Finite Group Databasebeta

← C22×#6200 #11722542

Subgroups — C22×#6200 #11722542

Order 864.

960 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 164 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×#272C211yesmaximal, characteristic, complemented
2C23×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C22×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C2×#6200C2112yesmaximal, direct product factor
5C22×#626—31nomaximal
6C22×#625—31nomaximal
7C23×H(3,𝔽3)C2211yescharacteristic
8C22×#272—24no—
9C22×#272C2216yescomplemented
10C22×(C3⋊D3)⋊C3—24no—
11C22×(C3⋊D3)⋊C3C2216yescomplemented
12C2×(C3⋊Dic3)⋊C3C2216yescomplemented
13#6200C22116yesdirect product factor
14C23×C3⋊D3D311yescharacteristic, complemented
15C23×C3×D3—31no—
16C23×C3×D3—31no—
17C23×C3×D3—31no—
18C23×C3×D3—61no—
19C22×C3⋊Dic3D311yescharacteristic, complemented
20C22×C3×Dic3—31no—
21C2×#626—312no—
22C2×#625—312no—
23C22×H(3,𝔽3)D414yescomplemented
24C22×H(3,𝔽3)C2313yes—
25C2×#272—212no—
26C2×#272C2314yescomplemented
27C2×(C3⋊D3)⋊C3—212no—
28C2×(C3⋊D3)⋊C3C2314yescomplemented
29(C3⋊Dic3)⋊C3C2314yescomplemented
30C22×#67—91no—
31C22×#67—91no—
32C22×D12—91no—
33C23×C32—21no—
34C23×C32C2×D311yescharacteristic
35C23×C32C2×D311yescharacteristic
36C22×C3⋊D3—24no—
37C22×C3⋊D3C2×D316yescomplemented
38C22×C3×D3—36no—
39C22×C3×D3—36no—
40C22×C3×D3—36no—
41C22×C3×D3—66no—
42C22×C3×D3—68no—
43C22×C3×D3—64no—
44C22×C3×D3—64no—
45C22×C3×D3—64no—
46C2×C3⋊Dic3C2×D316yescomplemented
47C2×C3×Dic3—36no—
48#626—316no—
49#625—316no—
50C2×H(3,𝔽3)C2×D416yescomplemented
51C2×H(3,𝔽3)C2411yescharacteristic
52#272—24no—
53(C3⋊D3)⋊C3—24no—
54C24×C3—91no—
55C24×C3—91no—
56C23×D3—31no—
57C23×D3—31no—
58C23×D3—31no—
59C23×D3—61no—
60C23×D3—91no—
61C22×C3×C4—91no—
62C2×#67—912no—
63C2×#67—912no—
64C22×Dic3—31no—
65C22×Dic3—91no—
66C2×D12—912no—
67C22×C32—24no—
68C22×C32—23no—
69C22×C32D1214yescomplemented
70C22×C32#6714yescomplemented
71C22×C32C22×D313yes—
72C22×C32C22×D313yes—
73C2×C3⋊D3—212no—
74C2×C3⋊D3C22×D314yescomplemented
75C2×C3×D3—34no—
76C2×C3×D3—34no—
77C2×C3×D3—34no—
78C2×C3×D3—64no—
79C2×C3×D3—624no—
80C2×C3×D3—612no—
81C2×C3×D3—612no—
82C2×C3×D3—612no—
83C3⋊Dic3C22×D314yescomplemented
84C3×Dic3—34no—
85C22×D4—271noSylow
86H(3,𝔽3)C22×D411yescharacteristic, complemented, Sylow
87C23×C3—31no—
88C23×C3—31no—
89C23×C3—61no—
90C23×C3—96no—
91C23×C3—96no—
92C23×C3—184no—
93C23×C3—184no—
94C23×C3D3211yescharacteristic
95C22×D3—36no—
96C22×D3—36no—
97C22×D3—36no—
98C22×D3—66no—
99C22×D3—64no—
100C22×D3—64no—
101C22×D3—68no—
102C22×D3—64no—
103C22×D3—96no—
104C22×D3—184no—
105C2×C3×C4—96no—
106#67—916no—
107#67—916no—
108C2×Dic3—36no—
