Finite Group Databasebeta

← #30638808

Subgroups — #30638808

Order 16256.

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

normal subgroups only

This group has 316 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×C4⋊Dic127C211yesmaximal, characteristic
2#22549840C211yesmaximal, characteristic
3#22549644C214yesmaximal, complemented
4#407×C127C211yesmaximal, characteristic
5C23×Dic127C412yes—
6C2×#14523232C2212yes—
7C2×C4⋊Dic127—21no—
8C2×C4⋊Dic127—21no—
9C2×C4⋊Dic127—21no—
10C2×C4⋊Dic127—21no—
11C2×C4⋊Dic127C412yes—
12C2×C4⋊Dic127C2212yes—
13C22⋊(C127⋊C8)—21no—
14C22⋊(C127⋊C8)—21no—
15C23×C4×C127C2211yescharacteristic
16C2×M16×C127C2212yes—
17C22⋊C8×C127—21no—
18C22⋊C8×C127—21no—
19C22×Dic127—22no—
20C22×Dic127—22no—
21C22×Dic127—44no—
22C22×Dic127C2×C414yes—
23#14523232—24no—
24C2×C127⋊C8—22no—
25C4⋊Dic127—22no—
26C4⋊Dic127—22no—
27C4⋊Dic127—22no—
28C4⋊Dic127—22no—
29C24×C127C2×C411yescharacteristic
30C22×C4×C127—24no—
31C22×C4×C127C2×C411yescharacteristic
32C22×C4×C127D411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37M16×C127—24no—
38C2×C8×C127—22no—
39C2×Dic127—24no—
40C2×Dic127—44no—
41C2×Dic127—44no—
42C2×Dic127—44no—
43C127⋊C8—42no—
44C23×C127—22no—
45C23×C127—22no—
46C23×C127—22no—
47C23×C127C22⋊C411yescharacteristic
48C23×C127C22⋊C411yescharacteristic
49C23×C127C22×C411yescharacteristic
50C2×C4×C127—22no—
51C2×C4×C127—22no—
52C2×C4×C127—22no—
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127C22⋊C411yescharacteristic
56C2×C4×C127C22⋊C411yescharacteristic
57C2×C4×C127C2×D411yescharacteristic
58C2×C4×C127C2×D411yescharacteristic
59C2×C4×C127#4114yes—
60C8×C127—42no—
61C22×C127—28no—
62C22×C127—21no—
63C22×C127—21no—
64C22×C127—21no—
65C22×C127—21no—
66C22×C127—21no—
67C22×C127—21no—
68C22×C127#10012yes—
69C22×C127#10112yes—
70C22×C127C2×C22⋊C411yescharacteristic
71C22×C127#11712yes—
72C4×C127—21no—
73C4×C127—21no—
74C4×C127#12312yes—
75C4×C127#12312yes—
76Dic127—44no—
77C2×C127—22no—
78C2×C127—22no—
79C2×C127—22no—
80C2×C127#38111yescharacteristic
81C2×C127C2×#10011yescharacteristic
82C2×C127C2×#10111yescharacteristic
83#1821—1271nomaximal, Sylow
84C127#182111yescharacteristic, complemented, Sylow
85C22×C4⋊C4—1271no—
86#407—1271no—
87#407D12711yescharacteristic
88#343—1274no—
89C23×C4—1272no—
90C23×C4C2×D12711yescharacteristic
91C2×M16—1272no—
92C2×M16C2×D12712yes—
93C2×C4⋊C4—1272no—
94C2×C4⋊C4—1272no—
95C2×C4⋊C4—2541no—
96C2×C4⋊C4—2541no—
97C2×C4⋊C4—2541no—
98C2×C4⋊C4—2541no—
99C22⋊C8—21no—
100C22⋊C8—21no—
101C22⋊C8—2541no—
102C22⋊C8—2541no—
103C24C4×D12711yescharacteristic
104C22×C4—24no—
105C22×C4—1274no—
106C22×C4—5084no—
107C22×C4—2542no—
108C22×C4—2542no—
109C22×C4C4×D12711yescharacteristic
110C22×C4C22⋊D12711yescharacteristic
111C22×C4C22⋊D12711yescharacteristic
112C22×C4D50811yescharacteristic
113C22×C4D50811yescharacteristic
114C22×C4C22×D12711yescharacteristic
115M16—24no—
116M16—2544no—
117C2×C8—22no—
118C2×C8—2542no—
119C4⋊C4—2542no—
120C4⋊C4—2542no—
121C4⋊C4—2542no—
122C4⋊C4—2542no—
123C23—22no—
124C23—22no—
125C23—22no—
126C23(C2×D127)⋊C411yescharacteristic
127C23(C2×D127)⋊C411yescharacteristic
128C23C2×C4×D12711yescharacteristic
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—5084no—
135C2×C4—5084no—
136C2×C4—5084no—
137C2×C4—2544no—
138C2×C4(C2×D127)⋊C411yescharacteristic
139C2×C4(C2×D127)⋊C411yescharacteristic
140C2×C4C2×C22⋊D12711yescharacteristic
141C2×C4C2×D50811yescharacteristic
142C2×C4#1452324714yes—
143C8—42no—
144C8—5082no—
145C22—28no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22#1686112512yes—
153C22#1686112712yes—
154C22C2×(C2×D127)⋊C411yescharacteristic
155C22#1686116512yes—
156C4—21no—
157C4—21no—
158C4—5084no—
159C4#1686119012yes—
160C4#1686119312yes—
161C2—22no—
162C2—22no—
163C2—22no—
164C2#2254973711yescharacteristic
165C2C2×#1686112511yescharacteristic
166C2C2×#1686112711yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC24×C127C2×C4
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1821
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #22549644C2semidirect
84. C127#1821semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1272
#225498402
#225496442
#407×C1272
#1821127