Finite Group Databasebeta

← #72220043

Subgroups — #72220043

Order 7776.

3794 conjugacy classes of subgroups; only the 457 normal subgroups are shown.

normal subgroups only

This group has 3794 conjugacy classes of subgroups — too many to draw. Only its 457 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#72220038—34nomaximal
2#72220037C311yesmaximal, characteristic, complemented
3#72219995C213yesmaximal, complemented
4C24×#6900C211yesmaximal, characteristic, complemented
5C2×#14517678C213yesmaximal, complemented
6C23×#6900—26no—
7C23×#6900C2213yes—
8C22×#92314C2211yescharacteristic, complemented
9#14517678—26no—
10C2×#11757618C2213yescomplemented
11C24×C3×3−1+2D314yes—
12C2×#10619980—312no—
13C24×C9⋊C9—24no—
14C24×C92C2×C311yescharacteristic, complemented
15#13047994—312no—
16C2×#10619673C2×C313yescomplemented
17#13046833C2×C313yescomplemented
18C22×#6900—28no—
19C22×#6900—26no—
20C22×#6900D416yescomplemented
21C22×#6900C2311yescharacteristic
22C2×#92314—23no—
23#11757618—23no—
24C3×#64162—91no—
25#11721874—34no—
26#11721704—912no—
27C23×C3×3−1+2—224no—
28C23×C3×3−1+2C2×D3112yes—
29C22×C3⋊AGL(1,ℤ/9)—34no—
30#10619980—624no—
31C2×C3⋊(3−1+2⋊C4)—312no—
32C23×C9⋊C9—248no—
33C23×C9⋊C9—212no—
34C23×C92—26no—
35C23×C92C22×C313yes—
36C22×C9⋊D9C22×C311yescharacteristic, complemented
37#10619673—26no—
38C2×C9⋊Dic9C22×C313yescomplemented
39C2×#6900—26no—
40C2×#6900C2×D413yes—
41#92314—41no—
42C24×C33C3⋊D311yescharacteristic
43C2×C3×#638—93no—
44C24×3−1+2—312no—
45C24×C3×C9—24no—
46C24×C3×C9C3×D314yes—
47C3×#3757—93no—
48C2×#6233—312no—
49C2×#611⋊C3—936no—
50#88551—312no—
51#3676⋊C3—936no—
52C22×C3×3−1+2—232no—
53C22×C3×3−1+2—224no—
54C22×C3×3−1+2#67124yes—
55C22×C3×3−1+2C22×D314yes—
56C2×C3⋊AGL(1,ℤ/9)—612no—
57C3⋊(3−1+2⋊C4)—612no—
58C22×C9⋊C9—264no—
59C22×C9⋊C9—248no—
60C22×C9⋊C9—224no—
61C22×C9⋊C9—24no—
62C22×C92—28no—
63C22×C92—26no—
64C22×C92C3×D416yescomplemented
65C22×C92C23×C311yescharacteristic
66C2×C9⋊D9—23no—
67C9⋊Dic9—23no—
68#64162—91no—
69C3×#953—274no—
70#63515—912no—
71#6900#12011yescharacteristic, complemented, Sylow
72C23×C33—26no—
73C23×C33C2×C3⋊D313yes—
74C22×C3×C3⋊D3—91no—
75C3×#638—186no—
76C2×C3×C3⋊Dic3—93no—
77C23×3−1+2—336no—
78C23×3−1+2—672no—
79C23×C3×C9—248no—
80C23×C3×C9—212no—
81C23×C3×C9—224no—
82C23×C3×C9C2×C3×D3112yes—
83C22×C3⋊D9—34no—
84C22×AGL(1,ℤ/9)—912no—
85#6233—624no—
86C2×C3⋊Dic9—312no—
87#611⋊C3—1872no—
88C2×3−1+2⋊C4—936no—
89C2×C3×3−1+2—224no—
90C2×C3×3−1+2C2×#67112yes—
91C3⋊AGL(1,ℤ/9)—124no—
92C2×C9⋊C9—248no—
93C2×C9⋊C9—212no—
94C2×C92—26no—
95C2×C92C2×C3×D413yes—
96C9⋊D9—41no—
97C24×C32—34no—
98C24×C32C3×C3⋊D311yescharacteristic
99C2×#638—93no—
100C2×C3×#67—2712no—
101C24×C9—64no—
102C24×C9AGL(1,ℤ/9)112yescomplemented
103#3757—93no—
104C3×#217—2712no—
105C2×#611—936no—
106#3676—936no—
107C22×C33—28no—
108C22×C33—26no—
109C22×C33#63816yes—
110C22×C33C22×C3⋊D311yescharacteristic
111C2×C3×C3⋊D3—183no—
112C3×C3⋊Dic3—183no—
113C22×3−1+2—312no—
114C22×3−1+2—372no—
115C22×3−1+2—672no—
116C22×3−1+2—696no—
117C22×C3×C9—232no—
118C22×C3×C9—264no—
119C22×C3×C9—248no—
120C22×C3×C9—224no—
121C22×C3×C9—24no—
122C22×C3×C9—224no—
123C22×C3×C9C3×#67124yes—
124C22×C3×C9C22×C3×D314yes—
125C2×C3⋊D9—612no—
126C2×AGL(1,ℤ/9)—1836no—
127C3⋊Dic9—612no—
1283−1+2⋊C4—1836no—
129C3×#120—811no—
130#953—274no—
131C3×3−1+2#95314yes—
132C9⋊C9—24no—
133C92C3×#12011yescharacteristic, complemented
