Finite Group Databasebeta

← C3×ΓL(1,𝔽8)×#1020 #72244916

Subgroups — C3×ΓL(1,𝔽8)×#1020 #72244916

Order 6048.

784 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 56 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×C7×#1020C311yesmaximal, characteristic, complemented
2C3×ΓL(1,𝔽8)×C22⋊A4C211yesmaximal, characteristic, complemented
3C3×ΓL(1,𝔽8)×#120—31nomaximal
4ΓL(1,𝔽8)×#1020C313yesmaximal, direct product factor
5C3×ΓL(1,𝔽8)×S4—43nomaximal
6C3×C22⋊(C22⋊ΓL(1,𝔽8))—21no—
7C3×C7×C22⋊A4C2×C311yescharacteristic, complemented
8C24×C3×ΓL(1,𝔽8)D311yescharacteristic, complemented
9ΓL(1,𝔽8)×C22⋊A4—23no—
10ΓL(1,𝔽8)×C22⋊A4C2×C313yescomplemented
11C2×C3×D4×ΓL(1,𝔽8)—33no—
12C3×C22⋊C4×ΓL(1,𝔽8)—33no—
13C32×#1020—71nomaximal, Hall
14C3×A4×ΓL(1,𝔽8)—43no—
15C3×A4×ΓL(1,𝔽8)—81no—
16C7×#1020C3211yescharacteristic, complemented
17C3×C7×#120—31no—
18ΓL(1,𝔽8)×#120—33no—
19C23×C3×ΓL(1,𝔽8)—33no—
20C23×C3×ΓL(1,𝔽8)—31no—
21C23×C3×ΓL(1,𝔽8)—61no—
22C3×C7×S4—43no—
23ΓL(1,𝔽8)×S4—49no—
24C3×D4×ΓL(1,𝔽8)—63no—
25C3×D4×ΓL(1,𝔽8)—63no—
26C2×C3×C4×ΓL(1,𝔽8)—33no—
27C32×C22⋊A4—71no—
28C3×D3×ΓL(1,𝔽8)—161no—
29C24×C3×C7C3×D311yescharacteristic, complemented
30C22⋊(C22⋊ΓL(1,𝔽8))—23no—
31C7×C22⋊A4—21no—
32C7×C22⋊A4C2×C3211yescharacteristic, complemented
33C24×ΓL(1,𝔽8)C3×D313yescomplemented
34C2×C3×C7×D4—33no—
35C2×D4×ΓL(1,𝔽8)—39no—
36C3×C7×C22⋊C4—33no—
37C22⋊C4×ΓL(1,𝔽8)—39no—
38C3×#1020—73no—
39C3×#1020ΓL(1,𝔽8)11yescharacteristic, direct product factor
40C32×#120—211no—
41C3×C22⋊ΓL(1,𝔽8)—82no—
42C3×C22⋊ΓL(1,𝔽8)—83no—
43C3×C7×A4—43no—
44C3×C7×A4—81no—
45C22×C3×ΓL(1,𝔽8)—21no—
46C22×C3×ΓL(1,𝔽8)—31no—
47C22×C3×ΓL(1,𝔽8)—33no—
48C22×C3×ΓL(1,𝔽8)—63no—
49C22×C3×ΓL(1,𝔽8)—63no—
50C22×C3×ΓL(1,𝔽8)S413yescomplemented
51A4×ΓL(1,𝔽8)—49no—
52A4×ΓL(1,𝔽8)—83no—
53A4×ΓL(1,𝔽8)—86no—
54A4×ΓL(1,𝔽8)—89no—
55C3×C4×ΓL(1,𝔽8)—63no—
56C7×#120—31noHall
57C32×S4—283no—
58C32×ΓL(1,𝔽8)—161noHall
59C23×C3×C7—33no—
60C23×C3×C7—31no—
61C23×C3×C7—61no—
62C23×ΓL(1,𝔽8)—39no—
63C23×ΓL(1,𝔽8)—33no—
64C23×ΓL(1,𝔽8)—63no—
65C7×S4—43no—
66C3×C7×D4—63no—
67C3×C7×D4—63no—
68C2×C3×C4×C7—33no—
69D4×ΓL(1,𝔽8)—69no—
70D4×ΓL(1,𝔽8)—69no—
71C2×C4×ΓL(1,𝔽8)—39no—
72C24×C32—71no—
73C3×C22⋊A4—73no—
74C3×C22⋊A4—143no—
75C3×C22⋊A4—141no—
76C3×C22⋊A4C2×ΓL(1,𝔽8)11yescharacteristic, complemented
77C2×C32×D4—213no—
78C32×C22⋊C4—213no—
79C3×D3×C7—161no—
80C2×C3×ΓL(1,𝔽8)—33no—
