Finite Group Databasebeta

← #30637228

Subgroups — #30637228

Order 16256.

240 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#22550433C211yesmaximal, characteristic
2#22549929C211yesmaximal, characteristic
3#22549612C214yesmaximal
4C4×C4⋊C4×C127C211yesmaximal, characteristic
5#16861209C411yescharacteristic, complemented
6#16861209C411yescharacteristic, complemented
7#16861201—21no—
8#16861188—21no—
9#16861188C2212yes—
10C4×Dic254—21no—
11C4×Dic254C412yescomplemented
12#16861168—21no—
13C4⋊(C127⋊C8)—21no—
14C22⋊(C127⋊C8)C2212yes—
15C4×C127⋊C8—21no—
16C2×C4⋊C4×C127—21no—
17C2×C42×C127—21no—
18C2×C42×C127C2211yescharacteristic
19#95×C127C2212yes—
20C2×Dic254—21no—
21C2×Dic254C2×C412yes—
22C2×C127⋊C8—22no—
23C4⋊Dic127—21no—
24C4⋊Dic127—21no—
25C4⋊Dic127—21no—
26C4⋊Dic127C2×C412yes—
27Dic127⋊C4—22no—
28Dic127⋊C4—22no—
29C22⋊Dic127—22no—
30C4×Dic127—22no—
31C22×C4×C127—22no—
32C22×C4×C127D411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127C2311yescharacteristic
35C4⋊C4×C127—24no—
36C42×C127—42no—
37C42×C127C2×C411yescharacteristic
38C42×C127C2×C411yescharacteristic
39C42×C127D411yescharacteristic
40C42×C127D411yescharacteristic
41Dic254—42no—
42Dic254—42no—
43C2×Dic127—22no—
44C2×Dic127—22no—
45C127⋊C8—42no—
46C23×C127C2×D411yescharacteristic
47C2×C4×C127—22no—
48C2×C4×C127—24no—
49C2×C4×C127—41no—
50C2×C4×C127—42no—
51C2×C4×C127—42no—
52C2×C4×C127C22⋊C411yescharacteristic
53C2×C4×C127C22⋊C411yescharacteristic
54C2×C4×C127C22⋊C411yescharacteristic
55C2×C4×C127C22⋊C411yescharacteristic
56C2×C4×C127SD1612yes—
57C2×C4×C127Q1612yes—
58C2×C4×C127C22×C411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C22×C127—21no—
61C22×C127—21no—
62C22×C127C2×C22⋊C411yescharacteristic
63C22×C127C2×SD1611yescharacteristic
64C22×C127C2×Q1611yescharacteristic
65C4×C127—21no—
66C4×C127—21no—
67C4×C127—21no—
68C4×C127—44no—
69C4×C127#10312yes—
70C4×C127#10312yes—
71C4×C127#10412yes—
72Dic127—42no—
73Dic127—42no—
74C2×C127—21no—
75C2×C127—21no—
76C2×C127#41011yescharacteristic
77C2×C127C2×#10311yescharacteristic
78C2×C127C2×#10411yescharacteristic
79#1503—1271nomaximal, Sylow
80C127#150311yescharacteristic, complemented, Sylow
81#533—1271no—
82#431—1271no—
83C4×C4⋊C4D12711yescharacteristic
84#328—1274no—
85C4⋊Q8—1271no—
86C4⋊Q8—1271no—
87#125—2541no—
88C22⋊Q8—1272no—
89C22⋊Q8—2541no—
90C4×Q8—1272no—
91C4×Q8—2541no—
92#117—2541no—
93C2×C4⋊C4—21no—
94C2×C42—21no—
95C2×C42C2×D12711yescharacteristic
96C4⋊C8—2541no—
97C22⋊C8—1272no—
98C4×C8—2541no—
99#95C2×D12712yes—
100C2×Q8—1272no—
101C2×Q8—2541no—
102C22×C4—22no—
103C22×C4C22⋊D12711yescharacteristic
104C22×C4C22⋊D12711yescharacteristic
105C22×C4C22×D12711yescharacteristic
106C2×C8—2542no—
107C4⋊C4—24no—
108C4⋊C4—1272no—
109C4⋊C4—2542no—
110C4⋊C4—2541no—
111C4⋊C4—2541no—
112C4⋊C4—2541no—
113C4⋊C4—2542no—
114C22⋊C4—2542no—
115C42—42no—
116C42—2542no—
117C42C4×D12711yescharacteristic
118C42C4×D12711yescharacteristic
119C42D50811yescharacteristic
120C42D50811yescharacteristic
121C23C2×C22⋊D12711yescharacteristic
122Q8—5082no—
123Q8—5082no—
124C2×C4—22no—
125C2×C4—24no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—2542no—
130C2×C4—2542no—
131C2×C4(C2×D127)⋊C411yescharacteristic
132C2×C4(C2×D127)⋊C411yescharacteristic
133C2×C4(C2×D127)⋊C411yescharacteristic
134C2×C4(C2×D127)⋊C411yescharacteristic
135C2×C4#1452323612yes—
136C2×C4#1452323812yes—
137C2×C4C2×C4×D12711yescharacteristic
138C2×C4C2×D50811yescharacteristic
139C8—5082no—
140C22—21no—
141C22—21no—
142C22C2×(C2×D127)⋊C411yescharacteristic
143C22C2×#1452323611yescharacteristic
144C22C2×#1452323811yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—44no—
149C4—5082no—
150C4—5082no—
151C4#1686113212yes—
152C4#1686113212yes—
153C4#1686113712yes—
154C2—21no—
155C2—21no—
156C2#2254984911yescharacteristic
157C2C2×#1686113211yescharacteristic
158C2C2×#1686113711yescharacteristic
subgroupG/H
centerC2×C4(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC4×C4⋊C4×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenterC4×C4⋊C4D127
hypocenterC127#1503
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #16861209C4semidirect
6. #16861209C4semidirect
11. C4×Dic254C4semidirect
80. C127#1503semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225499292
#225496122
C4×C4⋊C4×C1272
#1503127