Finite Group Databasebeta

← C3×#6379⋊C3 #72244598

Subgroups — C3×#6379⋊C3 #72244598

Order 2016.

264 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 86 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×#1157⋊C3C211yesmaximal, characteristic
2C3×C42×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C3×#1136⋊C3C211yesmaximal, characteristic, complemented
4C3×#6379C311yesmaximal, characteristic, complemented
5#6379⋊C3C312yesmaximal, direct product factor
6#6379⋊C3C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×ΓL(1,𝔽8)—21no—
8C2×C3×C4×ΓL(1,𝔽8)C2211yescharacteristic
9C3×(C22⋊D7)⋊C3—21no—
10C3×C4⋊AGL(1,𝔽7)C411yescharacteristic, complemented
11C3×C4×AGL(1,𝔽7)—21no—
12C3×ΓL(1,𝔽8)⋊Q8C411yescharacteristic, complemented
13C3×ΓL(1,𝔽8)⋊C8—21no—
14C3×#1157C2×C311yescharacteristic
15#1157⋊C3C2×C312yes—
16#1157⋊C3C2×C311yescharacteristic
17C3×C42×C7C2×C311yescharacteristic, complemented
18C3×#1136C2×C311yescharacteristic, complemented
19C42×ΓL(1,𝔽8)C2×C312yescomplemented
20C42×ΓL(1,𝔽8)C2×C311yescharacteristic, complemented
21#1136⋊C3C2×C312yescomplemented
22#1136⋊C3C2×C311yescharacteristic, complemented
23C32×#104—71nomaximal, Hall
24C22×C3×ΓL(1,𝔽8)D411yescharacteristic
25C2×C3×AGL(1,𝔽7)—21no—
26C3×C4×ΓL(1,𝔽8)—22no—
27C3×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
28C3×C4×ΓL(1,𝔽8)D411yescharacteristic
29C3×ΓL(1,𝔽8)⋊C4—21no—
30#6379C3211yescharacteristic, complemented, Hall
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C22×C311yescharacteristic
33C3×C22⋊D7—21no—
34C3×D28C3×C411yescharacteristic, complemented
35C3×C4×D7—21no—
36C3×Dic14C3×C411yescharacteristic, complemented
37C2×C4×ΓL(1,𝔽8)—22no—
38C2×C4×ΓL(1,𝔽8)—21no—
39C2×C4×ΓL(1,𝔽8)C22×C312yes—
40C2×C4×ΓL(1,𝔽8)C22×C311yescharacteristic
41(C22⋊D7)⋊C3—22no—
42(C22⋊D7)⋊C3—21no—
43C4⋊AGL(1,𝔽7)C3×C412yescomplemented
44C4⋊AGL(1,𝔽7)C3×C411yescharacteristic, complemented
45C4×AGL(1,𝔽7)—22no—
46C4×AGL(1,𝔽7)—21no—
47ΓL(1,𝔽8)⋊Q8C3×C412yescomplemented
48ΓL(1,𝔽8)⋊Q8C3×C411yescharacteristic, complemented
49C3×C7⋊C8—21no—
50ΓL(1,𝔽8)⋊C8—22no—
51ΓL(1,𝔽8)⋊C8—21no—
52C32×#41—71no—
53C32×M16—71no—
54C32×C42—141no—
55C2×C3×ΓL(1,𝔽8)—21no—
56C2×C3×ΓL(1,𝔽8)C22⋊C411yescharacteristic
57C3×AGL(1,𝔽7)—41no—
58#1157C2×C3211yescharacteristic
59C42×C7C2×C3211yescharacteristic, complemented
60#1136C2×C3211yescharacteristic, complemented
