Finite Group Databasebeta

← #13154274

Subgroups — #13154274

Order 1728.

1386 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 191 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
1H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
2#11721849C216yesmaximal, complemented
3#11721847C213yesmaximal, complemented
4#11721845C213yesmaximal, complemented
5#11721722C212yesmaximal, complemented
6C3×#5720—34nomaximal
7C23×#272C2211yescharacteristic
8C2×D4×H(3,𝔽3)C2213yes—
9C2×D4×H(3,𝔽3)C2213yescomplemented
10C2×#6235C2216yes—
11C22×#1084C2213yescomplemented
12C42×H(3,𝔽3)C2211yescharacteristic, complemented
13#88552C2213yes—
14#88552C2213yescomplemented
15#88542C2216yes—
16#88540C2216yescomplemented
17C32×#127D314yescomplemented
18C3×#939—324no—
19C3×#937—312no—
20C3×#935—312no—
21C3×#876—38no—
22C23×H(3,𝔽3)C2311yescharacteristic
23C23×H(3,𝔽3)C2313yes—
24C22×#272—43no—
25C22×#272C2312yes—
26D4×H(3,𝔽3)—43no—
27D4×H(3,𝔽3)—43no—
28C2×C4×H(3,𝔽3)C2313yes—
29#6235—46no—
30C2×#1084—43no—
31C2×#1084C2316yes—
32C3×#560—91no—
33#5720—34no—
34C23×C3×D3—34no—
35C2×C32×D4C2×D3112yes—
36C2×C32×D4C2×D3112yescomplemented
37C2×C3×#67—324no—
38C22×C3×Dic3—312no—
39C32×C42C2×D314yescomplemented
40C3×#217—312no—
41C3×#217—312no—
42C3×(C2×D3)⋊C4—324no—
43C3×Dic3⋊C4—324no—
44C22×H(3,𝔽3)—23no—
45C22×H(3,𝔽3)—26no—
46C22×H(3,𝔽3)—23no—
47C22×H(3,𝔽3)C2411yescharacteristic
48C2×#272—26no—
49C2×#272—44no—
50C4×H(3,𝔽3)—23no—
51#1084—26no—
52C3×#127—34no—
53C3×#127C3⋊D311yescharacteristic
54C3×#126—92no—
55C3×#123—93no—
56C3×#121—96no—
57C3×#120—93no—
58#939—324no—
59#937—312no—
60#935—312no—
61#876—38no—
62C23×C32C22×D314yes—
63C23×C32C22×D3112yes—
64C22×C3×D3—38no—
65C22×C3×D3—1212no—
66C32×D4—412no—
67C32×D4—412no—
68C2×C32×C4C22×D3112yes—
69C3×#67—1224no—
70C2×C3×Dic3—324no—
71C2×C3×Dic3—1212no—
72#560—91noSylow
73C2×H(3,𝔽3)—41no—
74C2×H(3,𝔽3)—43no—
75C2×H(3,𝔽3)H(5,𝔽2)13yes—
76#272—42no—
77C24×C3—91no—
78C23×D3—34no—
79C2×C3×D4—312no—
80C2×C3×D4—312no—
81C2×C3×D4—96no—
82C2×C3×D4C2×C3⋊D313yes—
83C2×C3×D4C2×C3⋊D313yes—
84C22×C3×C4—93no—
85C2×#67—324no—
86C22×Dic3—312no—
87C3×C4⋊C4—96no—
88C3×C22⋊C4—93no—
89C3×C22⋊C4—93no—
90C3×C22⋊C4—96no—
91C3×C42—34no—
92C3×C42C2×C3⋊D311yescharacteristic
93#217—312no—
94#217—312no—
95(C2×D3)⋊C4—324no—
96Dic3⋊C4—324no—
97C22×C32—212no—
98C22×C32—224no—
99C22×C32—212no—
100C22×C32C23×D314yes—
101C2×C3×D3—624no—
102C2×C3×D3—1216no—
103C32×C4—212no—
104C3×Dic3—624no—
105#127#27211yescharacteristic, complemented
106#126—92no—
107#123—93no—
108#121—96no—
109#120—93no—
110H(3,𝔽3)#56011yescharacteristic, complemented, Sylow
111C23×C3—34no—
112C23×C3—312no—
113C23×C3—92no—
114C23×C3—363no—
115C23×C3C22×C3⋊D311yescharacteristic
116C23×C3C22×C3⋊D313yes—
117C22×D3—38no—
118C22×D3—1212no—
119C3×D4—43no—
120C3×D4—43no—
121C3×D4—1212no—
122C3×D4—1212no—
123C3×D4—366no—
124C2×C3×C4—312no—
125C2×C3×C4—96no—
126C2×C3×C4—363no—
127C2×C3×C4C22×C3⋊D313yes—
128#67—1224no—
129C2×Dic3—324no—
130C2×Dic3—1212no—
131C2×C32—44no—
132C2×C32—412no—
133C2×C32#1004112yes—
134C3×D3—128no—
135C24—91no—
136C2×D4—96no—
137C2×D4C2×#27213yes—
138C2×D4C2×#27213yescomplemented
139C22×C4—93no—
140C4⋊C4—96no—
141C22⋊C4—93no—
142C22⋊C4—93no—
143C22⋊C4—96no—
144C42C2×#27211yescharacteristic, complemented
145C22×C3—23no—
146C22×C3—26no—
147C22×C3—23no—
148C22×C3—34no—
149C22×C3—612no—
150C22×C3—624no—
151C22×C3—612no—
152C22×C3—186no—
153C22×C3—364no—
154C22×C3C23×C3⋊D311yescharacteristic
155C2×D3—624no—
156C2×D3—1216no—
157C3×C4—23no—
158C3×C4—612no—
159C3×C4—186no—
160Dic3—624no—
161C32#572014yescomplemented
162C23—92no—
163C23—363no—
164C23C22×#27211yescharacteristic
165C23C22×#27213yes—
166D4—43no—
167D4—43no—
168D4—366no—
169C2×C4—96no—
170C2×C4—363no—
171C2×C4C22×#27213yes—
172C2×C3—312no—
173C2×C3—41no—
174C2×C3—43no—
175C2×C3—124no—
176C2×C3—1212no—
177C2×C3—362no—
178C2×C3#6436113yes—
179D3—128no—
180C22—23no—
181C22—26no—
182C22—23no—
183C22—186no—
184C22—364no—
185C22C23×#27211yescharacteristic
186C4—23no—
187C4—186no—
188C3—34no—
189C3#1059619111yescharacteristic
190C2—41no—
191C2—43no—
192C2—362no—
193C2#1172266413yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#127C2
hypercenterC3×#127C3⋊D3
hypocenterH(3,𝔽3)#560
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#127C2semidirect
2. #11721849C2semidirect
3. #11721847C2semidirect
4. #11721845C2semidirect
5. #11721722C2semidirect
9. C2×D4×H(3,𝔽3)C22semidirect
11. C22×#1084C22semidirect
12. C42×H(3,𝔽3)C22semidirect
14. #88552C22semidirect
16. #88540C22semidirect
17. C32×#127D3semidirect
36. C2×C32×D4C2×D3semidirect
39. C32×C42C2×D3semidirect
105. #127#272semidirect
110. H(3,𝔽3)#560semidirect
138. C2×D4C2×#272semidirect
144. C42C2×#272semidirect
161. C32#5720semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1272
#117218492
#117218472
#117218452
#117217222
C3×#57203