Finite Group Databasebeta

← #30638934

Subgroups — #30638934

Order 16256.

278 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#14523232C211yesmaximal, characteristic
2C2×#16861144C212yesmaximal, complemented
3C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
4#22549640C212yesmaximal, complemented
5#407×C127C211yesmaximal, characteristic
6C2×#14523232—24no—
7C2×#14523232—21no—
8C2×#14523232C2212yes—
9C22×C127⋊C8—21no—
10C22×C127⋊C8C2212yes—
11#16861144—24no—
12C22⋊(C127⋊C8)C412yes—
13C22⋊(C127⋊C8)C412yes—
14C23×C4×C127C2211yescharacteristic
15C2×M16×C127C2211yescharacteristic
16C2×M16×C127C2211yescharacteristic
17C22⋊C8×C127—21no—
18C22⋊C8×C127—21no—
19#14523232—24no—
20#14523232—22no—
21#14523232D414yes—
22C2×C127⋊C8—24no—
23C2×C127⋊C8—22no—
24C2×C127⋊C8—22no—
25C2×C127⋊C8—22no—
26C2×C127⋊C8C2×C414yes—
27C24×C127Q811yescharacteristic
28C22×C4×C127—22no—
29C22×C4×C127—22no—
30C22×C4×C127C2×C411yescharacteristic
31C22×C4×C127C2×C411yescharacteristic
32C22×C4×C127D411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127Q811yescharacteristic
35C22×C4×C127C2311yescharacteristic
36M16×C127—22no—
37M16×C127—22no—
38C2×C8×C127—21no—
39C2×C8×C127—21no—
40C127⋊C8—24no—
41C127⋊C8—24no—
42C23×C127—22no—
43C23×C127—22no—
44C23×C127—41no—
45C23×C127C4⋊C411yescharacteristic
46C23×C127C4⋊C411yescharacteristic
47C23×C127C2×Q811yescharacteristic
48C2×C4×C127—22no—
49C2×C4×C127—22no—
50C2×C4×C127—22no—
51C2×C4×C127—21no—
52C2×C4×C127—21no—
53C2×C4×C127—21no—
54C2×C4×C127—21no—
55C2×C4×C127C4⋊C411yescharacteristic
56C2×C4×C127C4⋊C411yescharacteristic
57C2×C4×C127C22×C411yescharacteristic
58C2×C4×C127C2×D412yes—
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127#4112yes—
61C8×C127—41no—
62C8×C127—41no—
63C22×C127—21no—
64C22×C127—21no—
65C22×C127—21no—
66C22×C127—21no—
67C22×C127—21no—
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—44no—
71C22×C127#10812yescomplemented
72C22×C127C2×C4⋊C411yescharacteristic
73C22×C127C22⋊Q812yes—
74C4×C127—21no—
75C4×C127—21no—
76C4×C127C4×D412yes—
77C4×C127#12011yescharacteristic
78C4×C127#12311yescharacteristic
79C2×C127—22no—
80C2×C127—22no—
81C2×C127—41no—
82C2×C127#38511yescharacteristic
83C2×C127C2×#10811yescharacteristic
84C2×C127#43011yescharacteristic
85#1855—1271nomaximal, Sylow
86C127#185511yescharacteristic, complemented, Sylow
87C22×M16—1271no—
88C2×#108—1272no—
89#407D12711yescharacteristic
90C2×C22⋊C8—1271no—
91#341—1272no—
92C23×C4C2×D12711yescharacteristic
93C2×M16—1272no—
94C2×M16—2544no—
95C2×M16—2541no—
96C2×M16C2×D12711yescharacteristic
97C2×M16C2×D12711yescharacteristic
98C22×C8—1272no—
99C22×C8—2541no—
100#108—2544no—
101C22⋊C8—21no—
102C22⋊C8—21no—
103C22⋊C8—1272no—
104C22⋊C8—1272no—
105C24Dic25411yescharacteristic
106C22×C4—22no—
107C22×C4—22no—
108C22×C4C4×D12711yescharacteristic
109C22×C4C4×D12711yescharacteristic
110C22×C4C22⋊D12711yescharacteristic
111C22×C4C22⋊D12711yescharacteristic
112C22×C4Dic25411yescharacteristic
113C22×C4C22×D12711yescharacteristic
114M16—22no—
115M16—22no—
116M16—1274no—
117M16—2544no—
118M16—2542no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—1274no—
122C2×C8—2544no—
123C2×C8—2542no—
124C2×C8—2542no—
125C2×C8—2542no—
126C23—22no—
127C23—22no—
128C23—41no—
129C23Dic127⋊C411yescharacteristic
130C23Dic127⋊C411yescharacteristic
131C23C2×Dic25411yescharacteristic
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4Dic127⋊C411yescharacteristic
140C2×C4Dic127⋊C411yescharacteristic
141C2×C4C2×C4×D12711yescharacteristic
142C2×C4D4×D12712yes—
143C2×C4C2×C22⋊D12711yescharacteristic
144C2×C4#1452324712yes—
145C8—41no—
146C8—41no—
147C8—2544no—
148C8—2544no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—44no—
157C22#1686114412yescomplemented
158C22C2×Dic127⋊C411yescharacteristic
159C22#1686118612yes—
160C4—21no—
161C4—21no—
162C4(C22⋊D127)⋊C412yes—
163C4#1686117711yescharacteristic
164C4#1686119011yescharacteristic
165C2—22no—
166C2—22no—
167C2—41no—
168C2#2254974811yescharacteristic
169C2C2×#1686114411yescharacteristic
170C2#2254992411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC22×C127C2×C4⋊C4
prime radicalC24×C127Q8
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1855
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16861144C2semidirect
4. #22549640C2semidirect
71. C22×C127#108semidirect
86. C127#1855semidirect
157. C22#16861144semidirect

Maximal subgroups

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

subgroupindex
C22×#145232322
C2×#168611442
C2×C22⋊(C127⋊C8)2
#225496402
#407×C1272
#1855127