61C3×#104—72no—
62C3×#104—71no—
63C3×#104—71no—
64C22×C3×C7C3×D411yescharacteristic
65C2×C3×D7—21no—
66C22×ΓL(1,𝔽8)C3×D412yes—
67C22×ΓL(1,𝔽8)C3×D411yescharacteristic
68C2×AGL(1,𝔽7)—22no—
69C2×AGL(1,𝔽7)—21no—
70C3×C4×C7—22no—
71C3×C4×C7C2×C3×C411yescharacteristic
72C3×C4×C7C3×D411yescharacteristic
73C3×Dic7—21no—
74C4×ΓL(1,𝔽8)—24no—
75C4×ΓL(1,𝔽8)—22no—
76C4×ΓL(1,𝔽8)C2×C3×C412yes—
77C4×ΓL(1,𝔽8)C2×C3×C411yescharacteristic
78C4×ΓL(1,𝔽8)C3×D412yes—
79C4×ΓL(1,𝔽8)C3×D411yescharacteristic
80ΓL(1,𝔽8)⋊C4—22no—
81ΓL(1,𝔽8)⋊C4—21no—
82C32×Q8—71no—
83C32×D4—71no—
84C32×D4—141no—
85C2×C32×C4—141no—
86C2×C32×C4—141no—
87C2×C32×C4—141no—
88C32×C8—141no—
89C3×ΓL(1,𝔽8)#10411yescharacteristic, complemented, Hall
90C2×C4×C7—21no—
91C2×C4×C7C22×C3211yescharacteristic
92C22⋊D7—21no—
93D28C32×C411yescharacteristic, complemented
94C4×D7—21no—
95Dic14C32×C411yescharacteristic, complemented
96C7⋊C8—21no—
97C3×#41—72no—
98C3×#41—71no—
99C3×#41—71no—
100C3×M16—72no—
101C3×M16—71no—
102C3×M16—71no—
103C3×C42—72no—
104C3×C42—141no—
105C3×C42AGL(1,𝔽7)11yescharacteristic, complemented
106C2×C3×C7—21no—
107C2×C3×C7C3×C22⋊C411yescharacteristic
108C3×D7—41no—
109C2×ΓL(1,𝔽8)—22no—
110C2×ΓL(1,𝔽8)—21no—
111C2×ΓL(1,𝔽8)C3×C22⋊C412yes—
112C2×ΓL(1,𝔽8)C3×C22⋊C411yescharacteristic
113AGL(1,𝔽7)—42no—
114AGL(1,𝔽7)—41no—
115C22×C32—141no—
116C22×C32—141no—
117C32×C4—142no—
118C32×C4—141no—
119C32×C4—141no—
120C32×C4—141no—
121#104—71noSylow
122C22×C7C32×D411yescharacteristic
123C2×D7—21no—
124C4×C7—22no—
125C4×C7C2×C32×C411yescharacteristic
126C4×C7C32×D411yescharacteristic
127Dic7—21no—
128C3×Q8—72no—
129C3×Q8—71no—
130C3×Q8—71no—
131C3×D4—72no—
132C3×D4—71no—
133C3×D4—71no—
134C3×D4—142no—
135C3×D4—141no—
136C3×D4—141no—
137C2×C3×C4—21no—
138C2×C3×C4—72no—
139C2×C3×C4—142no—
140C2×C3×C4—141no—
141C2×C3×C4—142no—
142C2×C3×C4—141no—
143C2×C3×C4—141no—
144C2×C3×C4—141no—
145C2×C3×C4C2×AGL(1,𝔽7)11yescharacteristic
146C3×C8—142no—
147C3×C8—141no—
148C3×C8—141no—
149C3×C7C3×#10411yescharacteristic, complemented
150ΓL(1,𝔽8)C3×#10412yescomplemented
151ΓL(1,𝔽8)C3×#10411yescharacteristic, complemented
152C2×C32—141no—
153C2×C32—141no—
154C2×C32—281no—
155#41—71no—
156M16—71no—
157C42C3×AGL(1,𝔽7)11yescharacteristic, complemented
158C2×C7—21no—
159C2×C7C32×C22⋊C411yescharacteristic
160D7—41no—
161C22×C3—72no—
162C22×C3—142no—
