Finite Group Databasebeta

← #44506751

Subgroups — #44506751

Order 30848.

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#29508112C211yesmaximal, characteristic, complemented
2#29508076C212yesmaximal, complemented
3#29507457C211yesmaximal, characteristic, complemented
4#29507457C212yesmaximal, complemented
5H(3,𝔽4)×C241C211yesmaximal, characteristic
6C2×#16838760—21no—
7C2×#16838758—21no—
8#21440642C2211yescharacteristic, complemented
9#21440566—22no—
10#21440563—22no—
11#21440553C2211yescharacteristic, complemented
12#21440553C2212yescomplemented
13#124×C241—22no—
14#124×C241—21no—
15#124×C241C2211yescharacteristic
16#124×C241C2212yes—
17#120×C241—22no—
18#120×C241—21no—
19C2×Dic482—21no—
20#16838760—42no—
21#16838758—42no—
22C2×#14515349—21no—
23C2×#14515349—22no—
24C4⋊Dic241—22no—
25C24×C241—21no—
26C2×Q8×C241D412yes—
27C2×Q8×C241D411yescharacteristic
28C2×D4×C241—21no—
29C2×D4×C241—22no—
30C2×D4×C241D412yes—
31C2×D4×C241D411yescharacteristic
32C22⋊C4×C241—22no—
33C22⋊C4×C241—21no—
34C22⋊C4×C241—22no—
35C22⋊C4×C241—22no—
36C22⋊C4×C241—22no—
37C42×C241—21no—
38C42×C241C2311yescharacteristic
39Dic482—41no—
40Dic482—41no—
41C2×Dic241—41no—
42#14515349—22no—
43#14515349—42no—
44C23×C241—21no—
45C23×C241—22no—
46C23×C241—81no—
47C23×C241—82no—
48Q8×C241—42no—
49Q8×C241—41no—
50D4×C241—42no—
51D4×C241—41no—
52D4×C241—81no—
53D4×C241—82no—
54C2×C4×C241—21no—
55C2×C4×C241—22no—
56C2×C4×C241C2×D412yes—
57C2×C4×C241C2×D411yescharacteristic
58C22×C241—41no—
59C22×C241—42no—
60C22×C241—42no—
61C22×C241—42no—
62C22×C241—42no—
63C22×C241—84no—
64C22×C241#12011yescharacteristic
65C4×C241—21no—
66C4×C241—22no—
67C4×C241—41no—
68C4×C241—42no—
69Dic241—81no—
70C2×C241—81no—
71C2×C241—82no—
72C2×C241#45311yescharacteristic
73C2×C241#45512yes—
74C241#165911yescharacteristic, complemented, Sylow
75#1659—2411nomaximal, Sylow
76H(3,𝔽4)D24111yescharacteristic
77#497—2411no—
78#489—2412no—
79#329—2411no—
80#329—2412no—
81C2×Q16—4821no—
82C2×SD16—4821no—
83C4⋊Q8—2411no—
84#124—22no—
85#124—21no—
86#124C2×D24112yes—
87#124C2×D24111yescharacteristic
88#120—22no—
89#120—21no—
90#103—4822no—
91#102—4822no—
92#97—2412no—
93#97—2411no—
94C24—21no—
95C2×Q8—4821no—
96C2×Q8C22⋊D24112yes—
97C2×Q8C22⋊D24111yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D24112yes—
101C2×D4C22⋊D24111yescharacteristic
102Q16—9642no—
103SD16—9642no—
104C2×C8—4822no—
105C2×C8—4821no—
106C4⋊C4—4822no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D24111yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—9641no—
121Q8—9641no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—9641no—
129C2×C4C2×C22⋊D24112yes—
130C2×C4C2×C22⋊D24111yescharacteristic
131C8—9642no—
132C8—4822no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#2144061111yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—19281no—
145C2—81no—
146C2—82no—
147C2#2950788511yescharacteristic
148C2#2950789512yes—
subgroupG/H
centerC22#21440611
commutator, cosocleC42×C241C23
FrattiniC42C22×D241
Fitting, prime radicalH(3,𝔽4)×C241C2
socleC22×C241#120
hypercenterH(3,𝔽4)D241
hypocenterC241#1659
LCSG ▷ C42×C241 ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #29508112C2semidirect
2. #29508076C2semidirect
3. #29507457C2semidirect
4. #29507457C2semidirect
8. #21440642C22semidirect
11. #21440553C22semidirect
12. #21440553C22semidirect
74. C241#1659semidirect

Maximal subgroups

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

subgroupindex
#295081122
#295080762
#295074572
H(3,𝔽4)×C2412
#1659241