Finite Group Databasebeta

← #72222243

Subgroups — #72222243

Order 6048.

456 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 35 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#72222240—32nomaximal
2#72222237C311yesmaximal, characteristic, complemented
3A42×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4#72214032—32nomaximal
5C3⋊(A4⋊AGL(1,𝔽7))—42nomaximal
6(C22⋊A4)⋊ΓL(1,𝔽8)—22no—
7A4×C22⋊ΓL(1,𝔽8)—22no—
8C7×A42C2×C311yescharacteristic, complemented
9ΓL(1,𝔽8)×C22⋊A4D312yescomplemented
10C22×A4×ΓL(1,𝔽8)D312yescomplemented
11C2×A4⋊AGL(1,𝔽7)—32no—
12A4⋊(ΓL(1,𝔽8)⋊C4)—32no—
13C3×A4⋊S4—71nomaximal, Hall
14C3×A4×ΓL(1,𝔽8)—42no—
15C7⋊#1020—32no—
16#10621506—32no—
17#10620789—91no—
18C3⋊(C7⋊S4)—42no—
19C2×A4×ΓL(1,𝔽8)—32no—
20A4⋊AGL(1,𝔽7)—62no—
21A4⋊AGL(1,𝔽7)—124no—
22A4⋊AGL(1,𝔽7)—122no—
23#4268⋊C3—122no—
24C3×A42—71no—
25C3⋊(C3⋊AGL(1,𝔽7))—161no—
26C22⋊(C22⋊ΓL(1,𝔽8))—22no—
27C7×C22⋊A4C3×D312yescomplemented
28C22×C22⋊ΓL(1,𝔽8)—22no—
29C22×C7×A4C3×D312yescomplemented
30C24×ΓL(1,𝔽8)C3⋊D311yescharacteristic, complemented
31C2×C7⋊S4—32no—
32C2×(C22⋊D7)⋊C3—92no—
33C2×(C22⋊D7)⋊C3—91no—
34C7⋊SL(2,ℤ/4)—32no—
35(C22⋊Dic7)⋊C3—92no—
36(C22⋊Dic7)⋊C3—91no—
37C3×#1020—212no—
38A4⋊S4—71no—
39C3×GL(2,ℤ/4)—212no—
40C3×C22⋊ΓL(1,𝔽8)—82no—
41C3×C7×A4—42no—
42C22×C3×ΓL(1,𝔽8)—42no—
43C2×C3⋊AGL(1,𝔽7)—122no—
44A4×ΓL(1,𝔽8)—44no—
45A4×ΓL(1,𝔽8)—84no—
46A4×ΓL(1,𝔽8)—82no—
47A4×ΓL(1,𝔽8)—122no—
48A4×ΓL(1,𝔽8)S412yescomplemented
49C3⋊(ΓL(1,𝔽8)⋊C4)—122no—
50#6516—91noHall
51C3×C3⋊S4—282no—
52C32×ΓL(1,𝔽8)—161noHall
53C2×C22⋊ΓL(1,𝔽8)—62no—
54C2×C7×A4—32no—
55C23×ΓL(1,𝔽8)—32no—
56C23×ΓL(1,𝔽8)—91no—
57C22×AGL(1,𝔽7)—91no—
58C7⋊S4—62no—
59C7⋊S4—124no—
60C7⋊S4—122no—
61#4268—122no—
62(C22⋊D7)⋊C3—182no—
63(C22⋊D7)⋊C3—182no—
64(C22⋊D7)⋊C3—182no—
65C2×ΓL(1,𝔽8)⋊C4—92no—
66C2×ΓL(1,𝔽8)⋊C4—91no—
67C3×C22⋊A4—72no—
68C22×C3×A4—72no—
69C2×C3×S4—212no—
70A42—142no—
71A42—142no—
72A42AGL(1,𝔽7)11yescharacteristic, complemented
73C3×SL(2,ℤ/4)—212no—
74C3⋊D21—161no—
75C2×C3×ΓL(1,𝔽8)—122no—
76C3⋊AGL(1,𝔽7)—242no—
77C3⋊AGL(1,𝔽7)—482no—
78C24×C7C3×C3⋊D311yescharacteristic, complemented
79C2×C22⋊D7—92no—
80C2×C22⋊D7—91no—
81C22⋊Dic7—92no—
82C22⋊Dic7—91no—
83C32×A4—282no—
84#1020—212no—
85GL(2,ℤ/4)—212no—
86C3×#120—631no—
87C22×C3×C7—42no—
88C2×D21—122no—
89C22⋊ΓL(1,𝔽8)—22no—
90C22⋊ΓL(1,𝔽8)—84no—
91C22⋊ΓL(1,𝔽8)—242no—
92C7×A4—44no—
93C7×A4—122no—
94C7×A4C3×S412yescomplemented
95C22×ΓL(1,𝔽8)—32no—
96C22×ΓL(1,𝔽8)—82no—
97C22×ΓL(1,𝔽8)—91no—
98C22×ΓL(1,𝔽8)—92no—
99C22×ΓL(1,𝔽8)C3⋊S412yescomplemented
100C2×AGL(1,𝔽7)—182no—
101C2×AGL(1,𝔽7)—181no—
102Dic21—122no—
103ΓL(1,𝔽8)⋊C4—182no—
104ΓL(1,𝔽8)⋊C4—181no—
105C2×C3×A4—212no—
106C3⋊S4—282no—
