Finite Group Databasebeta

← #30648853

Subgroups — #30648853

Order 16256.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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
2C2×C4×Dic254C211yesmaximal, characteristic, complemented
3#22550190C212yesmaximal, complemented
4#22550172C212yesmaximal, complemented
5#22550077C212yesmaximal, complemented
6#22550017C212yesmaximal, complemented
7#22549978C214yesmaximal, complemented
8#432×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—21no—
10C22×Dic254C2211yescharacteristic
11C2×#14523237C2214yes—
12#16861209C2211yescharacteristic
13#16861206C2211yescharacteristic
14#16861197—21no—
15#16861183—21no—
16#16861183C2212yes—
17C4×Dic254—21no—
18C4×Dic254—21no—
19C4×Dic254C2212yes—
20C4×Dic254C2212yes—
21C4×D508—21no—
22C4×D508C2212yes—
23C2×C4⋊Dic127C2211yescharacteristic
24C2×Dic127⋊C4—21no—
25C2×C4×Dic127—21no—
26#16861143C2214yes—
27#16861134C2214yes—
28#16861131C2214yes—
29C2×C42×C127C2211yescharacteristic
30C4⋊C8×C127C2212yes—
31C22⋊C8×C127C2212yes—
32#97×C127C2212yes—
33#14523248—44no—
34C2×Dic254—24no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254C2312yes—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508C2312yes—
41C2×C22⋊D127—22no—
42C22×Dic127—21no—
43C22×Dic127—21no—
44C2×C4×D127—22no—
45#14523237—28no—
46C4⋊Dic127—21no—
47C4⋊Dic127—21no—
48C4⋊Dic127C2312yes—
49C4⋊Dic127C2312yes—
50Dic127⋊C4—22no—
51Dic127⋊C4—22no—
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—22no—
54C4×Dic127—22no—
55C4×Dic127—22no—
56C22×C4×C127D411yescharacteristic
57C22×C4×C127D411yescharacteristic
58C22×C4×C127C2311yescharacteristic
59C2×C8×C127C2314yes—
60C42×C127D411yescharacteristic
61C42×C127D411yescharacteristic
62C42×C127C2311yescharacteristic
63C42×C127C2311yescharacteristic
64C22×D127—22no—
65Dic254—24no—
66Dic254—22no—
67Dic254—42no—
68Dic254#4114yescomplemented
69D508—24no—
70D508—42no—
71D508—42no—
72C22⋊D127—44no—
73C2×Dic127—24no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—42no—
78C4×D127—44no—
79C23×C127C2×D411yescharacteristic
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2411yescharacteristic
89C8×C127—24no—
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—21no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127#13712yes—
102C4×C127C2×#4112yes—
103Dic127—24no—
104Dic127—42no—
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×Q8—1271no—
115#480—1272no—
116#477—1272no—
117#460—1272no—
118#448—1272no—
119#440—1274no—
120#432D12711yescharacteristic, complemented
121C2×#41—2541no—
122C22×Q8—1271no—
123C2×SD16—1274no—
124C4⋊Q8—1271no—
125#127—1271no—
126#124—2541no—
127#121—1272no—
128#121—2541no—
129C4×Q8—1272no—
130C4×Q8—1272no—
131C4×Q8—2541no—
132C4×Q8—2541no—
133C4×D4—1272no—
134C4×D4—2541no—
135C2×C4⋊C4—1271no—
136C2×C4⋊C4—2541no—
137C2×C42—2541no—
138C2×C42C2×D12711yescharacteristic
139#107—1274no—
140C4⋊C8C2×D12712yes—
141#103—1274no—
142#102—1274no—
143C22⋊C8C2×D12712yes—
144#97C2×D12712yes—
145#41—5084no—
146C2×Q8—1272no—
147C2×Q8—2544no—
148C2×Q8—2541no—
149C2×Q8—2541no—
150C2×D4—1272no—
151C2×D4—2542no—
152C2×D4—2541no—
153C2×D4—2541no—
154C22×C4—2541no—
155C22×C4—2542no—
156C22×C4—2541no—
157C22×C4D50811yescharacteristic
158C22×C4D50811yescharacteristic
159C22×C4C22×D12711yescharacteristic
160SD16—2548no—
161C2×C8C22×D12714yes—
162C4⋊C4—1272no—
163C4⋊C4—1272no—
164C4⋊C4—2542no—
165C4⋊C4—2542no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C22⋊C4—2542no—
169C22⋊C4—2542no—
170C42—2542no—
171C42—2542no—
172C42D50811yescharacteristic
173C42D50811yescharacteristic
174C42C22×D12711yescharacteristic
175C42C22×D12711yescharacteristic
176C23—2542no—
177C23C2×D50811yescharacteristic
178Q8—1274no—
179Q8—5082no—
180Q8—2544no—
181Q8—2542no—
182D4—5082no—
183D4—5084no—
184D4—5082no—
185D4—2544no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—5084no—
190C2×C4—5082no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4—2544no—
194C2×C4—2542no—
195C2×C4C2×D50811yescharacteristic
196C2×C4C2×D50811yescharacteristic
197C2×C4C2×D50811yescharacteristic
198C2×C4C2×D50811yescharacteristic
199C2×C4C2×D50811yescharacteristic
200C2×C4C23×D12711yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—5082no—
205C22—5082no—
206C22—5082no—
207C22#1686123712yes—
208C22C22×D50811yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—5082no—
214C4—5082no—
215C4—2544no—
216C4#1686124212yes—
217C4#41×D12712yes—
218C2—22no—
219C2—10162no—
220C2#2255038811yescharacteristic
221C2C2×#1686123711yescharacteristic
222C2C2×#1686124211yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#432×C127C2
socleC22×C127C22×D4
prime radical#22550426C2
hypercenter#432D127
hypocenterC127#3112
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic254C2semidirect
3. #22550190C2semidirect
4. #22550172C2semidirect
5. #22550077C2semidirect
6. #22550017C2semidirect
7. #22549978C2semidirect
8. #432×C127C2semidirect
68. Dic254#41semidirect
112. C127#3112semidirect
120. #432D127semidirect

Maximal subgroups

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

subgroupindex
#225504262
C2×C4×Dic2542
#225501902
#225501722
#225500772
#225500172
#225499782
#432×C1272
#3112127