Finite Group Databasebeta

← #64091⋊C3 #11721422

Subgroups — #64091⋊C3 #11721422

Order 864.

489 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C23×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C211yesmaximal, characteristic, complemented
3C2×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
4C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
5#3752⋊C3C212yesmaximal, complemented
6#64091C311yesmaximal, characteristic, complemented
7C3×#882—31nomaximal
8C23×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊D3)⋊C3—22no—
10C22×(C3⋊D3)⋊C3—24no—
11C22×(C3⋊D3)⋊C3C2212yes—
12C22×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
13C2×C4×H(3,𝔽3)C2212yescomplemented
14#638⋊C3—22no—
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
16C4⋊((C3⋊D3)⋊C3)—24no—
17C23×C3⋊D3C2×C311yescharacteristic, complemented
18C23×C3×D3—31no—
19C2×#638C2×C311yescharacteristic, complemented
20C2×C4⋊(C3⋊D3)C2×C312yescomplemented
21C2×C3×#67—31no—
22C2×C3×D12—32no—
23C32×C22⋊C4—21no—
24C32×C22⋊C4D311yescharacteristic, complemented
25C32×C22⋊C4C2×C311yescharacteristic, complemented
26#3752C2×C312yescomplemented
27C3×(C2×D3)⋊C4—32no—
28C22×H(3,𝔽3)—22no—
29C22×H(3,𝔽3)D412yescomplemented
30C22×H(3,𝔽3)C2311yescharacteristic
31C2×(C3⋊D3)⋊C3—28no—
32C2×(C3⋊D3)⋊C3—22no—
33C2×(C3⋊D3)⋊C3—22no—
34C2×(C3⋊D3)⋊C3—22no—
35C2×(C3⋊D3)⋊C3—21no—
36C2×(C3⋊D3)⋊C3D414yescomplemented
37C4×H(3,𝔽3)—22no—
38(C3⋊Dic3)⋊C3—21no—
39C3×#120—91no—
40#882—31no—
41#882—91no—
42C23×C32—21no—
43C23×C32C2×D311yescharacteristic
44C23×C32C22×C311yescharacteristic
45C22×C3⋊D3—22no—
46C22×C3⋊D3—24no—
47C22×C3⋊D3C22×C312yes—
48C22×C3⋊D3C22×C311yescharacteristic, complemented
49C22×C3×D3—32no—
50C22×C3×D3—31no—
51C22×C3×D3—64no—
52C22×C3×D3—62no—
53C2×C32×C4—22no—
54C2×C32×C4C2×D312yescomplemented
55C2×C32×C4C22×C312yescomplemented
56#638—22no—
57C2×C3⋊Dic3C22×C311yescharacteristic, complemented
58C4⋊(C3⋊D3)—24no—
59C3×#67—62no—
60C2×C3×Dic3—31no—
61C3×D12—64no—
62C2×H(3,𝔽3)—22no—
63C2×H(3,𝔽3)C2×D411yescharacteristic
64C2×H(3,𝔽3)C2×D412yes—
65(C3⋊D3)⋊C3—24no—
66(C3⋊D3)⋊C3—41no—
67C24×C3—91no—
68C23×D3—31no—
69C23×D3—91no—
70C2×C3×D4—92no—
71C2×C3×D4—91no—
72C2×#67—31no—
73C2×#67—91no—
74C2×D12—32no—
75C2×D12—92no—
76C3×C22⋊C4—31no—
77C3×C22⋊C4—31no—
78C3×C22⋊C4—61no—
79C3×C22⋊C4—92no—
80C3×C22⋊C4C3×D311yescharacteristic
81(C2×D3)⋊C4—32no—
82(C2×D3)⋊C4—92no—
83C22×C32—22no—
84C22×C32—22no—
85C22×C32—22no—
86C22×C32—21no—
87C22×C32—42no—
88C22×C32D1212yescomplemented
89C22×C32C3×D412yescomplemented
90C22×C32C22×D311yescharacteristic
91C22×C32C23×C311yescharacteristic
92C2×C3⋊D3—28no—
93C2×C3⋊D3—22no—
94C2×C3⋊D3—22no—
95C2×C3⋊D3—22no—
96C2×C3⋊D3—21no—
97C2×C3⋊D3C3×D414yescomplemented
98C2×C3×D3—34no—
99C2×C3×D3—68no—
100C2×C3×D3—62no—
101C2×C3×D3—62no—
102C2×C3×D3—61no—
103C2×C3×D3—62no—
104C32×C4—22no—
105C32×C4—22no—
106C32×C4—42no—
107C3⋊Dic3—21no—
108C3×Dic3—61no—
109#120—91noSylow
110H(3,𝔽3)#12011yescharacteristic, complemented, Sylow
111C23×C3—31no—
112C23×C3—31no—
113C23×C3—61no—
114C23×C3—92no—
115C23×C3—91no—
116C23×C3—182no—
117C23×C3—184no—
118C23×C3C2×C3×D311yescharacteristic
119C22×D3—32no—
120C22×D3—31no—
121C22×D3—64no—
122C22×D3—62no—
123C22×D3—92no—
124C22×D3—91no—
125C22×D3—184no—
126C22×D3—182no—
127C3×D4—184no—
128C3×D4—182no—
