Finite Group Databasebeta

← ΓL(1,𝔽8)×#4184 #72249191

Subgroups — ΓL(1,𝔽8)×#4184 #72249191

Order 3402.

192 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 29 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
1ΓL(1,𝔽8)×#731C211yesmaximal, characteristic, complemented, Hall
2ΓL(1,𝔽8)×AGL(1,ℤ/9)—31nomaximal
3ΓL(1,𝔽8)×AGL(1,ℤ/9)—31nomaximal
4ΓL(1,𝔽8)×(C3⋊D3)⋊C3—31nomaximal
5C3×D9×ΓL(1,𝔽8)—31nomaximal
6C7×#4184C311yesmaximal, characteristic, complemented
7ΓL(1,𝔽8)×3−1+2D311yescharacteristic, complemented
8ΓL(1,𝔽8)×3−1+2D311yescharacteristic, complemented
9ΓL(1,𝔽8)×H(3,𝔽3)D311yescharacteristic
10C3×C9×ΓL(1,𝔽8)D311yescharacteristic, complemented
11C7×#731C2×C311yescharacteristic, complemented
12#10588045—21no—
13#10588045—21no—
14#10588041—21no—
15#10588037—21no—
16C3×#4184—71nomaximal, Hall
17C3⋊D3×ΓL(1,𝔽8)—31no—
18C3×D3×ΓL(1,𝔽8)—91no—
19C7×AGL(1,ℤ/9)—31no—
20C7×AGL(1,ℤ/9)—31no—
21C7×(C3⋊D3)⋊C3—31no—
22C3×C7×D9—31no—
23(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
24(C3⋊D3)⋊ΓL(1,𝔽8)—31no—
25D9×ΓL(1,𝔽8)—31no—
26D9×ΓL(1,𝔽8)—31no—
27D9×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—
34C3×#731—71noSylow
35C32×ΓL(1,𝔽8)—61no—
36C32×ΓL(1,𝔽8)C3⋊D311yescharacteristic
37C7×3−1+2C3×D311yescharacteristic, complemented
38C7×3−1+2C3×D311yescharacteristic, complemented
39C7×H(3,𝔽3)C3×D311yescharacteristic
40C3×C7×C9C3×D311yescharacteristic, complemented
41C7⋊H(3,𝔽3)—21no—
42C7⋊H(3,𝔽3)—31no—
43C7⋊H(3,𝔽3)—61no—
44#4430—21no—
45#4430—21no—
46C3×C7⋊C9—21no—
47#4428—31no—
48#4428—61no—
49#4427—31no—
50#4427—61no—
51C9×ΓL(1,𝔽8)—31no—
52C9×ΓL(1,𝔽8)—31no—
53C9×ΓL(1,𝔽8)—61no—
54C9×ΓL(1,𝔽8)—61no—
55C9×ΓL(1,𝔽8)—61no—
56C3×AGL(1,ℤ/9)—211no—
57C3×AGL(1,ℤ/9)—211no—
58C3×(C3⋊D3)⋊C3—211no—
59C32×D9—211no—
60#4184ΓL(1,𝔽8)11yescharacteristic, direct product factor
61C7×C3⋊D3—31no—
62C3×D3×C7—91no—
63C2×C3×ΓL(1,𝔽8)—271no—
64D3×ΓL(1,𝔽8)—91no—
65D3×ΓL(1,𝔽8)—91no—
66D3×ΓL(1,𝔽8)—91no—
67D3×ΓL(1,𝔽8)—271no—
68C7×D9—31no—
69C7×D9—31no—
70C7×D9—31no—
71C3×3−1+2—71no—
72C3×3−1+2—71no—
73C3×H(3,𝔽3)—71no—
74C32×C9—71no—
75#731—141no—
76#731—141no—
77#731—141no—
78#731—141no—
79#731C2×ΓL(1,𝔽8)11yescharacteristic, complemented
80C32×C7—31no—
81C32×C7C3×C3⋊D311yescharacteristic
82C3×ΓL(1,𝔽8)—61no—
83C3×ΓL(1,𝔽8)—61no—
84C3×ΓL(1,𝔽8)—61no—
85C3×ΓL(1,𝔽8)—181no—
86C3×ΓL(1,𝔽8)—181no—
87C3×ΓL(1,𝔽8)#27211yescharacteristic
88C3×ΓL(1,𝔽8)#27211yescharacteristic
89C3×ΓL(1,𝔽8)#27211yescharacteristic
90C7×C9—31no—
91C7×C9—31no—
92C7×C9—61no—
93C7⋊C9—61no—
94C7⋊C9—61no—
95C7⋊C9—61no—
96C3×C3⋊D3—211no—
97C32×D3—631no—
