Finite Group Databasebeta

← #11721766

Subgroups — #11721766

Order 864.

330 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#88542C211yesmaximal, characteristic
3#88542C211yesmaximal, characteristic
4#88542C211yesmaximal, characteristic
5C4⋊#1084C211yesmaximal, characteristic, complemented
6#88540C211yesmaximal, characteristic, complemented
7C4×#1084C211yesmaximal, characteristic, complemented
8C3×#898—32nomaximal
9C3×#898—32nomaximal
10C22×#272C2211yescharacteristic
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×C4×H(3,𝔽3)C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#1084C2211yescharacteristic
15C2×#1084C2211yescharacteristic
16C2×#1084C2211yescharacteristic
17C32×C4⋊C4D312yescomplemented
18C32×C4⋊C4D312yescomplemented
19C3×(C2×D3)⋊C4—32no—
20C3×(C2×D3)⋊C4—32no—
21C3×(C2×D3)⋊C4—32no—
22C3×(C2×D3)⋊C4—32no—
23C3×(C2×D3)⋊C4—32no—
24C3×(C2×D3)⋊C4—32no—
25C3×C4⋊Dic3—32no—
26C3×C4⋊Dic3—32no—
27C3×Dic3⋊C4—32no—
28C3×Dic3⋊C4—32no—
29C3×C4×Dic3—32no—
30C3×C4×Dic3—32no—
31C22×H(3,𝔽3)C2311yescharacteristic
32C2×#272—21no—
33C2×#272—21no—
34C2×#272—21no—
35C4×H(3,𝔽3)—21no—
36C4×H(3,𝔽3)—21no—
37C4×H(3,𝔽3)—21no—
38#1084—21no—
39#1084—21no—
40#1084—21no—
41C3×#126—91no—
42#898—32no—
43#898—32no—
44C22×C3×D3—32no—
45C22×C3×D3—32no—
46C2×C32×C4C2×D312yes—
47C2×C32×C4C2×D312yes—
48C2×C32×C4C2×D312yes—
49C2×C32×C4C2×D312yes—
50C2×C32×C4C2×D312yes—
51C2×C32×C4C2×D312yes—
52C2×C3×Dic3—32no—
53C2×C3×Dic3—32no—
54C2×C3×Dic3—32no—
55C2×C3×Dic3—32no—
56C2×C3×Dic3—32no—
57C2×C3×Dic3—32no—
58C2×H(3,𝔽3)#4111yescharacteristic
59C2×H(3,𝔽3)#4111yescharacteristic
60C2×H(3,𝔽3)#4111yescharacteristic
61#272—41no—
62C3×C4⋊C4—32no—
63C3×C4⋊C4—62no—
64C3×C4⋊C4—91no—
65C3×C4⋊C4—91no—
66C3×C4⋊C4C3⋊D311yescharacteristic
67C3×C22⋊C4—91no—
68C3×C22⋊C4—91no—
69C3×C22⋊C4—91no—
70C3×C42—91no—
71(C2×D3)⋊C4—32no—
72(C2×D3)⋊C4—32no—
73(C2×D3)⋊C4—32no—
74(C2×D3)⋊C4—32no—
75(C2×D3)⋊C4—32no—
76(C2×D3)⋊C4—32no—
77C4⋊Dic3—32no—
78C4⋊Dic3—32no—
79Dic3⋊C4—32no—
80Dic3⋊C4—32no—
81C4×Dic3—32no—
82C4×Dic3—32no—
83C22×C32C22×D312yes—
84C22×C32C22×D312yes—
85C2×C3×D3—62no—
86C2×C3×D3—62no—
87C2×C3×D3—62no—
88C2×C3×D3—62no—
89C2×C3×D3—62no—
90C2×C3×D3—62no—
91C32×C4—22no—
92C32×C4—22no—
93C32×C4—22no—
94C32×C4—22no—
95C32×C4—22no—
96C32×C4—22no—
97C3×Dic3—62no—
98C3×Dic3—62no—
99C3×Dic3—62no—
100C3×Dic3—62no—
101C3×Dic3—62no—
102C3×Dic3—62no—
103#126—91noSylow
104H(3,𝔽3)#12611yescharacteristic, complemented, Sylow
105C23×C3—91no—
106C22×D3—32no—
107C22×D3—32no—
108C2×C3×C4—32no—
109C2×C3×C4—32no—
110C2×C3×C4—32no—
111C2×C3×C4—62no—
112C2×C3×C4—62no—
113C2×C3×C4—62no—
114C2×C3×C4—91no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4C2×C3⋊D311yescharacteristic
118C2×C3×C4C2×C3⋊D311yescharacteristic
119C2×C3×C4C2×C3⋊D311yescharacteristic
120C2×Dic3—32no—
121C2×Dic3—32no—
122C2×Dic3—32no—
123C2×Dic3—32no—
124C2×Dic3—32no—
125C2×Dic3—32no—
126C2×C32#23512yes—
127C2×C32#23512yes—
128C2×C32#23712yes—
129C2×C32#23712yes—
130C2×C32#23912yes—
131C2×C32#23912yes—
132C3×D3—122no—
133C3×D3—122no—
134C4⋊C4—91no—
135C4⋊C4—91no—
136C4⋊C4#27211yescharacteristic, complemented
137C22⋊C4—91no—
138C22⋊C4—91no—
139C22⋊C4—91no—
140C42—91no—
141C22×C3—32no—
142C22×C3—62no—
143C22×C3—181no—
144C22×C3—181no—
145C22×C3—181no—
146C22×C3C22×C3⋊D311yescharacteristic
147C2×D3—62no—
148C2×D3—62no—
149C2×D3—62no—
150C2×D3—62no—
151C2×D3—62no—
152C2×D3—62no—
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—62no—
157C3×C4—62no—
158C3×C4—62no—
159C3×C4—62no—
160C3×C4—122no—
161C3×C4—122no—
162C3×C4—181no—
163C3×C4—181no—
164C3×C4—181no—
165Dic3—62no—
166Dic3—62no—
167Dic3—62no—
168Dic3—62no—
169Dic3—62no—
170Dic3—62no—
171C32#89812yescomplemented
172C32#89812yescomplemented
173C23—91no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4C2×#27211yescharacteristic
178C2×C4C2×#27211yescharacteristic
179C2×C4C2×#27211yescharacteristic
180C2×C3—32no—
181C2×C3—32no—
182C2×C3—32no—
183C2×C3—62no—
184C2×C3—62no—
185C2×C3—62no—
186C2×C3—361no—
187C2×C3#382811yescharacteristic
188C2×C3#383011yescharacteristic
189C2×C3#383211yescharacteristic
190D3—122no—
191D3—122no—
192C22—181no—
193C22—181no—
194C22—181no—
195C22C22×#27211yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4—181no—
200C4—181no—
201C4—181no—
202C3—32no—
203C3—62no—
204C3#6410711yescharacteristic
205C2—361no—
206C2#8874011yescharacteristic
207C2#8874411yescharacteristic
208C2#8874811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC4⋊C4×H(3,𝔽3)C2
prime radicalC22×#272C22
hypercenterC3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#126
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×H(3,𝔽3)C2semidirect
5. C4⋊#1084C2semidirect
6. #88540C2semidirect
7. C4×#1084C2semidirect
17. C32×C4⋊C4D3semidirect
18. C32×C4⋊C4D3semidirect
104. H(3,𝔽3)#126semidirect
136. C4⋊C4#272semidirect
171. C32#898semidirect
172. C32#898semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×H(3,𝔽3)2
#885422
C4⋊#10842
#885402
C4×#10842
C3×#8983