Finite Group Databasebeta

← #49123857

Subgroups — #49123857

Order 36224.

672 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C22×C22⋊Dic283C211yesmaximal, characteristic
2#31826359C214yesmaximal, complemented
3#31826353C214yesmaximal, complemented
4#31826323C212yesmaximal, complemented
5#31826321C211yesmaximal, characteristic
6C22⋊C4×Dic283C212yesmaximal, complemented
7#379×C283C211yesmaximal, characteristic
8C23×Dic283C2211yescharacteristic
9C23×Dic283C2211yescharacteristic
10C2×Dic283⋊C4C2214yescomplemented
11C2×C22⋊Dic283C418yescomplemented
12C2×C22⋊Dic283C2212yes—
13C2×C22⋊Dic283C2212yes—
14C2×C22⋊Dic283C2212yes—
15C2×C22⋊Dic283C2212yes—
16C2×C22⋊Dic283C2212yes—
17C2×C4×Dic283C2214yescomplemented
18C283⋊#95C2214yescomplemented
19C283⋊#95C2214yescomplemented
20C25×C283C2211yescharacteristic
21C2×C22⋊C4×C283C2212yes—
22C2×C22⋊C4×C283C2212yes—
23C2×C22⋊C4×C283C2212yes—
24C22×Dic283—42no—
25C22×Dic283—42no—
26C22×Dic283C2×C414yes—
27C22×Dic283C2×C414yes—
28C22×Dic283C2314yes—
29C22×Dic283C2312yes—
30C22×Dic283C2312yes—
31Dic283⋊C4—44no—
32C22⋊Dic283—24no—
33C22⋊Dic283—42no—
34C22⋊Dic283—42no—
35C22⋊Dic283C2×C418yes—
36C22⋊Dic283C2×C418yes—
37C4×Dic283—44no—
38C24×C283—24no—
39C24×C283—42no—
40C24×C283—42no—
41C24×C283C2×C414yes—
42C24×C283C2311yescharacteristic
43C24×C283C2311yescharacteristic
44C24×C283C2311yescharacteristic
45C22×C4×C283C2314yes—
46C22⋊C4×C283—24no—
47C22⋊C4×C283—42no—
48C22⋊C4×C283—42no—
49C2×Dic283—24no—
50C2×Dic283—24no—
51C2×Dic283—24no—
52C2×Dic283—24no—
53C2×Dic283—24no—
54C2×Dic283—44no—
55C2×Dic283—44no—
56C2×Dic283C22×C414yes—
57C2×Dic283C22×C414yes—
58C23×C283—22no—
59C23×C283—22no—
60C23×C283—28no—
61C23×C283—22no—
62C23×C283—22no—
63C23×C283—22no—
64C23×C283—22no—
65C23×C283—22no—
66C23×C283—22no—
67C23×C283—44no—
68C23×C283—44no—
69C23×C283—416no—
70C23×C283C22×C412yes—
71C23×C283C22×C412yes—
72C23×C283C22×C412yes—
73C23×C283#4114yes—
74C23×C283C2411yescharacteristic
75C2×C4×C283—24no—
76C2×C4×C283—24no—
77C2×C4×C283—24no—
78C22×C283—22no—
79C22×C283—22no—
80C22×C283—28no—
81C22×C283—22no—
82C22×C283—22no—
83C22×C283—22no—
84C22×C283—22no—
85C22×C283—22no—
86C22×C283—22no—
87C22×C283—416no—
88C22×C283—44no—
89C22×C283—44no—
90C22×C283#11714yes—
91C22×C283C23×C411yescharacteristic
92C22×C283C2×#4112yes—
93C22×C283H(5,𝔽2)12yes—
94C22×C283H(5,𝔽2)12yes—
95C4×C283—44no—
96Dic283—24no—
97Dic283—24no—
98Dic283—44no—
99C2×C283—24no—
100C2×C283—42no—
101C2×C283—42no—
102C2×C283C2×#11711yescharacteristic
103C2×C283#51811yescharacteristic
104C2×C283#51811yescharacteristic
105C2×C283#53814yes—
106C283#231211yescharacteristic, complemented, Sylow
107#2312—2831nomaximal, Sylow
108C22×C22⋊C4—2831no—
109#388—2834no—
110#388—2834no—
111#380—2832no—
112#379—2831no—
113#379D28311yescharacteristic
114C4×C22⋊C4—2832no—
115C25C2×D28311yescharacteristic
116C23×C4—2831no—
117C23×C4—2831no—
118C2×C4⋊C4—2834no—
119C2×C22⋊C4—2838no—
120C2×C22⋊C4—2832no—
121C2×C22⋊C4—2832no—
122C2×C22⋊C4—2832no—
123C2×C22⋊C4—2832no—
124C2×C22⋊C4—2832no—
125C2×C22⋊C4C2×D28312yes—
126C2×C22⋊C4C2×D28312yes—
127C2×C22⋊C4C2×D28312yes—
128C2×C42—2834no—
129#95—2834no—
130#95—2834no—
131C24—24no—
132C24—42no—
133C24—42no—
134C24C4×D28314yes—
135C24C22×D28311yescharacteristic
136C24C22×D28311yescharacteristic
137C24C22×D28311yescharacteristic
138C22×C4—2832no—
139C22×C4—2832no—
140C22×C4—2834no—
141C22×C4—2834no—
142C22×C4—2834no—
143C22×C4—11322no—
144C22×C4—11322no—
145C22×C4C22×D28314yes—
146C4⋊C4—11324no—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—2838no—
151C22⋊C4—2838no—
152C22⋊C4—5664no—
153C22⋊C4—11322no—
154C22⋊C4—11322no—
155C42—11324no—
156C23—22no—
157C23—22no—
158C23—28no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—44no—
166C23—44no—
167C23—416no—
168C23C2×C4×D28312yes—
169C23C2×C4×D28312yes—
170C23C2×C4×D28312yes—
171C23#1798539114yes—
172C23C23×D28311yescharacteristic
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—2834no—
177C2×C4—2834no—
178C2×C4—5664no—
179C2×C4—5664no—
180C2×C4—5664no—
181C2×C4—5664no—
182C2×C4—5664no—
183C2×C4—11324no—
184C2×C4—11324no—
185C22—28no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—416no—
195C22—44no—
196C22—44no—
197C22#2258356614yes—
198C22C22×C4×D28311yescharacteristic
199C22C2×#1798539112yes—
200C22#2258365512yes—
201C22#2258365512yes—
202C4—44no—
203C4—5664no—
204C4—5664no—
205C4—11324no—
206C2—24no—
207C2—42no—
208C2—42no—
209C2C2×#2258356611yescharacteristic
210C2#3182693811yescharacteristic
211C2#3182693811yescharacteristic
212C2#3182705014yes—
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#379×C283C2
socle, cosocleC23×C283C24
prime radicalC25×C283C22
hypercenter#379D283
hypocenterC283#2312
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #31826359C2semidirect
3. #31826353C2semidirect
4. #31826323C2semidirect
6. C22⋊C4×Dic283C2semidirect
10. C2×Dic283⋊C4C22semidirect
11. C2×C22⋊Dic283C4semidirect
17. C2×C4×Dic283C22semidirect
18. C283⋊#95C22semidirect
19. C283⋊#95C22semidirect
106. C283#2312semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Dic2832
#318263592
#318263532
#318263232
#318263212
C22⋊C4×Dic2832
#379×C2832
#2312283