Finite Group Databasebeta

← C2×C3⋊(#1020⋊C3) #13181815

Subgroups — C2×C3⋊(#1020⋊C3) #13181815

Order 1728.

422 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 34 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×C3×A42C211yesmaximal, characteristic, complemented
2C3⋊(#1020⋊C3)C212yesmaximal, direct product factor
3C2×C3⋊#1020C311yesmaximal, characteristic, complemented
4C2×#1020⋊C3—33nomaximal
5C2×C3⋊(#120⋊C3)—31nomaximal
6C3×A42C2211yescharacteristic, complemented
7C2×C3×C22⋊A4D311yescharacteristic, complemented
8C2×C3×C22⋊A4C2×C311yescharacteristic, complemented
9C23×C3×A4—21no—
10C3⋊#1020C2×C312yescomplemented
11C2×A42—23no—
12C2×A42D313yescomplemented
13#1020⋊C3—36no—
14C3⋊(#120⋊C3)—32no—
15C2×C32×A4—81no—
16C2×#1020—33no—
17C2×#953—31no—
18C2×#120⋊C3—91no—
19C3×C22⋊A4C2×D311yescharacteristic, complemented
20C3×C22⋊A4C22×C311yescharacteristic, complemented
21C22×C3×A4—21no—
22C22×C3×A4—61no—
23C22×C3×A4—61no—
24C2×D3×A4—121no—
25C2×C3⋊S4—121no—
26A42—23no—
27A42C2×D313yescomplemented
28C2×C3×C3⋊D3—161nomaximal
29C32×A4—81no—
30C25×C3C3×D311yescharacteristic, complemented
31C2×C22⋊A4—21no—
32C2×C22⋊A4C3×D313yescomplemented
33C2×C22⋊A4C3⋊D311yescharacteristic, complemented
34C23×A4—23no—
35#1020—36no—
36C22×#67—91no—
37#953—32no—
38#953—92no—
39C2×#217—91no—
40#120⋊C3—92no—
41C23×C32—81no—
42C2×C3×A4—21no—
43C2×C3×A4—61no—
44C2×C3×A4—61no—
45C2×C3×A4—83no—
46C2×C3×A4—81no—
47C2×C3×A4—81no—
48C2×C3×A4—86no—
49C2×C3×A4—121no—
50C2×C3×A4—121no—
51D3×A4—122no—
52C3⋊S4—122no—
53C2×#120—91noSylow
54C2×C33—161no—
55C3×C3⋊D3—162no—
56C24×C3—61no—
57C24×C3—61no—
58C24×C3—91no—
59C24×C3—91no—
60C24×C3C2×C3×D311yescharacteristic, complemented
61C23×D3—31no—
62C22⋊A4—21no—
63C22⋊A4C2×C3×D313yescomplemented
64C22⋊A4C2×C3⋊D311yescharacteristic, complemented
65C22×A4—23no—
66C22×A4—63no—
67C22×A4—63no—
68C22×A4—361no—
69C2×S4—363no—
70C2×#67—92no—
71C2×#67—92no—
72C2×#67—181no—
73C2×#67—181no—
74C2×#67—181no—
75C2×#67—181no—
76C22×Dic3—91no—
77#217—92no—
78#217—92no—
79C22×C32—81no—
80C22×C32—241no—
81C22×C32—241no—
82C2×C3⋊D3—161no—
83C2×C3×D3—481no—
84C2×C3×D3—483no—
85C3×A4—21no—
86C3×A4—83no—
87C3×A4—81no—
88C3×A4—81no—
89C3×A4—86no—
90C3×A4—121no—
91C3×A4—121no—
92C25C3×C3⋊D311yescharacteristic, complemented
93C22×D4—271no—
94#120—92no—
95#120—272no—
96C2×C22⋊C4—271no—
97C33—161noSylow
98C23×C3—21no—
99C23×C3—31no—
100C23×C3—31no—
101C23×C3—61no—
102C23×C3—61no—
103C23×C3—83no—
104C23×C3—91no—
105C23×C3—91no—
106C23×C3—91no—
107C23×C3—91no—
108C23×C3—91no—
109C23×C3—181no—
110C23×C3—181no—
111C23×C3—181no—
112C23×C3—182no—
113C22×D3—32no—
114C22×D3—181no—
115C22×D3—181no—
116C2×A4—23no—
117C2×A4—63no—
118C2×A4—63no—
119C2×A4—83no—
120C2×A4—81no—
121C2×A4—82no—
122C2×A4—123no—
123C2×A4—121no—
124C2×A4—241no—
125C2×A4—362no—
126S4—366no—
127#67—182no—
128#67—182no—
129#67—182no—
130#67—182no—
131C2×Dic3—92no—
132C2×Dic3—181no—
133C2×Dic3—181no—