109C2×Dic3—96no—
110D12—916no—
111C2×C32—21no—
112C2×C32—26no—
113C2×C32C2×D1216yescomplemented
114C2×C32C2×#6716yescomplemented
115C2×C32C23×D311yescharacteristic
116C2×C32C23×D311yescharacteristic
117C3⋊D3—24no—
118C3×D3—68no—
119C3×D3—64no—
120C3×D3—64no—
121C3×D3—64no—
122C24—91no—
123C24—91no—
124C2×D4—2712no—
125C22×C4—91no—
126C22×C3—34no—
127C22×C3—34no—
128C22×C3—33no—
129C22×C3—33no—
130C22×C3—64no—
131C22×C3—63no—
132C22×C3—94no—
133C22×C3—94no—
134C22×C3—1812no—
135C22×C3—1812no—
136C22×C3#62614yes—
137C22×C3C2×D3213yes—
138C2×D3—34no—
139C2×D3—34no—
140C2×D3—34no—
141C2×D3—64no—
142C2×D3—612no—
143C2×D3—612no—
144C2×D3—624no—
145C2×D3—612no—
146C2×D3—94no—
147C2×D3—1812no—
148C3×C4—94no—
149Dic3—34no—
150Dic3—94no—
151C32—21no—
152C32C22×D1211yescharacteristic, complemented
153C32C22×#6711yescharacteristic, complemented
154C23—96no—
155C23—96no—
156C23—184no—
157C23—184no—
158C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
159D4—2716no—
160C2×C4—96no—
161C2×C3—31no—
162C2×C3—31no—
163C2×C3—36no—
164C2×C3—36no—
165C2×C3—61no—
166C2×C3—66no—
167C2×C3—184no—
168C2×C3—184no—
169C2×C3C2×#62616yes—
170C2×C3C22×D3211yescharacteristic
171D3—64no—
172D3—64no—
173D3—68no—
174D3—64no—
175D3—184no—
176C22—94no—
177C22—94no—
178C22—1812no—
179C22—1812no—
180C22#620014yesdirect product factor
181C22C2×((C3⋊D3)⋊C3)⋊C213yes—
182C4—94no—
183C3—31no—
184C3—31no—
185C3—61no—
186C3C22×#62611yescharacteristic
187C2—184no—
188C2—184no—
189C2C2×#620016yesdirect product factor
190C2C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC23((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C24
FrattiniC2×C3C22×D32
FittingC23×H(3,𝔽3)C22
socleC23×C3D32
hypocenterH(3,𝔽3)C22×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×#272C2semidirect
2. C23×(C3⋊D3)⋊C3C2semidirect
3. C22×(C3⋊Dic3)⋊C3C2semidirect
4. C2×#6200C2direct (N × H)
9. C22×#272C22semidirect
11. C22×(C3⋊D3)⋊C3C22semidirect
12. C2×(C3⋊Dic3)⋊C3C22semidirect
13. #6200C22direct (N × H)
14. C23×C3⋊D3D3semidirect
19. C22×C3⋊Dic3D3semidirect
23. C22×H(3,𝔽3)D4semidirect
26. C2×#272C23semidirect
28. C2×(C3⋊D3)⋊C3C23semidirect
29. (C3⋊Dic3)⋊C3C23semidirect
37. C22×C3⋊D3C2×D3semidirect
46. C2×C3⋊Dic3C2×D3semidirect
50. C2×H(3,𝔽3)C2×D4semidirect
69. C22×C32D12semidirect
70. C22×C32#67semidirect
74. C2×C3⋊D3C22×D3semidirect
83. C3⋊Dic3C22×D3semidirect
86. H(3,𝔽3)C22×D4semidirect
113. C2×C32C2×D12semidirect
114. C2×C32C2×#67semidirect
152. C32C22×D12semidirect
153. C32C22×#67semidirect
180. C22#6200direct (N × H)
189. C2C2×#6200direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#2722
C23×(C3⋊D3)⋊C32
C22×(C3⋊Dic3)⋊C32
C2×#62002
C22×#6263
C22×#6253