Finite Group Databasebeta

← #49123885

Subgroups — #49123885

Order 36224.

600 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 269 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
1C2×#22583569C211yesmaximal, characteristic
2C2×#22583569C211yesmaximal, characteristic
3#31826361C214yesmaximal, complemented
4#31826338C214yesmaximal
5#31826334C212yesmaximal, complemented
6#31826330C212yesmaximal, complemented
7C22×C4⋊C4×C283C211yesmaximal, characteristic
8#22583569—24no—
9#22583569C418yescomplemented
10C2×C4⋊Dic283C2211yescharacteristic
11C2×C4⋊Dic283C2211yescharacteristic
12C2×C4⋊Dic283C2212yes—
13C2×Dic283⋊C4C2218yes—
14C2×C22⋊Dic283C2212yes—
15C2×C22⋊Dic283C2212yes—
16C2×C4×Dic283C2212yes—
17C2×C4×Dic283C2212yes—
18C283⋊#95C2218yes—
19C23×C4×C283C2212yes—
20C23×C4×C283C2211yescharacteristic
21C2×C4⋊C4×C283—24no—
22C2×C4⋊C4×C283C2212yes—
23C2×C4⋊C4×C283C2212yes—
24C22×Dic283C2312yes—
25C22×Dic283C2312yes—
26C22×Dic283C2314yes—
27C4⋊Dic283—22no—
28C4⋊Dic283—22no—
29C4⋊Dic283C2×C418yescomplemented
30Dic283⋊C4—48no—
31C22⋊Dic283—24no—
32C22⋊Dic283C2×C418yes—
33C4×Dic283—24no—
34C4×Dic283C2×C418yescomplemented
35C24×C283C2311yescharacteristic
36C22×C4×C283—24no—
37C22×C4×C283—44no—
38C22×C4×C283C2×C414yes—
39C22×C4×C283D414yes—
40C22×C4×C283Q814yes—
41C22×C4×C283C2314yes—
42C22×C4×C283C2311yescharacteristic
43C22×C4×C283C2311yescharacteristic
44C4⋊C4×C283—24no—
45C4⋊C4×C283—24no—
46C2×Dic283—24no—
47C2×Dic283—24no—
48C2×Dic283—22no—
49C2×Dic283—22no—
50C2×Dic283—24no—
51C2×Dic283—24no—
52C2×Dic283C22×C418yes—
53C23×C283—22no—
54C23×C283—22no—
55C23×C283C22×C412yes—
56C23×C283C2×D412yes—
57C23×C283C2×Q812yes—
58C23×C283C2411yescharacteristic
59C2×C4×C283—24no—
60C2×C4×C283—28no—
61C2×C4×C283—48no—
62C2×C4×C283C4⋊C4116yescomplemented
63C2×C4×C283C22×C412yes—
64C2×C4×C283C22×C412yes—
65C2×C4×C283C2×D412yes—
66C2×C4×C283C2×D412yes—
67C2×C4×C283C2×Q812yes—
68C2×C4×C283C2×Q812yes—
69C22×C283—24no—
70C22×C283—22no—
71C22×C283—22no—
72C22×C283—24no—
73C22×C283C2×C4⋊C414yes—
74C22×C283C23×C411yescharacteristic
75C22×C283C22×D411yescharacteristic
76C22×C283C22×Q811yescharacteristic
77C22×C283H(5,𝔽2)12yes—
78C22×C2832−1+412yes—
79C4×C283—44no—
80C4×C283C2×C4⋊C414yes—
81C4×C283C2×C4⋊C414yes—
82Dic283—28no—
83Dic283—42no—
84Dic283—42no—
85C2×C283—24no—
86C2×C283C22×C4⋊C411yescharacteristic
87C2×C283#52012yes—
88C2×C283#53511yescharacteristic
89C2×C283#53611yescharacteristic
90C2×C283#54412yes—
91C283#231711yescharacteristic, complemented, Sylow
92#2317—2831nomaximal, Sylow
93C2×#117—2831no—
94C2×#117—2831no—
95C22×C4⋊C4D28311yescharacteristic
96#389—2834no—
97#385—2834no—
98#384—2832no—
99#382—2832no—
100C23×C4C2×D28312yes—
101C23×C4C2×D28311yescharacteristic
102#117—2838no—
103#117—5664no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—2832no—
106C2×C4⋊C4—2831no—
107C2×C4⋊C4—2831no—
108C2×C4⋊C4—2838no—
109C2×C4⋊C4C2×D28312yes—
110C2×C4⋊C4C2×D28312yes—
111C2×C22⋊C4—2832no—
112C2×C22⋊C4—2832no—
113C2×C42—2832no—
114C2×C42—2832no—
115#95—2838no—
116C24C22×D28311yescharacteristic
117C22×C4—24no—
118C22×C4—44no—
119C22×C4—2834no—
120C22×C4—2832no—
121C22×C4—2832no—
122C22×C4C4×D28314yes—
123C22×C4C22⋊D28314yes—
124C22×C4Dic56614yes—
125C22×C4C22×D28314yes—
126C22×C4C22×D28311yescharacteristic
127C22×C4C22×D28311yescharacteristic
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—2838no—
131C4⋊C4—5662no—
132C4⋊C4—5662no—
133C4⋊C4—11328no—
134C22⋊C4—2838no—
135C22⋊C4—5664no—
136C42—2838no—
137C42—5664no—
138C23—22no—
139C23—22no—
140C23C2×C4×D28312yes—
141C23C2×C22⋊D28312yes—
142C23C2×Dic56612yes—
143C23C23×D28311yescharacteristic
144C2×C4—24no—
145C2×C4—28no—
146C2×C4—48no—
147C2×C4—2838no—
148C2×C4—5664no—
149C2×C4—5664no—
150C2×C4—5662no—
151C2×C4—5662no—
152C2×C4—5664no—
153C2×C4—5664no—
154C2×C4Dic283⋊C4116yescomplemented
155C2×C4C2×C4×D28312yes—
156C2×C4C2×C4×D28312yes—
157C2×C4C2×C22⋊D28312yes—
158C2×C4C2×C22⋊D28312yes—
159C2×C4C2×Dic56612yes—
160C2×C4C2×Dic56612yes—
161C22—24no—
162C22—22no—
163C22—22no—
164C22—24no—
165C22C2×Dic283⋊C414yes—
166C22C22×C4×D28311yescharacteristic
167C22C22×C22⋊D28311yescharacteristic
168C22C22×Dic56611yescharacteristic
169C22#2258365512yes—
170C22#2258365812yes—
171C4—44no—
172C4—5668no—
173C4—11322no—
174C4—11322no—
175C4C2×Dic283⋊C414yes—
176C4C2×Dic283⋊C414yes—
177C2—24no—
178C2C22×Dic283⋊C411yescharacteristic
179C2#3182694612yes—
180C2#3182703211yescharacteristic
181C2#3182703911yescharacteristic
182C2#3182709612yes—
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
FittingC22×C4⋊C4×C283C2
socle, cosocleC23×C283C24
prime radicalC24×C283C23
hypercenterC22×C4⋊C4D283
hypocenterC283#2317
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #31826361C2semidirect
5. #31826334C2semidirect
6. #31826330C2semidirect
9. #22583569C4semidirect
29. C4⋊Dic283C2×C4semidirect
34. C4×Dic283C2×C4semidirect
62. C2×C4×C283C4⋊C4semidirect
91. C283#2317semidirect
154. C2×C4Dic283⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#225835692
#318263612
#318263382
#318263342
#318263302
C22×C4⋊C4×C2832
#2317283