Finite Group Databasebeta

← #72249181

Subgroups — #72249181

Order 3402.

200 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 28 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
1#13133628C211yesmaximal, characteristic, complemented, Hall
2C3×(C3⋊D3)⋊ΓL(1,𝔽8)—31nomaximal
3ΓL(1,𝔽8)×AGL(1,ℤ/9)—31nomaximal
4ΓL(1,𝔽8)×AGL(1,ℤ/9)—31nomaximal
5ΓL(1,𝔽8)×(C3⋊D3)⋊C3—31nomaximal
6C7×#4183C311yesmaximal, characteristic, complemented
7C3×C7⋊H(3,𝔽3)D311yescharacteristic, complemented
8ΓL(1,𝔽8)×3−1+2D311yescharacteristic, complemented
9ΓL(1,𝔽8)×3−1+2D311yescharacteristic, complemented
10ΓL(1,𝔽8)×H(3,𝔽3)D311yescharacteristic, complemented
11C7×AΣL(1,𝔽27)C2×C311yescharacteristic, complemented
12#10588048—21no—
13#10588046—21no—
14#10588046—21no—
15#10588043—21no—
16#4183⋊C3—71nomaximal, Hall
17C3×C7×C3⋊D3—31no—
18C3⋊D3×ΓL(1,𝔽8)—31no—
19C3×D3×ΓL(1,𝔽8)—91no—
20C7×AGL(1,ℤ/9)—31no—
21C7×AGL(1,ℤ/9)—31no—
22C7×(C3⋊D3)⋊C3—31no—
23(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
24(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
25(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
26(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
27(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
28D9×ΓL(1,𝔽8)—31no—
29D9×ΓL(1,𝔽8)—31no—
30#67361—31no—
31#67361—31no—
32ΓL(1,𝔽64)—31no—
33ΓL(1,𝔽64)—31no—
34#6909—71noSylow
35C33×C7C3×D311yescharacteristic, complemented
36C32×ΓL(1,𝔽8)—21no—
37C32×ΓL(1,𝔽8)—61no—
38C32×ΓL(1,𝔽8)C3⋊D311yescharacteristic
39C7×3−1+2C3×D311yescharacteristic, complemented
40C7×3−1+2C3×D311yescharacteristic, complemented
41C7×H(3,𝔽3)C3×D311yescharacteristic
42C7⋊H(3,𝔽3)—21no—
43C7⋊H(3,𝔽3)—31no—
44C7⋊H(3,𝔽3)—31no—
45C7⋊H(3,𝔽3)—31no—
46C7⋊H(3,𝔽3)—31no—
47C7⋊H(3,𝔽3)—61no—
48#4430—21no—
49#4430—21no—
50#4428—31no—
51#4428—61no—
52#4427—31no—
53#4427—61no—
54C9×ΓL(1,𝔽8)—31no—
55C9×ΓL(1,𝔽8)—61no—
56C3×AGL(1,ℤ/9)—211no—
57C3×AGL(1,ℤ/9)—211no—
58C3×(C3⋊D3)⋊C3—211no—
59C3×(C3⋊D3)⋊C3—211no—
60#4183ΓL(1,𝔽8)11yescharacteristic, complemented
61C7×C3⋊D3—31no—
62C7×C3⋊D3—31no—
63C3×D3×C7—91no—
64C3×D3×C7—271no—
65C2×C3×ΓL(1,𝔽8)—271no—
66D3×ΓL(1,𝔽8)—91no—
67D3×ΓL(1,𝔽8)—91no—
68D3×ΓL(1,𝔽8)—91no—
69D3×ΓL(1,𝔽8)—271no—
70C7×D9—31no—
71C7×D9—31no—
72C3×3−1+2—71no—
73C3×3−1+2—71no—
74C3×H(3,𝔽3)—71no—
75C3×H(3,𝔽3)—71no—
76AΣL(1,𝔽27)—141no—
77AΣL(1,𝔽27)—141no—
78AΣL(1,𝔽27)—141no—
79AΣL(1,𝔽27)—141no—
80AΣL(1,𝔽27)C2×ΓL(1,𝔽8)11yescharacteristic, complemented
81C32×C7—31no—
82C32×C7—31no—
83C32×C7—181no—
84C32×C7C3×C3⋊D311yescharacteristic
85C3×ΓL(1,𝔽8)—61no—
86C3×ΓL(1,𝔽8)—61no—
87C3×ΓL(1,𝔽8)—91no—
88C3×ΓL(1,𝔽8)—181no—
89C3×ΓL(1,𝔽8)—181no—
90C3×ΓL(1,𝔽8)—181no—
91C3×ΓL(1,𝔽8)#27211yescharacteristic
92C3×ΓL(1,𝔽8)#27211yescharacteristic
93C3×ΓL(1,𝔽8)#27211yescharacteristic
94C7×C9—31no—
95C7×C9—31no—
96C7⋊C9—61no—
97C7⋊C9—61no—
98C3×C3⋊D3—31no—
99C3×C3⋊D3—211no—
100C32×D3—631no—
101AGL(1,ℤ/9)—31no—
102AGL(1,ℤ/9)—31no—
103AGL(1,ℤ/9)—211no—
104AGL(1,ℤ/9)—211no—
105AGL(1,ℤ/9)—211no—
106AGL(1,ℤ/9)—211no—
107(C3⋊D3)⋊C3—31no—
108(C3⋊D3)⋊C3—211no—
109(C3⋊D3)⋊C3—211no—
110(C3⋊D3)⋊C3—211no—
111(C3⋊D3)⋊C3—211no—
112(C3⋊D3)⋊C3—211no—
113C3×D9—211no—
114C3×D9—211no—
115C2×C3×C7—271no—
116D3×C7—91no—
117D3×C7—271no—
118D3×C7—271no—
119C2×ΓL(1,𝔽8)—271no—
120C2×ΓL(1,𝔽8)—271no—
121C2×ΓL(1,𝔽8)—271no—
122C33—71no—
123C33—141no—
124C33—211no—
125C33D3×ΓL(1,𝔽8)11yescharacteristic, complemented
1263−1+2—141no—
1273−1+2—141no—
1283−1+2—211no—
1293−1+2—211no—
1303−1+2—421no—
1313−1+2—421no—
1323−1+2D3×ΓL(1,𝔽8)11yescharacteristic, complemented
1333−1+2D3×ΓL(1,𝔽8)11yescharacteristic, complemented
134H(3,𝔽3)—141no—
135H(3,𝔽3)—211no—
136H(3,𝔽3)—421no—
137H(3,𝔽3)—421no—
138H(3,𝔽3)—421no—
139H(3,𝔽3)—421no—
140H(3,𝔽3)D3×ΓL(1,𝔽8)11yescharacteristic
141C3×C9—421no—
142C3×C9—421no—
143C3×C7—61no—
144C3×C7—181no—
145C3×C7—181no—
146C3×C7C3×#27211yescharacteristic
147ΓL(1,𝔽8)—31no—
148ΓL(1,𝔽8)—61no—
149ΓL(1,𝔽8)—61no—
150ΓL(1,𝔽8)—181no—
151ΓL(1,𝔽8)—181no—
152C2×C32—1891no—
153C3⋊D3—31no—
154C3⋊D3—31no—
155C3×D3—91no—
156C3×D3—271no—
157C3×D3—631no—
158C3×D3—631no—
159C3×D3—631no—
160C3×D3—1891no—
161D9—31no—
162D9—31no—
163C2×C7—271noHall
164C32—31no—
165C32—61no—
166C32—71no—
167C32—71no—
168C32—71no—
169C32—91no—
170C32—421no—
171C32—421no—
172C32—631no—
173C32—1261no—
174C32—1261no—
175C32—1261no—
176C32C3⋊D3×ΓL(1,𝔽8)11yescharacteristic
177C9—61no—
178C9—61no—
179C9—421no—
180C9—421no—
181C7#4183⋊C311yescharacteristic, complemented, Sylow
182C2×C3—271no—
183C2×C3—1891no—
184C2×C3—1891no—
185C2×C3—1891no—
186D3—91no—
187D3—271no—
188D3—271no—
189C3—61no—
190C3—181no—
191C3—181no—
192C3—211no—
193C3—421no—
194C3—421no—
195C3—1261no—
196C3—1261no—
197C3ΓL(1,𝔽8)×#27211yescharacteristic
198C2—271noSylow
subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×AΣL(1,𝔽27)C2×C3
FrattiniC32C3⋊D3×ΓL(1,𝔽8)
socleC3×C7C3×#272
LCSG ▷ C7×AΣL(1,𝔽27)
UCSC1

