Finite Group Databasebeta

← #126×#272 #13154250

Subgroups — #126×#272 #13154250

Order 1728.

1353 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1353 conjugacy classes of subgroups — too many to draw. Only its 201 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)×#126C211yesmaximal, characteristic, complemented
2#11721766C213yesmaximal, complemented
3C4⋊C4×#272C213yesmaximal, complemented
4C22⋊C4×#272C213yesmaximal
5#11721735C213yesmaximal, complemented
6#11721722C211yesmaximal, characteristic, complemented
7C42×#272C211yesmaximal, characteristic, complemented
8C3×D3×#126—34nomaximal
9C23×#272C2211yescharacteristic
10C2×C4×#272C2213yes—
11C2×C4×#272C2213yes—
12C4⋊C4×H(3,𝔽3)C2213yescomplemented
13C22⋊C4×H(3,𝔽3)C2213yes—
14C42×H(3,𝔽3)C2211yescharacteristic, complemented
15#88552C2213yes—
16#88542C2213yes—
17#88542C2213yes—
18C4⋊#1084C2213yescomplemented
19#88540C2213yescomplemented
20#88540C2213yescomplemented
21C4×#1084C2213yescomplemented
22C32×#126D314yescomplemented
23C3×#898—312no—
24C3×D3×C4⋊C4—312no—
25C3×D3×C22⋊C4—312no—
26C3×#879—312no—
27C3×#876—34no—
28C3×C42×D3—34no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×#272—26no—
31C22×#272C2311yescharacteristic
32C22×#272C2311yescharacteristic
33C2×C4×H(3,𝔽3)C2313yes—
34C2×C4×H(3,𝔽3)C2313yes—
35C2×#1084C2313yes—
36C2×#1084C2313yes—
37C4×#272—26no—
38C4×#272—26no—
39C2×C3×#126—91no—
40D3×#126—34no—
41C23×C3×D3—34no—
42C2×C3×C4×D3—312no—
43C2×C3×C4×D3—312no—
44C32×C4⋊C4C2×D3112yescomplemented
45C32×C22⋊C4C2×D3112yes—
46C32×C42C2×D314yescomplemented
47C3×#217—312no—
48C3×(C2×D3)⋊C4—312no—
49C3×(C2×D3)⋊C4—312no—
50C3×C4⋊Dic3—312no—
51C3×Dic3⋊C4—312no—
52C3×Dic3⋊C4—312no—
53C3×C4×Dic3—312no—
54C22×H(3,𝔽3)—23no—
55C22×H(3,𝔽3)C2411yescharacteristic
56C2×#272—23no—
57C2×#272—44no—
58C2×#272#4116yes—
59C4×H(3,𝔽3)—23no—
60C4×H(3,𝔽3)—23no—
61#1084—23no—
62#1084—23no—
63C3×#126—34no—
64C3×#126—93no—
65C3×#126—93no—
66C3×#126—91no—
67C3×#126C3⋊D311yescharacteristic
68C2×C3×C4⋊C4—93no—
69C2×C3×C22⋊C4—93no—
70C2×C3×C42—91no—
71#898—312no—
72D3×C4⋊C4—312no—
73D3×C22⋊C4—312no—
74#879—312no—
75#876—34no—
76C42×D3—34no—
77C23×C32C22×D314yes—
78C22×C3×D3—34no—
79C22×C3×D3—34no—
80C22×C3×D3—624no—
81C2×C32×C4C22×D3112yes—
82C2×C32×C4C22×D3112yes—
83C2×C3×Dic3—312no—
84C2×C3×Dic3—312no—
85C3×C4×D3—624no—
86C3×C4×D3—624no—
87C2×#126—91noSylow
88C2×H(3,𝔽3)—41no—
89C2×H(3,𝔽3)C2×#4113yes—
90#272—41no—
91#272#12614yesdirect product factor
92C24×C3—91no—
93C23×D3—34no—
94C22×C3×C4—93no—
95C22×C3×C4—93no—
96C2×C4×D3—312no—
97C2×C4×D3—312no—
98C3×C4⋊C4—312no—
99C3×C4⋊C4—93no—
100C3×C4⋊C4—93no—
101C3×C4⋊C4—93no—
102C3×C4⋊C4C2×C3⋊D313yes—
103C3×C22⋊C4—312no—
104C3×C22⋊C4—93no—
105C3×C22⋊C4—93no—
106C3×C22⋊C4—93no—
107C3×C22⋊C4C2×C3⋊D313yes—
108C3×C42—34no—
109C3×C42—93no—
110C3×C42C2×C3⋊D311yescharacteristic
111#217—312no—
112(C2×D3)⋊C4—312no—
113(C2×D3)⋊C4—312no—
114C4⋊Dic3—312no—
115Dic3⋊C4—312no—
116Dic3⋊C4—312no—
