Finite Group Databasebeta

← C2×#13091501 #72228902

Subgroups — C2×#13091501 #72228902

Order 2916.

2247 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 2247 conjugacy classes of subgroups — too many to draw. Only its 180 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
1#13091501C212yesmaximal, direct product factor
2C2×H(3,𝔽3)2C211yesmaximal, characteristic, complemented
3C2×C3⋊(C3⋊#272)C311yesmaximal, characteristic, complemented
4C2×C3×C3⋊#272—31nomaximal
5C2×C3×C3⋊((C3⋊D3)⋊C3)—34nomaximal
6C2×#92300—38nomaximal
7H(3,𝔽3)2C2211yescharacteristic, complemented, Sylow
8C2×C32×H(3,𝔽3)—21no—
9C2×C32×H(3,𝔽3)D311yescharacteristic, complemented
10C2×C32×H(3,𝔽3)D314yescomplemented
11C2×C32×H(3,𝔽3)C2×C311yescharacteristic, complemented
12C3⋊(C3⋊#272)C2×C312yescomplemented
13C3×C3⋊#272—32no—
14C3×C3⋊((C3⋊D3)⋊C3)—38no—
15C2×C32⋊H(3,𝔽3)—24no—
16C2×C32⋊H(3,𝔽3)—28no—
17C2×C32⋊H(3,𝔽3)D318yescomplemented
18#92300—316no—
19C2×C3×C3⋊(C3⋊D3)—34no—
20C2×C32×C3⋊D3—94no—
21C2×C3⋊#272—31no—
22C2×C3⋊#272—38no—
23C2×C3⋊#272—99no—
24C2×C3×#272—99no—
25C2×C3⋊((C3⋊D3)⋊C3)—312no—
26C2×C3×(C3⋊D3)⋊C3—91no—
27C2×C3×(C3⋊D3)⋊C3—98no—
28C32×H(3,𝔽3)—21no—
29C32×H(3,𝔽3)C2×D311yescharacteristic, complemented
30C32×H(3,𝔽3)C2×D314yescomplemented
31C32×H(3,𝔽3)C22×C311yescharacteristic, complemented
32C32⋊H(3,𝔽3)—24no—
33C32⋊H(3,𝔽3)—28no—
34C32⋊H(3,𝔽3)C2×D318yescomplemented
35C2×C34—24no—
36C2×C34C3×D314yescomplemented
37C2×C34C3⋊D314yescomplemented
38C3×C3⋊(C3⋊D3)—38no—
39C32×C3⋊D3—98no—
40C2×C3×H(3,𝔽3)—28no—
41C2×C3×H(3,𝔽3)—24no—
42C2×C3×H(3,𝔽3)—24no—
43C2×C3×H(3,𝔽3)—28no—
44C2×C3×H(3,𝔽3)—224no—
45C2×C3×H(3,𝔽3)—39no—
46C2×C3×H(3,𝔽3)—312no—
47C2×C3×H(3,𝔽3)—39no—
48C2×C3×H(3,𝔽3)—612no—
49C2×C3×H(3,𝔽3)—69no—
50C2×C3×H(3,𝔽3)C3×D311yescharacteristic
51C2×C3×H(3,𝔽3)C3×D318yescomplemented
52C2×C3×H(3,𝔽3)C3⋊D311yescharacteristic
53C2×C3×H(3,𝔽3)C3⋊D318yescomplemented
54C3⋊#272—32no—
55C3⋊#272—316no—
56C3⋊#272—918no—
57C3×#272—918no—
58C3⋊((C3⋊D3)⋊C3)—324no—
59C3×(C3⋊D3)⋊C3—92no—
60C3×(C3⋊D3)⋊C3—916no—
61C2×C3⋊(C3⋊D3)—34no—
62C2×C3×C3⋊D3—912no—
63C2×C3×C3⋊D3—94no—
64C2×C3×C3⋊D3—98no—
65C2×C3×C3⋊D3—91no—
66C2×C3×C3⋊D3—2712no—
67C2×C32×D3—2712no—
68C2×C32×D3—271no—
69C2×#272—99no—
70C2×#272—2724no—
71C2×(C3⋊D3)⋊C3—924no—
72C2×(C3⋊D3)⋊C3—93no—
73C34—24no—
74C34C2×C3×D314yescomplemented
75C34C2×C3⋊D314yescomplemented
76C3×H(3,𝔽3)—28no—
77C3×H(3,𝔽3)—28no—
78C3×H(3,𝔽3)—24no—
79C3×H(3,𝔽3)—24no—
80C3×H(3,𝔽3)—224no—
81C3×H(3,𝔽3)—39no—
82C3×H(3,𝔽3)—312no—
83C3×H(3,𝔽3)—39no—
84C3×H(3,𝔽3)—612no—
