Finite Group Databasebeta

← #44549449

Subgroups — #44549449

Order 32128.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#30632403C211yesmaximal, characteristic, complemented
2#30632061C211yesmaximal, characteristic, complemented
3#30631883C212yesmaximal, complemented
4#30631414C212yesmaximal, complemented
5#333×C251C211yesmaximal, characteristic, complemented
6C2×#16860797—22no—
7C2×D2008—22no—
8#22548712C2211yescharacteristic
9#22548704—22no—
10#22548690—22no—
11C4×D1004C2212yes—
12#22548673—22no—
13#22548637—22no—
14C4×C251⋊C8C2211yescharacteristic
15C4⋊Q8×C251C2211yescharacteristic
16C4⋊C8×C251C2212yes—
17C2×D1004—22no—
18C2×D1004—22no—
19C2×D1004—21no—
20C2×D1004D412yes—
21C2×C4×D251—22no—
22#16860797—44no—
23D2008—44no—
24C2×C251⋊C8—21no—
25C4⋊Dic251D412yes—
26(C2×D251)⋊C4—24no—
27(C2×D251)⋊C4—22no—
28C4×Dic251—22no—
29C2×Q8×C251D412yes—
30C2×C8×C251—22no—
31C4⋊C4×C251—22no—
32C42×C251C2311yescharacteristic
33C22×D251—22no—
34C22×D251—41no—
35D1004—22no—
36D1004—22no—
37D1004—42no—
38D1004—42no—
39D1004—42no—
40D1004—82no—
41C2×Dic251—22no—
42C4×D251—42no—
43C4×D251—42no—
44C251⋊C8—22no—
45Q8×C251—42no—
46C2×C4×C251—22no—
47C2×C4×C251C2×D412yes—
48C2×C4×C251C2×D411yescharacteristic
49C8×C251—42no—
50C22×C251#12011yescharacteristic
51C2×D251—42no—
52C2×D251—42no—
53C2×D251—42no—
54C2×D251—82no—
55C2×D251—81no—
56C4×C251—21no—
57C4×C251—42no—
58C4×C251#13512yes—
59C4×C251AGL(1,ℤ/8)12yes—
60Dic251—42no—
61C2×C251#44912yes—
62C2×C251#45311yescharacteristic
63D251—82no—
64D251—161no—
65C251#162411yescharacteristic, complemented, Sylow
66#1624—2511nomaximal, Sylow
67#555—2511no—
68#492—2511no—
69#459—2512no—
70#333D25111yescharacteristic, complemented
71D4⋊C8—2512no—
72C2×SD16—5022no—
73C2×D8—5022no—
74C4⋊Q8C2×D25111yescharacteristic
75#127—2511no—
76#124—5022no—
77#121—5022no—
78C4×D4—2512no—
79#117—5022no—
80C4⋊C8C2×D25112yes—
81#102—5022no—
82C4×C8—2511no—
83C2×Q8D100412yes—
84C2×D4—5022no—
85C2×D4—5022no—
86C2×D4—5021no—
87C2×D4—2512no—
88C22×C4—5022no—
89SD16—10044no—
90D8—10044no—
91C2×C8—22no—
92C2×C8—5021no—
93C4⋊C4—22no—
94C4⋊C4—2512no—
95C22⋊C4—5024no—
96C22⋊C4—5022no—
97C42—5022no—
98C42C22×D25111yescharacteristic
99C23—10041no—
100C23—5022no—
101Q8—42no—
102D4—20082no—
103D4—10042no—
104D4—10042no—
105D4—10042no—
106D4—5022no—
107D4—5022no—
108C2×C4—22no—
109C2×C4—10042no—
110C2×C4—10042no—
111C2×C4—5022no—
112C2×C4D4×D25112yes—
113C2×C4C2×D100411yescharacteristic
114C8—42no—
115C8—5022no—
116C22—20082no—
117C22—20081no—
118C22—10042no—
119C22—10042no—
120C22—10042no—
121C22#2254868511yescharacteristic
122C4—21no—
123C4—42no—
124C4—10042no—
125C4#2254873812yes—
126C4#2254874212yes—
127C2—40161no—
128C2—20082no—
129C2#3063183512yes—
130C2#3063185711yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#333×C251C2
socleC22×C251#120
hypercenter#333D251
hypocenterC251#1624
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #30632403C2semidirect
2. #30632061C2semidirect
3. #30631883C2semidirect
4. #30631414C2semidirect
5. #333×C251C2semidirect
65. C251#1624semidirect
70. #333D251semidirect

Maximal subgroups

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

subgroupindex
#306324032
#306320612
#306318832
#306314142
#333×C2512
#1624251