Finite Group Databasebeta

← #30638658

Subgroups — #30638658

Order 16256.

584 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16861128C212yesmaximal, complemented
2C2×#16861126C212yesmaximal, complemented
3#22549840C212yesmaximal, complemented
4C22×#41×C127C211yesmaximal, characteristic
5C2×#14523232C2214yes—
6#16861128—24no—
7#16861126—24no—
8C22⋊(C127⋊C8)—24no—
9C2×#41×C127—24no—
10C2×#41×C127—24no—
11C2×#41×C127—24no—
12C22×Q8×C127C2211yescharacteristic
13C22×D4×C127C412yes—
14C22×D4×C127C2211yescharacteristic
15C23×C4×C127C412yes—
16C23×C4×C127C2211yescharacteristic
17#14523232—24no—
18#14523232—24no—
19C2×C127⋊C8—24no—
20C24×C127C2×C412yes—
21C24×C127C2×C411yescharacteristic
22#41×C127—232no—
23C2×Q8×C127—22no—
24C2×Q8×C127—22no—
25C2×Q8×C127D414yes—
26C2×D4×C127—24no—
27C2×D4×C127—24no—
28C2×D4×C127—24no—
29C2×D4×C127—22no—
30C2×D4×C127—22no—
31C2×D4×C127D414yes—
32C22×C4×C127—24no—
33C22×C4×C127—24no—
34C22×C4×C127—24no—
35C22×C4×C127—22no—
36C22×C4×C127—22no—
37C22×C4×C127C2×C411yescharacteristic
38C22×C4×C127C2×C412yes—
39C22×C4×C127D414yes—
40C22×C4×C127C2311yescharacteristic
41C127⋊C8—44no—
42C23×C127—21no—
43C23×C127—22no—
44C23×C127—21no—
45C23×C127—22no—
46C23×C127—42no—
47C23×C127—44no—
48C23×C127C22⋊C412yes—
49C23×C127C22⋊C414yes—
50C23×C127C22×C411yescharacteristic
51Q8×C127—28no—
52D4×C127—216no—
53D4×C127—28no—
54C2×C4×C127—216no—
55C2×C4×C127—21no—
56C2×C4×C127—21no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—28no—
60C2×C4×C127C22⋊C412yes—
61C2×C4×C127C22⋊C414yes—
62C2×C4×C127C2×D412yes—
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D412yes—
65C22×C127—22no—
66C22×C127—22no—
67C22×C127—24no—
68C22×C127—24no—
69C22×C127—44no—
70C22×C127—42no—
71C22×C127—42no—
72C22×C127—44no—
73C22×C127—44no—
74C22×C127C2×C22⋊C411yescharacteristic
75C22×C127C2×C22⋊C411yescharacteristic
76C22×C127C2×C22⋊C411yescharacteristic
77C4×C127—22no—
78C4×C127—24no—
79C4×C127#12014yes—
80C2×C127—21no—
81C2×C127—21no—
82C2×C127—42no—
83C2×C127—44no—
84C2×C127#37911yescharacteristic
85C2×C127#41312yes—
86#1785—1271nomaximal, Sylow
87C127#178511yescharacteristic, complemented, Sylow
88C22×#41D12711yescharacteristic
89C2×#101—1272no—
90C2×#100—1272no—
91#407—1272no—
92C2×#41—24no—
93C2×#41—24no—
94C2×#41—24no—
95C22×Q8C2×D12711yescharacteristic
96C22×D4Dic12712yes—
97C22×D4C2×D12711yescharacteristic
98C23×C4Dic12712yes—
99C23×C4C2×D12711yescharacteristic
100C2×M16—1274no—
101#101—2544no—
102#100—2544no—
103C22⋊C8—2544no—
104C24C2×Dic12712yes—
105C24C2×Dic12711yescharacteristic
106#41—232no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8C22⋊D12714yes—
110C2×D4—24no—
111C2×D4—24no—
112C2×D4—24no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4C22⋊D12714yes—
116C22×C4—24no—
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4C2×Dic12711yescharacteristic
122C22×C4C2×Dic12712yes—
123C22×C4C22⋊D12714yes—
124C22×C4C22×D12711yescharacteristic
125M16—2544no—
126M16—2544no—
127C2×C8—2544no—
128C23—21no—
129C23—22no—
130C23—21no—
131C23—22no—
132C23—42no—
133C23—44no—
134C23C22⋊Dic12712yes—
135C23C22⋊Dic12714yes—
136C23C22×Dic12711yescharacteristic
137Q8—28no—
138D4—216no—
139D4—28no—
140C2×C4—216no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—28no—
146C2×C4C22⋊Dic12714yes—
147C2×C4C22⋊Dic12712yes—
148C2×C4C2×C22⋊D12712yes—
149C2×C4C2×C22⋊D12712yes—
150C2×C4C2×C22⋊D12712yes—
151C8—5084no—
152C22—22no—
153C22—24no—
154C22—22no—
155C22—24no—
156C22—44no—
157C22—42no—
158C22—42no—
159C22—44no—
160C22—44no—
161C22C2×C22⋊Dic12711yescharacteristic
162C22C2×C22⋊Dic12711yescharacteristic
163C22C2×C22⋊Dic12711yescharacteristic
164C4—22no—
165C4—24no—
166C4#1686117814yes—
167C2—21no—
168C2—21no—
169C2—42no—
170C2—44no—
171C2#2254973211yescharacteristic
172C2#2254986012yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC22×#41×C127C2
socleC22×C127C2×C22⋊C4
cosocleC22×C4×C127C23
hypercenterC22×#41D127
hypocenterC127#1785
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16861128C2semidirect
2. C2×#16861126C2semidirect
3. #22549840C2semidirect
87. C127#1785semidirect

Maximal subgroups

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

subgroupindex
C2×#168611282
C2×#168611262
#225498402
C22×#41×C1272
#1785127