Finite Group Databasebeta

← #4207⋊A4 #14519261

Subgroups — #4207⋊A4 #14519261

Order 1944.

431 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 58 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#11758106C211yesmaximal, characteristic, complemented
2C22×#4207C311yesmaximal, characteristic, complemented
3#10620070C312yesmaximal, complemented
4C3⋊(D9⋊A4)C311yesmaximal, characteristic, complemented
5A4×AGL(1,ℤ/9)—32nomaximal
6A4×(C3⋊D3)⋊C3—31nomaximal
7C3×D9⋊A4—31nomaximal
8#4207⋊C3—41nomaximal
9C22×#737C2×C311yescharacteristic, complemented
10C2×#4207—31no—
11C2×#4207—31no—
12#66440—22no—
13#66440—22no—
14#66440—22no—
15#66440—22no—
16A4×3−1+2—21no—
17A4×3−1+2—22no—
18A4×3−1+2D312yescomplemented
19A4×H(3,𝔽3)D311yescharacteristic
20#66437—21no—
21#66436—21no—
22#66436—22no—
23#66436C2×C312yescomplemented
24C3×C9⋊A4—21no—
25C3×C9⋊A4D311yescharacteristic, complemented
26C3×C9⋊A4C2×C311yescharacteristic, complemented
273−1+4—41noSylow
28A4×C3⋊D3—31no—
29C3×D3×A4—91no—
30C22×C3⋊D9C3211yescharacteristic, complemented
31C22×AGL(1,ℤ/9)—32no—
32C22×(C3⋊D3)⋊C3—31no—
33C22×C3×D9—31no—
34(C3⋊D3)⋊A4—32no—
35A4×D9—32no—
36D9⋊A4—31no—
37D9⋊A4—32no—
38D9⋊A4—32no—
39D9⋊A4—32no—
40C2×#737—31no—
41#4207—31no—
42#4207—42no—
43#4207A411yescharacteristic, complemented
44C3⋊AGL(1,ℤ/9)—41no—
45C3×AGL(1,ℤ/9)—121no—
46C3×AGL(1,ℤ/9)—122no—
47C3×(C3⋊D3)⋊C3—121no—
48C32×A4—21no—
49C32×A4C3×D311yescharacteristic
50C32×A4C3⋊D311yescharacteristic
51C22×3−1+2—21no—
52C22×3−1+2—22no—
53C22×3−1+2C3×D312yescomplemented
54C22×H(3,𝔽3)C3×D311yescharacteristic
55C22×C3×C9—21no—
56C22×C3×C9C3×D311yescharacteristic, complemented
57C22×C3×C9C2×C3211yescharacteristic, complemented
58C2×C3⋊D9—31no—
59C2×C3⋊D9—31no—
60C2×AGL(1,ℤ/9)—92no—
61C2×AGL(1,ℤ/9)—92no—
62C2×(C3⋊D3)⋊C3—91no—
63C2×(C3⋊D3)⋊C3—91no—
64C2×C3×D9—91no—
65C2×C3×D9—91no—
66C22⋊H(3,𝔽3)—21no—
67C22⋊H(3,𝔽3)—22no—
68C22⋊H(3,𝔽3)C3×D312yes—
69#1094—22no—
70#1094—22no—
71#1094—22no—
72#1094—22no—
73C3×C22⋊C9—21no—
74C3×C22⋊C9—21no—
75C3×C22⋊C9—21no—
76C3×C22⋊C9—21no—
77C9⋊A4—22no—
78C9⋊A4—22no—
79C9⋊A4—22no—
80C9⋊A4—22no—
81C9⋊A4—22no—
82C9⋊A4—21no—
83C9⋊A4—22no—
84C9⋊A4—21no—
85C9⋊A4—22no—
86C9⋊A4—22no—
87C9⋊A4—22no—
88C9⋊A4—21no—
89C9⋊A4—22no—
90C9⋊A4—22no—
91C9⋊A4C3×D311yescharacteristic
92C9⋊A4C3×D312yes—
93C9⋊A4C3×D312yescomplemented
94C9⋊A4C3×D312yescomplemented
95C9×A4—22no—
96C9×A4—22no—
97C9×A4—21no—
98C9×A4—22no—
99C9×A4—22no—
100C9×A4—22no—
101C9×A4C3×D312yescomplemented
102#737—42no—
103#737—82no—
104#737—82no—
105#737—82no—
106#737—82no—
107#737—82no—
108#737—81no—
109#737—81no—
110#737C2×A411yescharacteristic, complemented
111C3×3−1+2—42no—
112C3×3−1+2—41no—
113C3×3−1+2—41no—
114C3×3−1+2—81no—
115C3×3−1+2—82no—
116C3×3−1+2—81no—
117C3×H(3,𝔽3)—41no—
118C22×C3⋊D3—31no—
119C22×C3×D3—91no—
120C2×C3×A4—271no—
121D3×A4—91no—
122D3×A4—92no—
123D3×A4—271no—
124C22×D9—31no—
125C22×D9—32no—
126C3×C3⋊D3—121no—
127C32×D3—361no—
128C2×3−1+2—32no—
129C2×3−1+2—62no—
130C2×3−1+2—61no—
131C2×H(3,𝔽3)—31no—
132C2×C3×C9—31no—
133C2×C3×C9—31no—
134C2×C3×C9—61no—
135C3⋊D9—31no—
136C3⋊D9C3×A411yescharacteristic, complemented
