Finite Group Databasebeta

← C3×#6495⋊C3 #72244597

Subgroups — C3×#6495⋊C3 #72244597

Order 2016.

408 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 138 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×D4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C3×#1157⋊C3C211yesmaximal, characteristic, complemented
3C3×#1142⋊C3C212yesmaximal, complemented
4C3×#1141⋊C3C212yesmaximal, complemented
5C3×#1136⋊C3C211yesmaximal, characteristic, complemented
6C3×#6495C311yesmaximal, characteristic, complemented
7#6495⋊C3C313yesmaximal, direct product factor
8C23×C3×ΓL(1,𝔽8)—21no—
9C3×D4×ΓL(1,𝔽8)—21no—
10C3×D4×ΓL(1,𝔽8)C2212yes—
11C2×C3×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C3×(C22⋊D7)⋊C3—21no—
13C3×C4⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
14C3×C4×AGL(1,𝔽7)—21no—
15C3×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic, complemented
16C3×ΓL(1,𝔽8)⋊C8C2212yescomplemented
17C2×C3×C7×D4C2×C311yescharacteristic, complemented
18C3×#1157C2×C311yescharacteristic, complemented
19C2×D4×ΓL(1,𝔽8)C2×C313yescomplemented
20#1157⋊C3C2×C313yescomplemented
21C3×#1142C2×C312yescomplemented
22C3×#1141C2×C312yescomplemented
23C3×#1136C2×C311yescharacteristic, complemented
24#1142⋊C3C2×C316yescomplemented
25#1141⋊C3C2×C316yescomplemented
26#1136⋊C3C2×C313yescomplemented
27C32×AGL(1,ℤ/8)—71nomaximal, Hall
28C22×C3×ΓL(1,𝔽8)—22no—
29C22×C3×ΓL(1,𝔽8)—42no—
30C22×C3×ΓL(1,𝔽8)D411yescharacteristic
31C2×C3×AGL(1,𝔽7)—21no—
32C3×C4×ΓL(1,𝔽8)D411yescharacteristic
33C3×C4×ΓL(1,𝔽8)C2311yescharacteristic
34C3×ΓL(1,𝔽8)⋊C4—21no—
35#6495C3211yescharacteristic, complemented, Hall
36C23×C3×C7—21no—
37C23×ΓL(1,𝔽8)—23no—
38C3×C7×D4—21no—
39C3×C7×D4C22×C312yes—
40C2×C3×C4×C7C22×C311yescharacteristic
41C3×C22⋊D7—21no—
42C3×D28C22×C311yescharacteristic, complemented
43C3×C4×D7—21no—
44C3×Dic14C22×C311yescharacteristic, complemented
45D4×ΓL(1,𝔽8)—23no—
46D4×ΓL(1,𝔽8)C22×C316yes—
47C2×C4×ΓL(1,𝔽8)C22×C313yes—
48(C22⋊D7)⋊C3—23no—
49C4⋊AGL(1,𝔽7)C22×C313yescomplemented
50C4×AGL(1,𝔽7)—23no—
51ΓL(1,𝔽8)⋊Q8C22×C313yescomplemented
52C3×C7⋊C8C22×C312yescomplemented
53ΓL(1,𝔽8)⋊C8C22×C316yescomplemented
54C32×#41—71no—
55C2×C32×D4—141no—
56C32×SD16—72no—
57C32×D8—72no—
58C32×M16—71no—
59C2×C3×ΓL(1,𝔽8)—21no—
60C2×C3×ΓL(1,𝔽8)—42no—
61C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
62C3×AGL(1,𝔽7)—41no—
63C2×C7×D4C2×C3211yescharacteristic, complemented
64#1157C2×C3211yescharacteristic, complemented
65#1142C2×C3212yescomplemented
