Finite Group Databasebeta

← #49130498

Subgroups — #49130498

Order 36224.

738 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×Dic566C211yesmaximal, characteristic, complemented
2C2×#22583643C212yesmaximal, complemented
3#31826976C211yesmaximal, characteristic
4#31826623C214yesmaximal, complemented
5#31826606C214yesmaximal, complemented
6#31826475C212yesmaximal, complemented
7#407×C283C211yesmaximal, characteristic, complemented
8C2×#17985392C2211yescharacteristic
9C22×Dic566—28no—
10C22×Dic566—22no—
11C22×Dic566—21no—
12C22×Dic566C2214yes—
13C22×Dic566C2212yes—
14C23×Dic283—21no—
15#22583643—28no—
16C2×Dic1132C2214yescomplemented
17C2×#17985381C2214yescomplemented
18#22583593—21no—
19#22583589C2212yes—
20#22583584C2212yes—
21#22583579—21no—
22C4×C22⋊D283—22no—
23#22583569C2211yescharacteristic
24#22583535C2218yescomplemented
25C23×C4×C283C2211yescharacteristic
26C2×M16×C283C2212yes—
27C22⋊C8×C283C2214yes—
28#17985392—24no—
29C2×Dic566—28no—
30C2×Dic566—216no—
31C2×Dic566—216no—
32C2×Dic566—24no—
33C2×Dic566—24no—
34C2×Dic566—22no—
35C2×Dic566D418yescomplemented
36C2×Dic566C2311yescharacteristic
37C2×Dic566C2314yes—
38C2×D1132C2311yescharacteristic
39C2×C22⋊D283—22no—
40C2×C22⋊D283—21no—
41C22×Dic283—28no—
42C22×Dic283—22no—
43C22×Dic283—24no—
44C2×C4×D283—21no—
45Dic1132—28no—
46#17985381—28no—
47C4⋊Dic283C2312yes—
48Dic283⋊C4—22no—
49C22⋊Dic283—21no—
50C22⋊Dic283—22no—
51(C2×D283)⋊C4—22no—
52C4×Dic283—21no—
53C24×C283D411yescharacteristic
54C22×C4×C283—24no—
55C22×C4×C283D412yes—
56C22×C4×C283D411yescharacteristic
57C22×C4×C283C2312yes—
58C22×C4×C283C2311yescharacteristic
59M16×C283—24no—
60C2×C8×C283C2314yes—
61C22×D283—21no—
62Dic566—216no—
63Dic566—28no—
64Dic566—24no—
65Dic566—22no—
66Dic566C2×D418yes—
67D1132—22no—
68C22⋊D283—42no—
69C22⋊D283—44no—
70C2×Dic283—216no—
71C2×Dic283—28no—
72C2×Dic283—24no—
73C2×Dic283—22no—
74C2×Dic283—21no—
75C4×D283—42no—
76C23×C283—22no—
77C23×C283—24no—
78C23×C283C2×D411yescharacteristic
79C23×C283C2×D412yes—
80C2×C4×C283—22no—
81C2×C4×C283—24no—
82C2×C4×C283—24no—
83C2×C4×C283C2×D414yes—
84C2×C4×C283C2×D412yes—
85C2×C4×C283C2×D411yescharacteristic
86C2×C4×C283C2411yescharacteristic
87C8×C283—24no—
88C22×C283—28no—
89C22×C283—22no—
90C22×C283—22no—
91C22×C283—22no—
92C22×C283#12012yes—
93C22×C283#13714yescomplemented
94C22×C283C22×D411yescharacteristic
95C2×D283—41no—
96C2×D283—42no—
97C4×C283—22no—
98C4×C283#12012yes—
99C4×C283C22×D412yes—
100Dic283—28no—
101Dic283—41no—
102Dic283—42no—
103C2×C283—22no—
104C2×C283—24no—
105C2×C283C2×#12011yescharacteristic
106C2×C283C2×#13712yes—
107D283—81no—
108C283#300411yescharacteristic, complemented, Sylow
109#3004—2831nomaximal, Sylow
110C23×Q8—2831no—
