Finite Group Databasebeta

← C2×H(3,𝔽3)×#120 #13154546

Subgroups — C2×H(3,𝔽3)×#120 #13154546

Order 1728.

3641 conjugacy classes of subgroups; only the 735 normal subgroups are shown.

normal subgroups only

This group has 3641 conjugacy classes of subgroups — too many to draw. Only its 735 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
1C25×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×D4×H(3,𝔽3)C213yesmaximal, complemented
3H(3,𝔽3)×#120C218yesmaximal, direct product factor
4C2×C22⋊C4×H(3,𝔽3)C213yesmaximal, complemented
5C2×C32×#120C314yesmaximal, complemented
6C24×H(3,𝔽3)—212no—
7C24×H(3,𝔽3)C2213yes—
8C24×H(3,𝔽3)C2214yescomplemented
9C24×H(3,𝔽3)C2211yescharacteristic, complemented
10C2×D4×H(3,𝔽3)—212no—
11C2×D4×H(3,𝔽3)C22112yescomplemented
12C22×C4×H(3,𝔽3)C2213yescomplemented
13C22⋊C4×H(3,𝔽3)C22112yescomplemented
14C25×C32C2×C314yescomplemented
15C22×C32×D4C2×C3112yescomplemented
16C32×#120C2×C3132yescomplemented
17C2×C32×C22⋊C4C2×C3112yescomplemented
18C23×H(3,𝔽3)—232no—
19C23×H(3,𝔽3)—212no—
20C23×H(3,𝔽3)—224no—
21C23×H(3,𝔽3)—26no—
22C23×H(3,𝔽3)D4112yescomplemented
23C23×H(3,𝔽3)C2316yes—
24C23×H(3,𝔽3)C2311yescharacteristic
25C23×H(3,𝔽3)C2312yescomplemented
26D4×H(3,𝔽3)—224no—
27C2×C4×H(3,𝔽3)—26no—
28C2×C4×H(3,𝔽3)C2316yescomplemented
29C2×C3×#120—34no—
30C2×C3×#120C3211yescharacteristic
31C24×C32—248no—
32C24×C32C22×C3112yes—
33C24×C32C22×C3116yescomplemented
34C24×C32C22×C314yescomplemented
35C2×C32×D4—248no—
36C2×C32×D4C22×C3148yescomplemented
37C22×C32×C4C22×C3112yescomplemented
38C32×C22⋊C4C22×C3148yescomplemented
39C22×H(3,𝔽3)—232no—
40C22×H(3,𝔽3)—224no—
41C22×H(3,𝔽3)—212no—
42C22×H(3,𝔽3)—26no—
43C22×H(3,𝔽3)—44no—
44C22×H(3,𝔽3)C2×D416yes—
45C22×H(3,𝔽3)C2×D4112yescomplemented
46C22×H(3,𝔽3)C2411yescharacteristic
47C4×H(3,𝔽3)—26no—
48C25×C3—34no—
49C25×C3C2×C3211yescharacteristic
50C22×C3×D4—312no—
51C22×C3×D4C2×C3213yes—
52C3×#120—332no—
53C3×#120C2×C3218yes—
54C2×C3×C22⋊C4—312no—
55C2×C3×C22⋊C4C2×C3213yes—
56C23×C32—2128no—
57C23×C32—248no—
58C23×C32—296no—
59C23×C32—224no—
60C23×C32C3×D4148yescomplemented
61C23×C32C23×C3124yes—
62C23×C32C23×C314yes—
63C23×C32C23×C318yescomplemented
64C32×D4—296no—
65C2×C32×C4—224no—
66C2×C32×C4C23×C3124yescomplemented
67C2×#120H(3,𝔽3)11yescharacteristic, direct product factor, Sylow
68C2×H(3,𝔽3)—212no—
69C2×H(3,𝔽3)—42no—
70C2×H(3,𝔽3)#12014yesdirect product factor
71C2×H(3,𝔽3)C22×D413yes—
72C24×C3—212no—
73C24×C3—312no—
74C24×C3—316no—
75C24×C3—34no—
76C24×C3—648no—
77C24×C3C22×C3213yes—
78C24×C3C22×C3214yes—
79C24×C3C22×C3211yescharacteristic
80C2×C3×D4—212no—
81C2×C3×D4—348no—
82C2×C3×D4—648no—
83C2×C3×D4C22×C32112yes—
84C22×C3×C4—312no—
85C22×C3×C4C22×C3213yes—
86C3×C22⋊C4—348no—
87C3×C22⋊C4C22×C32112yes—
88C22×C32—2128no—
89C22×C32—296no—
90C22×C32—248no—
91C22×C32—224no—
92C22×C32—416no—
93C22×C32C2×C3×D4124yes—
94C22×C32C2×C3×D4148yescomplemented
95C22×C32C24×C314yes—
96C32×C4—224no—
97C25C2×H(3,𝔽3)11yescharacteristic, complemented
98C22×D4C2×H(3,𝔽3)13yescomplemented
99#120C2×H(3,𝔽3)18yesdirect product factor
100C2×C22⋊C4C2×H(3,𝔽3)13yescomplemented
101H(3,𝔽3)C2×#12011yescharacteristic, direct product factor, Sylow
102C23×C3—232no—
103C23×C3—212no—
104C23×C3—224no—
105C23×C3—26no—
106C23×C3—348no—
107C23×C3—34no—
108C23×C3—324no—
109C23×C3—38no—
110C23×C3—648no—
111C23×C3—696no—
112C23×C3—624no—
113C23×C3—6128no—
114C23×C3C32×D4112yes—
