Finite Group Databasebeta

← #30639547

Subgroups — #30639547

Order 16256.

344 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16861133C212yesmaximal, complemented
2C2×#16861130C212yesmaximal
3C2×#16861120C211yesmaximal, characteristic, complemented
4#22549746C211yesmaximal, characteristic, complemented
5C2×#124×C127C211yesmaximal, characteristic
6C22×C127⋊C8C2212yes—
7C2×C4⋊Dic127—22no—
8C2×C4⋊Dic127C2212yes—
9#16861133—24no—
10#16861130—24no—
11#16861120—22no—
12C22×Q8×C127C2211yescharacteristic
13C22×D4×C127C2211yescharacteristic
14#124×C127—22no—
15#124×C127C414yescomplemented
16C2×C22⋊C4×C127—22no—
17C2×C42×C127C2211yescharacteristic
18C22×Dic127—22no—
19C2×C127⋊C8—22no—
20C2×C127⋊C8—22no—
21C2×C127⋊C8D414yes—
22C4⋊Dic127—22no—
23C4⋊Dic127—22no—
24C4⋊Dic127—44no—
25C24×C127—21no—
26C2×Q8×C127—21no—
27C2×Q8×C127—22no—
28C2×Q8×C127—22no—
29C2×Q8×C127C2×C412yes—
30C2×D4×C127—22no—
31C2×D4×C127—21no—
32C2×D4×C127—22no—
33C2×D4×C127C2×C412yes—
34C22×C4×C127—21no—
35C22×C4×C127D412yes—
36C22×C4×C127C2311yescharacteristic
37C22⋊C4×C127—28no—
38C42×C127C2×C412yes—
39C42×C127D412yes—
40C2×Dic127—42no—
41C2×Dic127—44no—
42C127⋊C8—24no—
43C23×C127—22no—
44C23×C127—41no—
45C23×C127—41no—
46C23×C127—42no—
47C23×C127—42no—
48C23×C127C2×D411yescharacteristic
49Q8×C127—24no—
50Q8×C127—42no—
51D4×C127—24no—
52D4×C127—42no—
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—22no—
56C2×C4×C127—41no—
57C2×C4×C127—41no—
58C2×C4×C127C22⋊C414yes—
59C2×C4×C127C22×C411yescharacteristic
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127#4112yes—
63C22×C127—42no—
64C22×C127—44no—
65C22×C127—82no—
66C22×C127—81no—
67C22×C127—81no—
68C22×C127C2×C22⋊C411yescharacteristic
69C22×C127#12411yescharacteristic
70C22×C127#12711yescharacteristic
71C22×C127AGL(1,ℤ/8)12yes—
72C22×C127#13712yes—
73C4×C127—24no—
74C4×C127—42no—
75C4×C127C4×D412yes—
76C4×C127#12112yes—
77Dic127—82no—
78C2×C127—82no—
79C2×C127#39011yescharacteristic
80C2×C127#41712yes—
81C2×C127#48912yes—
82C2×C127#49611yescharacteristic
83C2×C127#49711yescharacteristic
84#1959—1271nomaximal, Sylow
85C127#195911yescharacteristic, complemented, Sylow
86C2×#124D12711yescharacteristic
87C2×#103—1272no—
88C2×#102—1272no—
89C2×#97—1271no—
90#384—1271no—
91C22×Q8C2×D12711yescharacteristic
92C22×D4C2×D12711yescharacteristic
93C22×C8—1272no—
94#124—22no—
95#124Dic12714yescomplemented
96C2×C4⋊C4—1272no—
97C2×C4⋊C4—2542no—
98C2×C22⋊C4—22no—
99C2×C42C2×D12711yescharacteristic
100#103—2544no—
101#102—2544no—
102#97—2542no—
103C24—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8C2×Dic12712yes—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—22no—
111C2×D4C2×Dic12712yes—
112C22×C4—21no—
113C22×C4—2542no—
114C22×C4C22⋊D12712yes—
115C22×C4C22×D12711yescharacteristic
116C2×C8—1274no—
117C2×C8—2542no—
118C2×C8—2542no—
119C4⋊C4—5084no—
120C4⋊C4—2542no—
121C4⋊C4—2542no—
122C22⋊C4—28no—
123C42C2×Dic12712yes—
124C42C22⋊D12712yes—
125C23—22no—
126C23—41no—
127C23—41no—
128C23—42no—
129C23—42no—
130C23C2×C22⋊D12711yescharacteristic
131Q8—24no—
132Q8—42no—
133D4—24no—
134D4—42no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—5082no—
141C2×C4—5084no—
142C2×C4C22⋊Dic12714yes—
143C2×C4C22×Dic12711yescharacteristic
144C2×C4D4×D12712yes—
145C2×C4C2×C22⋊D12711yescharacteristic
146C2×C4#1452324712yes—
147C8—2544no—
148C22—42no—
149C22—44no—
150C22—82no—
151C22—81no—
152C22—81no—
153C22C2×C22⋊Dic12711yescharacteristic
154C22#1686119611yescharacteristic
155C22#1686120511yescharacteristic
156C22#1686123412yes—
157C22#1686123912yes—
158C4—24no—
159C4—42no—
160C4—10162no—
161C4D4×Dic12712yes—
162C4#1686118012yes—
163C2—82no—
164C2#2254977911yescharacteristic
165C2#2254987712yes—
166C2#2255023512yes—
167C2#2255026611yescharacteristic
168C2#2255027111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#124×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#124D127
hypocenterC127#1959
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16861133C2semidirect
3. C2×#16861120C2semidirect
4. #22549746C2semidirect
15. #124×C127C4semidirect
85. C127#1959semidirect
95. #124Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611332
C2×#168611302
C2×#168611202
#225497462
C2×#124×C1272
#1959127