Finite Group Databasebeta

← #49120218

Subgroups — #49120218

Order 36224.

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

normal subgroups only

This group has 256 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#31826620C212yesmaximal, complemented
2#31826377C211yesmaximal, characteristic
3#31826202C212yesmaximal, complemented
4#31826189C211yesmaximal, characteristic
5#543×C283C211yesmaximal, characteristic
6C22×Dic566C2211yescharacteristic
7C2×#17985382—22no—
8C2×C22⋊Dic283—21no—
9#22583533—22no—
10C22⋊(C283⋊C8)C2212yes—
11C283⋊#95—21no—
12C283⋊#95C2211yescharacteristic
13C22⋊Q8×C283—22no—
14C22⋊Q8×C283—21no—
15C22⋊Q8×C283—22no—
16C22⋊Q8×C283C2212yes—
17C2×C22⋊C4×C283—21no—
18C2×C22⋊C4×C283C2211yescharacteristic
19C2×Dic566—22no—
20C2×Dic566—42no—
21C22×Dic283—21no—
22C22×Dic283—21no—
23#17985382—44no—
24C2×C283⋊C8—22no—
25C22⋊Dic283—42no—
26C24×C283D411yescharacteristic
27C2×Q8×C283—21no—
28C2×Q8×C283D412yes—
29C22×C4×C283—21no—
30C22×C4×C283—21no—
31C22×C4×C283D411yescharacteristic
32C22×C4×C283C2311yescharacteristic
33C4⋊C4×C283—21no—
34C4⋊C4×C283—22no—
35C4⋊C4×C283—22no—
36C4⋊C4×C283D412yes—
37C22⋊C4×C283—22no—
38C22⋊C4×C283—22no—
39C22⋊C4×C283—22no—
40Dic566—24no—
41Dic566—42no—
42C2×Dic283—22no—
43C2×Dic283—41no—
44C2×Dic283—41no—
45C2×Dic283—41no—
46C2×Dic283—41no—
47C2×Dic283—41no—
48C283⋊C8—42no—
49C23×C283—21no—
50C23×C283—21no—
51C23×C283—41no—
52C23×C283—41no—
53C23×C283Q1612yescomplemented
54C23×C283C2×D411yescharacteristic
55Q8×C283—42no—
56Q8×C283—81no—
57C2×C4×C283—22no—
58C2×C4×C283—21no—
59C2×C4×C283—22no—
60C2×C4×C283—41no—
61C2×C4×C283—41no—
62C2×C4×C283—81no—
63C2×C4×C283—81no—
64C2×C4×C283C2×D412yes—
65C22×C283—21no—
66C22×C283—21no—
67C22×C283—22no—
68C22×C283—42no—
69C22×C283—41no—
70C22×C283—41no—
71C22×C283—81no—
72C22×C283#12011yescharacteristic
73C22×C283C2×Q1611yescharacteristic
74C22×C283#13711yescharacteristic
75C4×C283—22no—
76C4×C283—41no—
77C4×C283—42no—
78C4×C283—42no—
79Dic283—42no—
80Dic283—81no—
81C2×C283—21no—
82C2×C283—21no—
83C2×C283—42no—
84C2×C283#45111yescharacteristic
85C2×C283#45511yescharacteristic
86C2×C283B(4,𝔽2)11yescharacteristic
87C283#160211yescharacteristic, complemented, Sylow
88#1602—2831nomaximal, Sylow
89#543D28311yescharacteristic
90#451—2832no—
91#393—2831no—
92#328—2832no—
93#323—2831no—
94C22×Q8—2831no—
95C2×Q16—5662no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8C2×D28312yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—5661no—
102C2×C22⋊C4C2×D28311yescharacteristic
103#103—5662no—
104C22⋊C8—2832no—
105#95—2831no—
106#95—5661no—
107C24C22⋊D28311yescharacteristic
108C2×Q8—21no—
109C2×Q8—5662no—
110C2×Q8—11322no—
111C2×Q8C22⋊D28312yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—5661no—
115C22×C4—5661no—
116C22×C4C22⋊D28311yescharacteristic
117C22×C4C22×D28311yescharacteristic
118Q16—11324no—
119C2×C8—5662no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4C22⋊D28312yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—11322no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1798538212yescomplemented
133C23C2×C22⋊D28311yescharacteristic
134Q8—42no—
135Q8—81no—
136Q8—5664no—
137Q8—11322no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—5662no—
146C2×C4—11321no—
147C2×C4—11321no—
148C2×C4—11321no—
149C2×C4—11321no—
150C2×C4—11321no—
151C2×C4C2×C22⋊D28312yes—
152C8—11322no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#2258357911yescharacteristic
161C22C2×#1798538211yescharacteristic
162C22#2258364111yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—11322no—
168C4—22641no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#3182662111yescharacteristic
173C2#3182664311yescharacteristic
174C2#3182665211yescharacteristic
subgroupG/H
centerC22#22583579
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#543×C283C2
socleC22×C283#120
prime radicalC24×C283D4
hypercenter#543D283
hypocenterC283#1602
LCSG ▷ C22×C4×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #31826620C2semidirect
3. #31826202C2semidirect
53. C23×C283Q16semidirect
87. C283#1602semidirect
132. C23#17985382semidirect

Maximal subgroups

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

subgroupindex
#318266202
#318263772
#318262022
#318261892
#543×C2832
#1602283