

Order 16256.

492 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14523236C211yesmaximal, characteristic, complemented
2C2×#16861217C211yesmaximal, characteristic, complemented
3#22550452C211yesmaximal, characteristic, complemented
4#22550443C211yesmaximal, characteristic
5#22550121C214yesmaximal, complemented
6#22550104C214yesmaximal, complemented
7#22549912C211yesmaximal, characteristic, complemented
8#22549886C211yesmaximal, characteristic, complemented
9#418×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×Dic254C2211yescharacteristic
12C2×#14523236—24no—
13C2×#14523236C2212yes—
14C2×#14523236C2212yes—
15#16861217—24no—
16C2×#14523231C2212yescomplemented
17C22×C127⋊C8C2211yescharacteristic
18C2×C8×D127C2212yescomplemented
19#16861209—21no—
20#16861205—21no—
21#16861197—21no—
22#16861196—21no—
23#16861193—22no—
24#16861189C2214yescomplemented
25#16861180C2214yes—
26#16861178—22no—
27#16861168C2211yescharacteristic
28#16861142C2212yescomplemented
29#16861140C2212yescomplemented
30#16861134C2214yescomplemented
31C22×D4×C127C2211yescharacteristic
32C2×M16×C127C2211yescharacteristic
33#117×C127C2211yescharacteristic
34#102×C127C2214yes—
35#14523248—22no—
36#14523248—22no—
37C2×Dic254—21no—
38C2×Dic254—22no—
39C2×Dic254—22no—
40C2×Dic254D411yescharacteristic
41C2×Dic254C2312yes—
42C2×D508D411yescharacteristic
43C2×C22⋊D127—24no—
44C2×C22⋊D127D412yes—
45C22×Dic127—21no—
46C2×C4×D127C2312yes—
47#14523236—24no—
48#14523236—24no—
49#14523231—24no—
50C2×C127⋊C8D414yescomplemented
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2311yescharacteristic
53C8×D127—24no—
54C4⋊Dic127—24no—
55C4⋊Dic127C2312yes—
56C22⋊Dic127—24no—
57C22⋊Dic127—21no—
58(C2×D127)⋊C4—24no—
59C4×Dic127—21no—
60C24×C127—21no—
61C2×D4×C127—22no—
62C2×D4×C127—21no—
63C2×D4×C127—22no—
64C2×D4×C127C2312yes—
65C22×C4×C127C2311yescharacteristic
66M16×C127—22no—
67C2×C8×C127C2312yes—
68C4⋊C4×C127C2312yes—
69C22⋊C4×C127—21no—
70C42×C127—21no—
71C22×D127C2×D412yes—
72Dic254—22no—
73Dic254—22no—
74Dic254—22no—
75Dic254—42no—
76D508—22no—
77C22⋊D127—24no—
78C22⋊D127—84no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—42no—
82C2×Dic127C2×D412yes—
83C4×D127—22no—
84C4×D127—22no—
85C127⋊C8C2×D412yes—
86C127⋊C8C2×D412yes—
87C23×C127—22no—
88C23×C127—42no—
89C23×C127—82no—
90C23×C127C2×D411yescharacteristic
91D4×C127—22no—
92D4×C127—22no—
93D4×C127—42no—
94C2×C4×C127—22no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D412yes—
97C2×C4×C127#4112yes—
98C2×C4×C127C2411yescharacteristic
99C8×C127—22no—
100C22×C127—22no—
101C22×C127—42no—
102C22×C127—44no—
103C22×C127—84no—
104C22×C127#12112yes—
105C22×C127C22×D411yescharacteristic
106C2×D127—22no—
107C2×D127—24no—
108C4×C127—42no—
109C4×C127#12112yes—
110C4×C127C22×D411yescharacteristic
111C4×C127C2×#4111yescharacteristic
112Dic127—22no—
113Dic127—42no—
114Dic127—42no—
115C2×C127—22no—
116C2×C127—82no—
117C2×C127C2×#12111yescharacteristic
118C2×C127#57712yes—
119D127—42no—
120#3057—1271nomaximal, Sylow