137AGL(1,ℤ/9)—32no—
138AGL(1,ℤ/9)—92no—
139AGL(1,ℤ/9)—121no—
140AGL(1,ℤ/9)—122no—
141AGL(1,ℤ/9)—122no—
142AGL(1,ℤ/9)—122no—
143(C3⋊D3)⋊C3—31no—
144(C3⋊D3)⋊C3—91no—
145(C3⋊D3)⋊C3—122no—
146C3×D9—31no—
147C3×D9—91no—
148C3×D9—122no—
149C22×C32—21no—
150C22×C32C32×D311yescharacteristic
151C22×C32C3×C3⋊D311yescharacteristic
152C2×C3⋊D3—91no—
153C2×C3⋊D3—91no—
154C2×C3×D3—271no—
155C2×C3×D3—271no—
156C3×A4—21no—
157C3×A4—22no—
158C3×A4—91no—
159C3×A4—91no—
160C3×A4—181no—
161C3×A4C3×C3⋊D312yes—
162C3×A4C3×C3⋊D311yescharacteristic
163C22×C9—22no—
164C22×C9—21no—
165C22×C9C32×D312yes—
166C22×C9C32×D311yescharacteristic
167C2×D9—91no—
168C2×D9—92no—
169C2×D9—91no—
170C2×D9—92no—
171C22⋊C9—22no—
172C22⋊C9—22no—
173C22⋊C9—22no—
174C22⋊C9—22no—
175C22⋊C9—21no—
176C33—41no—
177C33—41no—
178C33—81no—
1793−1+2—21no—
1803−1+2—22no—
1813−1+2—41no—
1823−1+2—42no—
1833−1+2—42no—
1843−1+2—42no—
1853−1+2—82no—
1863−1+2—81no—
1873−1+2—82no—
1883−1+2—82no—
1893−1+2—82no—
1903−1+2—82no—
1913−1+2—82no—
1923−1+2—82no—
1933−1+2—81no—
1943−1+2—82no—
1953−1+2—82no—
1963−1+2—82no—
1973−1+2—82no—
1983−1+2—82no—
1993−1+2—81no—
2003−1+2—82no—
2013−1+2—82no—
2023−1+2—82no—
2033−1+2D3×A412yescomplemented
204H(3,𝔽3)—42no—
205H(3,𝔽3)—82no—
206H(3,𝔽3)—81no—
207H(3,𝔽3)D3×A411yescharacteristic
208C3×C9—21no—
209C3×C9—42no—
210C3×C9—82no—
211C3×C9—82no—
212C3×C9—82no—
213C3×C9—82no—
214C3×C9—81no—
215C3×C9—81no—
216C3×C9—81no—
217C3×C9—81no—
218C3×C9—81no—
219C3×C9—82no—
220C3×C9D3×A411yescharacteristic, complemented
221C3×C9C2×C3×A411yescharacteristic, complemented
222C23×C3—271no—
223C22×D3—91no—
224C22×D3—271no—
225C2×A4—271no—
226C2×A4—272no—
227C2×C32—31no—
228C2×C32—31no—
229C2×C32—61no—
230C2×C32—1081no—
231C3⋊D3—31no—
232C3⋊D3—91no—
233C3×D3—91no—
234C3×D3—271no—
235C3×D3—362no—
236C3×D3—361no—
237C3×D3—1081no—
238C2×C9—31no—
239C2×C9—32no—
240C2×C9—61no—
241C2×C9—62no—
242D9—31no—
243D9—32no—
244D9—91no—
245D9—92no—
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—61no—
249C22×C3—811no—
250C22×C3—811no—
251C22×C3C32×C3⋊D311yescharacteristic
252C2×D3—271no—
253C2×D3—271no—
254C2×D3—811no—
255C2×D3—811no—
256A4—31no—
257A4—32no—
258A4—61no—
259A4—62no—
260C32—21no—
261C32—42no—
262C32—41no—
263C32—82no—
264C32—81no—
265C32—361no—
266C32—361no—
267C32—721no—
268C32C3×D3×A411yescharacteristic
269C32A4×C3⋊D311yescharacteristic
270C9—22no—
271C9—21no—
272C9—82no—
273C9—82no—
274C9—82no—
275C9—82no—
276C9—81no—
277C9C3×D3×A411yescharacteristic
278C9C3×D3×A412yes—
279C23—271noSylow
280C2×C3—31no—
281C2×C3—91no—
282C2×C3—91no—
283C2×C3—181no—
284C2×C3—271no—
285C2×C3—811no—
286C2×C3—1081no—
287C2×C3—1082no—
288D3—91no—
289D3—271no—
290D3—271no—
291D3—811no—
292C22—811no—
293C22—811no—
294C22#4207⋊C311yescharacteristic, complemented
295C3—31no—
296C3—31no—
297C3—61no—
298C3—121no—
299C3—122no—
300C3—241no—
301C3—242no—
302C3C3×A4×C3⋊D311yescharacteristic
303C2—31no—
304C2—271no—
305C2—811no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22×C3×C9C2×C32
FrattiniC3C3×A4×C3⋊D3
FittingC22×#737C2×C3
socleC22×C3C32×C3⋊D3
LCSG ▷ C22×C3×C9
UCSC1

