Finite Group Databasebeta

← #30639189

Subgroups — #30639189

Order 16256.

378 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16861168C211yesmaximal, characteristic, complemented
2C2×#16861124C212yesmaximal
3#22549739C211yesmaximal, characteristic, complemented
4#22549643C212yesmaximal, complemented
5C2×#123×C127C211yesmaximal, characteristic
6#16861168—24no—
7C2×C4⋊Dic127—21no—
8C2×C22⋊Dic127C2212yes—
9C2×C22⋊Dic127C2212yes—
10C2×C4×Dic127—21no—
11#16861124—24no—
12C127⋊#95—22no—
13C22×D4×C127C2211yescharacteristic
14C23×C4×C127C2211yescharacteristic
15#123×C127—22no—
16#123×C127—21no—
17#123×C127—21no—
18C2×C4⋊C4×C127C412yescomplemented
19C2×C22⋊C4×C127C412yescomplemented
20C2×C22⋊C4×C127C2211yescharacteristic
21C22×Dic127—22no—
22C22×Dic127—22no—
23C4⋊Dic127—24no—
24C22⋊Dic127—22no—
25C22⋊Dic127—22no—
26C22⋊Dic127—22no—
27C22⋊Dic127D414yes—
28C4×Dic127—24no—
29C24×C127C2×C411yescharacteristic
30C24×C127C2311yescharacteristic
31C2×D4×C127—21no—
32C2×D4×C127—21no—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C22×C4×C127—22no—
36C22×C4×C127—22no—
37C22×C4×C127C2×C411yescharacteristic
38C22×C4×C127C2×C412yes—
39C22×C4×C127C2×C412yes—
40C22×C4×C127D412yes—
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C4⋊C4×C127—22no—
44C4⋊C4×C127—22no—
45C22⋊C4×C127—21no—
46C22⋊C4×C127—21no—
47C22⋊C4×C127—22no—
48C22⋊C4×C127—22no—
49C2×Dic127—24no—
50C2×Dic127—44no—
51C2×Dic127—44no—
52C2×Dic127—42no—
53C23×C127—21no—
54C23×C127—22no—
55C23×C127—42no—
56C23×C127—42no—
57C23×C127C22×C411yescharacteristic
58C23×C127C2×D412yes—
59C23×C127C2×D411yescharacteristic
60C23×C127C2×D411yescharacteristic
61C23×C127#4112yes—
62D4×C127—41no—
63D4×C127—42no—
64D4×C127—41no—
65C2×C4×C127—24no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—24no—
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—22no—
72C2×C4×C127—21no—
73C2×C4×C127—22no—
74C2×C4×C127—42no—
75C2×C4×C127C22⋊C412yes—
76C2×C4×C127C22⋊C412yes—
77C22×C127—22no—
78C22×C127—22no—
79C22×C127—22no—
80C22×C127—42no—
81C22×C127—42no—
82C22×C127—41no—
83C22×C127—42no—
84C22×C127—82no—
85C22×C127C2×C22⋊C411yescharacteristic
86C22×C127C4×D412yes—
87C22×C127#12011yescharacteristic
88C22×C127#12112yes—
89C22×C127#12311yescharacteristic
90C4×C127—22no—
91C4×C127—22no—
92C4×C127—41no—
93C4×C127—42no—
94Dic127—44no—
95Dic127—82no—
96C2×C127—21no—
97C2×C127—22no—
98C2×C127—22no—
99C2×C127—21no—
100C2×C127—81no—
101C2×C127#38611yescharacteristic
102C2×C127#41012yes—
103#1899—1271nomaximal, Sylow
104C127#189911yescharacteristic, complemented, Sylow
105C2×#123D12711yescharacteristic
106C2×#117—1271no—
107C2×#99—1272no—
108#381—1271no—
109#342—1272no—
110C22×D4C2×D12711yescharacteristic
111C23×C4C2×D12711yescharacteristic
112#123—22no—
113#123—21no—
114#123—21no—
115#117—2544no—
116C2×C4⋊C4—2541no—
117C2×C4⋊C4Dic12712yescomplemented
118C2×C22⋊C4—1272no—
119C2×C22⋊C4—1272no—
120C2×C22⋊C4Dic12712yescomplemented
121C2×C22⋊C4C2×D12711yescharacteristic
122C2×C42—2541no—
123#99—2544no—
124#95—2542no—
125C24C2×Dic12711yescharacteristic
126C24C22×D12711yescharacteristic
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—2542no—
134C22×C4—2542no—
135C22×C4C2×Dic12711yescharacteristic
136C22×C4C2×Dic12712yes—
137C22×C4C2×Dic12712yes—
138C22×C4C22⋊D12712yes—
139C22×C4C22⋊D12711yescharacteristic
140C22×C4C22⋊D12711yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—2544no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—1274no—
149C22⋊C4—2542no—
150C22⋊C4—2542no—
151C22⋊C4—2542no—
152C42—2544no—
153C23—21no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23C22×Dic12711yescharacteristic
158C23D4×D12712yes—
159C23C2×C22⋊D12711yescharacteristic
160C23C2×C22⋊D12711yescharacteristic
161C23#1452324712yes—
162D4—41no—
163D4—41no—
164D4—42no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—5084no—
176C2×C4—5082no—
177C2×C4—5084no—
178C2×C4—2544no—
179C2×C4C22⋊Dic12712yes—
180C2×C4C22⋊Dic12712yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—41no—
188C22—82no—
189C22C2×C22⋊Dic12711yescharacteristic
190C22D4×Dic12712yes—
191C22#1686117711yescharacteristic
192C22#1686118112yes—
193C22#1686119011yescharacteristic
194C4—22no—
195C4—22no—
196C4—42no—
197C4—41no—
198C4—10162no—
199C4—5084no—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2—81no—
205C2#2254975611yescharacteristic
206C2#2254985012yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
FittingC2×#123×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C22
cosocleC24×C127C23
hypercenterC2×#123D127
hypocenterC127#1899
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16861168C2semidirect
3. #22549739C2semidirect
4. #22549643C2semidirect
18. C2×C4⋊C4×C127C4semidirect
19. C2×C22⋊C4×C127C4semidirect
104. C127#1899semidirect
117. C2×C4⋊C4Dic127semidirect
120. C2×C22⋊C4Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611682
C2×#168611242
#225497392
#225496432
C2×#123×C1272
#1899127