Finite Group Databasebeta

← #30648852

Subgroups — #30648852

Order 16256.

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

normal subgroups only

This group has 424 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
1#22550426C211yesmaximal, characteristic
2#22550189C212yesmaximal, complemented
3#22550171C212yesmaximal, complemented
4#22550076C212yesmaximal, complemented
5#22550015C212yesmaximal, complemented
6#22549974C214yesmaximal, complemented
7#22549933C211yesmaximal, characteristic, complemented
8C2×C4×Q8×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—21no—
10C2×#14523235C2214yes—
11#16861209C2211yescharacteristic
12#16861206C2211yescharacteristic
13#16861197—21no—
14#16861183—21no—
15#16861183C2212yes—
16C4×D508—21no—
17C4×D508C2212yes—
18#16861142C2214yes—
19C4⋊(C127⋊C8)C2212yes—
20#16861133C2214yes—
21#16861129C2214yes—
22C22⋊(C127⋊C8)C2212yes—
23#16861120C2212yes—
24C22×Q8×C127C2211yescharacteristic
25C4×Q8×C127—21no—
26C4×Q8×C127—21no—
27C4×Q8×C127C2212yes—
28C4×Q8×C127C2212yes—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127C2211yescharacteristic
31C2×C42×C127—21no—
32C2×C42×C127C2211yescharacteristic
33#14523248—44no—
34C2×Dic254—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508C2312yes—
38C2×C22⋊D127—22no—
39C2×C4×D127—22no—
40#14523235—28no—
41C2×C127⋊C8C2314yes—
42C4⋊Dic127—21no—
43C4⋊Dic127C2312yes—
44(C2×D127)⋊C4—22no—
45(C2×D127)⋊C4—22no—
46C2×Q8×C127—24no—
47C2×Q8×C127—21no—
48C2×Q8×C127C2312yes—
49C22×C4×C127—21no—
50C22×C4×C127—21no—
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—22no—
55C4⋊C4×C127—22no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127C2312yes—
58C42×C127—22no—
59C42×C127—22no—
60C42×C127D411yescharacteristic
61C42×C127D411yescharacteristic
62C42×C127C2311yescharacteristic
63C42×C127C2311yescharacteristic
64C22×D127—22no—
65Dic254—42no—
66D508—24no—
67D508—42no—
68D508—42no—
69C22⋊D127—44no—
70C2×Dic127—22no—
71C4×D127—44no—
72C127⋊C8—24no—
73C23×C127C2×D411yescharacteristic
74Q8×C127—24no—
75Q8×C127—22no—
76Q8×C127#4114yescomplemented
77C2×C4×C127—24no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—42no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2411yescharacteristic
90C22×C127—21no—
91C22×C127—21no—
92C22×C127AGL(1,ℤ/8)12yes—
93C22×C127C22×D411yescharacteristic
94C2×D127—42no—
95C2×D127—42no—
96C2×D127—42no—
97C4×C127—24no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—42no—
103C4×C127#13712yes—
104C4×C127C2×#4112yes—
105Dic127—42no—
106C2×C127—22no—
107C2×C127#52511yescharacteristic
108C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C127C2×#13711yescharacteristic
110D127—82no—
111#3112—1271nomaximal, Sylow
112C127#311211yescharacteristic, complemented, Sylow
113#532—1271no—
114C2×C4×Q8D12711yescharacteristic, complemented
115#480—1272no—
116#477—1272no—
117#460—1272no—
118#448—1272no—
119#440—1274no—
120#432—1271no—
121C2×#41—2541no—
122C22×Q8C2×D12711yescharacteristic
123C2×SD16—1274no—
124C4⋊Q8—1271no—
125#127—1271no—
126#124—2541no—
127#121—1272no—
128#121—2541no—
129C4×Q8—21no—
130C4×Q8—21no—
131C4×Q8C2×D12712yes—
132C4×Q8C2×D12712yes—
133C4×D4—1272no—
134C4×D4—2541no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4C2×D12711yescharacteristic
137C2×C42—21no—
138C2×C42C2×D12711yescharacteristic
139#107—1274no—
140C4⋊C8—1272no—
141#103—1274no—
142#102—1274no—
143C22⋊C8—1272no—
144#97—1272no—
145#41—5084no—
146C2×Q8—24no—
147C2×Q8—21no—
148C2×Q8—2541no—
149C2×Q8C22×D12712yes—
150C2×D4—1272no—
151C2×D4—2542no—
152C2×D4—2541no—
153C2×D4—2541no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—2542no—
157C22×C4C22⋊D12711yescharacteristic
158C22×C4C22⋊D12711yescharacteristic
159C22×C4C22×D12711yescharacteristic
160SD16—2548no—
161C2×C8—1274no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—1272no—
166C4⋊C4—2541no—
167C4⋊C4C22×D12712yes—
168C22⋊C4—2542no—
169C22⋊C4—2542no—
170C42—22no—
171C42—22no—
172C42C22⋊D12711yescharacteristic
173C42C22⋊D12711yescharacteristic
174C42C22×D12711yescharacteristic
175C42C22×D12711yescharacteristic
176C23—2542no—
177C23C2×C22⋊D12711yescharacteristic
178Q8—24no—
179Q8—22no—
180Q8—5082no—
181Q8#1452324814yescomplemented
182D4—5082no—
183D4—5084no—
184D4—5082no—
185D4—2544no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—5084no—
194C2×C4—2542no—
195C2×C4C2×C22⋊D12711yescharacteristic
196C2×C4C2×C22⋊D12711yescharacteristic
197C2×C4C2×C22⋊D12711yescharacteristic
198C2×C4C2×C22⋊D12711yescharacteristic
199C2×C4C2×C22⋊D12711yescharacteristic
200C2×C4C23×D12711yescharacteristic
201C8—2544no—
202C22—21no—
203C22—21no—
204C22—5082no—
205C22—5082no—
206C22—5082no—
207C22#1686123412yes—
208C22C22×C22⋊D12711yescharacteristic
209C4—24no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—5082no—
216C4#1686124012yes—
217C4C2×#1452324812yes—
218C2—22no—
219C2—10162no—
220C2#2255038711yescharacteristic
221C2C2×#1686123411yescharacteristic
222C2C2×#1686124011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×Q8×C127C2
socleC22×C127C22×D4
prime radical#22550426C2
hypercenterC2×C4×Q8D127
hypocenterC127#3112
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22550189C2semidirect
3. #22550171C2semidirect
4. #22550076C2semidirect
5. #22550015C2semidirect
6. #22549974C2semidirect
7. #22549933C2semidirect
8. C2×C4×Q8×C127C2semidirect
76. Q8×C127#41semidirect
112. C127#3112semidirect
114. C2×C4×Q8D127semidirect
181. Q8#14523248semidirect

Maximal subgroups

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

subgroupindex
#225504262
#225501892
#225501712
#225500762
#225500152
#225499742
#225499332
C2×C4×Q8×C1272
#3112127