107C3×S4—422no—
108C3×S4—844no—
109C3×S4—842no—
110C3×#67—842no—
111C32×C7—161no—
112C3×ΓL(1,𝔽8)—42no—
113C3×ΓL(1,𝔽8)—162no—
114C3×ΓL(1,𝔽8)—322no—
115C3×ΓL(1,𝔽8)—322no—
116C23×C7—32no—
117C23×C7—91no—
118C22×D7—91no—
119C22⋊D7—182no—
120C22⋊D7—182no—
121C22⋊D7—182no—
122C2×Dic7—92no—
123C2×Dic7—91no—
124C3×C3⋊D3—1121no—
125C24×C3—71no—
126C22⋊A4—142no—
127C22⋊A4C3⋊AGL(1,𝔽7)12yescomplemented
128C22×A4—142no—
129C22×A4C3⋊AGL(1,𝔽7)12yescomplemented
130C2×S4—212no—
131C2×C3×D4—632no—
132C2×C3×D4—631no—
133SL(2,ℤ/4)—212no—
134C3×C22⋊C4—632no—
135C3×C22⋊C4—631no—
136C2×C3×C7—122no—
137D21—242no—
138D21—482no—
139C2×ΓL(1,𝔽8)—32no—
140C2×ΓL(1,𝔽8)—91no—
141C2×ΓL(1,𝔽8)—242no—
142AGL(1,𝔽7)—361no—
143C22×C32—282no—
144C2×C3×D3—842no—
145C3×A4—42no—
146C3×A4—72no—
147C3×A4—284no—
148C3×A4—564no—
149C3×A4—562no—
150C3×A4—562no—
151C3×A4—842no—
152C3×Dic3—842no—
153#120—631noSylow
154C22×C7—32no—
155C22×C7—91no—
156C22×C7—92no—
157C22×C7C3×C3⋊S412yescomplemented
158C2×D7—182no—
159C2×D7—181no—
160Dic7—182no—
161Dic7—181no—
162C33—1121noSylow
163C23×C3—212no—
164C23×C3—631no—
165C23×C3—631no—
166C2×A4—32no—
167C2×A4—422no—
168S4—422no—
169S4—844no—
170S4—842no—
171C3×D4—1262no—
172C3×D4—1262no—
173C3×D4—1262no—
174C2×C3×C4—632no—
175C2×C3×C4—631no—
176#67—842no—
177C3×C7—42no—
178C3×C7—162no—
179ΓL(1,𝔽8)—82no—
180ΓL(1,𝔽8)—322no—
181ΓL(1,𝔽8)A4⋊S411yescharacteristic, complemented
182C2×C32—842no—
183C3⋊D3—1121no—
184C3×D3—3362no—
185C3×D3—1682no—
186C24C3⋊(C3⋊AGL(1,𝔽7))11yescharacteristic, complemented
187C2×D4—632no—
188C2×D4—631no—
189C22⋊C4—632no—
190C22⋊C4—631no—
191C2×C7—32no—
192C2×C7—91no—
193D7—361no—
194C22×C3—42no—
195C22×C3—72no—
196C22×C3—212no—
197C22×C3—562no—
198C22×C3—631no—
199C22×C3—632no—
200C22×C3—1262no—
201C22×C3—1261no—
202C2×D3—842no—
203A4—44no—
204A4—122no—
205A4—142no—
206A4—564no—
207A4—1682no—
208A4A4⋊AGL(1,𝔽7)12yescomplemented
209C3×C4—1262no—
210C3×C4—1261no—
211Dic3—842no—
212C32—161no—
213C32—282no—
214C32—1122no—
215C32—2242no—
216C32—2242no—
217C23—32no—
218C23—91no—
219C23—631no—
220D4—1262no—
221D4—1262no—
222D4—1262no—
223C2×C4—632no—
224C2×C4—631no—
225C7C3×A4⋊S411yescharacteristic, complemented, Sylow
226C2×C3—122no—
227C2×C3—212no—
228C2×C3—631no—
229C2×C3—1682no—
230C2×C3—2521no—
231D3—3362no—
232D3—1682no—
233C22—32no—
234C22—91no—
235C22—92no—
236C22—1262no—
237C22—1261no—
238C22C3⋊(A4⋊AGL(1,𝔽7))12yescomplemented
239C4—1262no—
240C4—1261no—
241C3—42no—
242C3—71no—
243C3—162no—
244C3—562no—
245C3—2242no—
246C2—32no—
247C2—91no—
248C2—2521no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC7×A42C2×C3
Fitting, socleC24×C7C3×C3⋊D3
LCSG ▷ C7×A42
UCSC1

