Finite Group Databasebeta

← #30637698

Subgroups — #30637698

Order 16256.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550457C211yesmaximal, characteristic
2#22550015C212yesmaximal, complemented
3#22549609C212yesmaximal, complemented
4#22549602C211yesmaximal, characteristic, complemented
5#395×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—21no—
7C2×#14523235—22no—
8#16861189—22no—
9#16861184—22no—
10#16861183—21no—
11#16861183C2212yes—
12#16861133—22no—
13C22⋊(C127⋊C8)C2212yes—
14C22×Q8×C127C2211yescharacteristic
15C2×C4⋊C4×C127—21no—
16C2×C4⋊C4×C127C2211yescharacteristic
17#95×C127—21no—
18#95×C127C2211yescharacteristic
19#14523248—42no—
20#14523248—42no—
21C2×Dic254—21no—
22C2×D508—21no—
23C2×D508—21no—
24C2×D508D412yes—
25C2×C22⋊D127—22no—
26C2×C4×D127—22no—
27#14523235—44no—
28C2×C127⋊C8—22no—
29C4⋊Dic127—21no—
30C4⋊Dic127D412yes—
31(C2×D127)⋊C4—22no—
32(C2×D127)⋊C4—22no—
33C2×Q8×C127—22no—
34C2×Q8×C127—42no—
35C22×C4×C127—21no—
36C22×C4×C127—21no—
37C22×C4×C127D411yescharacteristic
38C22×C4×C127D411yescharacteristic
39C22×C4×C127C2311yescharacteristic
40C4⋊C4×C127—22no—
41C4⋊C4×C127—42no—
42C22×D127—22no—
43Dic254—42no—
44D508—42no—
45D508—42no—
46D508—81no—
47C22⋊D127—42no—
48C22⋊D127—42no—
49C2×Dic127—22no—
50C4×D127—44no—
51C127⋊C8—42no—
52C23×C127C2×D411yescharacteristic
53Q8×C127—24no—
54Q8×C127—42no—
55C2×C4×C127—21no—
56C2×C4×C127—21no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—21no—
60C2×C4×C127—41no—
61C2×C4×C127—41no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127SD1612yescomplemented
67C2×C4×C127C2×D412yes—
68C22×C127—21no—
69C22×C127—21no—
70C22×C127#12011yescharacteristic
71C22×C127C2×SD1611yescharacteristic
72C22×C127AGL(1,ℤ/8)11yescharacteristic
73C2×D127—42no—
74C2×D127—42no—
75C2×D127—42no—
76C4×C127—22no—
77C4×C127—22no—
78C4×C127—42no—
79C4×C127—42no—
80C4×C127—41no—
81Dic127—42no—
82C2×C127—21no—
83C2×C127—21no—
84C2×C127#45011yescharacteristic
85C2×C127#45611yescharacteristic
86C2×C127B(4,𝔽2)11yescharacteristic
87D127—82no—
88#1604—1271nomaximal, Sylow
89C127#160411yescharacteristic, complemented, Sylow
90#537—1271no—
91#448—1272no—
92#395D12711yescharacteristic, complemented
93#327—1272no—
94#324—1271no—
95C2×#41—2541no—
96C22×Q8C2×D12711yescharacteristic
97C2×SD16—2542no—
98C22⋊Q8—2542no—
99#121—1272no—
100#121—2542no—
101#121—2541no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C2×D12711yescharacteristic
104#103—2542no—
105C22⋊C8—1272no—
106#95—21no—
107#95C2×D12711yescharacteristic
108#41—5082no—
109#41—5082no—
110C2×Q8—22no—
111C2×Q8—42no—
112C2×Q8—2541no—
113C2×D4—1272no—
114C2×D4—2541no—
115C2×D4—2541no—
116C2×D4—2542no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—2542no—
120C22×C4C22⋊D12711yescharacteristic
121C22×C4C22⋊D12711yescharacteristic
122C22×C4C22×D12711yescharacteristic
123SD16—5084no—
124C2×C8—2542no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—1272no—
128C4⋊C4—2541no—
129C22⋊C4—2542no—
130C22⋊C4—2542no—
131C23—2542no—
132C23C2×C22⋊D12711yescharacteristic
133Q8—24no—
134Q8—42no—
135Q8—5082no—
136D4—10161no—
137D4—5082no—
138D4—5082no—
139D4—5082no—
140D4—5082no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—5084no—
153C2×C4—2542no—
154C2×C4#1452323512yescomplemented
155C2×C4D4×D12712yes—
156C8—5082no—
157C22—21no—
158C22—21no—
159C22—5082no—
160C22—5082no—
161C22—5082no—
162C22#1686117711yescharacteristic
163C22C2×#1452323511yescharacteristic
164C22#1686123411yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—42no—
169C4—41no—
170C4—5082no—
171C2—21no—
172C2—21no—
173C2—10162no—
174C2#2255002611yescharacteristic
175C2#2255005811yescharacteristic
176C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#395×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#395D127
hypocenterC127#1604
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
2. #22550015C2semidirect
3. #22549609C2semidirect
4. #22549602C2semidirect
5. #395×C127C2semidirect
66. C2×C4×C127SD16semidirect
89. C127#1604semidirect
92. #395D127semidirect
154. C2×C4#14523235semidirect

Maximal subgroups

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

subgroupindex
#225504572
#225500152
#225496092
#225496022
#395×C1272
#1604127