Finite Group Databasebeta

← #30637667

Subgroups — #30637667

Order 16256.

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

normal subgroups only

This group has 310 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, complemented
2#22550012C212yesmaximal, complemented
3#22549793C211yesmaximal, characteristic, complemented
4#22549611C212yesmaximal, complemented
5#324×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—21no—
7C22×D508C2211yescharacteristic
8C2×D1016—22no—
9#16861189—22no—
10#16861184—22no—
11#16861183—21no—
12#16861183C2212yes—
13C2×(C2×D127)⋊C4—21no—
14C2×(C2×D127)⋊C4—21no—
15#16861131—22no—
16C127⋊#95C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C23×D127—21no—
20#14523248—42no—
21#14523248—42no—
22C2×Dic254—21no—
23C2×D508—21no—
24C2×D508—21no—
25C2×D508—22no—
26C2×D508—42no—
27C2×D508D412yes—
28C2×C22⋊D127—22no—
29C22×Dic127—21no—
30C2×C4×D127—22no—
31D1016—44no—
32C4⋊Dic127—21no—
33C4⋊Dic127D412yes—
34(C2×D127)⋊C4—22no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—42no—
37(C2×D127)⋊C4—42no—
38C22×C4×C127D411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C22×C4×C127C2311yescharacteristic
41C2×C8×C127—22no—
42C4⋊C4×C127—22no—
43C22×D127—22no—
44C22×D127—22no—
45C22×D127—41no—
46C22×D127—41no—
47C22×D127—41no—
48C22×D127—41no—
49C22×D127—41no—
50C22×D127—41no—
51Dic254—42no—
52D508—24no—
53D508—42no—
54D508—42no—
55D508—42no—
56D508—81no—
57C22⋊D127—42no—
58C22⋊D127—42no—
59C2×Dic127—22no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C2×Dic127—41no—
63C4×D127—44no—
64C23×C127C2×D411yescharacteristic
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127D812yescomplemented
71C2×C4×C127C2×D412yes—
72C8×C127—42no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127#12011yescharacteristic
76C22×C127C2×D811yescharacteristic
77C22×C127AGL(1,ℤ/8)11yescharacteristic
78C2×D127—42no—
79C2×D127—42no—
80C2×D127—42no—
81C2×D127—42no—
82C2×D127—42no—
83C2×D127—42no—
84C2×D127—81no—
85C2×D127—81no—
86C2×D127—81no—
87C2×D127—81no—
88C4×C127—22no—
89C4×C127—22no—
90C4×C127—41no—
91Dic127—42no—
92Dic127—81no—
93C2×C127—21no—
94C2×C127—21no—
95C2×C127#44711yescharacteristic
96C2×C127#45411yescharacteristic
97C2×C127B(4,𝔽2)11yescharacteristic
98D127—82no—
99D127—82no—
100#1598—1271nomaximal, Sylow
101C127#159811yescharacteristic, complemented, Sylow
102#537—1271no—
103#447—1272no—
104#394—1271no—
105#327—1272no—
106#324D12711yescharacteristic, complemented
107C2×#41—2541no—
108C22×D4—1271no—
109C2×D8—2542no—
110C22⋊Q8—2542no—
111#121—1272no—
112#121—2542no—
113#121—2541no—
114C2×C4⋊C4C2×D12711yescharacteristic
115C2×C22⋊C4—2541no—
116C2×C22⋊C4—2541no—
117#102—2542no—
118C22⋊C8C2×D12712yes—
119#95—1271no—
120C24—2541no—
121#41—5082no—
122#41—5082no—
123C2×Q8—2541no—
124C2×D4—1272no—
125C2×D4—5082no—
126C2×D4—2541no—
127C2×D4—2541no—
128C2×D4—2542no—
129C2×D4—2542no—
130C22×C4—2542no—
131C22×C4—2541no—
132C22×C4D50811yescharacteristic
133C22×C4D50811yescharacteristic
134C22×C4C22×D12711yescharacteristic
135D8—5084no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—1272no—
139C4⋊C4—2541no—
140C22⋊C4—5082no—
141C22⋊C4—5082no—
142C22⋊C4—2542no—
143C22⋊C4—2542no—
144C23—5081no—
145C23—5081no—
146C23—5081no—
147C23—5081no—
148C23—5081no—
149C23—5081no—
150C23—2542no—
151C23—2542no—
152C23C2×D50811yescharacteristic
153Q8—5082no—
154D4—10161no—
155D4—5082no—
156D4—5082no—
157D4—5082no—
158D4—5082no—
159D4—5082no—
160D4—2544no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—5084no—
167C2×C4—5081no—
168C2×C4—5081no—
169C2×C4—5081no—
170C2×C4—2542no—
171C2×C4D101612yescomplemented
172C2×C4D4×D12712yes—
173C8—42no—
174C22—21no—
175C22—21no—
176C22—10161no—
177C22—10161no—
178C22—10161no—
179C22—10161no—
180C22—5082no—
181C22—5082no—
182C22—5082no—
183C22—5082no—
184C22—5082no—
185C22—5082no—
186C22#1686117911yescharacteristic
187C22C2×D101611yescharacteristic
188C22#1686123711yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—10161no—
193C4—5082no—
194C2—21no—
195C2—21no—
196C2—10162no—
197C2—10162no—
198C2#2255001211yescharacteristic
199C2#2255005111yescharacteristic
200C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127#120
hypercenter#324D127
hypocenterC127#1598
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #22550457C2semidirect
2. #22550012C2semidirect
3. #22549793C2semidirect
4. #22549611C2semidirect
5. #324×C127C2semidirect
70. C2×C4×C127D8semidirect
101. C127#1598semidirect
106. #324D127semidirect
171. C2×C4D1016semidirect

Maximal subgroups

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

subgroupindex
#225504572
#225500122
#225497932
#225496112
#324×C1272
#1598127