Finite Group Databasebeta

← #30639556

Subgroups — #30639556

Order 16256.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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#22550422C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic, complemented
4C2×#16861136C211yesmaximal, characteristic, complemented
5C2×#16861135C211yesmaximal, characteristic, complemented
6C42×Dic127C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861207C411yescharacteristic, complemented
11#16861205C411yescharacteristic, complemented
12#16861198C411yescharacteristic, complemented
13#16861196C411yescharacteristic, complemented
14#16861181—22no—
15C4×C22⋊D127—22no—
16C2×C4×Dic127—22no—
17C2×C4×Dic127C2211yescharacteristic
18C2×C4×Dic127C2211yescharacteristic
19Dic127⋊C8—22no—
20#16861137—22no—
21#16861136—22no—
22#16861135—22no—
23C2×#41×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25C2×C42×C127C2211yescharacteristic
26#104×C127—22no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254C2×C411yescharacteristic
32C2×D508C2×C411yescharacteristic
33C2×C22⋊D127—22no—
34C2×C22⋊D127—21no—
35C22×Dic127—21no—
36C22×Dic127D412yes—
37C2×C4×D127—21no—
38#14523232—21no—
39#14523232—21no—
40C2×C127⋊C8—21no—
41Dic127⋊C4—22no—
42C22⋊Dic127—22no—
43(C2×D127)⋊C4—22no—
44C4×Dic127—28no—
45C4×Dic127—21no—
46C4×Dic127C2×C411yescharacteristic
47C4×Dic127C2×C411yescharacteristic
48C4×Dic127D412yes—
49C4×Dic127D412yescomplemented
50#41×C127—21no—
51#41×C127—21no—
52#41×C127—21no—
53#41×C127—21no—
54C2×Q8×C127C2×C411yescharacteristic
55C2×D4×C127—21no—
56C2×D4×C127C2×C411yescharacteristic
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60M16×C127—21no—
61M16×C127—21no—
62C2×C8×C127—21no—
63C42×C127—21no—
64C42×C127D412yescomplemented
65C22×D127—21no—
66Dic254—21no—
67Dic254—21no—
68D508—21no—
69D508—21no—
70C22⋊D127—41no—
71C22⋊D127—41no—
72C22⋊D127—44no—
73C2×Dic127—24no—
74C2×Dic127—22no—
75C2×Dic127—24no—
76C2×Dic127—21no—
77C2×Dic127C22⋊C412yes—
78C2×Dic127C22⋊C412yes—
79C4×D127—41no—
80C4×D127—41no—
81C127⋊C8—41no—
82C23×C127—21no—
83C23×C127C2×D411yescharacteristic
84Q8×C127—21no—
85Q8×C127—21no—
86D4×C127—21no—
87D4×C127—21no—
88D4×C127—41no—
89D4×C127—41no—
90C2×C4×C127—24no—
91C2×C4×C127—22no—
92C2×C4×C127—21no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127C22×C411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C8×C127—41no—
102C22×C127—22no—
103C22×C127—41no—
104C22×C127—42no—
105C22×C127C2×C22⋊C411yescharacteristic
106C22×C127#12111yescharacteristic
107C22×C127#12111yescharacteristic
108C2×D127—41no—
109C2×D127—42no—
110C4×C127—24no—
111C4×C127—41no—
112C4×C127C4×D411yescharacteristic
113C4×C127C4×D411yescharacteristic
114C4×C127#12411yescharacteristic
115C4×C127#12711yescharacteristic
116Dic127—24no—
117Dic127—22no—
118Dic127—41no—
119Dic127#10414yescomplemented
120C2×C127—21no—
121C2×C127—21no—
122C2×C127—81no—
123C2×C127#39011yescharacteristic
124C2×C127C2×#10412yes—
125D127—81no—
126#1962—1271nomaximal, Sylow
127C127#196211yescharacteristic, complemented, Sylow
128#532—1271no—
