Finite Group Databasebeta

← C3×#10620874 #72245848

Subgroups — C3×#10620874 #72245848

Order 2016.

326 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C3×#66728C211yesmaximal, characteristic, complemented
2C2×C32×D28C211yesmaximal, characteristic, complemented
3C3×C7×#208C211yesmaximal, characteristic, complemented
4C3×#66565C212yesmaximal, complemented
5C3×#66561C212yesmaximal, complemented
6C3×#6471—31nomaximal
7#10620874C311yesmaximal, characteristic, direct product factor
8C22×C32×D7—21no—
9C2×C32×C4×C7C2211yescharacteristic
10C3×#4268—21no—
11C3×D84C2211yescharacteristic, complemented
12C3×C4×D21—21no—
13C3×Dic42C2211yescharacteristic, complemented
14C32×D28—21no—
15C32×D28C2212yes—
16C3×C7×C3⋊C8C2212yescomplemented
17#66728C2×C311yescharacteristic, complemented
18C3×#1157—31no—
19C2×C3×D28—21no—
20C2×C3×D28D311yescharacteristic, complemented
21C2×C3×D28C2×C311yescharacteristic, complemented
22C3×C7×M16—31no—
23C7×#208C2×C311yescharacteristic, complemented
24C3×D56—32no—
25C3×#1132—32no—
26#66565C2×C312yescomplemented
27#66561C2×C312yescomplemented
28C3×#932—71nomaximal, Hall
29C22×C32×C7D411yescharacteristic
30C2×C3×D21—21no—
31C2×C32×D7—22no—
32C2×C32×D7—42no—
33C32×C4×C7D411yescharacteristic
34C32×C4×C7C2311yescharacteristic
35C3×Dic21—21no—
36#6471—31noHall
37C22×C3×D7—21no—
38C22×C3×D7—21no—
39C22×C3×D7—21no—
40C22×C3×D7—21no—
41C2×C3×C4×C7—21no—
42C2×C3×C4×C7C2×D311yescharacteristic
43C2×C3×C4×C7C22×C311yescharacteristic
44#4268—21no—
45D84C22×C311yescharacteristic, complemented
46C4×D21—21no—
47Dic42C22×C311yescharacteristic, complemented
48C3×C22⋊D7—61no—
49C3×D28—21no—
50C3×D28—21no—
51C3×D28—21no—
52C3×D28—21no—
53C3×D28—22no—
54C3×D28—31no—
55C3×D28C2×D312yes—
56C3×D28C22×C312yes—
57C3×C4×D7—61no—
58C3×Dic14—31no—
59C3×C7×C8—32no—
60C7×C3⋊C8C22×C312yescomplemented
61C2×C32×D4—71no—
62C3×#235—71no—
63C3×#214—72no—
64C3×#213—72no—
65C3×#208D711yescharacteristic, complemented
66C2×C32×C7—21no—
67C2×C32×C7C2×D411yescharacteristic
68C3×D21—41no—
69C32×D7—42no—
70#1157—31no—
71C2×D28C3×D311yescharacteristic, complemented
72C7×M16—31no—
73D56—32no—
74#1132—32no—
75C3×AGL(1,ℤ/8)—211no—
76#932—71no—
77C22×C3×C7—21no—
78C22×C3×C7#6711yescharacteristic
79C22×C3×C7C3×D411yescharacteristic
80C2×D21—21no—
81C2×C3×D7—22no—
82C2×C3×D7—22no—
83C2×C3×D7—22no—
84C2×C3×D7—22no—
85C2×C3×D7—42no—
86C2×C3×D7—42no—
87C2×C3×D7—42no—
88C2×C3×D7—42no—
89C2×C3×D7—61no—
90C3×C4×C7—21no—
91C3×C4×C7—21no—
92C3×C4×C7#6711yescharacteristic
93C3×C4×C7C3×D411yescharacteristic
94C3×C4×C7C22×D311yescharacteristic
95C3×C4×C7C23×C311yescharacteristic
96Dic21—21no—
97C3×Dic7—61no—
98C23×C32—141no—
99C32×D4—72no—
100C32×D4—141no—
101C2×C32×C4C2×D711yescharacteristic
102C3×#67—141no—
103C3×D12—71no—
104C3×C4×D3—141no—
105C3×Dic6—71no—
106C3×C3⋊C8C2×D712yescomplemented
107C32×C7AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
108C22×D7—21no—
109C2×C4×C7C2×C3×D311yescharacteristic
110C22⋊D7—61no—
111D28—21no—
112D28—31no—
113D28C2×C3×D312yes—
114C4×D7—61no—
115Dic14—31no—
116C7×C8—32no—
117C3×#41—211no—
118C2×C3×D4—71no—
119C2×C3×D4—71no—
120C2×C3×D4—141no—
121#235—71no—
122C3×SD16—212no—
123C3×D8—212no—
124C3×M16—31no—
125#214—72no—
126#213—72no—
127#208C3×D711yescharacteristic, complemented
128C2×C3×C7—21no—
129C2×C3×C7—21no—
130C2×C3×C7—21no—
131C2×C3×C7—21no—
132C2×C3×C7—21no—
133C2×C3×C7C2×#6711yescharacteristic
134C2×C3×C7C2×C3×D411yescharacteristic
135D21—41no—
136C3×D7—42no—
137C3×D7—42no—
138C3×D7—42no—
139C3×D7—42no—
140C3×D7—121no—
141C22×C32—142no—
142C22×C32—282no—
143C22×C32D2811yescharacteristic
144C2×C3×D3—141no—
