Finite Group Databasebeta

← #64095⋊C3 #11721426

Subgroups — #64095⋊C3 #11721426

Order 864.

307 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C2×#638⋊C3C211yesmaximal, characteristic
2C22×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#3752⋊C3C212yesmaximal, complemented
5C4⋊((C3⋊Dic3)⋊C3)C212yesmaximal, complemented
6#64095C311yesmaximal, characteristic, complemented
7C3×#886—31nomaximal
8C23×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×C4×H(3,𝔽3)C2212yes—
11#638⋊C3—22no—
12C2×(C3⋊Dic3)⋊C3—22no—
13C2×(C3⋊Dic3)⋊C3C2212yes—
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×#638C2×C311yescharacteristic
16C22×C3⋊Dic3C2×C311yescharacteristic, complemented
17C2×C3×#67—31no—
18C22×C3×Dic3—31no—
19C32×C22⋊C4—21no—
20C32×C22⋊C4D311yescharacteristic, complemented
21C32×C22⋊C4C2×C311yescharacteristic, complemented
22#3752C2×C312yescomplemented
23C4⋊(C3⋊Dic3)C2×C312yescomplemented
24C3×(C2×D3)⋊C4—32no—
25C3×C4⋊Dic3—32no—
26C22×H(3,𝔽3)—22no—
27C22×H(3,𝔽3)D412yes—
28C22×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊D3)⋊C3—21no—
30C2×(C3⋊D3)⋊C3—22no—
31C4×H(3,𝔽3)—22no—
32(C3⋊Dic3)⋊C3—22no—
33(C3⋊Dic3)⋊C3—21no—
34C3×#123—91no—
35#886—31no—
36#886—91no—
37C23×C32—21no—
38C23×C32C2×D311yescharacteristic
39C23×C32C22×C311yescharacteristic
40C22×C3⋊D3C22×C311yescharacteristic
41C22×C3×D3—31no—
42C2×C32×C4—22no—
43C2×C32×C4C2×D312yes—
44C2×C32×C4C22×C312yes—
45#638—22no—
46C2×C3⋊Dic3—22no—
47C2×C3⋊Dic3C22×C311yescharacteristic
48C2×C3⋊Dic3C22×C312yes—
49C3×#67—62no—
50C2×C3×Dic3—31no—
51C2×C3×Dic3—32no—
52C2×C3×Dic3—62no—
53C2×H(3,𝔽3)—22no—
54C2×H(3,𝔽3)C2×D411yescharacteristic
55C2×H(3,𝔽3)#4112yes—
56(C3⋊D3)⋊C3—41no—
57C2×C3×D4—91no—
58C22×C3×C4—91no—
59C2×#67—31no—
60C2×#67—91no—
61C22×Dic3—31no—
62C22×Dic3—91no—
63C3×C4⋊C4—92no—
64C3×C22⋊C4—31no—
65C3×C22⋊C4—31no—
66C3×C22⋊C4—61no—
67C3×C22⋊C4—92no—
68C3×C22⋊C4C3×D311yescharacteristic
69(C2×D3)⋊C4—32no—
70(C2×D3)⋊C4—92no—
71C4⋊Dic3—32no—
72C4⋊Dic3—92no—
73C22×C32—22no—
74C22×C32—22no—
75C22×C32—22no—
76C22×C32—21no—
77C22×C32—42no—
78C22×C32D1212yes—
79C22×C32C3×D412yes—
80C22×C32C22×D311yescharacteristic
81C22×C32C23×C311yescharacteristic
82C2×C3⋊D3—21no—
83C2×C3⋊D3—22no—
84C2×C3×D3—61no—
85C2×C3×D3—62no—
86C32×C4—22no—
87C32×C4—22no—
88C32×C4—42no—
89C3⋊Dic3—22no—
90C3⋊Dic3—21no—
91C3×Dic3—62no—
92C3×Dic3—61no—
93#123—91noSylow
94H(3,𝔽3)#12311yescharacteristic, complemented, Sylow
95C23×C3—31no—
96C23×C3—31no—
97C23×C3—61no—
98C23×C3—91no—
99C23×C3C2×C3×D311yescharacteristic
100C22×D3—31no—
101C22×D3—91no—
102C3×D4—182no—
103C2×C3×C4—32no—
