Finite Group Databasebeta

← #13147922

Subgroups — #13147922

Order 1728.

969 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 969 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C42×#272C211yesmaximal, characteristic, complemented
2#11721387C212yesmaximal, complemented
3#11721161C213yesmaximal
4#11721147C213yesmaximal, complemented
5#11721112C216yesmaximal, complemented
6D3×#876—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
8C2×C4×#272C2213yes—
9C2×#6199C2213yes—
10C42×H(3,𝔽3)C2211yescharacteristic, complemented
11C4×#1084C2213yescomplemented
12#3752⋊C3C2216yes—
13#3750⋊C3C2216yescomplemented
14#88450C2216yes—
15#88448C2216yescomplemented
16#64085D312yescomplemented
17C3×#876—32no—
18C3×C42×D3—32no—
19C3×C42×D3—61no—
20D3×(C2×D3)⋊C4—36no—
21D3×Dic3⋊C4—36no—
22#63868—36no—
23#63859—36no—
24C22×#272C2311yescharacteristic
25C22×(C3⋊D3)⋊C3C2312yes—
26C2×((C3⋊D3)⋊C3)⋊C2—26no—
27C2×C4×H(3,𝔽3)C2313yes—
28C2×#1084C2313yes—
29C4×#272—26no—
30C2×(C3⋊Dic3)⋊C3C2316yes—
31#6199—26no—
32D3×#126—92no—
33C2×#876—91no—
34C22×D32—32no—
35C2×C3×C4×D3—36no—
36C2×C3×C4×D3—63no—
37C2×D3×Dic3—36no—
38C32×C42—21no—
39C32×C42C2×D312yescomplemented
40#3752C2×D316yes—
41#3750C2×D316yescomplemented
42C3×(C2×D3)⋊C4—36no—
43C3×Dic3⋊C4—36no—
44C3×C4×Dic3—36no—
45C3×C4×Dic3—63no—
46#3724—36no—
47Dic3⋊Dic3—36no—
48#3722—36no—
49(C2×D3)⋊Dic3—36no—
50C22×H(3,𝔽3)C2411yescharacteristic
51C2×#272#4116yes—
52C2×(C3⋊D3)⋊C3—26no—
53((C3⋊D3)⋊C3)⋊C2—44no—
54C4×H(3,𝔽3)—23no—
55#1084—23no—
56(C3⋊Dic3)⋊C3—26no—
57C3×#126—92no—
58C2×C3×C42—91no—
59C2×(C2×D3)⋊C4—93no—
60C2×Dic3⋊C4—93no—
61#898—96no—
62D3×C4⋊C4—96no—
63D3×C22⋊C4—96no—
64#879—96no—
65#876—32no—
66#876—92no—
67#876—96no—
68C42×D3—32no—
69C42×D3—61no—
70C22×C3⋊D3C22×D312yes—
71C22×C3×D3—32no—
72C22×C3×D3—32no—
73C22×C3×D3—61no—
74C2×D32—612no—
75C2×C32×C4—23no—
76C2×C32×C4C22×D316yes—
77C2×C3⋊Dic3C22×D316yes—
78C2×C3×Dic3—36no—
79C2×C3×Dic3—36no—
80C2×C3×Dic3—63no—
81C3×C4×D3—612no—
82C3×C4×D3—612no—
83D3×Dic3—612no—
84C2×#126—271noSylow
85C2×H(3,𝔽3)C2×#4113yes—
86#272#12614yescomplemented
87(C3⋊D3)⋊C3—42no—
88C23×D3—91no—
89C23×D3—92no—
90C22×C3×C4—93no—
91C22×Dic3—93no—
92C2×C4×D3—36no—
93C2×C4×D3—63no—
94C2×C4×D3—96no—
95C3×C4⋊C4—96no—
96C3×C22⋊C4—96no—
97C3×C42—32no—
98C3×C42—61no—
99C3×C42—93no—
100C3×C42D3211yescharacteristic
101#217—96no—
102(C2×D3)⋊C4—36no—
103(C2×D3)⋊C4—96no—
104(C2×D3)⋊C4—96no—
105(C2×D3)⋊C4—96no—
106C4⋊Dic3—96no—
107Dic3⋊C4—36no—
108Dic3⋊C4—96no—
109Dic3⋊C4—96no—
110Dic3⋊C4—96no—
111C4×Dic3—36no—
112C4×Dic3—63no—
