Finite Group Databasebeta

← C2×#88654 #11722488

Subgroups — C2×#88654 #11722488

Order 864.

880 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C22×((C3⋊D3)⋊C3)⋊C2C212yesmaximal, complemented
2C2×C4⋊#272C211yesmaximal, characteristic, complemented
3C2×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4C2×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5C2×#6200C212yesmaximal, complemented
6#88654C218yesmaximal, direct product factor
7C2×#3802—31nomaximal
8C2×D3×D12—31nomaximal
9C22×#272C2212yescomplemented
10C22×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2212yescomplemented
12C2×((C3⋊D3)⋊C3)⋊C2—28no—
13C2×((C3⋊D3)⋊C3)⋊C2C2218yescomplemented
14C2×C4×H(3,𝔽3)C2211yescharacteristic, complemented
15C4⋊#272C2214yescomplemented
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
17C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
18C4×(C3⋊D3)⋊C3C2214yescomplemented
19#6200C2218yescomplemented
20C22×D32—32no—
21C22×D32—32no—
22C2×C4⋊(C3⋊D3)D311yescharacteristic, complemented
23C2×C4×C3⋊D3D311yescharacteristic, complemented
24C2×C3×D12—31no—
25C2×C3×D12—31no—
26C2×C3×D12—31no—
27C2×C3×D12—61no—
28C2×C3×C4×D3—31no—
29C2×#626—32no—
30C2×#625—32no—
31#3802—38no—
32D3×D12—38no—
33C22×H(3,𝔽3)C2311yescharacteristic
34C2×#272—24no—
35C2×#272C2314yescomplemented
36C2×(C3⋊D3)⋊C3—24no—
37C2×(C3⋊D3)⋊C3D414yescomplemented
38C2×(C3⋊D3)⋊C3C2312yes—
39C2×(C3⋊D3)⋊C3C2314yescomplemented
40((C3⋊D3)⋊C3)⋊C2—216no—
41C4×H(3,𝔽3)C2312yescomplemented
42(C3⋊Dic3)⋊C3C2312yescomplemented
43C2×D3×D4—91no—
44C2×D3×D4—91no—
45C22×D12—91no—
46C22×C3⋊D3C2×D311yescharacteristic
47C22×C3⋊D3C2×D312yescomplemented
48C22×C3×D3—31no—
49C22×C3×D3—32no—
50C22×C3×D3—32no—
51C22×C3×D3—32no—
52C22×C3×D3—62no—
53C2×D32—38no—
54C2×D32—38no—
55C2×D32—68no—
56C2×D32—68no—
57C2×C32×C4—21no—
58C2×C32×C4C2×D311yescharacteristic, complemented
59C2×C32×C4C2×D311yescharacteristic, complemented
60C2×C3⋊Dic3C2×D311yescharacteristic, complemented
61C4⋊(C3⋊D3)C2×D314yescomplemented
62C4×C3⋊D3C2×D314yescomplemented
63C2×C3×Dic3—31no—
64C3×D12—34no—
65C3×D12—34no—
66C3×D12—34no—
67C3×D12—64no—
68C3×C4×D3—34no—
69#626—38no—
70#625—38no—
71C2×H(3,𝔽3)C2×D412yescomplemented
72C2×H(3,𝔽3)C2411yescharacteristic
73#272—24no—
74(C3⋊D3)⋊C3—24no—
75(C3⋊D3)⋊C3C2×D414yescomplemented
76C23×D3—92no—
77C23×D3—92no—
78C23×D3—92no—
79C2×C3×D4—91no—
80C2×C3×D4—91no—
81C22×C3×C4—91no—
82C2×#67—92no—
83C2×#67—92no—
84D3×D4—98no—
85D3×D4—98no—
86C2×D12—31no—
87C2×D12—31no—
88C2×D12—31no—
89C2×D12—61no—
90C2×D12—91no—
91C2×D12—98no—
92C2×D12—92no—
93C2×C4×D3—31no—
94C2×C4×D3—91no—
95C22×C32—21no—
96C22×C32C22×D311yescharacteristic
97C22×C32C22×D311yescharacteristic
98C2×C3⋊D3—24no—
99C2×C3⋊D3D1214yescomplemented
100C2×C3⋊D3C22×D312yes—
101C2×C3⋊D3C22×D314yescomplemented
102C2×C3×D3—34no—
103C2×C3×D3—32no—
104C2×C3×D3—34no—
105C2×C3×D3—34no—
106C2×C3×D3—34no—
107C2×C3×D3—64no—
108C2×C3×D3—64no—
109C2×C3×D3—64no—
110C2×C3×D3—64no—
111C2×C3×D3—124no—
112D32—616no—
113D32—616no—
114C32×C4—22no—
115C32×C4C22×D312yescomplemented
116C32×C4C22×D312yescomplemented
117C3⋊Dic3C22×D312yescomplemented
118C3×Dic3—32no—
119C22×D4—271noSylow
120H(3,𝔽3)C22×D411yescharacteristic, complemented, Sylow
121C23×C3—91no—
122C23×C3—92no—
123C23×C3—92no—
124C22×D3—31no—
125C22×D3—32no—
126C22×D3—32no—
127C22×D3—32no—
128C22×D3—62no—
129C22×D3—91no—
130C22×D3—92no—
131C22×D3—98no—
132C22×D3—98no—
133C22×D3—98no—
134C22×D3—188no—
135C22×D3—188no—
136C22×D3—188no—
137C3×D4—94no—
138C3×D4—94no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—61no—
142C2×C3×C4—94no—
143C2×C3×C4—91no—
144C2×C3×C4D3211yescharacteristic
145#67—98no—
146#67—98no—
147C2×Dic3—31no—