111C2×#137—2832no—
112#525—2831no—
113#451—2834no—
114#448—2834no—
115#419—2832no—
116#407D28311yescharacteristic, complemented
117C2×#41—2831no—
118C22×Q8—2832no—
119C22×Q8—2834no—
120C22×Q8—5668no—
121C22×Q8—5662no—
122C22×Q8—5661no—
123C23×C4—5661no—
124C23×C4C2×D28311yescharacteristic
125#137—5668no—
126C2×Q16—2834no—
127C2×SD16—2834no—
128C2×M16C2×D28312yes—
129#123—5661no—
130C22⋊Q8—2832no—
131#121—2832no—
132#120—5661no—
133C4×D4—5662no—
134#117—2831no—
135#103—2838no—
136C22⋊C8C2×D28314yes—
137C24D113211yescharacteristic
138#41—5664no—
139C2×Q8—2838no—
140C2×Q8—2834no—
141C2×Q8—2831no—
142C2×Q8—56616no—
143C2×Q8—5668no—
144C2×Q8—56616no—
145C2×Q8—5664no—
146C2×Q8—5664no—
147C2×Q8—5662no—
148C2×D4—2831no—
149C2×D4—5662no—
150C2×D4—5661no—
151C22×C4—24no—
152C22×C4—5668no—
153C22×C4—5664no—
154C22×C4—5662no—
155C22×C4—5661no—
156C22×C4D113212yes—
157C22×C4D113211yescharacteristic
158C22×C4C22×D28312yes—
159C22×C4C22×D28311yescharacteristic
160Q16—5668no—
161SD16—5668no—
162M16—24no—
163C2×C8C22×D28314yes—
164C4⋊C4—2832no—
165C4⋊C4—5662no—
166C22⋊C4—5661no—
167C22⋊C4—5662no—
168C22⋊C4—5662no—
169C42—5661no—
170C23—22no—
171C23—24no—
172C23—5661no—
173C23C2×D113211yescharacteristic
174C23C2×D113212yes—
175Q8—2838no—
176Q8—56616no—
177Q8—5668no—
178Q8—5664no—
179Q8—5662no—
180D4—5662no—
181D4—11322no—
182D4—11324no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—56616no—
187C2×C4—5668no—
188C2×C4—5664no—
189C2×C4—5662no—
190C2×C4—5661no—
191C2×C4—11322no—
192C2×C4D4×D28314yes—
193C2×C4C2×D113212yes—
194C2×C4C2×D113211yescharacteristic
195C2×C4C23×D28311yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—11321no—
202C22—11322no—
203C22#2258358012yes—
204C22#2258364314yescomplemented
205C22C22×D113211yescharacteristic
206C4—22no—
207C4—5668no—
208C4—11321no—
209C4—11322no—
210C4#2258358012yes—
211C4C2×D4×D28312yes—
212C2—22no—
213C2—24no—
214C2—22641no—
215C2C2×#2258358011yescharacteristic
216C2C2×#2258364312yes—
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#407×C283C2
socleC22×C283C22×D4
prime radical#31826976C2
hypercenter#407D283
hypocenterC283#3004
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×Dic566C2semidirect
2. C2×#22583643C2semidirect
4. #31826623C2semidirect
5. #31826606C2semidirect
6. #31826475C2semidirect
7. #407×C283C2semidirect
16. C2×Dic1132C22semidirect
17. C2×#17985381C22semidirect
24. #22583535C22semidirect
35. C2×Dic566D4semidirect
93. C22×C283#137semidirect
108. C283#3004semidirect
116. #407D283semidirect
204. C22#22583643semidirect

Maximal subgroups

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

subgroupindex
C23×Dic5662
C2×#225836432
#318269762
#318266232
#318266062
#318264752
#407×C2832
#3004283