Finite Group Databasebeta

← #46810191

Subgroups — #46810191

Order 34432.

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#31771000C211yesmaximal, characteristic, complemented
2#31770964C212yesmaximal, complemented
3#31770345C211yesmaximal, characteristic, complemented
4#31770345C212yesmaximal, complemented
5H(3,𝔽4)×C269C211yesmaximal, characteristic
6C2×#16864392—21no—
7C2×#16864390—21no—
8#22575453C2211yescharacteristic, complemented
9#22575377—22no—
10#22575374—22no—
11#22575364C2212yescomplemented
12#22575364C2211yescharacteristic, complemented
13#124×C269—22no—
14#124×C269—21no—
15#124×C269C2212yes—
16#124×C269C2211yescharacteristic
17#120×C269—22no—
18#120×C269—21no—
19C2×Dic538—21no—
20#16864392—42no—
21#16864390—42no—
22C2×#14524130—21no—
23C2×#14524130—22no—
24C4⋊Dic269—22no—
25C24×C269—21no—
26C2×Q8×C269D412yes—
27C2×Q8×C269D411yescharacteristic
28C2×D4×C269—21no—
29C2×D4×C269—22no—
30C2×D4×C269D412yes—
31C2×D4×C269D411yescharacteristic
32C22⋊C4×C269—22no—
33C22⋊C4×C269—21no—
34C22⋊C4×C269—22no—
35C22⋊C4×C269—22no—
36C22⋊C4×C269—22no—
37C42×C269—21no—
38C42×C269C2311yescharacteristic
39Dic538—41no—
40Dic538—41no—
41C2×Dic269—41no—
42#14524130—22no—
43#14524130—42no—
44C23×C269—21no—
45C23×C269—22no—
46C23×C269—81no—
47C23×C269—82no—
48Q8×C269—42no—
49Q8×C269—41no—
50D4×C269—42no—
51D4×C269—41no—
52D4×C269—81no—
53D4×C269—82no—
54C2×C4×C269—21no—
55C2×C4×C269—22no—
56C2×C4×C269C2×D412yes—
57C2×C4×C269C2×D411yescharacteristic
58C22×C269—41no—
59C22×C269—42no—
60C22×C269—42no—
61C22×C269—42no—
62C22×C269—42no—
63C22×C269—84no—
64C22×C269#12011yescharacteristic
65C4×C269—21no—
66C4×C269—22no—
67C4×C269—41no—
68C4×C269—42no—
69Dic269—81no—
70C2×C269—81no—
71C2×C269—82no—
72C2×C269#45311yescharacteristic
73C2×C269#45512yes—
74C269#165911yescharacteristic, complemented, Sylow
75#1659—2691nomaximal, Sylow
76H(3,𝔽4)D26911yescharacteristic
77#497—2691no—
78#489—2692no—
79#329—2691no—
80#329—2692no—
81C2×Q16—5381no—
82C2×SD16—5381no—
83C4⋊Q8—2691no—
84#124—22no—
85#124—21no—
86#124C2×D26912yes—
87#124C2×D26911yescharacteristic
88#120—22no—
89#120—21no—
90#103—5382no—
91#102—5382no—
92#97—2692no—
93#97—2691no—
94C24—21no—
95C2×Q8—5381no—
96C2×Q8C22⋊D26912yes—
97C2×Q8C22⋊D26911yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D26912yes—
101C2×D4C22⋊D26911yescharacteristic
102Q16—10762no—
103SD16—10762no—
104C2×C8—5381no—
105C2×C8—5382no—
106C4⋊C4—5382no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D26911yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—10761no—
121Q8—10761no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—10761no—
129C2×C4C2×C22⋊D26912yes—
130C2×C4C2×C22⋊D26911yescharacteristic
131C8—5382no—
132C8—10762no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#2257542211yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—21521no—
145C2—81no—
146C2—82no—
147C2#3177077311yescharacteristic
148C2#3177078312yes—
subgroupG/H
centerC22#22575422
commutator, cosocleC42×C269C23
FrattiniC42C22×D269
Fitting, prime radicalH(3,𝔽4)×C269C2
socleC22×C269#120
hypercenterH(3,𝔽4)D269
hypocenterC269#1659
LCSG ▷ C42×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #31771000C2semidirect
2. #31770964C2semidirect
3. #31770345C2semidirect
4. #31770345C2semidirect
8. #22575453C22semidirect
11. #22575364C22semidirect
12. #22575364C22semidirect
74. C269#1659semidirect

Maximal subgroups

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

subgroupindex
#317710002
#317709642
#317703452
H(3,𝔽4)×C2692
#1659269