117C4×Dic3—312no—
118C22×C32—212no—
119C22×C32C23×D314yes—
120C2×C3×D3—324no—
121C2×C3×D3—612no—
122C2×C3×D3—1216no—
123C32×C4—212no—
124C32×C4—212no—
125C3×Dic3—612no—
126C3×Dic3—612no—
127#126—93no—
128#126—93no—
129#126—91no—
130#126#27211yescharacteristic, direct product factor
131C2×C4⋊C4—93no—
132C2×C22⋊C4—93no—
133C2×C42—91no—
134H(3,𝔽3)C2×#12611yescharacteristic, complemented, Sylow
135C23×C3—34no—
136C23×C3—91no—
137C23×C3—91no—
138C23×C3—186no—
139C23×C3C22×C3⋊D311yescharacteristic
140C22×D3—34no—
141C22×D3—34no—
142C22×D3—624no—
143C2×C3×C4—312no—
144C2×C3×C4—312no—
145C2×C3×C4—93no—
146C2×C3×C4—93no—
147C2×C3×C4—186no—
148C2×C3×C4—186no—
149C2×C3×C4C22×C3⋊D313yes—
150C2×C3×C4C22×C3⋊D313yes—
151C2×Dic3—312no—
152C2×Dic3—312no—
153C4×D3—624no—
154C4×D3—624no—
155C2×C32—44no—
156C2×C32D3×#41112yes—
157C3×D3—316no—
158C3×D3—124no—
159C24—91no—
160C22×C4—93no—
161C22×C4—93no—
162C4⋊C4—93no—
163C4⋊C4—93no—
164C4⋊C4—93no—
165C4⋊C4C2×#27213yescomplemented
166C22⋊C4—93no—
167C22⋊C4—93no—
168C22⋊C4—93no—
169C22⋊C4C2×#27213yes—
170C42—93no—
171C42C2×#27211yescharacteristic, complemented
172C22×C3—23no—
173C22×C3—34no—
174C22×C3—612no—
175C22×C3—96no—
176C22×C3—183no—
177C22×C3—364no—
178C22×C3C23×C3⋊D311yescharacteristic
179C2×D3—324no—
180C2×D3—612no—
181C2×D3—1216no—
182C3×C4—23no—
183C3×C4—23no—
184C3×C4—612no—
185C3×C4—612no—
186C3×C4—183no—
187C3×C4—183no—
188Dic3—612no—
189Dic3—612no—
190C32D3×#12614yescomplemented
191C23—91no—
192C23—91no—
193C23—186no—
194C23C22×#27211yescharacteristic
195C2×C4—93no—
196C2×C4—93no—
197C2×C4—186no—
198C2×C4—186no—
199C2×C4C22×#27213yes—
200C2×C4C22×#27213yes—
201C2×C3—312no—
202C2×C3—41no—
203C2×C3—94no—
204C2×C3—124no—
205C2×C3—361no—
206C2×C3#41×C3⋊D313yes—
207D3—316no—
208D3—124no—
209C22—23no—
210C22—96no—
211C22—183no—
212C22—364no—
213C22C23×#27211yescharacteristic
214C4—23no—
215C4—23no—
216C4—183no—
217C4—183no—
218C3—34no—
219C3C3⋊D3×#12611yescharacteristic
220C2—41no—
221C2—94no—
222C2—361no—
223C2#41×#27213yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#126C2
prime radicalC23×#272C22
hypercenterC3×#126C3⋊D3
hypocenterH(3,𝔽3)C2×#126
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#126

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#126C2semidirect
2. #11721766C2semidirect
3. C4⋊C4×#272C2semidirect
5. #11721735C2semidirect
6. #11721722C2semidirect
7. C42×#272C2semidirect
12. C4⋊C4×H(3,𝔽3)C22semidirect
14. C42×H(3,𝔽3)C22semidirect
18. C4⋊#1084C22semidirect
19. #88540C22semidirect
20. #88540C22semidirect
21. C4×#1084C22semidirect
22. C32×#126D3semidirect
44. C32×C4⋊C4C2×D3semidirect
46. C32×C42C2×D3semidirect
91. #272#126direct (N × H)
130. #126#272direct (N × H)
134. H(3,𝔽3)C2×#126semidirect
165. C4⋊C4C2×#272semidirect
171. C42C2×#272semidirect
190. C32D3×#126semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1262
#117217662
C4⋊C4×#2722
C22⋊C4×#2722
#117217352
#117217222
C42×#2722
C3×D3×#1263