Finite Group Databasebeta

← C2×C3×#1020⋊C3 #13181814

Subgroups — C2×C3×#1020⋊C3 #13181814

Order 1728.

422 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 46 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⋊C3C212yesmaximal, direct product factor
3C2×C3×#1020C311yesmaximal, characteristic, complemented
4C2×#1020⋊C3C313yesmaximal, direct product factor
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×A42C2×C313yescomplemented
13#1020⋊C3C2×C316yesdirect product factor
14C3×#120⋊C3—32no—
15C2×C32×A4—81no—
16C2×#1020C3211yescharacteristic, complemented
17C2×C3×#120—31no—
18C2×#120⋊C3—33no—
19C3×C22⋊A4C2×D311yescharacteristic, complemented
20C3×C22⋊A4C22×C311yescharacteristic, complemented
21C22×C3×A4—21no—
22C22×C3×A4—61no—
23C22×C3×A4—61no—
24C22×C3×A4—121no—
25C2×C3×S4—121no—
26A42—23no—
27A42C22×C313yescomplemented
28C2×C32×D3—161nomaximal
29C32×A4—81no—
30C25×C3C3×D311yescharacteristic, complemented
31C2×C22⋊A4—21no—
32C2×C22⋊A4C3×D313yescomplemented
33C2×C22⋊A4C2×C3211yescharacteristic, complemented
34C23×A4—23no—
35#1020C2×C3212yescomplemented
36C22×C3×D4—91no—
37C3×#120—32no—
38C3×#120—92no—
39C2×C3×C22⋊C4—91no—
40#120⋊C3—36no—
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—
51C2×C3×A4—122no—
52C3×S4—122no—
53C2×#120—31noSylow
54C2×C33—161no—
55C32×D3—162no—
56C24×C3—31no—
57C24×C3—61no—
58C24×C3—61no—
59C24×C3—91no—
60C24×C3—91no—
61C24×C3C2×C3×D311yescharacteristic, complemented
62C22⋊A4—21no—
63C22⋊A4C2×C3×D313yescomplemented
64C22⋊A4C22×C3211yescharacteristic, complemented
65C22×A4—23no—
66C22×A4—63no—
67C22×A4—63no—
68C22×A4—123no—
69C2×S4—121no—
70C2×C3×D4—92no—
71C2×C3×D4—92no—
72C2×C3×D4—181no—
73C2×C3×D4—181no—
74C2×C3×D4—181no—
75C2×C3×D4—181no—
76C22×C3×C4—91no—
77C3×C22⋊C4—92no—
78C3×C22⋊C4—92no—
79C22×C32—81no—
80C22×C32—241no—
81C22×C32—241no—
82C22×C32—481no—
83C2×C3×D3—161no—
84C2×C3×D3—163no—
85C3×A4—21no—
86C3×A4—83no—
87C3×A4—81no—
88C3×A4—81no—
89C3×A4—86no—
90C3×A4—121no—
91C3×A4—121no—
92C25C32×D311yescharacteristic, complemented
93C22×D4—91no—
94#120—32no—
95#120—92no—
96C2×C22⋊C4—91no—
97C33—161noSylow
98C23×C3—21no—
99C23×C3—31no—
100C23×C3—31no—
101C23×C3—32no—
102C23×C3—61no—
103C23×C3—61no—
104C23×C3—83no—
105C23×C3—91no—
106C23×C3—91no—
107C23×C3—91no—
108C23×C3—91no—
109C23×C3—91no—
110C23×C3—181no—
111C23×C3—181no—
112C23×C3—181no—
113C23×C3—182no—
114C23×C3—181no—
115C23×C3—181no—
116C2×A4—23no—
117C2×A4—63no—
118C2×A4—63no—
119C2×A4—83no—
120C2×A4—81no—
121C2×A4—82no—
122C2×A4—121no—
123C2×A4—123no—
124C2×A4—126no—
125C2×A4—241no—
126S4—122no—
127C3×D4—182no—
128C3×D4—182no—
129C3×D4—182no—
130C3×D4—182no—
131C2×C3×C4—92no—
132C2×C3×C4—181no—
133C2×C3×C4—181no—
134C2×C32—81no—
135C2×C32—163no—
136C2×C32—161no—
137C2×C32—161no—
138C2×C32—241no—
139C2×C32—241no—
