Finite Group Databasebeta

← #30639862

Subgroups — #30639862

Order 16256.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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#22550438C211yesmaximal, characteristic
2C2×#16861193C212yesmaximal, complemented
3C2×#16861123C212yesmaximal, complemented
4#22549737C211yesmaximal, characteristic, complemented
5#342×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127C2212yes—
7C23×Dic127C2211yescharacteristic
8#16861193—22no—
9#16861193—24no—
10#16861193—24no—
11#16861183—22no—
12#16861179—22no—
13C2×C4⋊Dic127—22no—
14C2×C22⋊Dic127C2211yescharacteristic
15C2×(C2×D127)⋊C4—22no—
16#16861123—24no—
17C127⋊#95—22no—
18C2×C22⋊C4×C127C2211yescharacteristic
19C2×C22⋊C4×C127C2212yes—
20C23×D127D412yes—
21C2×D508—22no—
22C2×C22⋊D127—22no—
23C2×C22⋊D127—22no—
24C2×C22⋊D127—22no—
25C2×C22⋊D127—22no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127—22no—
28C22×Dic127—22no—
29C22×Dic127—21no—
30C22×Dic127—22no—
31C22×Dic127—21no—
32C22×Dic127—21no—
33C22×Dic127D412yes—
34C22×Dic127D412yescomplemented
35C4⋊Dic127—28no—
36C4⋊Dic127—22no—
37C22⋊Dic127—22no—
38(C2×D127)⋊C4—28no—
39(C2×D127)⋊C4—24no—
40C24×C127C2311yescharacteristic
41C22×C4×C127—22no—
42C22×C4×C127D412yes—
43C22⋊C4×C127—22no—
44C22⋊C4×C127—22no—
45C22⋊C4×C127D414yescomplemented
46C22×D127—22no—
47C22×D127—22no—
48C22×D127—22no—
49C22×D127—44no—
50D508—82no—
51C22⋊D127—42no—
52C22⋊D127—42no—
53C22⋊D127—42no—
54C22⋊D127—42no—
55C2×Dic127—24no—
56C2×Dic127—22no—
57C2×Dic127—22no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127—41no—
61C2×Dic127—42no—
62C2×Dic127—41no—
63C2×Dic127—42no—
64C2×Dic127—41no—
65C23×C127—42no—
66C23×C127C2×D412yes—
67C23×C127C2×D411yescharacteristic
68C23×C127C2×D411yescharacteristic
69C23×C127C2×D412yes—
70C23×C127#4111yescharacteristic
71C2×C4×C127—22no—
72C2×C4×C127—24no—
73C2×C4×C127—44no—
74C2×C4×C127—42no—
75C22×C127—22no—
76C22×C127—24no—
77C22×C127—41no—
78C22×C127—42no—
79C22×C127—41no—
80C22×C127#12012yes—
81C22×C127#12011yescharacteristic
82C22×C127#12111yescharacteristic
83C22×C127#12112yes—
84C22×C127#12711yescharacteristic
85C2×D127—44no—
86C2×D127—42no—
87C2×D127—44no—
88C2×D127—82no—
89C4×C127—44no—
90C4×C127—42no—
91Dic127—42no—
92Dic127—42no—
93Dic127—81no—
94C2×C127—21no—
95C2×C127—22no—
96C2×C127—22no—
97C2×C127—21no—
98C2×C127#39211yescharacteristic
99C2×C127#45812yes—
100D127—82no—
101#2022—1271nomaximal, Sylow
102C127#202211yescharacteristic, complemented, Sylow
103#534—1271no—
104C2×#123—1272no—
105C2×#99—1272no—
106#381—1271no—
107#342D12711yescharacteristic, complemented
108C22×D4—1272no—
109C23×C4—1271no—
110#123—2544no—
111#123—2542no—
112#123—2544no—
113#121—2542no—
114#120—2542no—
115C2×C4⋊C4—2542no—
116C2×C22⋊C4—1271no—
117C2×C22⋊C4—2542no—
118C2×C22⋊C4C2×D12711yescharacteristic
119C2×C22⋊C4C2×D12712yes—
120#99—2544no—
121#95—2542no—
122C24—1272no—
123C24C22×D12711yescharacteristic
124C2×D4—2542no—
125C2×D4—2542no—
126C2×D4—2542no—
127C2×D4—2542no—
128C2×D4—2542no—
129C2×D4—2542no—
130C2×D4—2542no—
131C22×C4—22no—
132C22×C4—1272no—
133C22×C4—1272no—
134C22×C4—2541no—
135C22×C4—2542no—
136C22×C4—2541no—
137C22×C4—2542no—
138C22×C4—2541no—
139C22×C4D50812yes—
140C4⋊C4—2548no—
141C4⋊C4—2542no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—2548no—
145C22⋊C4—2544no—
146C22⋊C4—2542no—
147C22⋊C4D50814yescomplemented
148C23—42no—
149C23—5084no—
150C23—2542no—
151C23—2542no—
152C23—2542no—
153C23D4×D12712yes—
154C23D4×D12711yescharacteristic
155C23C2×D50811yescharacteristic
156C23C2×D50812yes—
157C23#1452324911yescharacteristic
158D4—10162no—
159D4—5082no—
160D4—5082no—
161D4—5082no—
162D4—5082no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—5082no—
168C2×C4—5081no—
169C2×C4—5082no—
170C2×C4—5081no—
171C2×C4—5081no—
172C2×C4—2542no—
173C2×C4—2542no—
174C2×C4—2544no—
175C2×C4—2542no—
176C2×C4—2542no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—10162no—
183C22—5084no—
184C22—5082no—
185C22—5084no—
186C22#1686117912yes—
187C22#1686117911yescharacteristic
188C22#1686118412yes—
189C22#1686118411yescharacteristic
190C22#1686120611yescharacteristic
191C4—44no—
192C4—42no—
193C4—10161no—
194C4—5082no—
195C4—5082no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—10162no—
201C2#2254978611yescharacteristic
202C2#2255006712yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127#120
prime radical#22550438C2
hypercenter#342D127
hypocenterC127#2022
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#16861193C2semidirect
3. C2×#16861123C2semidirect
4. #22549737C2semidirect
5. #342×C127C2semidirect
34. C22×Dic127D4semidirect
45. C22⋊C4×C127D4semidirect
102. C127#2022semidirect
107. #342D127semidirect
147. C22⋊C4D508semidirect

Maximal subgroups

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

subgroupindex
#225504382
C2×#168611932
C2×#168611232
#225497372
#342×C1272
#2022127