Finite Group Databasebeta

← #49120568

Subgroups — #49120568

Order 36224.

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

normal subgroups only

This group has 228 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#31827207C211yesmaximal, characteristic
2#31826860C211yesmaximal, characteristic, complemented
3#31826825C212yesmaximal, complemented
4#31826204C212yesmaximal, complemented
5#329×C283C211yesmaximal, characteristic, complemented
6C2×#17985382—21no—
7C2×#17985380—21no—
8#22583610C2211yescharacteristic, complemented
9#22583599—21no—
10#22583598C2212yes—
11#22583596—22no—
12#22583580—22no—
13#22583578—21no—
14#22583535—22no—
15#22583532—22no—
16#22583521C2211yescharacteristic, complemented
17#124×C283C2211yescharacteristic
18#97×C283C2212yescomplemented
19C23×D283—21no—
20C2×Dic566—21no—
21C2×Dic566D412yes—
22C2×D1132—21no—
23C2×D1132D412yes—
24C2×C22⋊D283—22no—
25#17985382—42no—
26#17985380—42no—
27C2×C283⋊C8—21no—
28C4⋊Dic283—22no—
29C22⋊Dic283—21no—
30(C2×D283)⋊C4—22no—
31(C2×D283)⋊C4—22no—
32(C2×D283)⋊C4—22no—
33C4×Dic283—21no—
34C2×Q8×C283D411yescharacteristic
35C2×D4×C283D411yescharacteristic
36C2×C8×C283—22no—
37C22⋊C4×C283—22no—
38C42×C283C2311yescharacteristic
39C22×D283—22no—
40C22×D283—81no—
41C22×D283—82no—
42Dic566—42no—
43Dic566—41no—
44Dic566—41no—
45D1132—42no—
46D1132—81no—
47C22⋊D283—82no—
48C2×Dic283—22no—
49C2×Dic283—41no—
50C283⋊C8—22no—
51C23×C283—21no—
52Q8×C283—41no—
53D4×C283—41no—
54C2×C4×C283—21no—
55C2×C4×C283C2×D412yes—
56C2×C4×C283C2×D411yescharacteristic
57C8×C283—42no—
58C22×C283—41no—
59C22×C283—42no—
60C22×C283#12011yescharacteristic
61C2×D283—42no—
62C2×D283—42no—
63C2×D283—42no—
64C2×D283—84no—
65C4×C283—21no—
66C4×C283—22no—
67C4×C283—41no—
68Dic283—42no—
69Dic283—81no—
70C2×C283—81no—
71C2×C283#45311yescharacteristic
72C2×C283#45512yes—
73D283—82no—
74C283#165911yescharacteristic, complemented, Sylow
75#1659—2831nomaximal, Sylow
76H(3,𝔽4)—2831no—
77#497—2831no—
78#489—2832no—
79#329—2832no—
80#329D28311yescharacteristic, complemented
81C2×Q16—5661no—
82C2×SD16—5661no—
83C4⋊Q8—2831no—
84#124—2832no—
85#124—5662no—
86#124—5661no—
87#124C2×D28311yescharacteristic
88#120—5662no—
89#120—5661no—
90#103—5662no—
91#102—5662no—
92#97—2831no—
93#97C2×D28312yescomplemented
94C24—5661no—
95C2×Q8—2832no—
96C2×Q8—5661no—
97C2×Q8D113211yescharacteristic
98C2×D4—2832no—
99C2×D4—5661no—
100C2×D4—5662no—
101C2×D4D113211yescharacteristic
102Q16—11322no—
103SD16—11322no—
104C2×C8—22no—
105C2×C8—5661no—
106C4⋊C4—5662no—
107C22⋊C4—22no—
108C22⋊C4—5662no—
109C22⋊C4—5661no—
110C22⋊C4—5662no—
111C22⋊C4—5662no—
112C42—5661no—
113C42C22×D28311yescharacteristic
114C23—21no—
115C23—5662no—
116C23—22641no—
117C23—22642no—
118Q8—41no—
119Q8—11322no—
120Q8—11321no—
121Q8—11321no—
122D4—41no—
123D4—11322no—
124D4—22641no—
125D4—22642no—
126C2×C4—21no—
127C2×C4—5662no—
128C2×C4—11321no—
129C2×C4D4×D28312yes—
130C2×C4C2×D113211yescharacteristic
131C8—42no—
132C8—5662no—
133C22—41no—
134C22—42no—
135C22—11322no—
136C22—11322no—
137C22—11322no—
138C22—22644no—
139C22#2258358011yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—11322no—
144C4—22641no—
145C2—81no—
146C2—22642no—
147C2#3182663511yescharacteristic
148C2#3182664412yes—
subgroupG/H
centerC22#22583580
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#329×C283C2
socleC22×C283#120
prime radical#31827207C2
hypercenter#329D283
hypocenterC283#1659
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #31826860C2semidirect
3. #31826825C2semidirect
4. #31826204C2semidirect
5. #329×C283C2semidirect
8. #22583610C22semidirect
16. #22583521C22semidirect
18. #97×C283C22semidirect
74. C283#1659semidirect
80. #329D283semidirect
93. #97C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272072
#318268602
#318268252
#318262042
#329×C2832
#1659283