Finite Group Databasebeta

← #27140261

Subgroups — #27140261

Order 12416.

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#20260792C211yesmaximal, characteristic, complemented
2C4×C4⋊Dic97C211yesmaximal, characteristic, complemented
3#20260472C212yesmaximal, complemented
4#20260469C212yesmaximal
5#529×C97C211yesmaximal, characteristic
6C2×C4⋊Dic97—21no—
7C2×C4×Dic97—21no—
8C4⋊#11715895—21no—
9C22⋊#11715895C2212yes—
10#15690721—21no—
11#15690717C2212yes—
12#126×C97—21no—
13#125×C97C411yescharacteristic, complemented
14#124×C97C411yescharacteristic, complemented
15#123×C97—21no—
16C22⋊Q8×C97C2211yescharacteristic
17#121×C97C2211yescharacteristic
18C4×Q8×C97C411yescharacteristic, complemented
19C4×D4×C97—21no—
20C4×D4×C97C411yescharacteristic, complemented
21#117×C97—21no—
22C2×C42×C97C2211yescharacteristic
23C22×Dic97—22no—
24C2×#11715895—22no—
25C4⋊Dic97—24no—
26C4×Dic97—42no—
27C2×Q8×C97C2×C411yescharacteristic
28C2×D4×C97—21no—
29C2×D4×C97C2×C411yescharacteristic
30C22×C4×C97—21no—
31C22×C4×C97D411yescharacteristic
32C22×C4×C97D411yescharacteristic
33C22×C4×C97C2311yescharacteristic
34C4⋊C4×C97—21no—
35C4⋊C4×C97—21no—
36C4⋊C4×C97—21no—
37C4⋊C4×C97—21no—
38C4⋊C4×C97C2×C411yescharacteristic
39C4⋊C4×C97C2×C411yescharacteristic
40C22⋊C4×C97—21no—
41C22⋊C4×C97—21no—
42C22⋊C4×C97—21no—
43C22⋊C4×C97—21no—
44C22⋊C4×C97—21no—
45C42×C97—21no—
46C42×C97C2×C411yescharacteristic
47C42×C97C2×C411yescharacteristic
48C42×C97D411yescharacteristic
49C42×C97D411yescharacteristic
50C23×C97—21no—
51C23×C97C2×D411yescharacteristic
52Q8×C97—41no—
53D4×C97—41no—
54D4×C97—41no—
55D4×C97—41no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—22no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—41no—
63C2×C4×C97—42no—
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22⋊C411yescharacteristic
68C2×C4×C97C22×C411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×Dic97—24no—
71C2×Dic97—42no—
72C2×Dic97—42no—
73#11715895—42no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97C2×C22⋊C411yescharacteristic
80C22×C97AGL(1,ℤ/8)11yescharacteristic
81C22×C97#13711yescharacteristic
82C4×C97—21no—
83C4×C97—21no—
84C4×C97—21no—
85C4×C97—21no—
86C4×C97—21no—
87C4×C97—41no—
88C4×C97—41no—
89C4×C97—41no—
90C4×C97#10412yes—
91Dic97—44no—
92C2×C97—21no—
93C2×C97—21no—
94C2×C97—81no—
95C2×C97#41011yescharacteristic
96C2×C97#41711yescharacteristic
97C2×C97C2×#10411yescharacteristic
98#1509—971nomaximal, Sylow
99C97#150911yescharacteristic, complemented, Sylow
100#529D9711yescharacteristic
101#432—971no—
102C4×C4⋊C4—971no—
103#328—972no—
104#327—972no—
105#126—21no—
106#125Dic9711yescharacteristic, complemented
107#124Dic9711yescharacteristic, complemented
108#123—21no—
109C22⋊Q8C2×D9711yescharacteristic
110#121C2×D9711yescharacteristic
111C4×Q8Dic9711yescharacteristic, complemented
112C4×D4—21no—
113C4×D4Dic9711yescharacteristic, complemented
114#117—21no—
115C2×C4⋊C4—1941no—
116C2×C42—1941no—
117C2×C42C2×D9711yescharacteristic
118C4⋊C8—1941no—
119C22⋊C8—972no—
120#97—1941no—
121#95—972no—
122C2×Q8C2×Dic9711yescharacteristic
123C2×D4—21no—
124C2×D4C2×Dic9711yescharacteristic
125C22×C4—21no—
126C22×C4—1942no—
127C22×C4C22⋊D9711yescharacteristic
128C22×C4C22⋊D9711yescharacteristic
129C22×C4C22×D9711yescharacteristic
130C2×C8—1942no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—1944no—
136C4⋊C4C2×Dic9711yescharacteristic
137C4⋊C4C2×Dic9711yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C22⋊C4—21no—
143C42—21no—
144C42—3882no—
145C42C2×Dic9711yescharacteristic
146C42C2×Dic9711yescharacteristic
147C42C22⋊D9711yescharacteristic
148C42C22⋊D9711yescharacteristic
149C23—21no—
150C23C2×C22⋊D9711yescharacteristic
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—3882no—
164C2×C4—3882no—
165C2×C4—1944no—
166C2×C4C22⋊Dic9711yescharacteristic
167C2×C4C22⋊Dic9711yescharacteristic
168C2×C4C22⋊Dic9711yescharacteristic
169C2×C4C22⋊Dic9711yescharacteristic
170C2×C4C22×Dic9711yescharacteristic
171C2×C4C2×C22⋊D9711yescharacteristic
172C8—3882no—
173C22—21no—
174C22—21no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22C2×C22⋊Dic9711yescharacteristic
179C22#1569083511yescharacteristic
180C22#1569084011yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—3884no—
190C4#1569073712yes—
191C2—21no—
192C2—21no—
193C2—81no—
194C2#2026070911yescharacteristic
195C2#2026073611yescharacteristic
196C2C2×#1569073711yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#529×C97C2
socleC22×C97C2×C22⋊C4
prime radical#121×C97C22
cosocleC22×C4×C97C23
hypercenter#529D97
hypocenterC97#1509
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20260792C2semidirect
2. C4×C4⋊Dic97C2semidirect
3. #20260472C2semidirect
13. #125×C97C4semidirect
14. #124×C97C4semidirect
18. C4×Q8×C97C4semidirect
20. C4×D4×C97C4semidirect
99. C97#1509semidirect
106. #125Dic97semidirect
107. #124Dic97semidirect
111. C4×Q8Dic97semidirect
113. C4×D4Dic97semidirect

Maximal subgroups

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

subgroupindex
#202607922
C4×C4⋊Dic972
#202604722
#202604692
#529×C972
#150997