81C2×C3×ΓL(1,𝔽8)—61no—
82C2×C3×ΓL(1,𝔽8)—121no—
83D3×ΓL(1,𝔽8)—163no—
84C24×C7C32×D311yescharacteristic, complemented
85C2×C7×D4—33no—
86C7×C22⋊C4—33no—
87C32×A4—283no—
88C32×A4—561no—
89#1020C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor
90C3×#120—31no—
91C3×#120—213no—
92C22×C3×C7—21no—
93C22×C3×C7—31no—
94C22×C3×C7—33no—
95C22×C3×C7—63no—
96C22×C3×C7—63no—
97C22×C3×C7C3×S413yescomplemented
98C22⋊ΓL(1,𝔽8)—86no—
99C22⋊ΓL(1,𝔽8)—89no—
100C7×A4—43no—
101C7×A4—81no—
102C7×A4—82no—
103C7×A4—83no—
104C22×ΓL(1,𝔽8)—23no—
105C22×ΓL(1,𝔽8)—33no—
106C22×ΓL(1,𝔽8)—39no—
107C22×ΓL(1,𝔽8)—69no—
108C22×ΓL(1,𝔽8)—69no—
109C22×ΓL(1,𝔽8)C3×S419yescomplemented
110C3×C4×C7—63no—
111C4×ΓL(1,𝔽8)—69no—
112C23×C32—213no—
113C23×C32—211no—
114C23×C32—421no—
115C3×S4—43no—
116C3×S4—289no—
117C32×D4—423no—
118C32×D4—423no—
119C2×C32×C4—213no—
120C32×C7—161no—
121C3×ΓL(1,𝔽8)—163no—
122C3×ΓL(1,𝔽8)—321no—
123C3×ΓL(1,𝔽8)—323no—
124C3×ΓL(1,𝔽8)#102011yescharacteristic, direct product factor
125C23×C7—33no—
126C23×C7—31no—
127C23×C7—61no—
128C7×D4—63no—
129C7×D4—63no—
130C2×C4×C7—33no—
131C32×D3—1121no—
132C24×C3—73no—
133C24×C3D3×ΓL(1,𝔽8)11yescharacteristic, complemented
134C22⋊A4—21no—
135C22⋊A4—143no—
136C22⋊A4C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
137C2×C3×D4—33no—
138C2×C3×D4—219no—
139C3×C22⋊C4—33no—
140C3×C22⋊C4—219no—
141C2×C3×C7—33no—
142C2×C3×C7—61no—
143C2×C3×C7—121no—
144D3×C7—161no—
145C2×ΓL(1,𝔽8)—39no—
146C2×ΓL(1,𝔽8)—63no—
147C2×ΓL(1,𝔽8)—123no—
148C22×C32—73no—
149C22×C32—141no—
150C22×C32—211no—
151C22×C32—213no—
152C22×C32—423no—
153C22×C32—423no—
154C3×A4—43no—
155C3×A4—81no—
156C3×A4—289no—
157C3×A4—563no—
158C3×A4—569no—
159C3×A4—566no—
160C3×A4—562no—
161C3×A4—563no—
162C32×C4—423no—
163#120—31noSylow
164C22×C7—21no—
165C22×C7—31no—
166C22×C7—33no—
167C22×C7—63no—
168C22×C7—63no—
169C22×C7C32×S413yescomplemented
170C4×C7—63no—
171C33—1121noSylow
172C23×C3—33no—
173C23×C3—31no—
174C23×C3—61no—
175C23×C3—219no—
176C23×C3—213no—
177C23×C3—423no—
178S4—43no—
179C3×D4—63no—
180C3×D4—63no—
181C3×D4—429no—
182C3×D4—429no—
183C2×C3×C4—33no—
184C2×C3×C4—219no—
185C3×C7—161no—
186C3×C7—321no—
187C3×C7C3×#102011yescharacteristic, complemented
188ΓL(1,𝔽8)—323no—
189ΓL(1,𝔽8)C3×#102013yesdirect product factor
190C2×C32—213no—
191C2×C32—421no—
192C2×C32—841no—
