Finite Group Databasebeta

← #27169657

Subgroups — #27169657

Order 12928.

242 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 81 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#20265234C211yesmaximal, characteristic, complemented
2C4×C4⋊Dic101C211yesmaximal, characteristic, complemented
3#20264914C212yesmaximal, complemented
4#20264911C212yesmaximal
5#529×C101C211yesmaximal, characteristic
6C2×C4⋊Dic101—21no—
7C2×C4×Dic101—21no—
8C4⋊#11717579—21no—
9C22⋊#11717579C2212yes—
10#15692166—21no—
11#15692162C2212yes—
12#126×C101—21no—
13#125×C101C411yescharacteristic, complemented
14#124×C101C411yescharacteristic, complemented
15#123×C101—21no—
16C22⋊Q8×C101C2211yescharacteristic
17#121×C101C2211yescharacteristic
18C4×Q8×C101C411yescharacteristic, complemented
19C4×D4×C101—21no—
20C4×D4×C101C411yescharacteristic, complemented
21#117×C101—21no—
22C2×C42×C101C2211yescharacteristic
23C22×Dic101—22no—
24C2×#11717579—22no—
25C4⋊Dic101—24no—
26C4×Dic101—42no—
27C2×Q8×C101C2×C411yescharacteristic
28C2×D4×C101—21no—
29C2×D4×C101C2×C411yescharacteristic
30C22×C4×C101—21no—
31C22×C4×C101D411yescharacteristic
32C22×C4×C101D411yescharacteristic
33C22×C4×C101C2311yescharacteristic
34C4⋊C4×C101—21no—
35C4⋊C4×C101—21no—
36C4⋊C4×C101—21no—
37C4⋊C4×C101—21no—
38C4⋊C4×C101C2×C411yescharacteristic
39C4⋊C4×C101C2×C411yescharacteristic
40C22⋊C4×C101—21no—
41C22⋊C4×C101—21no—
42C22⋊C4×C101—21no—
43C22⋊C4×C101—21no—
44C22⋊C4×C101—21no—
45C42×C101—21no—
46C42×C101C2×C411yescharacteristic
47C42×C101C2×C411yescharacteristic
48C42×C101D411yescharacteristic
49C42×C101D411yescharacteristic
50C23×C101—21no—
51C23×C101C2×D411yescharacteristic
52Q8×C101—41no—
53D4×C101—41no—
54D4×C101—41no—
55D4×C101—41no—
56C2×C4×C101—21no—
57C2×C4×C101—21no—
58C2×C4×C101—22no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101—41no—
63C2×C4×C101—42no—
64C2×C4×C101C22⋊C411yescharacteristic
65C2×C4×C101C22⋊C411yescharacteristic
66C2×C4×C101C22⋊C411yescharacteristic
67C2×C4×C101C22⋊C411yescharacteristic
68C2×C4×C101C22×C411yescharacteristic
69C2×C4×C101C2×D411yescharacteristic
70C2×Dic101—24no—
71C2×Dic101—42no—
72C2×Dic101—42no—
73#11717579—42no—
74C22×C101—21no—
75C22×C101—21no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101C2×C22⋊C411yescharacteristic
80C22×C101AGL(1,ℤ/8)11yescharacteristic
81C22×C101#13711yescharacteristic
82C4×C101—21no—
83C4×C101—21no—
84C4×C101—21no—
85C4×C101—21no—
86C4×C101—21no—
87C4×C101—41no—
88C4×C101—41no—
89C4×C101—41no—
90C4×C101#10412yes—
91Dic101—44no—
92C2×C101—21no—
93C2×C101—21no—
94C2×C101—81no—
95C2×C101#41011yescharacteristic
96C2×C101#41711yescharacteristic
97C2×C101C2×#10411yescharacteristic
98#1509—1011nomaximal, Sylow
99C101#150911yescharacteristic, complemented, Sylow
100#529D10111yescharacteristic
101#432—1011no—
102C4×C4⋊C4—1011no—
103#328—1012no—
104#327—1012no—
105#126—21no—
106#125Dic10111yescharacteristic, complemented
107#124Dic10111yescharacteristic, complemented
108#123—21no—
109C22⋊Q8C2×D10111yescharacteristic
110#121C2×D10111yescharacteristic
111C4×Q8Dic10111yescharacteristic, complemented
112C4×D4—21no—
113C4×D4Dic10111yescharacteristic, complemented
114#117—21no—
115C2×C4⋊C4—2021no—
116C2×C42—2021no—
117C2×C42C2×D10111yescharacteristic
118C4⋊C8—2021no—
119C22⋊C8—1012no—
120#97—2021no—
121#95—1012no—
122C2×Q8C2×Dic10111yescharacteristic
123C2×D4—21no—
124C2×D4C2×Dic10111yescharacteristic
125C22×C4—21no—
126C22×C4—2022no—
127C22×C4C22⋊D10111yescharacteristic
128C22×C4C22⋊D10111yescharacteristic
129C22×C4C22×D10111yescharacteristic
130C2×C8—2022no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—2024no—
136C4⋊C4C2×Dic10111yescharacteristic
137C4⋊C4C2×Dic10111yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C22⋊C4—21no—
143C42—21no—
144C42—4042no—
145C42C2×Dic10111yescharacteristic
146C42C2×Dic10111yescharacteristic
147C42C22⋊D10111yescharacteristic
148C42C22⋊D10111yescharacteristic
149C23—21no—
150C23C2×C22⋊D10111yescharacteristic
151Q8—41no—
152D4—41no—
153D4—41no—
154D4—41no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—4042no—
164C2×C4—4042no—
165C2×C4—2024no—
166C2×C4C22⋊Dic10111yescharacteristic
167C2×C4C22⋊Dic10111yescharacteristic
168C2×C4C22⋊Dic10111yescharacteristic
169C2×C4C22⋊Dic10111yescharacteristic
170C2×C4C22×Dic10111yescharacteristic
171C2×C4C2×C22⋊D10111yescharacteristic
172C8—4042no—
173C22—21no—
174C22—21no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22C2×C22⋊Dic10111yescharacteristic
179C22#1569228011yescharacteristic
180C22#1569228511yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—4044no—
190C4#1569218212yes—
191C2—21no—
192C2—21no—
193C2—81no—
194C2#2026515111yescharacteristic
195C2#2026517811yescharacteristic
196C2C2×#1569218211yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#529×C101C2
socleC22×C101C2×C22⋊C4
prime radical#121×C101C22
cosocleC22×C4×C101C23
hypercenter#529D101
hypocenterC101#1509
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20265234C2semidirect
2. C4×C4⋊Dic101C2semidirect
3. #20264914C2semidirect
13. #125×C101C4semidirect
14. #124×C101C4semidirect
18. C4×Q8×C101C4semidirect
20. C4×D4×C101C4semidirect
99. C101#1509semidirect
106. #125Dic101semidirect
107. #124Dic101semidirect
111. C4×Q8Dic101semidirect
113. C4×D4Dic101semidirect

Maximal subgroups

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

subgroupindex
#202652342
C4×C4⋊Dic1012
#202649142
#202649112
#529×C1012
#1509101