Finite Group Databasebeta

← #30639673

Subgroups — #30639673

Order 16256.

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

normal subgroups only

This group has 260 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
1#22550433C211yesmaximal, characteristic
2C4×#14523232C211yesmaximal, characteristic
3#22549636C214yesmaximal
4C4×C4⋊C4×C127C211yesmaximal, characteristic
5C2×#14523232C2212yes—
6#16861209C411yescharacteristic, complemented
7#16861209C411yescharacteristic, complemented
8#16861201C412yescomplemented
9#16861188—21no—
10#16861188—21no—
11C4×Dic254—21no—
12C4×Dic254—21no—
13#16861168C2212yes—
14#16861120—21no—
15C4×C127⋊C8—21no—
16C2×C4⋊C4×C127—21no—
17C2×C42×C127C2212yes—
18C2×C42×C127C2211yescharacteristic
19#95×C127—21no—
20C2×Dic254—21no—
21C2×Dic254—21no—
22#14523232—22no—
23#14523232Q814yescomplemented
24C2×C127⋊C8—22no—
25C4⋊Dic127—21no—
26C4⋊Dic127—21no—
27C4⋊Dic127C2×C412yes—
28C4⋊Dic127C2×C412yes—
29Dic127⋊C4—22no—
30Dic127⋊C4—22no—
31C22⋊Dic127—22no—
32C4×Dic127—22no—
33C22×C4×C127—22no—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37C4⋊C4×C127—24no—
38C42×C127—24no—
39C42×C127C2×C411yescharacteristic
40C42×C127C2×C411yescharacteristic
41C42×C127D411yescharacteristic
42C42×C127D411yescharacteristic
43Dic254—42no—
44Dic254—42no—
45C2×Dic127—22no—
46C2×Dic127—22no—
47C127⋊C8—24no—
48C23×C127C2×D411yescharacteristic
49C2×C4×C127—22no—
50C2×C4×C127—24no—
51C2×C4×C127—41no—
52C2×C4×C127—42no—
53C2×C4×C127—42no—
54C2×C4×C127C22⋊C411yescharacteristic
55C2×C4×C127C22⋊C411yescharacteristic
56C2×C4×C127C22⋊C411yescharacteristic
57C2×C4×C127C22⋊C411yescharacteristic
58C2×C4×C127C22×C411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127C2×Q812yes—
61C2×C4×C127#4112yes—
62C22×C127—21no—
63C22×C127—21no—
64C22×C127C2×C22⋊C411yescharacteristic
65C22×C127#12411yescharacteristic
66C22×C127C4⋊Q811yescharacteristic
67C4×C127—21no—
68C4×C127—21no—
69C4×C127—21no—
70C4×C127—44no—
71C4×C127#10412yes—
72C4×C127C4×Q812yes—
73C4×C127C22⋊Q812yes—
74Dic127—42no—
75Dic127—42no—
76C2×C127—21no—
77C2×C127—21no—
78C2×C127#39111yescharacteristic
79C2×C127C2×#10411yescharacteristic
80C2×C127#42111yescharacteristic
81#1986—1271nomaximal, Sylow
82C127#198611yescharacteristic, complemented, Sylow
83#533—1271no—
84C4×M16—1271no—
85C4×C4⋊C4D12711yescharacteristic
86#339—1274no—
87C2×M16—1272no—
88C4⋊Q8—1271no—
89C4⋊Q8—1271no—
90#125—1272no—
91C22⋊Q8—2541no—
92C22⋊Q8—2541no—
93C4×Q8—2541no—
94C4×Q8—2541no—
95#117—1272no—
96C2×C4⋊C4—21no—
97C2×C42C2×D12712yes—
98C2×C42C2×D12711yescharacteristic
99#97—2541no—
100C4×C8—2541no—
101#95—21no—
102C2×Q8—2541no—
103C2×Q8—2541no—
104C22×C4—22no—
105C22×C4C22⋊D12711yescharacteristic
106C22×C4C22⋊D12711yescharacteristic
107C22×C4C22×D12711yescharacteristic
108M16—1274no—
109M16—2542no—
110C2×C8—2542no—
111C4⋊C4—24no—
112C4⋊C4—1272no—
113C4⋊C4—1272no—
114C4⋊C4—2542no—
115C4⋊C4—2542no—
116C4⋊C4—2541no—
117C4⋊C4—2541no—
118C22⋊C4—2542no—
119C42—24no—
120C42—2542no—
121C42C4×D12711yescharacteristic
122C42C4×D12711yescharacteristic
123C42D50811yescharacteristic
124C42D50811yescharacteristic
125C23C2×C22⋊D12711yescharacteristic
126Q8—5082no—
127Q8—5082no—
128C2×C4—22no—
129C2×C4—24no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—2542no—
134C2×C4—2542no—
135C2×C4(C2×D127)⋊C411yescharacteristic
136C2×C4(C2×D127)⋊C411yescharacteristic
137C2×C4(C2×D127)⋊C411yescharacteristic
138C2×C4(C2×D127)⋊C411yescharacteristic
139C2×C4C2×C4×D12711yescharacteristic
140C2×C4C2×D50811yescharacteristic
141C2×C4Q8×D12712yes—
142C2×C4#1452324712yes—
143C8—2544no—
144C22—21no—
145C22—21no—
146C22C2×(C2×D127)⋊C411yescharacteristic
147C22#1686119611yescharacteristic
148C22#1686120711yescharacteristic
149C4—21no—
150C4—21no—
151C4—21no—
152C4—44no—
153C4—5082no—
154C4—5082no—
155C4#1686113712yes—
156C4#1686117412yes—
157C4#1686118912yes—
158C2—21no—
159C2—21no—
160C2#2254978111yescharacteristic
161C2C2×#1686113711yescharacteristic
162C2#2254989411yescharacteristic
subgroupG/H
centerC2×C4C2×D508
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#1986
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #16861209C4semidirect
7. #16861209C4semidirect
8. #16861201C4semidirect
23. #14523232Q8semidirect
82. C127#1986semidirect

Maximal subgroups

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

subgroupindex
#225504332
C4×#145232322
#225496362
C4×C4⋊C4×C1272
#1986127