Finite Group Databasebeta

← #30637280

Subgroups — #30637280

Order 16256.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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#22549933C211yesmaximal, characteristic
3#22549612C214yesmaximal
4#383×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—
15#16861120—21no—
16C2×C42×C127C2211yescharacteristic
17#95×C127—21no—
18#95×C127—21no—
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
35C42×C127C2×C411yescharacteristic
36C42×C127C2×C411yescharacteristic
37C42×C127D411yescharacteristic
38C42×C127D411yescharacteristic
39Dic254—42no—
40Dic254—42no—
41C2×Dic127—22no—
42C2×Dic127—22no—
43C127⋊C8—42no—
44C23×C127C2×D411yescharacteristic
45C2×C4×C127—22no—
46C2×C4×C127—41no—
47C2×C4×C127—41no—
48C2×C4×C127—42no—
49C2×C4×C127—42no—
50C2×C4×C127—42no—
51C2×C4×C127C22⋊C411yescharacteristic
52C2×C4×C127C22⋊C411yescharacteristic
53C2×C4×C127C22⋊C411yescharacteristic
54C2×C4×C127C22⋊C411yescharacteristic
55C2×C4×C127C22×C411yescharacteristic
56C2×C4×C127C2×D411yescharacteristic
57C22×C127—21no—
58C22×C127—21no—
59C22×C127C2×C22⋊C411yescharacteristic
60C22×C127#13712yes—
61C4×C127—21no—
62C4×C127—21no—
63C4×C127—21no—
64C4×C127—21no—
65C4×C127—21no—
66C4×C127—21no—
67C4×C127—82no—
68Dic127—42no—
69Dic127—42no—
70C2×C127—22no—
71C2×C127#41011yescharacteristic
72C2×C127#41911yescharacteristic
73C2×C127#42111yescharacteristic
74#1515—1271nomaximal, Sylow
75C127#151511yescharacteristic, complemented, Sylow
76#533—1271no—
77#432—1271no—
78#383D12711yescharacteristic
79#328—1274no—
80C4⋊Q8—1271no—
81C4⋊Q8—1271no—
82#125—2541no—
83C22⋊Q8—1272no—
84C22⋊Q8—2541no—
85C4×Q8—1272no—
86C4×Q8—2541no—
87#117—2541no—
88C2×C42C2×D12711yescharacteristic
89C4⋊C8—2541no—
90C22⋊C8—1272no—
91#97—2541no—
92#95—21no—
93#95—21no—
94#95C2×D12712yes—
95C2×Q8—1272no—
96C2×Q8—2541no—
97C22×C4—22no—
98C22×C4C22⋊D12711yescharacteristic
99C22×C4C22⋊D12711yescharacteristic
100C22×C4C22×D12711yescharacteristic
101C2×C8—2542no—
102C4⋊C4—1272no—
103C4⋊C4—2542no—
104C4⋊C4—2541no—
105C4⋊C4—2541no—
106C4⋊C4—2541no—
107C4⋊C4—2542no—
108C22⋊C4—2542no—
109C42—2542no—
110C42C4×D12711yescharacteristic
111C42C4×D12711yescharacteristic
112C42D50811yescharacteristic
113C42D50811yescharacteristic
114C23C2×C22⋊D12711yescharacteristic
115Q8—5082no—
116Q8—5082no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—2542no—
124C2×C4—2542no—
125C2×C4(C2×D127)⋊C411yescharacteristic
126C2×C4(C2×D127)⋊C411yescharacteristic
127C2×C4(C2×D127)⋊C411yescharacteristic
128C2×C4(C2×D127)⋊C411yescharacteristic
129C2×C4C2×C4×D12711yescharacteristic
130C2×C4C2×D50811yescharacteristic
131C8—5082no—
132C22—21no—
133C22—21no—
134C22C2×(C2×D127)⋊C411yescharacteristic
135C22#1686123912yes—
136C4—21no—
137C4—21no—
138C4—21no—
139C4—21no—
140C4—21no—
141C4—21no—
142C4—82no—
143C4—5082no—
144C4—5082no—
145C2—22no—
146C2#2254984911yescharacteristic
147C2#2254988411yescharacteristic
148C2#2254989411yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#383×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#383D127
hypocenterC127#1515
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #383

Complements

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

NH ≅ G/Ntype
5. #16861209C4semidirect
6. #16861209C4semidirect
11. C4×Dic254C4semidirect
75. C127#1515semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225499332
#225496122
#383×C1272
#1515127