Finite Group Databasebeta

← #30648094

Subgroups — #30648094

Order 16256.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#22550674C212yesmaximal, complemented
2#22550450C211yesmaximal, characteristic, complemented
3#22550059C214yesmaximal, complemented
4#22550049C214yesmaximal, complemented
5#22549893C212yesmaximal, complemented
6C2×#16861128C211yesmaximal, characteristic, complemented
7C2×2−1+4×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861239C2214yescomplemented
11#16861235—22no—
12#16861234C2214yescomplemented
13#16861229—24no—
14C2×#14523232C2212yes—
15#16861207C2212yescomplemented
16#16861198C2212yescomplemented
17#16861192—22no—
18#16861185—22no—
19#16861168C2211yescharacteristic
20#16861136C2218yescomplemented
21#16861128C2214yescomplemented
222−1+4×C127—24no—
232−1+4×C127—22no—
242−1+4×C127C2214yes—
25C2×#41×C127—22no—
26C2×#41×C127—22no—
27C2×#41×C127C2212yes—
28C22×Q8×C127—22no—
29C22×Q8×C127C2211yescharacteristic
30#14523248—22no—
31#14523248—22no—
32C2×Dic254C2311yescharacteristic
33C2×D508C2311yescharacteristic
34C2×C22⋊D127—21no—
35C2×C4×D127—21no—
36#14523238—24no—
37#14523236—24no—
38#14523235—24no—
39#14523233—24no—
40#14523232—22no—
41#14523232C2314yes—
42C2×C127⋊C8—22no—
43C4⋊Dic127—21no—
44Dic127⋊C4—24no—
45C22⋊Dic127—21no—
46(C2×D127)⋊C4—24no—
47C4×Dic127C2312yes—
48#41×C127—28no—
49#41×C127—28no—
50#41×C127—28no—
51#41×C127—28no—
52#41×C127—22no—
53#41×C127—22no—
54#41×C127D414yes—
55#41×C127C2314yes—
56C2×Q8×C127—22no—
57C2×Q8×C127—28no—
58C2×Q8×C127—28no—
59C2×Q8×C127—22no—
60C2×Q8×C127—22no—
61C2×Q8×C127D412yes—
62C2×Q8×C127D412yes—
63C2×Q8×C127C2312yes—
64C2×D4×C127—22no—
65C2×D4×C127—22no—
66C2×D4×C127D412yes—
67C22×C4×C127—24no—
68C22×C4×C127—22no—
69C22×C4×C127D412yes—
70C22×C4×C127C2311yescharacteristic
71C22×D127—21no—
72Dic254—22no—
73D508—22no—
74C22⋊D127—41no—
75C22⋊D127—41no—
76C2×Dic127—22no—
77C2×Dic127—21no—
78C4×D127—42no—
79C127⋊C8—24no—
80C23×C127—22no—
81C23×C127C2×D411yescharacteristic
82Q8×C127—28no—
83Q8×C127—22no—
84Q8×C127—24no—
85Q8×C127—24no—
86Q8×C127C2×D414yes—
87D4×C127—28no—
88D4×C127—24no—
89D4×C127—24no—
90D4×C127—22no—
91D4×C127C2×D414yes—
92C2×C4×C127—216no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—24no—
97C2×C4×C127—24no—
98C2×C4×C127C2×D414yes—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D412yes—
101C2×C4×C127C2×D412yes—
102C2×C4×C127C2411yescharacteristic
103C22×C127—24no—
104C22×C127—41no—
105C22×C127—44no—
106C22×C127#12011yescharacteristic
107C22×C127#12011yescharacteristic
108C22×C127C22×D411yescharacteristic
109C2×D127—42no—
110C2×D127—41no—
111C4×C127—24no—
112C4×C127—24no—
113C4×C127#12012yes—
114C4×C127C22×D412yes—
115Dic127—42no—
116Dic127—41no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127—44no—
121C2×C127C2×#12011yescharacteristic
122D127—81no—
123#3020—1271nomaximal, Sylow
124C127#302011yescharacteristic, complemented, Sylow
125C2×2−1+4D12711yescharacteristic, complemented
126#575—1272no—
127#536—1271no—
128#456—1274no—
129#454—1274no—
130#421—1272no—
131C2×#101—1271no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D12714yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—1271no—
138C2×#41C2×D12712yes—
139C22×Q8—22no—
140C22×Q8C2×D12711yescharacteristic
141#137—1274no—
142#137—2542no—
143AGL(1,ℤ/8)—1274no—
144AGL(1,ℤ/8)—2542no—
145#135—2544no—
146C2×M16—1272no—
147C4⋊Q8—1272no—
148#124—1272no—
149#123—2542no—
150C22⋊Q8—2542no—
151#117—1271no—
152#104—1278no—
153#101—1274no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—2542no—
161#41—2542no—
162#41C22⋊D12714yes—
163#41C22×D12714yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—1271no—
170C2×Q8C22⋊D12712yes—
171C2×Q8C22⋊D12712yes—
172C2×Q8C22×D12712yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—1271no—
176C2×D4—2541no—
177C2×D4C22⋊D12712yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—2541no—
181C22×C4C22⋊D12712yes—
182C22×C4C22×D12711yescharacteristic
183Q16—2544no—
184SD16—2544no—
185SD16—2544no—
186D8—2544no—
187M16—1274no—
188M16—2542no—
189C2×C8—2542no—
190C4⋊C4—2544no—
191C4⋊C4—2541no—
192C22⋊C4—2541no—
193C22⋊C4—2544no—
194C42—1272no—
195C23—22no—
196C23—2541no—
197C23C2×C22⋊D12711yescharacteristic
198Q8—28no—
199Q8—24no—
200Q8—22no—
201Q8—24no—
202Q8—2542no—
203Q8C2×C22⋊D12714yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—5081no—
209D4—5081no—
210D4—2542no—
211D4C2×C22⋊D12714yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—5082no—
219C2×C4—2542no—
220C2×C4—2541no—
221C2×C4C2×C22⋊D12714yes—
222C2×C4C2×C22⋊D12712yes—
223C2×C4C2×C22⋊D12712yes—
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—2544no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—5082no—
231C22—5081no—
232C22#1686117811yescharacteristic
233C22#1686117811yescharacteristic
234C22C22×C22⋊D12711yescharacteristic
235C4—24no—
236C4—24no—
237C4—5082no—
238C4—5081no—
239C4#1686117812yes—
240C4C22×C22⋊D12712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—10161no—
246C2C2×#1686117811yescharacteristic
subgroupG/H
centerC2C2×#16861178
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×2−1+4×C127C2
socleC2×C127C2×#120
hypercenterC2×2−1+4D127
hypocenterC127#3020
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. #22550450C2semidirect
3. #22550059C2semidirect
4. #22550049C2semidirect
5. #22549893C2semidirect
6. C2×#16861128C2semidirect
7. C2×2−1+4×C127C2semidirect
10. #16861239C22semidirect
12. #16861234C22semidirect
15. #16861207C22semidirect
16. #16861198C22semidirect
20. #16861136C22semidirect
21. #16861128C22semidirect
124. C127#3020semidirect
125. C2×2−1+4D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225504502
#225500592
#225500492
#225498932
C2×#168611282
C2×2−1+4×C1272
#3020127