Finite Group Databasebeta

← #3743⋊A4 #13147239

Subgroups — #3743⋊A4 #13147239

Order 1728.

414 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C4×(C3⋊D3)⋊A4C211yesmaximal, characteristic
2C8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3(C3⋊(C3⋊C8))⋊A4C211yesmaximal, characteristic, complemented
4C22×#3743C311yesmaximal, characteristic, complemented
5A4×ΓL(1,𝔽25)—31nomaximal
6C2×(C3⋊D3)⋊A4C411yescharacteristic
7C4×C22⋊H(3,𝔽3)C2211yescharacteristic
8(C3⋊Dic3)⋊A4C411yescharacteristic
9#3743⋊C3—41nomaximal
10C22×C4×C3⋊D3C2×C311yescharacteristic
11C4×D3×A4—31no—
12C22×C32×C8C2×C311yescharacteristic, complemented
13C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
14C2×#3743—31no—
15C2×#3743—31no—
16C2×#3743—31no—
17C2×#3743—31no—
18C3×C8×A4—21no—
19C3×C8×A4D311yescharacteristic, complemented
20A4×C3⋊C8—31no—
21C2×C22⋊H(3,𝔽3)C2×C411yescharacteristic
22(C3⋊D3)⋊A4—21no—
23C4×(C3⋊D3)⋊C3—41no—
24C8×H(3,𝔽3)—41no—
25(C3⋊(C3⋊C8))⋊C3—41no—
26C22×ΓL(1,𝔽25)—31no—
27C22×ΓL(1,𝔽25)—91no—
28A4×M16—91no—
29C23×C3⋊D3C3×C411yescharacteristic
30C2×D3×A4—31no—
31C22×C32×C4C22×C311yescharacteristic
32C22×C3⋊Dic3C3×C411yescharacteristic
33C2×C4×C3⋊D3—31no—
34C2×C4×C3⋊D3—31no—
35C2×C4×C3⋊D3—31no—
36C2×C4×C3⋊D3—31no—
37C3×C4×A4—21no—
38C3×C4×A4C2×D311yescharacteristic
39Dic3×A4—31no—
40C2×C32×C8—31no—
41C2×C32×C8—31no—
42C2×C3⋊(C3⋊C8)—31no—
43C2×C3⋊(C3⋊C8)—31no—
44#3743—32no—
45#3743—31no—
46#3743—31no—
47#3743—31no—
48#3743A411yescharacteristic, complemented
49C3×ΓL(1,𝔽25)—121no—
50C2×(C3⋊D3)⋊C3—41no—
51C22⋊H(3,𝔽3)M1611yescharacteristic, complemented
52C4×H(3,𝔽3)—41no—
53(C3⋊Dic3)⋊C3—41no—
54C22×C4×D3—31no—
55C22×C4×D3—91no—
56C2×C4×A4—91no—
57C22×C3×C8—31no—
58C22×C3×C8C3×D311yescharacteristic
59C22×C3⋊C8—31no—
60C22×C3⋊C8—91no—
61C2×ΓL(1,𝔽25)—91no—
62C2×ΓL(1,𝔽25)—93no—
63C2×ΓL(1,𝔽25)—93no—
64C2×ΓL(1,𝔽25)—93no—
65C2×ΓL(1,𝔽25)—93no—
66C2×ΓL(1,𝔽25)—91no—
67C2×ΓL(1,𝔽25)—91no—
68C2×ΓL(1,𝔽25)—91no—
69C8×A4—31no—
70C8×A4—61no—
71C8×A4—91no—
72C23×C32C2×C3×C411yescharacteristic
73C22×C3⋊D3—21no—
74C22×C3⋊D3—31no—
75C22×C3⋊D3—31no—
76C22×C3⋊D3—61no—
77C2×C3×A4—21no—
78C2×C3×A4C4×D311yescharacteristic
79D3×A4—61no—
80C2×C32×C4—31no—
81C2×C32×C4—31no—
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—31no—
84C4×C3⋊D3—32no—
85C4×C3⋊D3—31no—
86C4×C3⋊D3—31no—
87C4×C3⋊D3—31no—
88C4×C3⋊D3C2×A411yescharacteristic
89C3×C4×D3—121no—
90C32×C8—31no—
91C32×C8—41no—
92C32×C8—81no—
93C32×C8C2×A411yescharacteristic, complemented
94C3⋊(C3⋊C8)—31no—
95C3⋊(C3⋊C8)C2×A411yescharacteristic, complemented