140C2×C32—323no—
141C2×C32—482no—
142C3×D3—162no—
143C3×D3—166no—
144C24—31no—
145C24—61no—
146C24—61no—
147C24—91no—
148C24—91no—
149C24C2×C32×D311yescharacteristic, complemented
150C2×D4—92no—
151C2×D4—92no—
152C2×D4—181no—
153C2×D4—181no—
154C2×D4—181no—
155C2×D4—181no—
156C22×C4—91no—
157C22⋊C4—92no—
158C22⋊C4—92no—
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—121no—
171C22×C3—181no—
172C22×C3—181no—
173C22×C3—181no—
174C22×C3—182no—
175C22×C3—182no—
176C22×C3—243no—
177C22×C3—243no—
178C22×C3—361no—
179C22×C3—483no—
180C2×D3—161no—
181A4—23no—
182A4—83no—
183A4—81no—
184A4—82no—
185A4—121no—
186A4—123no—
187A4—241no—
188C3×C4—182no—
189C32—81no—
190C32—163no—
191C32—161no—
192C32—161no—
193C32—323no—
194C23—21no—
195C23—31no—
196C23—31no—
197C23—32no—
198C23—61no—
199C23—61no—
200C23—91no—
201C23—91no—
202C23—91no—
203C23—91no—
204C23—91no—
205C23—181no—
206C23—181no—
207C23—181no—
208C23—182no—
209C23—181no—
210C23—181no—
211D4—182no—
212D4—182no—
213D4—182no—
214D4—182no—
215C2×C4—92no—
216C2×C4—181no—
217C2×C4—181no—
218C2×C3—61no—
219C2×C3—61no—
220C2×C3—83no—
221C2×C3—91no—
222C2×C3—91no—
223C2×C3—122no—
224C2×C3—161no—
225C2×C3—163no—
226C2×C3—243no—
227C2×C3—243no—
228C2×C3—321no—
229C2×C3—486no—
230C2×C3#1020⋊C311yescharacteristic, direct product factor
231D3—162no—
232C22—21no—
233C22—31no—
234C22—31no—
235C22—61no—
236C22—61no—
237C22—91no—
238C22—91no—
239C22—91no—
240C22—91no—
241C22—91no—
242C22—121no—
243C22—181no—
244C22—181no—
245C22—181no—
246C22—182no—
247C22—182no—
248C22—361no—
249C4—182no—
250C3—83no—
251C3—161no—
252C3—163no—
253C3—321no—
254C3C2×#1020⋊C311yescharacteristic, direct product factor
255C2—61no—
256C2—61no—
257C2—91no—
258C2—91no—
259C2—122no—
260C2C3×#1020⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#1020⋊C3
commutator, cosocle, hypocenterC22⋊A4C22×C32
FrattiniC1G
Fitting, socleC25×C3C3×D3
LCSG ▷ C22⋊A4
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×A42C2semidirect
2. C3×#1020⋊C3C2direct (N × H)
3. C2×C3×#1020C3semidirect
4. C2×#1020⋊C3C3direct (N × H)
6. C3×A42C22semidirect
7. C2×C3×C22⋊A4D3semidirect
8. C2×C3×C22⋊A4C2×C3semidirect
10. C3×#1020C2×C3semidirect
12. C2×A42C2×C3semidirect
13. #1020⋊C3C2×C3direct (N × H)
16. C2×#1020C32semidirect
19. C3×C22⋊A4C2×D3semidirect
20. C3×C22⋊A4C22×C3semidirect
27. A42C22×C3semidirect
30. C25×C3C3×D3semidirect
32. C2×C22⋊A4C3×D3semidirect
33. C2×C22⋊A4C2×C32semidirect
35. #1020C2×C32semidirect
61. C24×C3C2×C3×D3semidirect
63. C22⋊A4C2×C3×D3semidirect
64. C22⋊A4C22×C32semidirect
92. C25C32×D3semidirect
149. C24C2×C32×D3semidirect
230. C2×C3#1020⋊C3direct (N × H)
254. C3C2×#1020⋊C3direct (N × H)
260. C2C3×#1020⋊C3direct (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⋊C32
C2×C3×#10203
C2×#1020⋊C33
C2×C3×#120⋊C33
C2×C32×D316