115C23×C3C23×C3216yes—
116C23×C3C23×C3212yes—
117C23×C3C23×C3211yescharacteristic
118C3×D4—224no—
119C3×D4—696no—
120C2×C3×C4—26no—
121C2×C3×C4—324no—
122C2×C3×C4—624no—
123C2×C3×C4C23×C3216yes—
124C2×C32—248no—
125C2×C32—48no—
126C2×C32C3×#120116yescomplemented
127C2×C32C22×C3×D4112yes—
128C24—212no—
129C24C22×H(3,𝔽3)13yes—
130C24C22×H(3,𝔽3)14yescomplemented
131C24C22×H(3,𝔽3)11yescharacteristic, complemented
132C2×D4—212no—
133C2×D4C22×H(3,𝔽3)112yescomplemented
134C22×C4C22×H(3,𝔽3)13yescomplemented
135C22⋊C4C22×H(3,𝔽3)112yescomplemented
136C22×C3—232no—
137C22×C3—212no—
138C22×C3—224no—
139C22×C3—26no—
140C22×C3—34no—
141C22×C3—324no—
142C22×C3—348no—
143C22×C3—44no—
144C22×C3—696no—
145C22×C3—648no—
146C22×C3—624no—
147C22×C3—6128no—
148C22×C3—1216no—
149C22×C3C2×C32×D416yes—
150C22×C3C2×C32×D4112yes—
151C22×C3C24×C3211yescharacteristic
152C3×C4—26no—
153C3×C4—624no—
154C32C2×C3×#12014yescomplemented
155C23—232no—
156C23—212no—
157C23—224no—
158C23—26no—
159C23D4×H(3,𝔽3)112yescomplemented
160C23C23×H(3,𝔽3)16yes—
161C23C23×H(3,𝔽3)11yescharacteristic
162C23C23×H(3,𝔽3)12yescomplemented
163D4—224no—
164C2×C4—26no—
165C2×C4C23×H(3,𝔽3)16yescomplemented
166C2×C3—212no—
167C2×C3—316no—
168C2×C3—312no—
169C2×C3—42no—
170C2×C3—648no—
171C2×C3—128no—
172C2×C3C32×#12014yes—
173C2×C3C22×C32×D413yes—
174C22—232no—
175C22—224no—
176C22—212no—
177C22—26no—
178C22—44no—
179C22C2×D4×H(3,𝔽3)16yes—
180C22C2×D4×H(3,𝔽3)112yescomplemented
181C22C24×H(3,𝔽3)11yescharacteristic
182C4—26no—
183C3—34no—
184C3C2×C32×#12011yescharacteristic
185C2—212no—
186C2—42no—
187C2H(3,𝔽3)×#12014yesdirect product factor
188C2C22×D4×H(3,𝔽3)13yes—
subgroupG/H
center, socleC23×C3C23×C32
commutator, Frattini, cosocleC22×C3C24×C32
LCSG ▷ C22×C3 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C25×H(3,𝔽3)C2semidirect
2. C22×D4×H(3,𝔽3)C2semidirect
3. H(3,𝔽3)×#120C2direct (N × H)
4. C2×C22⋊C4×H(3,𝔽3)C2semidirect
5. C2×C32×#120C3semidirect
8. C24×H(3,𝔽3)C22semidirect
9. C24×H(3,𝔽3)C22semidirect
11. C2×D4×H(3,𝔽3)C22semidirect
12. C22×C4×H(3,𝔽3)C22semidirect
13. C22⋊C4×H(3,𝔽3)C22semidirect
14. C25×C32C2×C3semidirect
15. C22×C32×D4C2×C3semidirect
16. C32×#120C2×C3semidirect
17. C2×C32×C22⋊C4C2×C3semidirect
22. C23×H(3,𝔽3)D4semidirect
25. C23×H(3,𝔽3)C23semidirect
28. C2×C4×H(3,𝔽3)C23semidirect
33. C24×C32C22×C3semidirect
34. C24×C32C22×C3semidirect
36. C2×C32×D4C22×C3semidirect
37. C22×C32×C4C22×C3semidirect
38. C32×C22⋊C4C22×C3semidirect
45. C22×H(3,𝔽3)C2×D4semidirect
60. C23×C32C3×D4semidirect
63. C23×C32C23×C3semidirect
66. C2×C32×C4C23×C3semidirect
67. C2×#120H(3,𝔽3)direct (N × H)
70. C2×H(3,𝔽3)#120direct (N × H)
94. C22×C32C2×C3×D4semidirect
97. C25C2×H(3,𝔽3)semidirect
98. C22×D4C2×H(3,𝔽3)semidirect
99. #120C2×H(3,𝔽3)direct (N × H)
100. C2×C22⋊C4C2×H(3,𝔽3)semidirect
101. H(3,𝔽3)C2×#120direct (N × H)
126. C2×C32C3×#120semidirect
130. C24C22×H(3,𝔽3)semidirect
131. C24C22×H(3,𝔽3)semidirect
133. C2×D4C22×H(3,𝔽3)semidirect
134. C22×C4C22×H(3,𝔽3)semidirect
135. C22⋊C4C22×H(3,𝔽3)semidirect
154. C32C2×C3×#120semidirect
159. C23D4×H(3,𝔽3)semidirect
162. C23C23×H(3,𝔽3)semidirect
165. C2×C4C23×H(3,𝔽3)semidirect
180. C22C2×D4×H(3,𝔽3)semidirect
187. C2H(3,𝔽3)×#120direct (N × H)

Maximal subgroups

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

subgroupindex
C25×H(3,𝔽3)2
C22×D4×H(3,𝔽3)2
H(3,𝔽3)×#1202
C2×C22⋊C4×H(3,𝔽3)2
C2×C32×#1203