145C32×C4D2811yescharacteristic
146C32×C4C22×D711yescharacteristic
147C3×Dic3—141no—
148AGL(1,ℤ/8)—211noSylow
149C22×C7C3×#6711yescharacteristic
150C2×D7—22no—
151C2×D7—42no—
152C2×D7—61no—
153C4×C7C3×#6711yescharacteristic
154C4×C7C22×C3×D311yescharacteristic
155Dic7—61no—
156C23×C3—141no—
157C23×C3—141no—
158C23×C3—141no—
159C23×C3—141no—
160C3×Q8—211no—
161C3×D4—72no—
162C3×D4—72no—
163C3×D4—142no—
164C3×D4—141no—
165C3×D4—141no—
166C3×D4—141no—
167C3×D4—141no—
168C3×D4—211no—
169C3×D4—421no—
170C2×C3×C4—21no—
171C2×C3×C4—421no—
172C2×C3×C4D3×D711yescharacteristic
173C2×C3×C4C2×C3×D711yescharacteristic
174#67—141no—
175D12—71no—
176C4×D3—141no—
177Dic6—71no—
178C3×C8—32no—
179C3⋊C8C2×C3×D712yescomplemented
180C3×C7—21no—
181C3×C7#93211yescharacteristic, complemented
182C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
183C2×C32—21no—
184C2×C32—282no—
185C2×C32C2×D2811yescharacteristic
186C3×D3—281no—
187#41—211no—
188C2×D4—71no—
189SD16—212no—
190D8—212no—
191M16—31no—
192C2×C7—21no—
193C2×C7C2×C3×#6711yescharacteristic
194D7—42no—
195D7—121no—
196C22×C3—21no—
197C22×C3—142no—
198C22×C3—142no—
199C22×C3—142no—
200C22×C3—142no—
201C22×C3—282no—
202C22×C3—282no—
203C22×C3—282no—
204C22×C3—282no—
205C22×C3—421no—
206C22×C3#424611yescharacteristic
207C22×C3C3×D2811yescharacteristic
208C2×D3—141no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—421no—
212C3×C4#424611yescharacteristic
213C3×C4C3×D2811yescharacteristic
214C3×C4C2×D3×D711yescharacteristic
215C3×C4C22×C3×D711yescharacteristic
216Dic3—141no—
217C32#647111yescharacteristic, complemented, Sylow
218C23—141no—
219Q8—211no—
220D4—72no—
221D4—141no—
222D4—211no—
223D4—421no—
224C2×C4—421no—
225C2×C4C3×D3×D711yescharacteristic
226C8—32no—
227C7C3×#93211yescharacteristic, complemented, Sylow
228C2×C3—21no—
229C2×C3—21no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—282no—
234C2×C3—282no—
235C2×C3—282no—
236C2×C3—282no—
237C2×C3—841no—
238C2×C3C2×#424611yescharacteristic
239C2×C3C2×C3×D2811yescharacteristic
240D3—281no—
241C22—142no—
242C22—282no—
243C22—421no—
244C22C3×#424611yescharacteristic
245C4—421no—
246C4C3×#424611yescharacteristic
247C4C2×C3×D3×D711yescharacteristic
248C3—21no—
249C3#1062087411yescharacteristic, direct product factor
250C3C3×#647111yescharacteristic, complemented
251C2—21no—
252C2—282no—
253C2—841no—
254C2C2×C3×#424611yescharacteristic
subgroupG/H
centerC2×C3C2×#4246
commutator, cosocleC3×C4×C7C23×C3
FrattiniC4C2×C3×D3×D7
FittingC2×C32×C4×C7C22
socleC2×C32×C7C2×D4
hypercenterC2×C3×C4D3×D7
hypocenterC3×C7C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#66728C2semidirect
2. C2×C32×D28C2semidirect
3. C3×C7×#208C2semidirect
4. C3×#66565C2semidirect
5. C3×#66561C2semidirect
7. #10620874C3direct (N × H)
11. C3×D84C22semidirect
13. C3×Dic42C22semidirect
16. C3×C7×C3⋊C8C22semidirect
17. #66728C2×C3semidirect
20. C2×C3×D28D3semidirect
21. C2×C3×D28C2×C3semidirect
23. C7×#208C2×C3semidirect
26. #66565C2×C3semidirect
27. #66561C2×C3semidirect
45. D84C22×C3semidirect
47. Dic42C22×C3semidirect
60. C7×C3⋊C8C22×C3semidirect
65. C3×#208D7semidirect
71. C2×D28C3×D3semidirect
106. C3×C3⋊C8C2×D7semidirect
107. C32×C7AGL(1,ℤ/8)semidirect
127. #208C3×D7semidirect
179. C3⋊C8C2×C3×D7semidirect
181. C3×C7#932semidirect
182. C3×C7C3×AGL(1,ℤ/8)semidirect
217. C32#6471semidirect
227. C7C3×#932semidirect
249. C3#10620874direct (N × H)
250. C3C3×#6471semidirect

Maximal subgroups

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

subgroupindex
C3×#667282
C2×C32×D282
C3×C7×#2082
C3×#665652
C3×#665612
C3×#64713
#106208743
C3×#9327