96C3×C3⋊C8—121no—
97C22×M16—91noSylow
98C2×H(3,𝔽3)—41no—
99(C3⋊D3)⋊C3—81no—
100C23×D3—31no—
101C23×D3—91no—
102C22×A4—91no—
103C22×C3×C4—31no—
104C22×C3×C4C2×C3×D311yescharacteristic
105C22×Dic3—31no—
106C22×Dic3—91no—
107C2×C4×D3—91no—
108C2×C4×D3—93no—
109C2×C4×D3—93no—
110C2×C4×D3—93no—
111C2×C4×D3—93no—
112C2×C4×D3—91no—
113C2×C4×D3—91no—
114C2×C4×D3—91no—
115C4×A4—31no—
116C4×A4—61no—
117C4×A4—91no—
118C3×M16—361no—
119C2×C3×C8—33no—
120C2×C3×C8—33no—
121C2×C3×C8—31no—
122C2×C3×C8—31no—
123C2×C3⋊C8—91no—
124C2×C3⋊C8—93no—
125C2×C3⋊C8—93no—
126C2×C3⋊C8—91no—
127ΓL(1,𝔽25)—31no—
128ΓL(1,𝔽25)—91no—
129ΓL(1,𝔽25)—91no—
130ΓL(1,𝔽25)—91no—
131ΓL(1,𝔽25)—93no—
132ΓL(1,𝔽25)—93no—
133ΓL(1,𝔽25)—96no—
134ΓL(1,𝔽25)—92no—
135ΓL(1,𝔽25)—93no—
136ΓL(1,𝔽25)—91no—
137C22×C32—31no—
138C22×C32—31no—
139C22×C32C3×M1611yescharacteristic, complemented
140C2×C3⋊D3—31no—
141C2×C3⋊D3—61no—
142C2×C3⋊D3—61no—
143C2×C3⋊D3—61no—
144C2×C3⋊D3—61no—
145C2×C3⋊D3C4×A411yescharacteristic
146C2×C3×D3—121no—
147C3×A4—21no—
148C3×A4ΓL(1,𝔽25)11yescharacteristic, complemented
149C32×C4—31no—
150C32×C4—41no—
151C32×C4—81no—
152C32×C4C22×A411yescharacteristic
153C3⋊Dic3—31no—
154C3⋊Dic3C4×A411yescharacteristic
155C3×Dic3—121no—
156C23×C4—91no—
157C2×M16—271no—
158C2×M16—271no—
159C2×M16—271no—
160C2×M16—271no—
161C22×C8—91no—
162C22×C8(C3⋊D3)⋊C311yescharacteristic, complemented
163H(3,𝔽3)—41noSylow
164C23×C3—31no—
165C23×C3C3×C4×D311yescharacteristic
166C22×D3—61no—
167C22×D3—91no—
168C22×D3—93no—
169C22×D3—93no—
170C22×D3—91no—
171C22×D3—181no—
172C22×D3—183no—
173C22×D3—181no—
174C2×A4—31no—
175C2×A4—61no—
176C2×A4—181no—
177C2×C3×C4—33no—
178C2×C3×C4—33no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—361no—
182C2×Dic3—91no—
183C2×Dic3—93no—
184C2×Dic3—93no—
185C2×Dic3—91no—
186C4×D3—31no—
187C4×D3—91no—
188C4×D3—91no—
189C4×D3—91no—
190C4×D3—93no—
191C4×D3—93no—
192C4×D3—93no—
193C4×D3—96no—
194C4×D3—92no—
195C4×D3—91no—
196C3×C8—33no—
197C3×C8—31no—
198C3×C8—31no—
199C3×C8—121no—
200C3×C8—241no—
201C3×C8—361no—
202C3×C8D3×A411yescharacteristic
203C3⋊C8—31no—
204C3⋊C8—91no—
205C3⋊C8—93no—
206C3⋊C8—91no—
207C2×C32—31no—
208C2×C32—31no—
209C2×C32—41no—
210C2×C32—81no—
211C2×C32C2×C4×A411yescharacteristic
212C3⋊D3—21no—
213C3⋊D3—61no—
214C3×D3—241no—
215C24—91no—
216C22×C4—91no—
217C22×C4—271no—
218C22×C4—271no—
219C22×C4—271no—
220C22×C4—271no—
221C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
222M16—91no—
223M16—271no—
224M16—271no—
225M16—272no—
