Finite Group Databasebeta

← #30637725

Subgroups — #30637725

Order 16256.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550473C211yesmaximal, characteristic
2#22550206C211yesmaximal, characteristic, complemented
3#22550189C211yesmaximal, characteristic
4#22549612C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic
6#22549602C211yesmaximal, characteristic, complemented
7#398×C127C211yesmaximal, characteristic, complemented
8#16861204—21no—
9#16861199—21no—
10#16861194—21no—
11#16861192—21no—
12#16861188C2211yescharacteristic
13#16861187—21no—
14#16861183C2211yescharacteristic
15C4×C22⋊D127—21no—
16#16861142—21no—
17#16861140—21no—
18#16861132—21no—
19#16861129—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21C22⋊(C127⋊C8)C2211yescharacteristic
22C2×C4⋊C4×C127—21no—
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25#95×C127—21no—
26#95×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C4×D127—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C4⋊Dic127D411yescharacteristic
34C4⋊Dic127D411yescharacteristic
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39C22⋊Dic127—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44C4×Dic127—21no—
45C22×C4×C127—21no—
46C22×C4×C127—21no—
47C22×C4×C127D411yescharacteristic
48C22×C4×C127D411yescharacteristic
49C22×C4×C127C2311yescharacteristic
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—42no—
55C22×D127—21no—
56Dic254—41no—
57D508—41no—
58C22⋊D127—41no—
59C22⋊D127—41no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C4×D127—81no—
64C127⋊C8—41no—
65C127⋊C8—41no—
66C23×C127C2×D411yescharacteristic
67C2×C4×C127—22no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C22×C127—21no—
82C22×C127—21no—
83C22×C127#12011yescharacteristic
84C22×C127#13511yescharacteristic
85C22×C127AGL(1,ℤ/8)11yescharacteristic
86C2×D127—41no—
87C2×D127—41no—
88C2×D127—41no—
89C4×C127—21no—
90C4×C127—21no—
91C4×C127—42no—
92C4×C127—41no—
93C4×C127—41no—
94C4×C127—81no—
95Dic127—41no—
96Dic127—41no—
97Dic127—41no—
98C2×C127—21no—
99C2×C127—21no—
100C2×C127#44911yescharacteristic
101C2×C127#45611yescharacteristic
102C2×C127#45811yescharacteristic
103D127—81no—
104#1609—1271nomaximal, Sylow
105C127#160911yescharacteristic, complemented, Sylow
106#539—1271no—
107#483—1271no—
108#480—1271no—
109#398D12711yescharacteristic, complemented
110#328—1271no—
111#327—1271no—
112#324—1271no—
113#126—2541no—
114#125—2541no—
115#123—2541no—
116#123—2541no—
117C22⋊Q8—1271no—
118C22⋊Q8—2541no—
119#121—1271no—
120C4×D4—2541no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×D12711yescharacteristic
123C2×C4⋊C4C2×D12711yescharacteristic
124#107—2541no—
125#106—2541no—
126#103—2541no—
127#102—2541no—
128C22⋊C8—1271no—
129C22⋊C8—1271no—
130#95—21no—
131#95C2×D12711yescharacteristic
132C2×Q8—1271no—
133C2×D4—1271no—
134C2×D4—2541no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—2541no—
138C22×C4C22⋊D12711yescharacteristic
139C22×C4C22⋊D12711yescharacteristic
140C22×C4C22×D12711yescharacteristic
141C2×C8—2541no—
142C2×C8—2541no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—42no—
148C4⋊C4—1271no—
149C4⋊C4—1271no—
150C4⋊C4—2541no—
151C4⋊C4—2541no—
152C4⋊C4—2541no—
153C4⋊C4—2541no—
154C22⋊C4—2541no—
155C22⋊C4—2541no—
156C22⋊C4—2541no—
157C22⋊C4—2541no—
158C22⋊C4—2541no—
159C42—2541no—
160C23—2541no—
161C23C2×C22⋊D12711yescharacteristic
162Q8—5081no—
163D4—5081no—
164D4—5081no—
165D4—5081no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—10161no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4—2541no—
182C2×C4D4×D12711yescharacteristic
183C2×C4D4×D12711yescharacteristic
184C8—5081no—
185C8—5081no—
186C22—21no—
187C22—21no—
188C22—5081no—
189C22—5081no—
190C22—5081no—
191C22#1686117711yescharacteristic
192C22#1686122911yescharacteristic
193C22#1686123511yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—81no—
200C4—5081no—
201C4—5081no—
202C4—5081no—
203C2—21no—
204C2—21no—
205C2—10161no—
206C2#2255001911yescharacteristic
207C2#2255005811yescharacteristic
208C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#398×C127C2
socleC22×C127#120
prime radical#16861183C22
hypercenter#398D127
hypocenterC127#1609
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #22550206C2semidirect
4. #22549612C2semidirect
6. #22549602C2semidirect
7. #398×C127C2semidirect
105. C127#1609semidirect
109. #398D127semidirect

Maximal subgroups

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

subgroupindex
#225504732
#225502062
#225501892
#225496122
#225496092
#225496022
#398×C1272
#1609127