113C22×C32—21no—
114C22×C32C23×D312yes—
115C2×C3⋊D3—26no—
116C2×C3×D3—312no—
117C2×C3×D3—66no—
118C2×C3×D3—66no—
119D32—128no—
120C32×C4—26no—
121C32×C4—26no—
122C3⋊Dic3—26no—
123C3×Dic3—66no—
124C3×Dic3—66no—
125C3×Dic3—66no—
126#126—92no—
127#126—276no—
128C2×C4⋊C4—273no—
129C2×C22⋊C4—273no—
130C2×C42—91no—
131H(3,𝔽3)C2×#12611yescharacteristic, complemented, Sylow
132C23×C3—92no—
133C23×C3—91no—
134C22×D3—32no—
135C22×D3—32no—
136C22×D3—61no—
137C22×D3—92no—
138C22×D3—1812no—
139C22×D3—186no—
140C2×C3×C4—36no—
141C2×C3×C4—63no—
142C2×C3×C4—96no—
143C2×C3×C4—93no—
144C2×C3×C4—186no—
145C2×C3×C4C2×D3213yes—
146C2×Dic3—36no—
147C2×Dic3—36no—
148C2×Dic3—63no—
149C2×Dic3—96no—
150C2×Dic3—186no—
151C4×D3—612no—
152C4×D3—612no—
153C4×D3—1812no—
154C2×C32—23no—
155C2×C32D3×#4116yes—
156C3⋊D3—42no—
157C3×D3—38no—
158C3×D3—64no—
159C3×D3—122no—
160C24—271no—
161C22×C4—93no—
162C22×C4—273no—
163C4⋊C4—96no—
164C4⋊C4—276no—
165C22⋊C4—96no—
166C22⋊C4—276no—
167C42—93no—
168C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
169C22×C3—32no—
170C22×C3—61no—
171C22×C3—96no—
172C22×C3—186no—
173C22×C3C22×D3211yescharacteristic
174C2×D3—312no—
175C2×D3—66no—
176C2×D3—66no—
177C2×D3—186no—
178C2×D3—368no—
179C2×D3—364no—
180C3×C4—23no—
181C3×C4—66no—
182C3×C4—66no—
183C3×C4—186no—
184C3×C4—183no—
185Dic3—66no—
186Dic3—66no—
187Dic3—66no—
188Dic3—186no—
189C32—21no—
190C32D3×#12612yescomplemented
191C23—91no—
192C23—92no—
193C23—546no—
194C2×C4—93no—
195C2×C4—96no—
196C2×C4—186no—
197C2×C4—546no—
198C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
199C2×C3—36no—
200C2×C3—63no—
201C2×C3—94no—
202C2×C3—362no—
203C2×C3#6430613yes—
204D3—38no—
205D3—64no—
206D3—122no—
207D3—362no—
208C22—96no—
209C22—186no—
210C22—1084no—
211C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
212C4—23no—
213C4—183no—
214C4—186no—
215C3—32no—
216C3—61no—
217C3#1059402011yescharacteristic
218C2—94no—
219C2—362no—
220C2#1172249113yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#126
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C42×#272C2semidirect
2. #11721387C2semidirect
4. #11721147C2semidirect
5. #11721112C2semidirect
10. C42×H(3,𝔽3)C22semidirect
11. C4×#1084C22semidirect
13. #3750⋊C3C22semidirect
15. #88448C22semidirect
16. #64085D3semidirect
39. C32×C42C2×D3semidirect
41. #3750C2×D3semidirect
86. #272#126semidirect
131. H(3,𝔽3)C2×#126semidirect
168. C42((C3⋊D3)⋊C3)⋊C2semidirect
190. C32D3×#126semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
#117213872
#117211612
#117211472
#117211122
D3×#8763