Finite Group Databasebeta

← #44549677

Subgroups — #44549677

Order 32128.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#30632437C211yesmaximal, characteristic
2#30632096C212yesmaximal, complemented
3#30632082C211yesmaximal, characteristic
4#30631428C212yesmaximal, complemented
5#329×C251C211yesmaximal, characteristic
6C2×#16860800—21no—
7C2×#16860798—21no—
8#22548715C2211yescharacteristic
9#22548713—21no—
10#22548708—22no—
11#22548707C2212yes—
12#22548696—21no—
13#22548692—22no—
14#22548649—22no—
15#22548647—22no—
16#22548626C2211yescharacteristic
17#124×C251C2211yescharacteristic
18#97×C251C2212yes—
19C2×Dic502—21no—
20C2×Dic502—21no—
21C22×Dic251—21no—
22#16860800—42no—
23#16860798—42no—
24C2×C251⋊C8—21no—
25C4⋊Dic251—22no—
26C4⋊Dic251D412yes—
27C4⋊Dic251D412yes—
28Dic251⋊C4—22no—
29Dic251⋊C4—22no—
30Dic251⋊C4—22no—
31C22⋊Dic251—22no—
32C22⋊Dic251—21no—
33C4×Dic251—21no—
34C2×Q8×C251D411yescharacteristic
35C2×D4×C251D411yescharacteristic
36C2×C8×C251—22no—
37C22⋊C4×C251—22no—
38C42×C251C2311yescharacteristic
39Dic502—41no—
40Dic502—41no—
41Dic502—81no—
42C2×Dic251—22no—
43C2×Dic251—22no—
44C2×Dic251—41no—
45C2×Dic251—81no—
46C251⋊C8—22no—
47C23×C251—21no—
48Q8×C251—41no—
49D4×C251—41no—
50C2×C4×C251—21no—
51C2×C4×C251C2×D412yes—
52C2×C4×C251C2×D411yescharacteristic
53C8×C251—24no—
54C22×C251—41no—
55C22×C251—42no—
56C22×C251#12011yescharacteristic
57C4×C251—21no—
58C4×C251—22no—
59C4×C251—41no—
60Dic251—42no—
61Dic251—42no—
62Dic251—81no—
63C2×C251—81no—
64C2×C251#45311yescharacteristic
65C2×C251#45612yes—
66C251#166011yescharacteristic, complemented, Sylow
67#1660—2511nomaximal, Sylow
68#563—2511no—
69#501—2512no—
70#497—2511no—
71#330—2512no—
72#329D25111yescharacteristic
73C2×Q16—5021no—
74C2×SD16—5021no—
75C4⋊Q8—5021no—
76C4⋊Q8—2511no—
77#126—5022no—
78#125—2512no—
79#124C2×D25111yescharacteristic
80#123—5021no—
81C22⋊Q8—5022no—
82#107—5022no—
83#106—5022no—
84#97—2511no—
85#97C2×D25112yes—
86C2×Q8—5021no—
87C2×Q8—5021no—
88C2×Q8D100411yescharacteristic
89C2×D4D100411yescharacteristic
90C22×C4—5021no—
91Q16—10042no—
92SD16—10042no—
93C2×C8—22no—
94C2×C8—5021no—
95C4⋊C4—5022no—
96C4⋊C4—5022no—
97C4⋊C4—5022no—
98C4⋊C4—5022no—
99C4⋊C4—2512no—
100C4⋊C4—2512no—
101C22⋊C4—22no—
102C22⋊C4—5022no—
103C22⋊C4—5021no—
104C42—5021no—
105C42C22×D25111yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—20081no—
109Q8—10041no—
110Q8—10041no—
111D4—41no—
112C2×C4—21no—
113C2×C4—20081no—
114C2×C4—10041no—
115C2×C4—5022no—
116C2×C4—5022no—
117C2×C4D4×D25112yes—
118C2×C4C2×D100411yescharacteristic
119C8—24no—
120C8—5022no—
121C22—41no—
122C22—42no—
123C22#2254868511yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—20081no—
128C4—10042no—
129C4—10042no—
130C2—81no—
131C2#3063185711yescharacteristic
132C2#3063187112yes—
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#329×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenter#329D251
hypocenterC251#1660
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #30632096C2semidirect
4. #30631428C2semidirect
66. C251#1660semidirect

Maximal subgroups

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

subgroupindex
#306324372
#306320962
#306320822
#306314282
#329×C2512
#1660251