Finite Group Databasebeta

← #49119985

Subgroups — #49119985

Order 36224.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C4⋊Q8×D283C211yesmaximal, characteristic
2#31826521C212yesmaximal
3#31826214C211yesmaximal, characteristic, complemented
4#31826213C212yesmaximal, complemented
5#333×C283C211yesmaximal, characteristic, complemented
6C2×Q8×D283C412yes—
7#22583610C2211yescharacteristic
8#22583609—22no—
9#22583608C412yes—
10C4⋊C4×D283—22no—
11C42×D283C2211yescharacteristic
12C4⋊(C283⋊C8)C2212yes—
13(C2×D283)⋊C8—22no—
14#22583520—22no—
15C4⋊Q8×C283C2211yescharacteristic
16C4⋊C8×C283C2212yes—
17C2×Dic566—22no—
18C2×Dic566C2×C412yes—
19Q8×D283—44no—
20C2×C4×D283—22no—
21C2×C4×D283C2×C411yescharacteristic
22C2×C4×D283D412yes—
23C2×C283⋊C8—22no—
24C4⋊Dic283—22no—
25Dic283⋊C4—24no—
26C4×Dic283C2×C411yescharacteristic
27C4×Dic283D412yes—
28C2×Q8×C283C2×C412yes—
29C2×C8×C283—22no—
30C4⋊C4×C283—22no—
31C42×C283C2311yescharacteristic
32C22×D283C22⋊C411yescharacteristic
33Dic566—44no—
34Dic566—42no—
35C2×Dic283—22no—
36C2×Dic283C22⋊C412yes—
37C2×Dic283C22⋊C411yescharacteristic
38C4×D283—22no—
39C4×D283—22no—
40C4×D283—22no—
41C4×D283—44no—
42C283⋊C8—42no—
43Q8×C283—42no—
44C2×C4×C283—22no—
45C2×C4×C283C22×C411yescharacteristic
46C2×C4×C283C2×D412yes—
47C8×C283—42no—
48C22×C283C2×C22⋊C411yescharacteristic
49C2×D283—22no—
50C2×D283#10012yes—
51C4×C283—21no—
52C4×C283—42no—
53C4×C283#13712yes—
54C4×C283#13712yes—
55Dic283—21no—
56Dic283—22no—
57Dic283—42no—
58C2×C283C2×#10011yescharacteristic
59C2×C283#41912yes—
60D283—22no—
61C283#155811yescharacteristic, complemented, Sylow
62#1558—2831nomaximal, Sylow
63C2×C4⋊Q8—2831no—
64#432—2832no—
65#333—2831no—
66#333—2832no—
67#333D28311yescharacteristic, complemented
68C22×Q8—2832no—
69C4⋊Q8—2832no—
70C4⋊Q8—2831no—
71C4⋊Q8—5662no—
72C4⋊Q8C2×D28311yescharacteristic
73C2×C4⋊C4—5662no—
74C2×C42—2831no—
75C4⋊C8—2832no—
76C4⋊C8C2×D28312yes—
77C22⋊C8—5662no—
78#97—5662no—
79C2×Q8—2832no—
80C2×Q8—5662no—
81C2×Q8—11324no—
82C2×Q8C4×D28312yes—
83C22×C4—2831no—
84C22×C4—2832no—
85C22×C4—5662no—
86C2×C8—22no—
87C2×C8—5662no—
88C4⋊C4—22no—
89C4⋊C4—5664no—
90C4⋊C4—5662no—
91C42—2831no—
92C42—2832no—
93C42C22×D28311yescharacteristic
94C23—2831no—
95Q8—42no—
96Q8—11322no—
97Q8—11324no—
98C2×C4—22no—
99C2×C4—2832no—
100C2×C4—2831no—
101C2×C4—5662no—
102C2×C4—5662no—
103C2×C4—5662no—
104C2×C4—5662no—
105C2×C4—11324no—
106C2×C4C2×C4×D28311yescharacteristic
107C2×C4D4×D28312yes—
108C8—42no—
109C8—11322no—
110C22—2832no—
111C22—5662no—
112C22C22⋊C4×D28311yescharacteristic
113C4—21no—
114C4—42no—
115C4—5661no—
116C4—5662no—
117C4—11322no—
118C4#2258363912yes—
119C4#2258364212yes—
120C2—5662no—
121C2#100×D28311yescharacteristic
122C2#3182647212yes—
subgroupG/H
centerC22C22⋊C4×D283
commutatorC2×C4×C283C22×C4
FrattiniC42C22×D283
Fitting#333×C283C2
socleC22×C283C2×C22⋊C4
prime radicalC22×D283C22⋊C4
cosocleC42×C283C23
hypercenter#333D283
hypocenterC283#1558
LCSG ▷ C2×C4×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
3. #31826214C2semidirect
4. #31826213C2semidirect
5. #333×C283C2semidirect
61. C283#1558semidirect
67. #333D283semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D2832
#318265212
#318262142
#318262132
#333×C2832
#1558283