129#423—1271no—
130C2×#104—1271no—
131C2×#104—1271no—
132C2×#104—1271no—
133C2×#104D12711yescharacteristic, complemented
134C43—1271no—
135C2×#41—1271no—
136C2×#41C2×D12711yescharacteristic
137C2×M16—1271no—
138C2×M16C2×D12711yescharacteristic
139C4⋊Q8—1271no—
140#127—1271no—
141#124—1271no—
142#124—1271no—
143#121—2542no—
144C4×D4—2542no—
145C2×C42—1271no—
146C2×C42—1271no—
147C2×C42—2542no—
148C2×C42C2×D12711yescharacteristic
149C4⋊C8—2542no—
150#104—22no—
151#104—2542no—
152#104—2542no—
153#104—2542no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—2541no—
159#41—2541no—
160#41—2541no—
161#41—2541no—
162C2×Q8—1271no—
163C2×Q8C4×D12711yescharacteristic
164C2×D4—21no—
165C2×D4—1271no—
166C2×D4—2542no—
167C2×D4—2541no—
168C2×D4C4×D12711yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1272no—
172C22×C4—2541no—
173C22×C4—2541no—
174C22×C4C22×D12711yescharacteristic
175M16—21no—
176M16—21no—
177M16—2541no—
178M16—2541no—
179C2×C8—21no—
180C2×C8—2541no—
181C4⋊C4—2542no—
182C22⋊C4—2542no—
183C22⋊C4—2542no—
184C42—21no—
185C42—1271no—
186C42—1271no—
187C42—1272no—
188C42—1272no—
189C42—2548no—
190C42—2541no—
191C42C22⋊D12712yescomplemented
192C23—21no—
193C23—2541no—
194C23D4×D12711yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—2541no—
198Q8—2541no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—5084no—
204D4—5081no—
205D4—5081no—
206D4—2541no—
207D4—2541no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—1272no—
214C2×C4—1272no—
215C2×C4—5081no—
216C2×C4—5081no—
217C2×C4—2544no—
218C2×C4—2541no—
219C2×C4—2544no—
220C2×C4—2542no—
221C2×C4C2×C4×D12711yescharacteristic
222C2×C4D4×D12711yescharacteristic
223C2×C4D4×D12711yescharacteristic
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4#1452324711yescharacteristic
226C2×C4#1452324811yescharacteristic
227C8—41no—
228C8—5081no—
229C22—22no—
230C22—41no—
231C22—42no—
232C22—5081no—
233C22—5082no—
234C22C22⋊C4×D12711yescharacteristic
235C22#1686118111yescharacteristic
236C22#1686118211yescharacteristic
237C4—24no—
238C4—41no—
239C4—1274no—
240C4—5081no—
241C4—2544no—
242C4—2542no—
243C4(C22⋊D127)⋊C411yescharacteristic
244C4C4×C22⋊D12711yescharacteristic
245C4#1686119511yescharacteristic
246C4#1686120511yescharacteristic
247C2—21no—
248C2—21no—
249C2—81no—
250C2—10161no—
251C2#2254977611yescharacteristic
252C2#104×D12712yes—
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550422C2
cosocleC22×C4×C127C23
hypercenterC2×#104D127
hypocenterC127#1962
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
3. C2×#16861137C2semidirect
4. C2×#16861136C2semidirect
5. C2×#16861135C2semidirect
6. C42×Dic127C2semidirect
7. C2×#104×C127C2semidirect
10. #16861207C4semidirect
11. #16861205C4semidirect
12. #16861198C4semidirect
13. #16861196C4semidirect
49. C4×Dic127D4semidirect
64. C42×C127D4semidirect
119. Dic127#104semidirect
127. C127#1962semidirect
133. C2×#104D127semidirect
191. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504222
#225498992
C2×#168611372
C2×#168611362
C2×#168611352
C42×Dic1272
C2×#104×C1272
#1962127