Finite Group Databasebeta

← #49120345

Subgroups — #49120345

Order 36224.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#31827181C211yesmaximal, characteristic, complemented
2#31826839C211yesmaximal, characteristic, complemented
3#31826661C212yesmaximal, complemented
4#31826192C212yesmaximal, complemented
5#333×C283C211yesmaximal, characteristic, complemented
6C2×#17985379—22no—
7C2×D2264—22no—
8#22583607C2211yescharacteristic
9#22583599—22no—
10#22583585—22no—
11C4×D1132C2212yes—
12#22583568—22no—
13#22583532—22no—
14C4×C283⋊C8C2211yescharacteristic
15C4⋊Q8×C283C2211yescharacteristic
16C4⋊C8×C283C2212yes—
17C2×D1132—22no—
18C2×D1132—22no—
19C2×D1132—21no—
20C2×D1132D412yes—
21C2×C4×D283—22no—
22#17985379—44no—
23D2264—44no—
24C2×C283⋊C8—21no—
25C4⋊Dic283D412yes—
26(C2×D283)⋊C4—24no—
27(C2×D283)⋊C4—22no—
28C4×Dic283—22no—
29C2×Q8×C283D412yes—
30C2×C8×C283—22no—
31C4⋊C4×C283—22no—
32C42×C283C2311yescharacteristic
33C22×D283—22no—
34C22×D283—41no—
35D1132—22no—
36D1132—22no—
37D1132—42no—
38D1132—42no—
39D1132—42no—
40D1132—82no—
41C2×Dic283—22no—
42C4×D283—42no—
43C4×D283—42no—
44C283⋊C8—22no—
45Q8×C283—42no—
46C2×C4×C283—22no—
47C2×C4×C283C2×D412yes—
48C2×C4×C283C2×D411yescharacteristic
49C8×C283—42no—
50C22×C283#12011yescharacteristic
51C2×D283—42no—
52C2×D283—42no—
53C2×D283—42no—
54C2×D283—82no—
55C2×D283—81no—
56C4×C283—21no—
57C4×C283—42no—
58C4×C283#13512yes—
59C4×C283AGL(1,ℤ/8)12yes—
60Dic283—42no—
61C2×C283#44912yes—
62C2×C283#45311yescharacteristic
63D283—82no—
64D283—161no—
65C283#162411yescharacteristic, complemented, Sylow
66#1624—2831nomaximal, Sylow
67#555—2831no—
68#492—2831no—
69#459—2832no—
70#333D28311yescharacteristic, complemented
71D4⋊C8—2832no—
72C2×SD16—5662no—
73C2×D8—5662no—
74C4⋊Q8C2×D28311yescharacteristic
75#127—2831no—
76#124—5662no—
77#121—5662no—
78C4×D4—2832no—
79#117—5662no—
80C4⋊C8C2×D28312yes—
81#102—5662no—
82C4×C8—2831no—
83C2×Q8D113212yes—
84C2×D4—2832no—
85C2×D4—5662no—
86C2×D4—5662no—
87C2×D4—5661no—
88C22×C4—5662no—
89SD16—11324no—
90D8—11324no—
91C2×C8—22no—
92C2×C8—5661no—
93C4⋊C4—22no—
94C4⋊C4—2832no—
95C22⋊C4—5664no—
96C22⋊C4—5662no—
97C42—5662no—
98C42C22×D28311yescharacteristic
99C23—5662no—
100C23—11321no—
101Q8—42no—
102D4—5662no—
103D4—5662no—
104D4—11322no—
105D4—11322no—
106D4—11322no—
107D4—22642no—
108C2×C4—22no—
109C2×C4—5662no—
110C2×C4—11322no—
111C2×C4—11322no—
112C2×C4D4×D28312yes—
113C2×C4C2×D113211yescharacteristic
114C8—42no—
115C8—5662no—
116C22—11322no—
117C22—11322no—
118C22—11322no—
119C22—22642no—
120C22—22641no—
121C22#2258358011yescharacteristic
122C4—21no—
123C4—42no—
124C4—11322no—
125C4#2258363312yes—
126C4#2258363712yes—
127C2—45281no—
128C2—22642no—
129C2#3182661312yes—
130C2#3182663511yescharacteristic
subgroupG/H
centerC22#22583580
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#333×C283C2
socleC22×C283#120
hypercenter#333D283
hypocenterC283#1624
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #31827181C2semidirect
2. #31826839C2semidirect
3. #31826661C2semidirect
4. #31826192C2semidirect
5. #333×C283C2semidirect
65. C283#1624semidirect
70. #333D283semidirect

Maximal subgroups

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

subgroupindex
#318271812
#318268392
#318266612
#318261922
#333×C2832
#1624283