148C2×Dic3—91no—
149D12—34no—
150D12—34no—
151D12—34no—
152D12—64no—
153D12—94no—
154D12—98no—
155C4×D3—34no—
156C4×D3—94no—
157C2×C32—21no—
158C2×C32—22no—
159C2×C32C2×D1212yescomplemented
160C2×C32D3×D412yescomplemented
161C2×C32C23×D311yescharacteristic
162C2×C32C23×D311yescharacteristic
163C3⋊D3—24no—
164C3⋊D3C2×D1214yescomplemented
165C3×D3—34no—
166C3×D3—64no—
167C3×D3—64no—
168C3×D3—64no—
169C3×D3—124no—
170C24—272no—
171C2×D4—91no—
172C2×D4—91no—
173C2×D4—278no—
174C2×D4—272no—
175C22×C4—91no—
176C22×C3—31no—
177C22×C3—31no—
178C22×C3—61no—
179C22×C3—92no—
180C22×C3—94no—
181C22×C3—94no—
182C22×C3—94no—
183C22×C3—184no—
184C22×C3—184no—
185C22×C3C2×D3211yescharacteristic
186C2×D3—32no—
187C2×D3—34no—
188C2×D3—34no—
189C2×D3—34no—
190C2×D3—34no—
191C2×D3—64no—
192C2×D3—64no—
193C2×D3—64no—
194C2×D3—64no—
195C2×D3—92no—
196C2×D3—94no—
197C2×D3—94no—
198C2×D3—124no—
199C2×D3—184no—
200C2×D3—1816no—
201C2×D3—1816no—
202C2×D3—1816no—
203C3×C4—32no—
204C3×C4—32no—
205C3×C4—62no—
206C3×C4—92no—
207C3×C4C2×D3212yes—
208Dic3—32no—
209Dic3—92no—
210C32—21no—
211C32C22×D1211yescharacteristic, complemented
212C32C2×D3×D411yescharacteristic, complemented
213C23—91no—
214C23—92no—
215C23—92no—
216C23—278no—
217C23—548no—
218D4—94no—
219D4—94no—
220D4—278no—
221C2×C4—91no—
222C2×C4—94no—
223C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—32no—
227C2×C3—32no—
228C2×C3—61no—
229C2×C3—62no—
230C2×C3—94no—
231C2×C3—184no—
232C2×C3—184no—
233C2×C3D3×D1212yes—
234C2×C3C22×D3211yescharacteristic
235D3—34no—
236D3—64no—
237D3—64no—
238D3—64no—
239D3—94no—
240D3—124no—
241D3—184no—
242C22—92no—
243C22—94no—
244C22—94no—
245C22—94no—
246C22—184no—
247C22—184no—
248C22—5416no—
249C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
250C4—92no—
251C4C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
252C3—31no—
253C3—31no—
254C3—61no—
255C3C2×D3×D1211yescharacteristic
256C2—94no—
257C2—184no—
258C2—184no—
259C2#8865412yesdirect product factor
260C2C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C24
FrattiniC2×C3C22×D32
FittingC2×C4×H(3,𝔽3)C22
socleC22×C3C2×D32
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×((C3⋊D3)⋊C3)⋊C2C2semidirect
2. C2×C4⋊#272C2semidirect
3. C2×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C2×C4×(C3⋊D3)⋊C3C2semidirect
5. C2×#6200C2semidirect
6. #88654C2direct (N × H)
9. C22×#272C22semidirect
11. C22×(C3⋊D3)⋊C3C22semidirect
13. C2×((C3⋊D3)⋊C3)⋊C2C22semidirect
14. C2×C4×H(3,𝔽3)C22semidirect
15. C4⋊#272C22semidirect
16. C2×(C3⋊Dic3)⋊C3C22semidirect
17. C4⋊((C3⋊D3)⋊C3)C22semidirect
18. C4×(C3⋊D3)⋊C3C22semidirect
19. #6200C22semidirect
22. C2×C4⋊(C3⋊D3)D3semidirect
23. C2×C4×C3⋊D3D3semidirect
35. C2×#272C23semidirect
37. C2×(C3⋊D3)⋊C3D4semidirect
39. C2×(C3⋊D3)⋊C3C23semidirect
41. C4×H(3,𝔽3)C23semidirect
42. (C3⋊Dic3)⋊C3C23semidirect
47. C22×C3⋊D3C2×D3semidirect
58. C2×C32×C4C2×D3semidirect
59. C2×C32×C4C2×D3semidirect
60. C2×C3⋊Dic3C2×D3semidirect
61. C4⋊(C3⋊D3)C2×D3semidirect
62. C4×C3⋊D3C2×D3semidirect
71. C2×H(3,𝔽3)C2×D4semidirect
75. (C3⋊D3)⋊C3C2×D4semidirect
99. C2×C3⋊D3D12semidirect
101. C2×C3⋊D3C22×D3semidirect
115. C32×C4C22×D3semidirect
116. C32×C4C22×D3semidirect
117. C3⋊Dic3C22×D3semidirect
120. H(3,𝔽3)C22×D4semidirect
159. C2×C32C2×D12semidirect
160. C2×C32D3×D4semidirect
164. C3⋊D3C2×D12semidirect
211. C32C22×D12semidirect
212. C32C2×D3×D4semidirect
223. C2×C4((C3⋊D3)⋊C3)⋊C2semidirect
251. C4C2×((C3⋊D3)⋊C3)⋊C2semidirect
259. C2#88654direct (N × H)

Maximal subgroups

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

subgroupindex
C22×((C3⋊D3)⋊C3)⋊C22
C2×C4⋊#2722
C2×C4⋊((C3⋊D3)⋊C3)2
C2×C4×(C3⋊D3)⋊C32
C2×#62002
#886542
C2×#38023
C2×D3×D123