121C127#305711yescharacteristic, complemented, Sylow
122C22×SD16—1271no—
123C2×#131—1271no—
124#536—1271no—
125#535—1271no—
126#468—1274no—
127#465—1274no—
128#427—1271no—
129#419—1271no—
130#418D12711yescharacteristic, complemented
131C2×#41—1271no—
132C22×Q8—1271no—
133C22×D4C2×D12711yescharacteristic
134C2×SD16—1272no—
135C2×SD16—1272no—
136C2×SD16—2544no—
137#131—2544no—
138C2×M16—1272no—
139C2×M16C2×D12711yescharacteristic
140C22×C8—1271no—
141C22×C8—1272no—
142C4⋊Q8—2541no—
143#127—2541no—
144#124—2541no—
145#124—2541no—
146#123—2542no—
147C22⋊Q8—1274no—
148#121—1274no—
149#120—2542no—
150#117—1271no—
151#117C2×D12711yescharacteristic
152#107—1272no—
153#106—1272no—
154#103—1274no—
155#102C2×D12714yes—
156C24—21no—
157#41—2542no—
158#41—2542no—
159C2×Q8—1271no—
160C2×Q8—1272no—
161C2×Q8—2542no—
162C2×Q8—2542no—
163C2×Q8—2541no—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—1272no—
168C2×D4—1271no—
169C2×D4—2544no—
170C2×D4C22×D12712yes—
171C22×C4—1272no—
172C22×C4—2541no—
173C22×C4C22×D12711yescharacteristic
174SD16—2544no—
175SD16—2544no—
176M16—22no—
177M16—2544no—
178C2×C8—1274no—
179C2×C8—1271no—
180C2×C8—1271no—
181C2×C8—2544no—
182C2×C8C22×D12712yes—
183C4⋊C4—1272no—
184C4⋊C4—2544no—
185C4⋊C4C22×D12712yes—
186C22⋊C4—21no—
187C22⋊C4—2544no—
188C22⋊C4—2544no—
189C22⋊C4—2541no—
190C42—21no—
191C42—2541no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—1272no—
196C23D4×D12711yescharacteristic
197Q8—5082no—
198Q8—2542no—
199Q8—2542no—
200Q8—2542no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—10164no—
205D4—2544no—
206D4—2542no—
207C2×C4—22no—
208C2×C4—1272no—
209C2×C4—5082no—
210C2×C4—2542no—
211C2×C4—2542no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4D4×D12712yes—
215C2×C4D4×D12711yescharacteristic
216C2×C4#1452324812yes—
217C2×C4C23×D12711yescharacteristic
218C8—22no—
219C8—1272no—
220C8—1272no—
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—2542no—
226C22—2544no—
227C22#1686118212yes—
228C22C2×D4×D12711yescharacteristic
229C4—42no—
230C4—5082no—
231C4—5082no—
232C4—2542no—
233C4#1686118212yes—
234C4C2×D4×D12711yescharacteristic
235C4C2×#1452324811yescharacteristic
236C2—22no—
237C2—82no—
238C2—5082no—
239C2C2×#1686118211yescharacteristic
240C2#2255068612yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
prime radical#22550443C2
hypercenter#418D127
hypocenterC127#3057
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
2. C2×#16861217C2semidirect
3. #22550452C2semidirect
5. #22550121C2semidirect
6. #22550104C2semidirect
7. #22549912C2semidirect
8. #22549886C2semidirect
9. #418×C127C2semidirect
16. C2×#14523231C22semidirect
18. C2×C8×D127C22semidirect
24. #16861189C22semidirect
28. #16861142C22semidirect
29. #16861140C22semidirect
30. #16861134C22semidirect
50. C2×C127⋊C8D4semidirect
121. C127#3057semidirect
130. #418D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168612172
#225504522
#225504432
#225501212
#225501042
#225499122
#225498862
#418×C1272
#3057127