66#1141C2×C3212yescomplemented
67#1136C2×C3211yescharacteristic, complemented
68C3×AGL(1,ℤ/8)—73no—
69C3×AGL(1,ℤ/8)—71no—
70C22×C3×C7—22no—
71C22×C3×C7—42no—
72C22×C3×C7C3×D411yescharacteristic
73C2×C3×D7—21no—
74C22×ΓL(1,𝔽8)—26no—
75C22×ΓL(1,𝔽8)—46no—
76C22×ΓL(1,𝔽8)C3×D413yes—
77C2×AGL(1,𝔽7)—23no—
78C3×C4×C7C3×D411yescharacteristic
79C3×C4×C7C23×C311yescharacteristic
80C3×Dic7—21no—
81C4×ΓL(1,𝔽8)C3×D413yes—
82C4×ΓL(1,𝔽8)C23×C313yes—
83ΓL(1,𝔽8)⋊C4—23no—
84C23×C32—141no—
85C32×Q8—71no—
86C32×D4—71no—
87C32×D4—141no—
88C32×D4—141no—
89C32×D4—142no—
90C2×C32×C4—141no—
91C2×C32×C4—141no—
92C32×C8—72no—
93C3×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
94C23×C7—21no—
95C7×D4—21no—
96C7×D4C22×C3212yes—
97C2×C4×C7C22×C3211yescharacteristic
98C22⋊D7—21no—
99D28C22×C3211yescharacteristic, complemented
100C4×D7—21no—
101Dic14C22×C3211yescharacteristic, complemented
102C7⋊C8C22×C3212yescomplemented
103C3×#41—73no—
104C3×#41—71no—
105C2×C3×D4—143no—
106C2×C3×D4AGL(1,𝔽7)11yescharacteristic, complemented
107C3×SD16—76no—
108C3×SD16—72no—
109C3×D8—76no—
110C3×D8—72no—
111C3×M16—73no—
112C3×M16—71no—
113C2×C3×C7—21no—
114C2×C3×C7—42no—
115C2×C3×C7C2×C3×D411yescharacteristic
116C3×D7—41no—
117C2×ΓL(1,𝔽8)—23no—
118C2×ΓL(1,𝔽8)—46no—
119C2×ΓL(1,𝔽8)C2×C3×D413yes—
120AGL(1,𝔽7)—43no—
121C22×C32—142no—
122C22×C32—141no—
123C22×C32—141no—
124C22×C32—282no—
125C32×C4—141no—
126C32×C4—141no—
127C32×C4—141no—
128AGL(1,ℤ/8)—71noSylow
129C22×C7—22no—
130C22×C7—42no—
131C22×C7C32×D411yescharacteristic
132C2×D7—21no—
133C4×C7C32×D411yescharacteristic
134C4×C7C23×C3211yescharacteristic
135Dic7—21no—
136C23×C3—21no—
137C23×C3—143no—
138C3×Q8—73no—
139C3×Q8—71no—
140C3×D4—21no—
141C3×D4—73no—
142C3×D4—71no—
143C3×D4—143no—
144C3×D4—143no—
145C3×D4—141no—
146C3×D4—146no—
147C3×D4C2×AGL(1,𝔽7)12yes—
148C2×C3×C4—143no—
149C2×C3×C4—141no—
150C2×C3×C4—143no—
151C2×C3×C4C2×AGL(1,𝔽7)11yescharacteristic
152C3×C8—76no—
153C3×C8—72no—
154C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
155ΓL(1,𝔽8)C3×AGL(1,ℤ/8)13yescomplemented
156C2×C32—141no—
157C2×C32—141no—
158C2×C32—282no—
159C2×C32—281no—
160#41—71no—
161C2×D4C3×AGL(1,𝔽7)11yescharacteristic, complemented
162SD16—72no—
163D8—72no—
164M16—71no—
165C2×C7—21no—
166C2×C7—42no—
167C2×C7C2×C32×D411yescharacteristic
168D7—41no—
169C22×C3—22no—
170C22×C3—42no—
171C22×C3—146no—
172C22×C3—143no—
173C22×C3—141no—
174C22×C3—143no—
175C22×C3—286no—
176C22×C3(C22⋊D7)⋊C311yescharacteristic
177C3×C4—143no—
178C3×C4—141no—