163C22×C3—141no—
164C22×C3—141no—
165C22×C3—141no—
166C22×C3(C22⋊D7)⋊C311yescharacteristic
167C3×C4—22no—
168C3×C4—72no—
169C3×C4—72no—
170C3×C4—144no—
171C3×C4—142no—
172C3×C4—142no—
173C3×C4—141no—
174C3×C4—141no—
175C3×C4—141no—
176C3×C4—141no—
177C3×C4C4×AGL(1,𝔽7)11yescharacteristic
178C3×C4C4⋊AGL(1,𝔽7)11yescharacteristic
179C32—141noSylow
180Q8—71no—
181D4—71no—
182D4—141no—
183C2×C4—21no—
184C2×C4—141no—
185C2×C4C2×C3×AGL(1,𝔽7)11yescharacteristic
186C8—141no—
187C7C32×#10411yescharacteristic, complemented, Sylow
188C2×C3—21no—
189C2×C3—72no—
190C2×C3—142no—
191C2×C3—141no—
192C2×C3—141no—
193C2×C3—282no—
194C2×C3—281no—
195C2×C3—281no—
196C2×C3(C2×AGL(1,𝔽7))⋊C411yescharacteristic
197C22—141no—
198C22C3×(C22⋊D7)⋊C311yescharacteristic
199C4—22no—
200C4—141no—
201C4C3×C4×AGL(1,𝔽7)11yescharacteristic
202C4C3×C4⋊AGL(1,𝔽7)11yescharacteristic
203C3—72no—
204C3—141no—
205C3#6379⋊C311yescharacteristic, direct product factor
206C2—21no—
207C2—281no—
208C2C3×(C2×AGL(1,𝔽7))⋊C411yescharacteristic
subgroupG/H
centerC3×C4C4⋊AGL(1,𝔽7)
commutatorC4×C7C2×C32×C4
FrattiniC2×C4C2×C3×AGL(1,𝔽7)
FittingC3×C42×C7C2×C3
socleC2×C3×C7C3×C22⋊C4
prime radicalC3×#1157⋊C3C2
cosocleC2×C4×C7C22×C32
hypercenterC3×C42AGL(1,𝔽7)
hypocenterC7C32×#104
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C3×C42×ΓL(1,𝔽8)C2semidirect
3. C3×#1136⋊C3C2semidirect
4. C3×#6379C3semidirect
5. #6379⋊C3C3direct (N × H)
6. #6379⋊C3C3direct (N × H)
10. C3×C4⋊AGL(1,𝔽7)C4semidirect
12. C3×ΓL(1,𝔽8)⋊Q8C4semidirect
17. C3×C42×C7C2×C3semidirect
18. C3×#1136C2×C3semidirect
19. C42×ΓL(1,𝔽8)C2×C3semidirect
20. C42×ΓL(1,𝔽8)C2×C3semidirect
21. #1136⋊C3C2×C3semidirect
22. #1136⋊C3C2×C3semidirect
30. #6379C32semidirect
34. C3×D28C3×C4semidirect
36. C3×Dic14C3×C4semidirect
43. C4⋊AGL(1,𝔽7)C3×C4semidirect
44. C4⋊AGL(1,𝔽7)C3×C4semidirect
47. ΓL(1,𝔽8)⋊Q8C3×C4semidirect
48. ΓL(1,𝔽8)⋊Q8C3×C4semidirect
59. C42×C7C2×C32semidirect
60. #1136C2×C32semidirect
89. C3×ΓL(1,𝔽8)#104semidirect
93. D28C32×C4semidirect
95. Dic14C32×C4semidirect
105. C3×C42AGL(1,𝔽7)semidirect
149. C3×C7C3×#104semidirect
150. ΓL(1,𝔽8)C3×#104semidirect
151. ΓL(1,𝔽8)C3×#104semidirect
157. C42C3×AGL(1,𝔽7)semidirect
187. C7C32×#104semidirect
205. C3#6379⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#1157⋊C32
C3×C42×ΓL(1,𝔽8)2
C3×#1136⋊C32
C3×#63793
#6379⋊C33
C32×#1047