134C23×C32—26no—
135C23×C32—312no—
136C23×C32—624no—
137C23×C32C2×C3×C3⋊D313yes—
138C22×C3⋊D3—91no—
139C22×C3×D3—274no—
140#638—186no—
141C2×C3⋊Dic3—93no—
142C3×#67—5424no—
143C2×C3×Dic3—2712no—
144C23×C9—272no—
145C23×C9—648no—
146C23×C9—612no—
147C23×C9C2×AGL(1,ℤ/9)136yes—
148C22×D9—912no—
149#611—1872no—
150C2×Dic9—936no—
151C2×C33—26no—
152C2×C33C2×#63813yes—
153C3×C3⋊D3—361no—
154C2×3−1+2—336no—
155C2×3−1+2—672no—
156C2×C3×C9—224no—
157C2×C3×C9—248no—
158C2×C3×C9—212no—
159C2×C3×C9C2×C3×#67112yes—
160C3⋊D9—124no—
161AGL(1,ℤ/9)—3612no—
162C24×C3—91no—
163C24×C3C3⋊AGL(1,ℤ/9)14yes—
164C2×C3×D4—813no—
165C2×#67—2712no—
166C3×C22⋊C4—813no—
167#217—2712no—
168C22×C32—28no—
169C22×C32—26no—
170C22×C32—324no—
171C22×C32—34no—
172C22×C32—624no—
173C22×C32—632no—
174C22×C32C3×#63816yes—
175C22×C32C22×C3×C3⋊D311yescharacteristic
176C2×C3⋊D3—183no—
177C2×C3×D3—5412no—
178C3⋊Dic3—183no—
179C3×Dic3—5412no—
180C22×C9—296no—
181C22×C9—272no—
182C22×C9—664no—
183C22×C9—648no—
184C22×C9—624no—
185C22×C9—64no—
186C22×C9#611⋊C3172yescomplemented
187C22×C9C22×AGL(1,ℤ/9)112yes—
188C2×D9—1836no—
189Dic9—1836no—
190#120—811noSylow
191C33#6416211yescharacteristic
1923−1+2—312no—
193C3×C9—24no—
194C3×C9C3×#95314yes—
195C23×C3—224no—
196C23×C3—93no—
197C23×C3—186no—
198C23×C3—811no—
199C23×C3C2×C3⋊AGL(1,ℤ/9)112yes—
200C22×D3—274no—
201C3×D4—1626no—
202C2×C3×C4—813no—
203#67—5424no—
204C2×Dic3—2712no—
205C2×C32—26no—
206C2×C32—312no—
207C2×C32—624no—
208C2×C32C2×C3×#63813yes—
209C3⋊D3—361no—
210C3×D3—1084no—
211C2×C9—272no—
212C2×C9—648no—
213C2×C9—612no—
214C2×C9C2×#611⋊C3136yes—
215D9—3612no—
216C24#9231411yescharacteristic, complemented
217C2×D4—813no—
218C22⋊C4—813no—
219C22×C3—232no—
220C22×C3—224no—
221C22×C3—96no—
222C22×C3—91no—
223C22×C3—186no—
224C22×C3—188no—
225C22×C3—1623no—
226C22×C3#10619980124yes—
227C22×C3C22×C3⋊AGL(1,ℤ/9)14yes—
228C2×D3—5412no—
229C3×C4—1623no—
230Dic3—5412no—
231C32—34no—
232C32C3×#6416211yescharacteristic
233C9—64no—
234C9#11721704112yescomplemented
235C23—26no—
236C23—811no—
237C23C2×#9231413yes—
238D4—1626no—
239C2×C4—813no—
240C2×C3—224no—
241C2×C3—93no—
242C2×C3—186no—
243C2×C3—3241no—
244C2×C3C2×#10619980112yes—
245D3—1084no—
246C22—28no—
247C22—26no—
248C22—1623no—
249C22#1451767816yescomplemented
250C22C22×#9231411yescharacteristic
251C4—1623no—
252C3—91no—
253C3#7222003814yes—
254C2—26no—
255C2—3241no—
256C2C2×#1451767813yes—
subgroupG/H
centerC22C22×#92314
commutator, cosocleC22×C92C23×C3
Frattini, socleC22×C32C22×C3×C3⋊D3
FittingC24×#6900C2
hypercenterC24#92314
hypocenterC92C3×#120
LCSG ▷ C22×C92 ▷ C92
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
2. #72220037C3semidirect
3. #72219995C2semidirect
4. C24×#6900C2semidirect
5. C2×#14517678C2semidirect
8. C22×#92314C22semidirect
10. C2×#11757618C22semidirect
14. C24×C92C2×C3semidirect
16. C2×#10619673C2×C3semidirect
17. #13046833C2×C3semidirect
20. C22×#6900D4semidirect
36. C22×C9⋊D9C22×C3semidirect
38. C2×C9⋊Dic9C22×C3semidirect
64. C22×C92C3×D4semidirect
71. #6900#120semidirect
102. C24×C9AGL(1,ℤ/9)semidirect
133. C92C3×#120semidirect
186. C22×C9#611⋊C3semidirect
216. C24#92314semidirect
234. C9#11721704semidirect
249. C22#14517678semidirect

Maximal subgroups

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

subgroupindex
#722200383
#722200373
#722199952
C24×#69002
C2×#145176782