Complements

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

NH ≅ G/Ntype
1. #11758106C2semidirect
2. C22×#4207C3semidirect
3. #10620070C3semidirect
4. C3⋊(D9⋊A4)C3semidirect
9. C22×#737C2×C3semidirect
18. A4×3−1+2D3semidirect
23. #66436C2×C3semidirect
25. C3×C9⋊A4D3semidirect
26. C3×C9⋊A4C2×C3semidirect
30. C22×C3⋊D9C32semidirect
43. #4207A4semidirect
53. C22×3−1+2C3×D3semidirect
56. C22×C3×C9C3×D3semidirect
57. C22×C3×C9C2×C32semidirect
93. C9⋊A4C3×D3semidirect
94. C9⋊A4C3×D3semidirect
101. C9×A4C3×D3semidirect
110. #737C2×A4semidirect
136. C3⋊D9C3×A4semidirect
203. 3−1+2D3×A4semidirect
220. C3×C9D3×A4semidirect
221. C3×C9C2×C3×A4semidirect
294. C22#4207⋊C3semidirect

Maximal subgroups

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

subgroupindex
#117581062
C22×#42073
#106200703
C3⋊(D9⋊A4)3
A4×AGL(1,ℤ/9)3
A4×(C3⋊D3)⋊C33
C3×D9⋊A43
#4207⋊C34