134C2×C32—81no—
135C2×C32—163no—
136C2×C32—161no—
137C2×C32—161no—
138C2×C32—241no—
139C2×C32—241no—
140C2×C32—323no—
141C3⋊D3—162no—
142C3×D3—482no—
143C3×D3—486no—
144C24—61no—
145C24—61no—
146C24—91no—
147C24—91no—
148C24—91no—
149C24C2×C3×C3⋊D311yescharacteristic, complemented
150C2×D4—272no—
151C2×D4—272no—
152C2×D4—541no—
153C2×D4—541no—
154C2×D4—541no—
155C2×D4—541no—
156C22×C4—271no—
157C22⋊C4—272no—
158C22⋊C4—272no—
159C22×C3—21no—
160C22×C3—31no—
161C22×C3—31no—
162C22×C3—61no—
163C22×C3—61no—
164C22×C3—83no—
165C22×C3—91no—
166C22×C3—91no—
167C22×C3—91no—
168C22×C3—91no—
169C22×C3—91no—
170C22×C3—181no—
171C22×C3—181no—
172C22×C3—181no—
173C22×C3—182no—
174C22×C3—243no—
175C22×C3—243no—
176C22×C3—1441no—
177C2×D3—121no—
178C2×D3—182no—
179C2×D3—361no—
180C2×D3—483no—
181A4—23no—
182A4—83no—
183A4—81no—
184A4—82no—
185A4—123no—
186A4—121no—
187A4—241no—
188Dic3—182no—
189C32—81no—
190C32—163no—
191C32—161no—
192C32—161no—
193C32—323no—
194C23—21no—
195C23—31no—
196C23—31no—
197C23—61no—
198C23—61no—
199C23—91no—
200C23—91no—
201C23—91no—
202C23—92no—
203C23—91no—
204C23—91no—
205C23—181no—
206C23—181no—
207C23—181no—
208C23—182no—
209C23—541no—
210C23—541no—
211D4—542no—
212D4—542no—
213D4—542no—
214D4—542no—
215C2×C4—272no—
216C2×C4—541no—
217C2×C4—541no—
218C2×C3—61no—
219C2×C3—61no—
220C2×C3—83no—
221C2×C3—91no—
222C2×C3—91no—
223C2×C3—163no—
224C2×C3—161no—
225C2×C3—243no—
226C2×C3—243no—
227C2×C3—321no—
228C2×C3—1442no—
229C2×C3#1020⋊C311yescharacteristic, complemented
230D3—122no—
231D3—486no—
232C22—21no—
233C22—31no—
234C22—31no—
235C22—61no—
236C22—61no—
237C22—91no—
238C22—91no—
239C22—91no—
240C22—91no—
241C22—91no—
242C22—181no—
243C22—181no—
244C22—181no—
245C22—182no—
246C22—361no—
247C22—542no—
248C22—1081no—
249C4—542no—
250C3—83no—
251C3—163no—
252C3—161no—
253C3—321no—
254C3C2×#1020⋊C311yescharacteristic, complemented
255C2—61no—
256C2—61no—
257C2—91no—
258C2—91no—
259C2—362no—
260C2C3⋊(#1020⋊C3)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊(#1020⋊C3)
commutator, cosocle, hypocenterC3×C22⋊A4C22×C3
FrattiniC1G
Fitting, socleC25×C3C3×D3
LCSG ▷ C3×C22⋊A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×A42C2semidirect
2. C3⋊(#1020⋊C3)C2direct (N × H)
3. C2×C3⋊#1020C3semidirect
6. C3×A42C22semidirect
7. C2×C3×C22⋊A4D3semidirect
8. C2×C3×C22⋊A4C2×C3semidirect
10. C3⋊#1020C2×C3semidirect
12. C2×A42D3semidirect
19. C3×C22⋊A4C2×D3semidirect
20. C3×C22⋊A4C22×C3semidirect
27. A42C2×D3semidirect
30. C25×C3C3×D3semidirect
32. C2×C22⋊A4C3×D3semidirect
33. C2×C22⋊A4C3⋊D3semidirect
60. C24×C3C2×C3×D3semidirect
63. C22⋊A4C2×C3×D3semidirect
64. C22⋊A4C2×C3⋊D3semidirect
92. C25C3×C3⋊D3semidirect
149. C24C2×C3×C3⋊D3semidirect
229. C2×C3#1020⋊C3semidirect
254. C3C2×#1020⋊C3semidirect
260. C2C3⋊(#1020⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×A422
C3⋊(#1020⋊C3)2
C2×C3⋊#10203
C2×#1020⋊C33
C2×C3⋊(#120⋊C3)3
C2×C3×C3⋊D316