226M16—271no—
227C2×C8—31no—
228C2×C8—31no—
229C2×C8—271no—
230C2×C8—271no—
231C22×C3—31no—
232C22×C3—33no—
233C22×C3—33no—
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—361no—
237C22×C3C3×ΓL(1,𝔽25)11yescharacteristic
238C2×D3—31no—
239C2×D3—91no—
240C2×D3—93no—
241C2×D3—91no—
242C2×D3—181no—
243C2×D3—181no—
244C2×D3—183no—
245C2×D3—183no—
246C2×D3—183no—
247C2×D3—183no—
248C2×D3—181no—
249C2×D3—181no—
250A4—31no—
251A4—61no—
252C3×C4—33no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—121no—
256C3×C4—241no—
257C3×C4—361no—
258C3×C4C2×D3×A411yescharacteristic
259Dic3—31no—
260Dic3—91no—
261Dic3—93no—
262Dic3—91no—
263C32—41no—
264C32—81no—
265C32A4×M1611yescharacteristic, complemented
266C23—181no—
267C23—271no—
268C23—271no—
269C23—541no—
270C23C4×(C3⋊D3)⋊C311yescharacteristic
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—91no—
274C2×C4—271no—
275C2×C4—271no—
276C2×C4—272no—
277C2×C4—271no—
278C2×C4—271no—
279C2×C4—271no—
280C8—31no—
281C8—91no—
282C8—271no—
283C8(C3⋊D3)⋊A411yescharacteristic, complemented
284C2×C3—33no—
285C2×C3—33no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—121no—
290C2×C3—241no—
291C2×C3—721no—
292C2×C3C4×D3×A411yescharacteristic
293D3—61no—
294D3—181no—
295D3—183no—
296D3—181no—
297C22—31no—
298C22—31no—
299C22—91no—
300C22—271no—
301C22—541no—
302C22—541no—
303C22—541no—
304C22—541no—
305C22#3743⋊C311yescharacteristic, complemented
306C4—31no—
307C4—91no—
308C4—271no—
309C4C2×(C3⋊D3)⋊A411yescharacteristic
310C3—31no—
311C3—121no—
312C3—241no—
313C3A4×ΓL(1,𝔽25)11yescharacteristic
314C2—31no—
315C2—31no—
316C2—181no—
317C2—541no—
318C2C4×(C3⋊D3)⋊A411yescharacteristic
subgroupG/H
centerC4C2×(C3⋊D3)⋊A4
commutatorC23×C32C2×C3×C4
FrattiniC3×C4C2×D3×A4
FittingC22×C32×C8C2×C3
socleC23×C3C3×C4×D3
prime radicalC2×(C3⋊D3)⋊A4C4
cosocleC22×C32×C4C22×C3
hypercenterC8(C3⋊D3)⋊A4
hypocenterC22×C32C3×M16
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C8×C22⋊H(3,𝔽3)C2semidirect
3. (C3⋊(C3⋊C8))⋊A4C2semidirect
4. C22×#3743C3semidirect
12. C22×C32×C8C2×C3semidirect
13. C22×C3⋊(C3⋊C8)C2×C3semidirect
19. C3×C8×A4D3semidirect
48. #3743A4semidirect
51. C22⋊H(3,𝔽3)M16semidirect
93. C32×C8C2×A4semidirect
95. C3⋊(C3⋊C8)C2×A4semidirect
139. C22×C32C3×M16semidirect
148. C3×A4ΓL(1,𝔽25)semidirect
162. C22×C8(C3⋊D3)⋊C3semidirect
265. C32A4×M16semidirect
283. C8(C3⋊D3)⋊A4semidirect
305. C22#3743⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×(C3⋊D3)⋊A42
C8×C22⋊H(3,𝔽3)2
(C3⋊(C3⋊C8))⋊A42
C22×#37433
A4×ΓL(1,𝔽25)3
#3743⋊C34