193C3×D3—161no—
194C3×D3—1123no—
195C24C3×D3×ΓL(1,𝔽8)11yescharacteristic, complemented
196C2×D4—33no—
197C22⋊C4—33no—
198C2×C7—33no—
199C2×C7—61no—
200C2×C7—121no—
201C22×C3—21no—
202C22×C3—31no—
203C22×C3—33no—
204C22×C3—63no—
205C22×C3—63no—
206C22×C3—79no—
207C22×C3—143no—
208C22×C3—213no—
209C22×C3—219no—
210C22×C3—429no—
211C22×C3—429no—
212C22×C3ΓL(1,𝔽8)×S413yescomplemented
213A4—43no—
214A4—81no—
215A4—82no—
216A4—83no—
217A4—566no—
218A4—569no—
219C3×C4—63no—
220C3×C4—429no—
221C32—71no—
222C32—161no—
223C32—1123no—
224C32—2243no—
225C32—2241no—
226C23—33no—
227C23—31no—
228C23—61no—
229D4—63no—
230D4—63no—
231C2×C4—33no—
232C7C32×#102011yescharacteristic, complemented, Sylow
233C2×C3—33no—
234C2×C3—61no—
235C2×C3—121no—
236C2×C3—219no—
237C2×C3—423no—
238C2×C3—843no—
239D3—161no—
240C22—21no—
241C22—31no—
242C22—33no—
243C22—63no—
244C22—63no—
245C22C3×ΓL(1,𝔽8)×S413yescomplemented
246C4—63no—
247C3—73no—
248C3—161no—
249C3—321no—
250C3—2243no—
251C3ΓL(1,𝔽8)×#102011yescharacteristic, direct product factor
252C2—33no—
253C2—61no—
254C2—121no—
subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×#1020
commutator, cosocle, hypocenterC7×C22⋊A4C2×C32
FrattiniC1G
Fitting, socleC24×C3×C7C3×D3
LCSG ▷ C7×C22⋊A4
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×C7×#1020C3semidirect
2. C3×ΓL(1,𝔽8)×C22⋊A4C2semidirect
4. ΓL(1,𝔽8)×#1020C3direct (N × H)
7. C3×C7×C22⋊A4C2×C3semidirect
8. C24×C3×ΓL(1,𝔽8)D3semidirect
10. ΓL(1,𝔽8)×C22⋊A4C2×C3semidirect
16. C7×#1020C32semidirect
29. C24×C3×C7C3×D3semidirect
32. C7×C22⋊A4C2×C32semidirect
33. C24×ΓL(1,𝔽8)C3×D3semidirect
39. C3×#1020ΓL(1,𝔽8)direct (N × H)
50. C22×C3×ΓL(1,𝔽8)S4semidirect
76. C3×C22⋊A4C2×ΓL(1,𝔽8)semidirect
84. C24×C7C32×D3semidirect
89. #1020C3×ΓL(1,𝔽8)direct (N × H)
97. C22×C3×C7C3×S4semidirect
109. C22×ΓL(1,𝔽8)C3×S4semidirect
124. C3×ΓL(1,𝔽8)#1020direct (N × H)
133. C24×C3D3×ΓL(1,𝔽8)semidirect
136. C22⋊A4C2×C3×ΓL(1,𝔽8)semidirect
169. C22×C7C32×S4semidirect
187. C3×C7C3×#1020semidirect
189. ΓL(1,𝔽8)C3×#1020direct (N × H)
195. C24C3×D3×ΓL(1,𝔽8)semidirect
212. C22×C3ΓL(1,𝔽8)×S4semidirect
232. C7C32×#1020semidirect
245. C22C3×ΓL(1,𝔽8)×S4semidirect
251. C3ΓL(1,𝔽8)×#1020direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C7×#10203
C3×ΓL(1,𝔽8)×C22⋊A42
C3×ΓL(1,𝔽8)×#1203
ΓL(1,𝔽8)×#10203
C3×ΓL(1,𝔽8)×S44
C32×#10207