85C3×H(3,𝔽3)—69no—
86C3×H(3,𝔽3)C2×C3×D311yescharacteristic
87C3×H(3,𝔽3)C2×C3×D318yescomplemented
88C3×H(3,𝔽3)C2×C3⋊D311yescharacteristic
89C3×H(3,𝔽3)C2×C3⋊D318yescomplemented
90C2×C33—21no—
91C2×C33—24no—
92C2×C33—28no—
93C2×C33—34no—
94C2×C33—312no—
95C2×C33—312no—
96C2×C33—312no—
97C2×C33—64no—
98C2×C33—612no—
99C2×C33—612no—
100C2×C33—184no—
101C2×C33—184no—
102C2×C33—188no—
103C2×C33C3×C3⋊D311yescharacteristic
104C2×C33C3×C3⋊D318yes—
105C2×C33C3×C3⋊D314yes—
106C2×C33C3⋊(C3⋊D3)11yescharacteristic
107C3⋊(C3⋊D3)—38no—
108C3×C3⋊D3—924no—
109C3×C3⋊D3—98no—
110C3×C3⋊D3—916no—
111C3×C3⋊D3—92no—
112C3×C3⋊D3—2724no—
113C32×D3—2724no—
114C32×D3—272no—
115C2×H(3,𝔽3)—23no—
116C2×H(3,𝔽3)—324no—
117C2×H(3,𝔽3)—324no—
118C2×H(3,𝔽3)—624no—
119C2×H(3,𝔽3)—68no—
120C2×H(3,𝔽3)—612no—
121C2×H(3,𝔽3)—624no—
122C2×H(3,𝔽3)—64no—
123C2×H(3,𝔽3)—1824no—
124C2×H(3,𝔽3)(C3⋊D3)⋊C319yescomplemented
125C2×H(3,𝔽3)#27213yescomplemented
126#272—918no—
127#272—2748no—
128(C3⋊D3)⋊C3—948no—
129(C3⋊D3)⋊C3—96no—
130C22×C32—811no—
131C2×C3⋊D3—94no—
132C2×C3⋊D3—98no—
133C2×C3⋊D3—91no—
134C2×C3⋊D3—2712no—
135C2×C3×D3—2736no—
136C2×C3×D3—2712no—
137C2×C3×D3—273no—
138C2×C3×D3—271no—
139C2×C3×D3—818no—
140C2×C3×D3—811no—
141C33—21no—
142C33—24no—
143C33—28no—
144C33—34no—
145C33—312no—
146C33—312no—
147C33—312no—
148C33—64no—
149C33—612no—
150C33—612no—
151C33—184no—
152C33—184no—
153C33—188no—
154C33C2×C3×C3⋊D311yescharacteristic
155C33C2×C3×C3⋊D318yes—
156C33C2×C3×C3⋊D314yes—
157C33C2×C3⋊(C3⋊D3)11yescharacteristic
158H(3,𝔽3)—23no—
159H(3,𝔽3)—324no—
160H(3,𝔽3)—324no—
161H(3,𝔽3)—624no—
162H(3,𝔽3)—68no—
163H(3,𝔽3)—612no—
164H(3,𝔽3)—624no—
165H(3,𝔽3)—64no—
166H(3,𝔽3)—1824no—
167H(3,𝔽3)C2×(C3⋊D3)⋊C319yescomplemented
168H(3,𝔽3)C2×#27213yescomplemented
169C2×C32—21no—
170C2×C32—23no—
171C2×C32—31no—
172C2×C32—34no—
173C2×C32—31no—
174C2×C32—38no—
175C2×C32—38no—
176C2×C32—64no—
177C2×C32—61no—
178C2×C32—61no—
179C2×C32—62no—
180C2×C32—68no—
181C2×C32—68no—
182C2×C32—616no—
183C2×C32—61no—
184C2×C32—64no—
185C2×C32—68no—
186C2×C32—68no—
187C2×C32—64no—
188C2×C32—912no—
189C2×C32—936no—
190C2×C32—1812no—
191C2×C32—1836no—
192C2×C32—812no—
193C2×C32C3⋊((C3⋊D3)⋊C3)112yescomplemented
194C2×C32C3×#27211yescharacteristic, complemented
195C2×C32C3⋊#27213yescomplemented
196C2×C32C3×C3⋊(C3⋊D3)11yescharacteristic
197C3⋊D3—98no—
198C3⋊D3—916no—
199C3⋊D3—92no—
200C3⋊D3—2724no—
201C3×D3—2772no—
202C3×D3—2724no—
203C3×D3—276no—
204C3×D3—272no—
205C3×D3—8116no—
206C3×D3—812no—
207C22×C3—813no—
208C22×C3—811no—
209C2×D3—2712no—