129C2×C3×C4—32no—
130C2×C3×C4—32no—
131C2×C3×C4—62no—
132C2×C3×C4—91no—
133C2×C3×C4C2×C3×D312yes—
134#67—62no—
135#67—182no—
136C2×Dic3—31no—
137C2×Dic3—91no—
138D12—64no—
139D12—184no—
140C2×C32—22no—
141C2×C32—22no—
142C2×C32—21no—
143C2×C32—22no—
144C2×C32—42no—
145C2×C32C2×D1211yescharacteristic
146C2×C32D3×D412yes—
147C2×C32C2×C3×D411yescharacteristic
148C2×C32C2×C3×D412yes—
149C3⋊D3—24no—
150C3⋊D3—41no—
151C3×D3—64no—
152C3×D3—121no—
153C24—91no—
154C2×D4—91no—
155C2×D4—92no—
156C22⋊C4—92no—
157C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
158C22×C3—22no—
159C22×C3—32no—
160C22×C3—32no—
161C22×C3—31no—
162C22×C3—31no—
163C22×C3—62no—
164C22×C3—62no—
165C22×C3—62no—
166C22×C3—61no—
167C22×C3—94no—
168C22×C3—122no—
169C22×C3—182no—
170C22×C3—182no—
171C22×C3—181no—
172C22×C3—182no—
173C22×C3—188no—
174C22×C3C3×D1212yes—
175C22×C3C22×C3×D311yescharacteristic
176C2×D3—34no—
177C2×D3—68no—
178C2×D3—62no—
179C2×D3—62no—
180C2×D3—61no—
181C2×D3—62no—
182C2×D3—94no—
183C2×D3—182no—
184C2×D3—182no—
185C2×D3—188no—
186C2×D3—182no—
187C2×D3—181no—
188C3×C4—22no—
189C3×C4—62no—
190C3×C4—62no—
191C3×C4—122no—
192C3×C4—181no—
193Dic3—61no—
194Dic3—181no—
195C32—21no—
196C32#88211yescharacteristic, complemented
197C32C3×#12011yescharacteristic, complemented
198C23—92no—
199C23—91no—
200C23—184no—
201C23—182no—
202C23C2×(C3⋊D3)⋊C311yescharacteristic
203D4—184no—
204D4—182no—
205C2×C4—91no—
206C2×C4C2×(C3⋊D3)⋊C312yescomplemented
207C2×C3—22no—
208C2×C3—31no—
209C2×C3—31no—
210C2×C3—32no—
211C2×C3—32no—
212C2×C3—62no—
213C2×C3—62no—
214C2×C3—61no—
215C2×C3—62no—
216C2×C3—122no—
217C2×C3—184no—
218C2×C3—361no—
219C2×C3C2×C3×D1211yescharacteristic
220C2×C3C3×D3×D412yes—
221D3—64no—
222D3—121no—
223D3—184no—
224D3—361no—
225C22—22no—
226C22—94no—
227C22—188no—
228C22—182no—
229C22—182no—
230C22—181no—
231C22—182no—
232C22C4⋊((C3⋊D3)⋊C3)12yescomplemented
233C22C22×(C3⋊D3)⋊C311yescharacteristic
234C4—22no—
235C4—181no—
236C3—31no—
237C3—31no—
238C3—61no—
239C3C3×#88211yescharacteristic
240C2—22no—
241C2—184no—
242C2—361no—
243C2C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
244C2D4×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC22⋊C4×H(3,𝔽3)C2
hypercenterC22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×(C3⋊D3)⋊C3C2semidirect
2. C2×#638⋊C3C2semidirect
3. C2×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C22⋊C4×H(3,𝔽3)C2semidirect
5. #3752⋊C3C2semidirect
6. #64091C3semidirect
12. C22×(C3⋊D3)⋊C3C22semidirect
13. C2×C4×H(3,𝔽3)C22semidirect
15. C2×(C3⋊Dic3)⋊C3C22semidirect
17. C23×C3⋊D3C2×C3semidirect
19. C2×#638C2×C3semidirect
20. C2×C4⋊(C3⋊D3)C2×C3semidirect
24. C32×C22⋊C4D3semidirect
25. C32×C22⋊C4C2×C3semidirect
26. #3752C2×C3semidirect
29. C22×H(3,𝔽3)D4semidirect
36. C2×(C3⋊D3)⋊C3D4semidirect
48. C22×C3⋊D3C22×C3semidirect
54. C2×C32×C4C2×D3semidirect
55. C2×C32×C4C22×C3semidirect
57. C2×C3⋊Dic3C22×C3semidirect
88. C22×C32D12semidirect
89. C22×C32C3×D4semidirect
97. C2×C3⋊D3C3×D4semidirect
110. H(3,𝔽3)#120semidirect
157. C22⋊C4(C3⋊D3)⋊C3semidirect
196. C32#882semidirect
197. C32C3×#120semidirect
206. C2×C4C2×(C3⋊D3)⋊C3semidirect
232. C22C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C23×(C3⋊D3)⋊C32
C2×#638⋊C32
C2×C4⋊((C3⋊D3)⋊C3)2
C22⋊C4×H(3,𝔽3)2
#3752⋊C32
#640913
C3×#8823