Finite Group Databasebeta

← C3×#4245⋊C3 #13094707

Subgroups — C3×#4245⋊C3 #13094707

Order 1512.

272 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 64 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×D3×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C2×C32×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C3×C3⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic, complemented
4C3×#4245C311yesmaximal, characteristic, complemented
5C3×(C22⋊D7)⋊C3—31nomaximal
6#4245⋊C3C313yesmaximal, direct product factor
7C2×C32×ΓL(1,𝔽8)C2211yescharacteristic
8C3×D3×ΓL(1,𝔽8)—21no—
9C32×AGL(1,𝔽7)—21no—
10C2×C3×D3×C7C2×C311yescharacteristic, complemented
11C2×C32×D7C2×C311yescharacteristic, complemented
12C22×C3×ΓL(1,𝔽8)—31no—
13C2×D3×ΓL(1,𝔽8)C2×C313yescomplemented
14C2×C3×AGL(1,𝔽7)—21no—
15C2×C3×AGL(1,𝔽7)—23no—
16C2×C3×AGL(1,𝔽7)D311yescharacteristic, complemented
17C2×C3×AGL(1,𝔽7)C2×C313yescomplemented
18C3×Dic21C2×C311yescharacteristic, complemented
19C3⋊(ΓL(1,𝔽8)⋊C4)C2×C313yescomplemented
20C3×ΓL(1,𝔽8)⋊C4—31no—
21C32×#67—71nomaximal, Hall
22C32×ΓL(1,𝔽8)D411yescharacteristic, complemented, Hall
23C3×C22⋊D7—31no—
24#4245C3211yescharacteristic, complemented
25(C22⋊D7)⋊C3—33no—
26C2×C32×C7C22×C311yescharacteristic
27C3×D3×C7—21no—
28C32×D7—21no—
29C2×C3×ΓL(1,𝔽8)—21no—
30C2×C3×ΓL(1,𝔽8)—23no—
31C2×C3×ΓL(1,𝔽8)—61no—
32C2×C3×ΓL(1,𝔽8)C2×D311yescharacteristic
33C2×C3×ΓL(1,𝔽8)C22×C313yes—
34D3×ΓL(1,𝔽8)—23no—
35C3×AGL(1,𝔽7)—22no—
36C3×AGL(1,𝔽7)—26no—
37C3×AGL(1,𝔽7)—21no—
38C3×AGL(1,𝔽7)—23no—
39C22×C33—71no—
40C2×C32×D3—71no—
41C32×Dic3—71no—
42C22×C3×C7—31no—
43C2×D3×C7C2×C3211yescharacteristic, complemented
44C2×C3×D7—21no—
45C2×C3×D7C3×D311yescharacteristic, complemented
46C2×C3×D7C2×C3211yescharacteristic, complemented
47C22×ΓL(1,𝔽8)—33no—
48C2×AGL(1,𝔽7)—23no—
49C2×AGL(1,𝔽7)C3×D313yescomplemented
50Dic21C2×C3211yescharacteristic, complemented
51C3×Dic7—31no—
52ΓL(1,𝔽8)⋊C4—33no—
53C32×D4—211no—
54C3×#67—73no—
55C3×#67—71no—
56C32×C7C3×D411yescharacteristic, complemented
57C3×ΓL(1,𝔽8)—21no—
58C3×ΓL(1,𝔽8)—23no—
59C3×ΓL(1,𝔽8)#6711yescharacteristic, complemented
60C3×ΓL(1,𝔽8)C3×D413yescomplemented
61C22⋊D7—31noHall
62C2×C33—71no—
63C2×C33—141no—
64C32×D3—141no—
65C2×C3×C7—21no—
66C2×C3×C7—61no—
67C2×C3×C7C2×C3×D311yescharacteristic
68C2×C3×C7C22×C3211yescharacteristic
69C3×D7—22no—
70C3×D7—21no—
71C3×D7—21no—
72D3×C7—21no—
73C2×ΓL(1,𝔽8)—23no—
74C2×ΓL(1,𝔽8)—63no—
75C2×ΓL(1,𝔽8)C2×C3×D313yes—
76AGL(1,𝔽7)—26no—
77AGL(1,𝔽7)—23no—
78C22×C32—71no—
79C22×C32—73no—
80C22×C32—71no—
81C22×C32—143no—
82C22×C32—141no—
83C22×C32—211no—
84C2×C3×D3—73no—
85C2×C3×D3AGL(1,𝔽7)11yescharacteristic, complemented
86C32×C4—211no—
87C3×Dic3—73no—
88C3×Dic3—71no—
89C22×C7—31no—
90C2×D7C32×D311yescharacteristic, complemented