Complements

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

NH ≅ G/Ntype
2. #72222237C3semidirect
3. A42×ΓL(1,𝔽8)C2semidirect
8. C7×A42C2×C3semidirect
9. ΓL(1,𝔽8)×C22⋊A4D3semidirect
10. C22×A4×ΓL(1,𝔽8)D3semidirect
27. C7×C22⋊A4C3×D3semidirect
29. C22×C7×A4C3×D3semidirect
30. C24×ΓL(1,𝔽8)C3⋊D3semidirect
48. A4×ΓL(1,𝔽8)S4semidirect
72. A42AGL(1,𝔽7)semidirect
78. C24×C7C3×C3⋊D3semidirect
94. C7×A4C3×S4semidirect
99. C22×ΓL(1,𝔽8)C3⋊S4semidirect
127. C22⋊A4C3⋊AGL(1,𝔽7)semidirect
129. C22×A4C3⋊AGL(1,𝔽7)semidirect
157. C22×C7C3×C3⋊S4semidirect
181. ΓL(1,𝔽8)A4⋊S4semidirect
186. C24C3⋊(C3⋊AGL(1,𝔽7))semidirect
208. A4A4⋊AGL(1,𝔽7)semidirect
225. C7C3×A4⋊S4semidirect
238. C22C3⋊(A4⋊AGL(1,𝔽7))semidirect

Maximal subgroups

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

subgroupindex
#722222403
#722222373
A42×ΓL(1,𝔽8)2
#722140323
C3⋊(A4⋊AGL(1,𝔽7))4
C3×A4⋊S47