Finite Group Databasebeta

← #49120378

Subgroups — #49120378

Order 36224.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#31827100C211yesmaximal, characteristic
2#31826837C211yesmaximal, characteristic, complemented
3#31826756C212yesmaximal, complemented
4#31826192C212yesmaximal, complemented
5#331×C283C211yesmaximal, characteristic, complemented
6C2×D4×D283—22no—
7C2×#17985380—22no—
8#22583610C2211yescharacteristic, complemented
9#22583581—22no—
10#22583578—22no—
11C4×D1132C2212yes—
12#22583542—22no—
13#22583532—22no—
14C4×C283⋊C8C2211yescharacteristic, complemented
15#127×C283C2211yescharacteristic
16C4⋊C8×C283C2212yescomplemented
17C23×D283—22no—
18C2×Dic566—21no—
19C2×D1132D412yes—
20C2×C22⋊D283—24no—
21D4×D283—44no—
22D4×D283—44no—
23C2×C4×D283—22no—
24#17985380—44no—
25C2×C283⋊C8—21no—
26C4⋊Dic283—22no—
27C4⋊Dic283D412yes—
28C22⋊Dic283—22no—
29(C2×D283)⋊C4—22no—
30C2×D4×C283—22no—
31C2×D4×C283D412yes—
32C2×C8×C283—22no—
33C42×C283C2311yescharacteristic
34C22×D283—22no—
35C22×D283—48no—
36C22×D283—82no—
37Dic566—42no—
38D1132—22no—
39D1132SD1614yescomplemented
40C22⋊D283—48no—
41C2×Dic283—22no—
42C2×Dic283—41no—
43C4×D283—42no—
44C4×D283—42no—
45C283⋊C8—22no—
46C23×C283—22no—
47D4×C283—42no—
48D4×C283—42no—
49D4×C283—42no—
50C2×C4×C283C2×D412yes—
51C2×C4×C283C2×D411yescharacteristic
52C8×C283—42no—
53C22×C283—44no—
54C22×C283—42no—
55C22×C283#12011yescharacteristic
56C2×D283—24no—
57C2×D283—42no—
58C2×D283—42no—
59C2×D283—88no—
60C4×C283—21no—
61C4×C283C2×SD1612yes—
62C4×C283AGL(1,ℤ/8)12yes—
63Dic283—42no—
64Dic283—81no—
65C2×C283—82no—
66C2×C283#45012yes—
67C2×C283#45311yescharacteristic
68D283—44no—
69C283#162911yescharacteristic, complemented, Sylow
70#1629—2831nomaximal, Sylow
71D42—2831no—
72#492—2831no—
73#476—2832no—
74#331D28311yescharacteristic, complemented
75D4⋊C8—2832no—
76C22×D4—5662no—
77C2×SD16—5662no—
78C4⋊Q8—2831no—
79#127C2×D28311yescharacteristic
80#121—5662no—
81#120—5662no—
82C4×D4—2832no—
83#106—5662no—
84C4⋊C8C2×D28312yescomplemented
85#102—5662no—
86C4×C8—2831no—
87C24—5662no—
88C2×Q8—5661no—
89C2×D4—22no—
90C2×D4—2832no—
91C2×D4—5664no—
92C2×D4—11324no—
93C2×D4—11324no—
94C2×D4D113212yes—
95C22×C4—5662no—
96SD16—11324no—
97C2×C8—22no—
98C2×C8—5661no—
99C4⋊C4—2832no—
100C4⋊C4—5662no—
101C22⋊C4—5662no—
102C22⋊C4—5662no—
103C42C22×D28311yescharacteristic
104C23—22no—
105C23—5662no—
106C23—11328no—
107C23—22642no—
108Q8—11322no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—2834no—
113D4—5662no—
114D4—11328no—
115C2×C4—5662no—
116C2×C4—11322no—
117C2×C4—11322no—
118C2×C4—11321no—
119C2×C4D4×D28312yes—
120C2×C4C2×D113211yescharacteristic
121C8—42no—
122C8—5662no—
123C22—44no—
124C22—42no—
125C22—5664no—
126C22—11322no—
127C22—11322no—
128C22—22648no—
129C22#2258358011yescharacteristic
130C4—21no—
131C4—11322no—
132C4—22641no—
133C4SD16×D28312yes—
134C4#2258363412yes—
135C2—82no—
136C2—11324no—
137C2#3182661712yes—
138C2#3182663511yescharacteristic
subgroupG/H
centerC22#22583580
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#331×C283C2
socleC22×C283#120
prime radical#31827100C2
hypercenter#331D283
hypocenterC283#1629
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #31826837C2semidirect
3. #31826756C2semidirect
4. #31826192C2semidirect
5. #331×C283C2semidirect
8. #22583610C22semidirect
14. C4×C283⋊C8C22semidirect
16. C4⋊C8×C283C22semidirect
39. D1132SD16semidirect
69. C283#1629semidirect
74. #331D283semidirect
84. C4⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271002
#318268372
#318267562
#318261922
#331×C2832
#1629283