Finite Group Databasebeta

← #49119799

Subgroups — #49119799

Order 36224.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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#31826522C211yesmaximal, characteristic, complemented
2#31826331C211yesmaximal, characteristic, complemented
3#31826202C212yesmaximal, complemented
4#31826199C212yesmaximal
5#529×C283C211yesmaximal, characteristic
6C2×C4⋊Dic283—21no—
7C4⋊(C283⋊C8)—21no—
8C22⋊(C283⋊C8)C2212yes—
9#22583521—21no—
10C283⋊#95—21no—
11C283⋊#95C2212yes—
12#126×C283—21no—
13#125×C283C411yescharacteristic, complemented
14#124×C283C411yescharacteristic, complemented
15#123×C283—21no—
16C22⋊Q8×C283C2211yescharacteristic
17#121×C283C2211yescharacteristic
18C4×Q8×C283C411yescharacteristic, complemented
19C4×D4×C283—21no—
20C4×D4×C283C411yescharacteristic, complemented
21#117×C283—21no—
22C2×C42×C283C2211yescharacteristic
23C22×Dic283—22no—
24C2×C283⋊C8—22no—
25C4⋊Dic283—42no—
26C2×Q8×C283C2×C411yescharacteristic
27C2×D4×C283—21no—
28C2×D4×C283C2×C411yescharacteristic
29C22×C4×C283—21no—
30C22×C4×C283D411yescharacteristic
31C22×C4×C283D411yescharacteristic
32C22×C4×C283C2311yescharacteristic
33C4⋊C4×C283—21no—
34C4⋊C4×C283—21no—
35C4⋊C4×C283—21no—
36C4⋊C4×C283—21no—
37C4⋊C4×C283C2×C411yescharacteristic
38C4⋊C4×C283C2×C411yescharacteristic
39C22⋊C4×C283—21no—
40C22⋊C4×C283—21no—
41C22⋊C4×C283—21no—
42C22⋊C4×C283—21no—
43C22⋊C4×C283—21no—
44C42×C283—21no—
45C42×C283C2×C411yescharacteristic
46C42×C283C2×C411yescharacteristic
47C42×C283D411yescharacteristic
48C42×C283D411yescharacteristic
49C2×Dic283—42no—
50C2×Dic283—42no—
51C2×Dic283—42no—
52C283⋊C8—42no—
53C23×C283—21no—
54C23×C283C2×D411yescharacteristic
55Q8×C283—41no—
56D4×C283—41no—
57D4×C283—41no—
58D4×C283—41no—
59C2×C4×C283—21no—
60C2×C4×C283—21no—
61C2×C4×C283—21no—
62C2×C4×C283—21no—
63C2×C4×C283—21no—
64C2×C4×C283—41no—
65C2×C4×C283—41no—
66C2×C4×C283—42no—
67C2×C4×C283C22⋊C411yescharacteristic
68C2×C4×C283C22⋊C411yescharacteristic
69C2×C4×C283C22⋊C411yescharacteristic
70C2×C4×C283C22⋊C411yescharacteristic
71C2×C4×C283C22×C411yescharacteristic
72C2×C4×C283C2×D411yescharacteristic
73C22×C283—21no—
74C22×C283—21no—
75C22×C283—41no—
76C22×C283—41no—
77C22×C283—41no—
78C22×C283C2×C22⋊C411yescharacteristic
79C22×C283AGL(1,ℤ/8)11yescharacteristic
80C22×C283#13711yescharacteristic
81C4×C283—21no—
82C4×C283—21no—
83C4×C283—21no—
84C4×C283—21no—
85C4×C283—21no—
86C4×C283—21no—
87C4×C283—41no—
88C4×C283—41no—
89C4×C283—41no—
90Dic283—82no—
91C2×C283—21no—
92C2×C283—21no—
93C2×C283—81no—
94C2×C283#41011yescharacteristic
95C2×C283#41711yescharacteristic
96C2×C283#42111yescharacteristic
97C283#151211yescharacteristic, complemented, Sylow
98#1512—2831nomaximal, Sylow
99#529D28311yescharacteristic
100#432—2831no—
101#382—2831no—
102#328—2832no—
103#327—2832no—
104#126—21no—
105#125Dic28311yescharacteristic, complemented
106#124Dic28311yescharacteristic, complemented
107#123—21no—
108C22⋊Q8C2×D28311yescharacteristic
109#121C2×D28311yescharacteristic
110C4×Q8Dic28311yescharacteristic, complemented
111C4×D4—21no—
112C4×D4Dic28311yescharacteristic, complemented
113#117—21no—
114C2×C4⋊C4—5661no—
115C2×C42C2×D28311yescharacteristic
116C4⋊C8—5661no—
117C22⋊C8—2832no—
118#97—5661no—
119#95—2832no—
120#95—5661no—
121C2×Q8C2×Dic28311yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic28311yescharacteristic
124C22×C4—21no—
125C22×C4—5662no—
126C22×C4C22⋊D28311yescharacteristic
127C22×C4C22⋊D28311yescharacteristic
128C22×C4C22×D28311yescharacteristic
129C2×C8—5662no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—11322no—
135C4⋊C4C2×Dic28311yescharacteristic
136C4⋊C4C2×Dic28311yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic28311yescharacteristic
144C42C2×Dic28311yescharacteristic
145C42C22⋊D28311yescharacteristic
146C42C22⋊D28311yescharacteristic
147C23—21no—
148C23C2×C22⋊D28311yescharacteristic
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—11322no—
162C2×C4—11322no—
163C2×C4—11322no—
164C2×C4C22⋊Dic28311yescharacteristic
165C2×C4C22⋊Dic28311yescharacteristic
166C2×C4C22⋊Dic28311yescharacteristic
167C2×C4C22⋊Dic28311yescharacteristic
168C2×C4C22×Dic28311yescharacteristic
169C2×C4C2×C22⋊D28311yescharacteristic
170C8—11322no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C2×C22⋊Dic28311yescharacteristic
177C22#2258363511yescharacteristic
178C22#2258364011yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—22642no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#3182643911yescharacteristic
193C2#3182646611yescharacteristic
194C2#3182648211yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic283
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
Fitting#529×C283C2
socleC22×C283C2×C22⋊C4
prime radical#121×C283C22
cosocleC22×C4×C283C23
hypercenter#529D283
hypocenterC283#1512
LCSG ▷ C2×C4×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #31826522C2semidirect
2. #31826331C2semidirect
3. #31826202C2semidirect
13. #125×C283C4semidirect
14. #124×C283C4semidirect
18. C4×Q8×C283C4semidirect
20. C4×D4×C283C4semidirect
97. C283#1512semidirect
105. #125Dic283semidirect
106. #124Dic283semidirect
110. C4×Q8Dic283semidirect
112. C4×D4Dic283semidirect

Maximal subgroups

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

subgroupindex
#318265222
#318263312
#318262022
#318261992
#529×C2832
#1512283