98AGL(1,ℤ/9)—31no—
99AGL(1,ℤ/9)—31no—
100AGL(1,ℤ/9)—211no—
101AGL(1,ℤ/9)—211no—
102AGL(1,ℤ/9)—211no—
103AGL(1,ℤ/9)—211no—
104(C3⋊D3)⋊C3—31no—
105(C3⋊D3)⋊C3—211no—
106(C3⋊D3)⋊C3—211no—
107C3×D9—31no—
108C3×D9—211no—
109C3×D9—211no—
110C3×D9—211no—
111C3×D9—211no—
112C3×D9—211no—
113C2×C3×C7—271no—
114D3×C7—91no—
115D3×C7—271no—
116C2×ΓL(1,𝔽8)—271no—
117C2×ΓL(1,𝔽8)—271no—
118C2×ΓL(1,𝔽8)—271no—
119C33—71no—
120C33—211no—
1213−1+2—141no—
1223−1+2—141no—
1233−1+2—211no—
1243−1+2—421no—
1253−1+2—421no—
1263−1+2—421no—
1273−1+2D3×ΓL(1,𝔽8)11yescharacteristic, complemented
1283−1+2D3×ΓL(1,𝔽8)11yescharacteristic, complemented
129H(3,𝔽3)—141no—
130H(3,𝔽3)—421no—
131H(3,𝔽3)—421no—
132H(3,𝔽3)D3×ΓL(1,𝔽8)11yescharacteristic
133C3×C9—141no—
134C3×C9—211no—
135C3×C9—211no—
136C3×C9—421no—
137C3×C9—421no—
138C3×C9—421no—
139C3×C9D3×ΓL(1,𝔽8)11yescharacteristic, complemented
140C3×C7—61no—
141C3×C7—181no—
142C3×C7C3×#27211yescharacteristic
143ΓL(1,𝔽8)—21no—
144ΓL(1,𝔽8)—31no—
145ΓL(1,𝔽8)—61no—
146ΓL(1,𝔽8)—181no—
147ΓL(1,𝔽8)#418411yescharacteristic, direct product factor
148C2×C32—1891no—
149C3⋊D3—31no—
150C3×D3—91no—
151C3×D3—631no—
152C3×D3—631no—
153C3×D3—631no—
154C3×D3—1891no—
155D9—31no—
156D9—31no—
157D9—31no—
158C2×C7—271noHall
159C32—61no—
160C32—71no—
161C32—71no—
162C32—71no—
163C32—421no—
164C32—421no—
165C32—421no—
166C32—1261no—
167C32—1261no—
168C32C3⋊D3×ΓL(1,𝔽8)11yescharacteristic
169C9—31no—
170C9—61no—
171C9—61no—
172C9—421no—
173C9—421no—
174C9—421no—
175C7C3×#418411yescharacteristic, complemented, Sylow
176C2×C3—271no—
177C2×C3—1891no—
178C2×C3—1891no—
179C2×C3—1891no—
180D3—91no—
181D3—271no—
182C3—61no—
183C3—71no—
184C3—141no—
185C3—181no—
186C3—421no—
187C3—421no—
188C3—1261no—
189C3ΓL(1,𝔽8)×#27211yescharacteristic
190C2—271noSylow
subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×#731C2×C3
FrattiniC32C3⋊D3×ΓL(1,𝔽8)
socleC3×C7C3×#272
LCSG ▷ C7×#731
UCSC1

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#731C2semidirect
6. C7×#4184C3semidirect
7. ΓL(1,𝔽8)×3−1+2D3semidirect
8. ΓL(1,𝔽8)×3−1+2D3semidirect
10. C3×C9×ΓL(1,𝔽8)D3semidirect
11. C7×#731C2×C3semidirect
37. C7×3−1+2C3×D3semidirect
38. C7×3−1+2C3×D3semidirect
40. C3×C7×C9C3×D3semidirect
60. #4184ΓL(1,𝔽8)direct (N × H)
79. #731C2×ΓL(1,𝔽8)semidirect
127. 3−1+2D3×ΓL(1,𝔽8)semidirect
128. 3−1+2D3×ΓL(1,𝔽8)semidirect
139. C3×C9D3×ΓL(1,𝔽8)semidirect
147. ΓL(1,𝔽8)#4184direct (N × H)
175. C7C3×#4184semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#7312
ΓL(1,𝔽8)×AGL(1,ℤ/9)3
ΓL(1,𝔽8)×(C3⋊D3)⋊C33
C3×D9×ΓL(1,𝔽8)3
C7×#41843
C3×#41847