Finite Group Databasebeta

← #49120567

Subgroups — #49120567

Order 36224.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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#31826860C211yesmaximal, characteristic, complemented
2#31826824C212yesmaximal, complemented
3#31826205C212yesmaximal, complemented
4#31826205C211yesmaximal, characteristic, complemented
5H(3,𝔽4)×C283C211yesmaximal, characteristic
6C2×#17985382—21no—
7C2×#17985380—21no—
8#22583610C2211yescharacteristic, complemented
9#22583534—22no—
10#22583531—22no—
11#22583521C2212yescomplemented
12#22583521C2211yescharacteristic, complemented
13#124×C283—21no—
14#124×C283—22no—
15#124×C283C2212yes—
16#124×C283C2211yescharacteristic
17#120×C283—21no—
18#120×C283—22no—
19C2×Dic566—21no—
20#17985382—42no—
21#17985380—42no—
22C2×C283⋊C8—21no—
23C2×C283⋊C8—22no—
24C4⋊Dic283—22no—
25C24×C283—21no—
26C2×Q8×C283D411yescharacteristic
27C2×Q8×C283D412yes—
28C2×D4×C283—21no—
29C2×D4×C283—22no—
30C2×D4×C283D411yescharacteristic
31C2×D4×C283D412yes—
32C22⋊C4×C283—22no—
33C22⋊C4×C283—21no—
34C22⋊C4×C283—22no—
35C22⋊C4×C283—22no—
36C22⋊C4×C283—22no—
37C42×C283—21no—
38C42×C283C2311yescharacteristic
39Dic566—41no—
40Dic566—41no—
41C2×Dic283—41no—
42C283⋊C8—22no—
43C283⋊C8—42no—
44C23×C283—21no—
45C23×C283—22no—
46C23×C283—81no—
47C23×C283—82no—
48Q8×C283—42no—
49Q8×C283—41no—
50D4×C283—42no—
51D4×C283—41no—
52D4×C283—81no—
53D4×C283—82no—
54C2×C4×C283—21no—
55C2×C4×C283—22no—
56C2×C4×C283C2×D412yes—
57C2×C4×C283C2×D411yescharacteristic
58C22×C283—41no—
59C22×C283—42no—
60C22×C283—42no—
61C22×C283—42no—
62C22×C283—42no—
63C22×C283—84no—
64C22×C283#12011yescharacteristic
65C4×C283—21no—
66C4×C283—22no—
67C4×C283—41no—
68C4×C283—42no—
69Dic283—81no—
70C2×C283—81no—
71C2×C283—82no—
72C2×C283#45311yescharacteristic
73C2×C283#45512yes—
74C283#165911yescharacteristic, complemented, Sylow
75#1659—2831nomaximal, Sylow
76H(3,𝔽4)D28311yescharacteristic
77#497—2831no—
78#489—2832no—
79#329—2831no—
80#329—2832no—
81C2×Q16—5661no—
82C2×SD16—5661no—
83C4⋊Q8—2831no—
84#124—22no—
85#124—21no—
86#124C2×D28312yes—
87#124C2×D28311yescharacteristic
88#120—22no—
89#120—21no—
90#103—5662no—
91#102—5662no—
92#97—2832no—
93#97—2831no—
94C24—21no—
95C2×Q8—5661no—
96C2×Q8C22⋊D28312yes—
97C2×Q8C22⋊D28311yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D28312yes—
101C2×D4C22⋊D28311yescharacteristic
102Q16—11322no—
103SD16—11322no—
104C2×C8—5661no—
105C2×C8—5662no—
106C4⋊C4—5662no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D28311yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—11321no—
121Q8—11321no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—11321no—
129C2×C4C2×C22⋊D28312yes—
130C2×C4C2×C22⋊D28311yescharacteristic
131C8—5662no—
132C8—11322no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#2258357911yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—22641no—
145C2—81no—
146C2—82no—
147C2#3182663311yescharacteristic
148C2#3182664312yes—
subgroupG/H
centerC22#22583579
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting, prime radicalH(3,𝔽4)×C283C2
socleC22×C283#120
hypercenterH(3,𝔽4)D283
hypocenterC283#1659
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #31826860C2semidirect
2. #31826824C2semidirect
3. #31826205C2semidirect
4. #31826205C2semidirect
8. #22583610C22semidirect
11. #22583521C22semidirect
12. #22583521C22semidirect
74. C283#1659semidirect

Maximal subgroups

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

subgroupindex
#318268602
#318268242
#318262052
H(3,𝔽4)×C2832
#1659283