Complements

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

NH ≅ G/Ntype
1. #13133628C2semidirect
6. C7×#4183C3semidirect
7. C3×C7⋊H(3,𝔽3)D3semidirect
8. ΓL(1,𝔽8)×3−1+2D3semidirect
9. ΓL(1,𝔽8)×3−1+2D3semidirect
10. ΓL(1,𝔽8)×H(3,𝔽3)D3semidirect
11. C7×AΣL(1,𝔽27)C2×C3semidirect
35. C33×C7C3×D3semidirect
39. C7×3−1+2C3×D3semidirect
40. C7×3−1+2C3×D3semidirect
60. #4183ΓL(1,𝔽8)semidirect
80. AΣL(1,𝔽27)C2×ΓL(1,𝔽8)semidirect
125. C33D3×ΓL(1,𝔽8)semidirect
132. 3−1+2D3×ΓL(1,𝔽8)semidirect
133. 3−1+2D3×ΓL(1,𝔽8)semidirect
181. C7#4183⋊C3semidirect

Maximal subgroups

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

subgroupindex
#131336282
C3×(C3⋊D3)⋊ΓL(1,𝔽8)3
ΓL(1,𝔽8)×AGL(1,ℤ/9)3
ΓL(1,𝔽8)×(C3⋊D3)⋊C33
C7×#41833
#4183⋊C37