Finite Group Databasebeta

← #44549479

Subgroups — #44549479

Order 32128.

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

normal subgroups only

This group has 314 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#30632322C211yesmaximal, characteristic
2#30631974C212yesmaximal, complemented
3#30631431C211yesmaximal, characteristic, complemented
4#30631416C212yesmaximal, complemented
5#492×C251C211yesmaximal, characteristic, complemented
6C2×D4×D251—22no—
7#22548686—22no—
8#22548683—22no—
9C4×D1004C2212yes—
10#22548646—22no—
11C4⋊(C251⋊C8)C2212yescomplemented
12#22548635—22no—
13C2×SD16×C251—22no—
14C4⋊Q8×C251C2211yescharacteristic, complemented
15#127×C251C2211yescharacteristic
16C4×C8×C251C2211yescharacteristic, complemented
17C23×D251—22no—
18C2×D1004D412yes—
19C2×C22⋊D251—24no—
20D4×D251—44no—
21D4×D251—44no—
22C2×C4×D251—22no—
23C2×C251⋊C8—22no—
24C4⋊Dic251D412yes—
25C22⋊Dic251—22no—
26(C2×D251)⋊C4—22no—
27C2×Q8×C251—21no—
28C2×D4×C251—22no—
29C2×D4×C251D412yes—
30SD16×C251—44no—
31C2×C8×C251—21no—
32C4⋊C4×C251—22no—
33C42×C251C2311yescharacteristic
34C22×D251—22no—
35C22×D251—48no—
36C22×D251—82no—
37D1004—22no—
38D1004SD1614yescomplemented
39C22⋊D251—48no—
40C2×Dic251—22no—
41C4×D251—42no—
42C4×D251—42no—
43C251⋊C8—42no—
44C23×C251—22no—
45Q8×C251—42no—
46D4×C251—42no—
47D4×C251—42no—
48D4×C251—42no—
49C2×C4×C251—41no—
50C2×C4×C251C2×D412yes—
51C2×C4×C251C2×D411yescharacteristic
52C8×C251—22no—
53C22×C251—44no—
54C22×C251—42no—
55C22×C251#12011yescharacteristic
56C2×D251—24no—
57C2×D251—42no—
58C2×D251—42no—
59C2×D251—88no—
60C4×C251—21no—
61C4×C251—81no—
62C4×C251C2×SD1612yes—
63C4×C251AGL(1,ℤ/8)12yes—
64Dic251—42no—
65C2×C251—82no—
66C2×C251#45012yes—
67C2×C251#45311yescharacteristic
68D251—44no—
69C251#162911yescharacteristic, complemented, Sylow
70#1629—2511nomaximal, Sylow
71D42—2511no—
72#492D25111yescharacteristic, complemented
73#476—2512no—
74#331—2511no—
75D4⋊C8—2512no—
76C22×D4—5022no—
77C2×SD16—22no—
78C4⋊Q8C2×D25111yescharacteristic, complemented
79#127C2×D25111yescharacteristic
80#121—5022no—
81#120—5022no—
82C4×D4—2512no—
83#106—5022no—
84C4⋊C8—2512no—
85#102—5022no—
86C4×C8C2×D25111yescharacteristic, complemented
87C24—5022no—
88C2×Q8—21no—
89C2×D4—22no—
90C2×D4—10044no—
91C2×D4—10044no—
92C2×D4—5024no—
93C2×D4—2512no—
94C2×D4C22⋊D25112yes—
95C22×C4—5022no—
96SD16—44no—
97C2×C8—21no—
98C2×C8—5022no—
99C4⋊C4—22no—
100C4⋊C4—2512no—
101C22⋊C4—5022no—
102C22⋊C4—5022no—
103C42C22×D25111yescharacteristic
104C23—22no—
105C23—20082no—
106C23—10048no—
107C23—5022no—
108Q8—42no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—10048no—
113D4—5022no—
114D4—2514no—
115C2×C4—41no—
116C2×C4—10042no—
117C2×C4—10042no—
118C2×C4—5022no—
119C2×C4D4×D25112yes—
120C2×C4C2×C22⋊D25111yescharacteristic
121C8—22no—
122C8—10042no—
123C22—44no—
124C22—42no—
125C22—20088no—
126C22—10042no—
127C22—10042no—
128C22—5024no—
129C22#2254868311yescharacteristic
130C4—21no—
131C4—81no—
132C4—10042no—
133C4SD16×D25112yes—
134C4#2254874212yes—
135C2—82no—
136C2—10044no—
137C2#3063183612yes—
138C2#3063185411yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#492×C251C2
socleC22×C251#120
prime radical#30632322C2
hypercenter#492D251
hypocenterC251#1629
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #30631974C2semidirect
3. #30631431C2semidirect
4. #30631416C2semidirect
5. #492×C251C2semidirect
11. C4⋊(C251⋊C8)C22semidirect
14. C4⋊Q8×C251C22semidirect
16. C4×C8×C251C22semidirect
38. D1004SD16semidirect
69. C251#1629semidirect
72. #492D251semidirect
78. C4⋊Q8C2×D251semidirect
86. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323222
#306319742
#306314312
#306314162
#492×C2512
#1629251