179C3×C4—143no—
180C3×C4—143no—
181C3×C4(C22⋊D7)⋊C311yescharacteristic
182C3×C4C22×AGL(1,𝔽7)11yescharacteristic
183C32—141noSylow
184C23—21no—
185Q8—71no—
186D4—21no—
187D4—71no—
188D4—141no—
189D4C2×C3×AGL(1,𝔽7)12yes—
190C2×C4—141no—
191C2×C4C2×C3×AGL(1,𝔽7)11yescharacteristic
192C8—72no—
193C7C32×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
194C2×C3—21no—
195C2×C3—42no—
196C2×C3—143no—
197C2×C3—143no—
198C2×C3—286no—
199C2×C3—283no—
200C2×C3—281no—
201C2×C3C2×(C22⋊D7)⋊C311yescharacteristic
202C22—22no—
203C22—42no—
204C22—141no—
205C22C3×(C22⋊D7)⋊C311yescharacteristic
206C4—141no—
207C4C3×(C22⋊D7)⋊C311yescharacteristic
208C4C22×C3×AGL(1,𝔽7)11yescharacteristic
209C3—143no—
210C3#6495⋊C311yescharacteristic, direct product factor
211C2—21no—
212C2—42no—
213C2—281no—
214C2C2×C3×(C22⋊D7)⋊C311yescharacteristic
subgroupG/H
centerC2×C3C2×(C22⋊D7)⋊C3
commutator, cosocleC4×C7C23×C32
FrattiniC4C22×C3×AGL(1,𝔽7)
FittingC2×C3×C7×D4C2×C3
socleC2×C3×C7C2×C3×D4
hypercenterC2×C3×D4AGL(1,𝔽7)
hypocenterC7C32×AGL(1,ℤ/8)
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×ΓL(1,𝔽8)C2semidirect
2. C3×#1157⋊C3C2semidirect
3. C3×#1142⋊C3C2semidirect
4. C3×#1141⋊C3C2semidirect
5. C3×#1136⋊C3C2semidirect
6. C3×#6495C3semidirect
7. #6495⋊C3C3direct (N × H)
13. C3×C4⋊AGL(1,𝔽7)C22semidirect
15. C3×ΓL(1,𝔽8)⋊Q8C22semidirect
16. C3×ΓL(1,𝔽8)⋊C8C22semidirect
17. C2×C3×C7×D4C2×C3semidirect
18. C3×#1157C2×C3semidirect
19. C2×D4×ΓL(1,𝔽8)C2×C3semidirect
20. #1157⋊C3C2×C3semidirect
21. C3×#1142C2×C3semidirect
22. C3×#1141C2×C3semidirect
23. C3×#1136C2×C3semidirect
24. #1142⋊C3C2×C3semidirect
25. #1141⋊C3C2×C3semidirect
26. #1136⋊C3C2×C3semidirect
35. #6495C32semidirect
42. C3×D28C22×C3semidirect
44. C3×Dic14C22×C3semidirect
49. C4⋊AGL(1,𝔽7)C22×C3semidirect
51. ΓL(1,𝔽8)⋊Q8C22×C3semidirect
52. C3×C7⋊C8C22×C3semidirect
53. ΓL(1,𝔽8)⋊C8C22×C3semidirect
63. C2×C7×D4C2×C32semidirect
64. #1157C2×C32semidirect
65. #1142C2×C32semidirect
66. #1141C2×C32semidirect
67. #1136C2×C32semidirect
93. C3×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
99. D28C22×C32semidirect
101. Dic14C22×C32semidirect
102. C7⋊C8C22×C32semidirect
106. C2×C3×D4AGL(1,𝔽7)semidirect
154. C3×C7C3×AGL(1,ℤ/8)semidirect
155. ΓL(1,𝔽8)C3×AGL(1,ℤ/8)semidirect
161. C2×D4C3×AGL(1,𝔽7)semidirect
193. C7C32×AGL(1,ℤ/8)semidirect
210. C3#6495⋊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×D4×ΓL(1,𝔽8)2
C3×#1157⋊C32
C3×#1142⋊C32
C3×#1141⋊C32
C3×#1136⋊C32
C3×#64953
#6495⋊C33
C32×AGL(1,ℤ/8)7