91Dic7—31no—
92C33—71noSylow
93C3×D4—211no—
94C3×D4—213no—
95#67—71no—
96C3×C7—21no—
97C3×C7C3×#6711yescharacteristic, complemented
98C3×C7C32×D411yescharacteristic, complemented
99ΓL(1,𝔽8)—23no—
100ΓL(1,𝔽8)C3×#6713yescomplemented
101C2×C32—71no—
102C2×C32—73no—
103C2×C32—143no—
104C2×C32—141no—
105C2×C32—143no—
106C2×C32—146no—
107C2×C32—142no—
108C2×C32—141no—
109C2×C32—141no—
110C2×C32—421no—
111C2×C32C2×AGL(1,𝔽7)11yescharacteristic
112C3×D3—21no—
113C3×D3—143no—
114C2×C7—61no—
115C2×C7C2×C32×D311yescharacteristic
116D7—21no—
117C22×C3—31no—
118C22×C3—73no—
119C22×C3—71no—
120C22×C3—71no—
121C22×C3—141no—
122C22×C3—143no—
123C22×C3—213no—
124C2×D3C3×AGL(1,𝔽7)11yescharacteristic, complemented
125C3×C4—213no—
126C3×C4—211no—
127Dic3—71no—
128C32—71no—
129C32—73no—
130C32—143no—
131C32—141no—
132C32(C22⋊D7)⋊C311yescharacteristic, complemented
133D4—211noSylow
134C7C32×#6711yescharacteristic, complemented, Sylow
135C2×C3—21no—
136C2×C3—61no—
137C2×C3—73no—
138C2×C3—142no—
139C2×C3—143no—
140C2×C3—146no—
141C2×C3—141no—
142C2×C3—141no—
143C2×C3—143no—
144C2×C3—423no—
145C2×C3D3×AGL(1,𝔽7)11yescharacteristic
146C2×C3C2×C3×AGL(1,𝔽7)11yescharacteristic
147D3—21no—
148C22—31no—
149C22—71no—
150C4—211no—
151C3—21no—
152C3—73no—
153C3—143no—
154C3#4245⋊C311yescharacteristic, direct product factor
155C3C3×(C22⋊D7)⋊C311yescharacteristic, complemented
156C2—61no—
157C2—141no—
158C2C3×D3×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3D3×AGL(1,𝔽7)
commutator, cosocleC2×C3×C7C22×C32
FrattiniC2C3×D3×AGL(1,𝔽7)
Fitting, socleC2×C32×C7C22×C3
hypocenterC3×C7C32×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×ΓL(1,𝔽8)C2semidirect
2. C2×C32×AGL(1,𝔽7)C2semidirect
3. C3×C3⋊(ΓL(1,𝔽8)⋊C4)C2semidirect
4. C3×#4245C3semidirect
6. #4245⋊C3C3direct (N × H)
10. C2×C3×D3×C7C2×C3semidirect
11. C2×C32×D7C2×C3semidirect
13. C2×D3×ΓL(1,𝔽8)C2×C3semidirect
16. C2×C3×AGL(1,𝔽7)D3semidirect
17. C2×C3×AGL(1,𝔽7)C2×C3semidirect
18. C3×Dic21C2×C3semidirect
19. C3⋊(ΓL(1,𝔽8)⋊C4)C2×C3semidirect
22. C32×ΓL(1,𝔽8)D4semidirect
24. #4245C32semidirect
43. C2×D3×C7C2×C32semidirect
45. C2×C3×D7C3×D3semidirect
46. C2×C3×D7C2×C32semidirect
49. C2×AGL(1,𝔽7)C3×D3semidirect
50. Dic21C2×C32semidirect
56. C32×C7C3×D4semidirect
59. C3×ΓL(1,𝔽8)#67semidirect
60. C3×ΓL(1,𝔽8)C3×D4semidirect
85. C2×C3×D3AGL(1,𝔽7)semidirect
90. C2×D7C32×D3semidirect
97. C3×C7C3×#67semidirect
98. C3×C7C32×D4semidirect
100. ΓL(1,𝔽8)C3×#67semidirect
124. C2×D3C3×AGL(1,𝔽7)semidirect
132. C32(C22⋊D7)⋊C3semidirect
134. C7C32×#67semidirect
154. C3#4245⋊C3direct (N × H)
155. C3C3×(C22⋊D7)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×ΓL(1,𝔽8)2
C2×C32×AGL(1,𝔽7)2
C3×C3⋊(ΓL(1,𝔽8)⋊C4)2
C3×#42453
C3×(C22⋊D7)⋊C33
#4245⋊C33
C32×#677