104C2×C3×C4—32no—
105C2×C3×C4—62no—
106C2×C3×C4—92no—
107C2×C3×C4—91no—
108C2×C3×C4—182no—
109C2×C3×C4C2×C3×D312yes—
110#67—62no—
111#67—182no—
112C2×Dic3—31no—
113C2×Dic3—32no—
114C2×Dic3—62no—
115C2×Dic3—91no—
116C2×Dic3—92no—
117C2×Dic3—182no—
118C2×C32—22no—
119C2×C32—22no—
120C2×C32—21no—
121C2×C32—22no—
122C2×C32—42no—
123C2×C32C2×D1211yescharacteristic
124C2×C32#23712yes—
125C2×C32C2×C3×D411yescharacteristic
126C2×C32C3×#4112yes—
127C3⋊D3—41no—
128C3×D3—121no—
129C2×D4—91no—
130C22×C4—91no—
131C4⋊C4—92no—
132C22⋊C4—92no—
133C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
134C22×C3—22no—
135C22×C3—32no—
136C22×C3—32no—
137C22×C3—31no—
138C22×C3—31no—
139C22×C3—62no—
140C22×C3—62no—
141C22×C3—62no—
142C22×C3—61no—
143C22×C3—122no—
144C22×C3—181no—
145C22×C3—182no—
146C22×C3C3×D1212yes—
147C22×C3C22×C3×D311yescharacteristic
148C2×D3—61no—
149C2×D3—62no—
150C2×D3—181no—
151C2×D3—182no—
152C3×C4—22no—
153C3×C4—62no—
154C3×C4—62no—
155C3×C4—122no—
156C3×C4—182no—
157C3×C4—181no—
158Dic3—62no—
159Dic3—61no—
160Dic3—182no—
161Dic3—181no—
162C32—21no—
163C32#88611yescharacteristic, complemented
164C32C3×#12311yescharacteristic, complemented
165C23—91no—
166C23C2×(C3⋊D3)⋊C311yescharacteristic
167D4—182no—
168C2×C4—92no—
169C2×C4—91no—
170C2×C4—182no—
171C2×C4C2×(C3⋊D3)⋊C312yes—
172C2×C3—22no—
173C2×C3—31no—
174C2×C3—31no—
175C2×C3—32no—
176C2×C3—32no—
177C2×C3—62no—
178C2×C3—62no—
179C2×C3—61no—
180C2×C3—62no—
181C2×C3—122no—
182C2×C3—361no—
183C2×C3C2×C3×D1211yescharacteristic
184C2×C3C3×#23712yes—
185D3—121no—
186D3—361no—
187C22—22no—
188C22—181no—
189C22—182no—
190C22C4⋊((C3⋊D3)⋊C3)12yes—
191C22C22×(C3⋊D3)⋊C311yescharacteristic
192C4—22no—
193C4—182no—
194C4—181no—
195C3—31no—
196C3—31no—
197C3—61no—
198C3C3×#88611yescharacteristic
199C2—22no—
200C2—361no—
201C2C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
202C2#3830⋊C312yes—
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC22⋊C4×H(3,𝔽3)C2
prime radicalC2×#638⋊C3C2
hypercenterC22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×(C3⋊Dic3)⋊C3C2semidirect
3. C22⋊C4×H(3,𝔽3)C2semidirect
4. #3752⋊C3C2semidirect
5. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
6. #64095C3semidirect
16. C22×C3⋊Dic3C2×C3semidirect
20. C32×C22⋊C4D3semidirect
21. C32×C22⋊C4C2×C3semidirect
22. #3752C2×C3semidirect
23. C4⋊(C3⋊Dic3)C2×C3semidirect
94. H(3,𝔽3)#123semidirect
133. C22⋊C4(C3⋊D3)⋊C3semidirect
163. C32#886semidirect
164. C32C3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×#638⋊C32
C22×(C3⋊Dic3)⋊C32
C22⋊C4×H(3,𝔽3)2
#3752⋊C32
C4⋊((C3⋊Dic3)⋊C3)2
#640953
C3×#8863