210C2×D3—271no—
211C2×D3—811no—
212C2×D3—818no—
213C32—21no—
214C32—23no—
215C32—31no—
216C32—34no—
217C32—31no—
218C32—38no—
219C32—38no—
220C32—64no—
221C32—61no—
222C32—61no—
223C32—62no—
224C32—68no—
225C32—68no—
226C32—616no—
227C32—61no—
228C32—64no—
229C32—68no—
230C32—68no—
231C32—64no—
232C32—912no—
233C32—936no—
234C32—1812no—
235C32—1836no—
236C32C2×C3⋊((C3⋊D3)⋊C3)112yescomplemented
237C32C2×C3×#27211yescharacteristic, complemented
238C32C2×C3⋊#27213yescomplemented
239C32C2×C3×C3⋊(C3⋊D3)11yescharacteristic
240C2×C3—21no—
241C2×C3—312no—
242C2×C3—31no—
243C2×C3—33no—
244C2×C3—61no—
245C2×C3—63no—
246C2×C3—98no—
247C2×C3—184no—
248C2×C3—188no—
249C2×C3—816no—
250C2×C3—812no—
251C2×C3C3⋊(C3⋊((C3⋊D3)⋊C3))11yescharacteristic
252C2×C3C3×C3⋊#27211yescharacteristic
253D3—2724no—
254D3—272no—
255D3—812no—
256D3—8116no—
257C22—811noSylow
258C3—21no—
259C3—312no—
260C3—31no—
261C3—33no—
262C3—61no—
263C3—63no—
264C3—98no—
265C3—184no—
266C3—188no—
267C3C2×C3⋊(C3⋊((C3⋊D3)⋊C3))11yescharacteristic
268C3C2×C3×C3⋊#27211yescharacteristic
269C2—812no—
270C2#1309150111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C3⋊(C3⋊((C3⋊D3)⋊C3))
commutator, cosocle, hypocenterC32×H(3,𝔽3)C22×C3
FrattiniC32C2×C3×C3⋊(C3⋊D3)
FittingC2×H(3,𝔽3)2C2
socleC2×C32C3×C3⋊(C3⋊D3)
LCSG ▷ C32×H(3,𝔽3)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. #13091501C2direct (N × H)
2. C2×H(3,𝔽3)2C2semidirect
3. C2×C3⋊(C3⋊#272)C3semidirect
7. H(3,𝔽3)2C22semidirect
9. C2×C32×H(3,𝔽3)D3semidirect
10. C2×C32×H(3,𝔽3)D3semidirect
11. C2×C32×H(3,𝔽3)C2×C3semidirect
12. C3⋊(C3⋊#272)C2×C3semidirect
17. C2×C32⋊H(3,𝔽3)D3semidirect
29. C32×H(3,𝔽3)C2×D3semidirect
30. C32×H(3,𝔽3)C2×D3semidirect
31. C32×H(3,𝔽3)C22×C3semidirect
34. C32⋊H(3,𝔽3)C2×D3semidirect
36. C2×C34C3×D3semidirect
37. C2×C34C3⋊D3semidirect
51. C2×C3×H(3,𝔽3)C3×D3semidirect
53. C2×C3×H(3,𝔽3)C3⋊D3semidirect
74. C34C2×C3×D3semidirect
75. C34C2×C3⋊D3semidirect
87. C3×H(3,𝔽3)C2×C3×D3semidirect
89. C3×H(3,𝔽3)C2×C3⋊D3semidirect
124. C2×H(3,𝔽3)(C3⋊D3)⋊C3semidirect
125. C2×H(3,𝔽3)#272semidirect
167. H(3,𝔽3)C2×(C3⋊D3)⋊C3semidirect
168. H(3,𝔽3)C2×#272semidirect
193. C2×C32C3⋊((C3⋊D3)⋊C3)semidirect
194. C2×C32C3×#272semidirect
195. C2×C32C3⋊#272semidirect
236. C32C2×C3⋊((C3⋊D3)⋊C3)semidirect
237. C32C2×C3×#272semidirect
238. C32C2×C3⋊#272semidirect
270. C2#13091501direct (N × H)

Maximal subgroups

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

subgroupindex
#130915012
C2×H(3,𝔽3)22
C2×C3⋊(C3⋊#272)3
C2×C3×C3⋊#2723
C2×C3×C3⋊((C3⋊D3)⋊C3)3
C2×#923003