Finite Group Databasebeta

← #30638381

Subgroups — #30638381

Order 16256.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550279C211yesmaximal, characteristic, complemented
2#22550228C211yesmaximal, characteristic, complemented
3#22550227C211yesmaximal, characteristic, complemented
4#22550222C211yesmaximal, characteristic
5#22550222C211yesmaximal, characteristic
6#22549598C211yesmaximal, characteristic, complemented
7#492×C127C211yesmaximal, characteristic
8#16861209C2211yescharacteristic
9#16861209C2211yescharacteristic
10#16861142—22no—
11#16861134—22no—
12#16861132—22no—
13#16861130—22no—
14#16861130—22no—
15C4×C127⋊C8C2211yescharacteristic
16C4×C127⋊C8C2211yescharacteristic
17C2×SD16×C127—22no—
18C4⋊Q8×C127C2211yescharacteristic
19#127×C127C2211yescharacteristic
20C4×C8×C127C2211yescharacteristic
21C2×Dic254—21no—
22C2×Dic254—21no—
23C2×C127⋊C8D412yes—
24C2×C127⋊C8D412yes—
25C4⋊Dic127—21no—
26C4⋊Dic127—21no—
27C4⋊Dic127—21no—
28C4⋊Dic127—21no—
29C2×Q8×C127—21no—
30C2×D4×C127—21no—
31C2×D4×C127—21no—
32C2×D4×C127—21no—
33SD16×C127—44no—
34C2×C8×C127D412yes—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—21no—
37C42×C127C2311yescharacteristic
38Dic254—42no—
39Dic254—42no—
40C2×Dic127—41no—
41C2×Dic127—41no—
42C127⋊C8—22no—
43C127⋊C8—22no—
44C23×C127—41no—
45Q8×C127—42no—
46D4×C127—42no—
47D4×C127—42no—
48D4×C127—42no—
49C2×C4×C127—41no—
50C2×C4×C127C2×D411yescharacteristic
51C2×C4×C127C2×D411yescharacteristic
52C2×C4×C127C2×D411yescharacteristic
53C8×C127—22no—
54C22×C127—81no—
55C22×C127—81no—
56C22×C127—81no—
57C22×C127#12711yescharacteristic
58C4×C127—81no—
59C4×C127#13512yes—
60C4×C127AGL(1,ℤ/8)12yes—
61C4×C127#13712yes—
62Dic127—81no—
63Dic127—81no—
64C2×C127—161no—
65C2×C127#49511yescharacteristic
66C2×C127#49611yescharacteristic
67C2×C127#49711yescharacteristic
68#1719—1271nomaximal, Sylow
69C127#171911yescharacteristic, complemented, Sylow
70#500—1271no—
71#492D12711yescharacteristic
72#487—1271no—
73#487—1271no—
74#486—1271no—
75#486—1271no—
76#322—1271no—
77C2×SD16—22no—
78C4⋊Q8—1271no—
79C4⋊Q8—1271no—
80C4⋊Q8C2×D12711yescharacteristic
81#127C2×D12711yescharacteristic
82#107—2542no—
83#103—2542no—
84#103—2542no—
85#102—2542no—
86#102—2542no—
87C4×C8—1271no—
88C4×C8—1271no—
89C4×C8C2×D12711yescharacteristic
90C2×Q8—21no—
91C2×Q8—2541no—
92C2×Q8—2541no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96SD16—44no—
97C2×C8—1272no—
98C2×C8—1272no—
99C2×C8C22⋊D12712yes—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—2541no—
103C4⋊C4—2541no—
104C4⋊C4—2541no—
105C4⋊C4—2541no—
106C42C22×D12711yescharacteristic
107C23—41no—
108Q8—42no—
109Q8—5082no—
110Q8—5082no—
111D4—42no—
112D4—42no—
113D4—42no—
114C2×C4—41no—
115C2×C4—5081no—
116C2×C4—5081no—
117C2×C4D4×D12711yescharacteristic
118C2×C4D4×D12711yescharacteristic
119C2×C4C2×C22⋊D12711yescharacteristic
120C8—22no—
121C8—2542no—
122C8—2542no—
123C22—81no—
124C22—81no—
125C22—81no—
126C22#1686120511yescharacteristic
127C4—81no—
128C4—10161no—
129C4—10161no—
130C4#1686123012yes—
131C4#1686123512yes—
132C4#1686123812yes—
133C2—161no—
134C2#2255025711yescharacteristic
135C2#2255026311yescharacteristic
136C2#2255026711yescharacteristic
subgroupG/H
centerC22#16861205
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#492×C127C2
socleC22×C127#127
prime radical#127×C127C22
hypercenter#492D127
hypocenterC127#1719
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #22550279C2semidirect
2. #22550228C2semidirect
3. #22550227C2semidirect
6. #22549598C2semidirect
69. C127#1719semidirect

Maximal subgroups

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

subgroupindex
#225502792
#225502282
#225502272
#225502222
#225495982
#492×C1272
#1719127