Finite Group Databasebeta

← A4×C3⋊SL(2,ℤ/4) #13180326

Subgroups — A4×C3⋊SL(2,ℤ/4) #13180326

Order 1728.

470 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 51 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
1C2×C3×A42C211yesmaximal, characteristic
2C22×C3⋊SL(2,ℤ/4)C311yesmaximal, characteristic, complemented
3A4×SL(2,ℤ/4)—33nomaximal
4A4×#217—31nomaximal
5C3×A42C411yescharacteristic, complemented
6A4×C3⋊Dic3—41nomaximal
7C3×C3⋊SL(2,ℤ/4)—41nomaximal
8C2×C3×C22⋊A4—21no—
9C23×C3×A4D311yescharacteristic
10C23×C3×A4C2×C311yescharacteristic
11C2×A42—23no—
12C2×A42D313yes—
13C2×Dic3×A4—31no—
14C2×Dic3×A4—31no—
15C2×C3⋊SL(2,ℤ/4)—31no—
16C2×C3⋊SL(2,ℤ/4)—31no—
17C2×C32×A4—41no—
18C2×C32×A4—41no—
19C22×SL(2,ℤ/4)—33no—
20C22×#217—31no—
21A4×C22⋊C4—91no—
22C3×C22⋊A4—21no—
23C22×C3×A4—31no—
24C22×C3×A4—31no—
25C22×C3×A4—31no—
26C22×C3×A4—31no—
27C22×C3×A4Dic311yescharacteristic, complemented
28C22×C3×A4C3×C411yescharacteristic, complemented
29A42—23no—
30A42Dic313yescomplemented
31C22×C3⋊Dic3—41no—
32Dic3×A4—61no—
33Dic3×A4—61no—
34Dic3×A4—123no—
35C3⋊SL(2,ℤ/4)—31no—
36C3⋊SL(2,ℤ/4)A411yescharacteristic, direct product factor
37C3×SL(2,ℤ/4)—123no—
38C3×#217—121no—
39C32×A4—41no—
40C32×A4—41no—
41C3×C3⋊Dic3—161no—
42C25×C3C3×D311yescharacteristic
43C2×C22⋊A4—23no—
44C23×A4—21no—
45C23×A4C3×D313yes—
46C23×A4C3⋊D311yescharacteristic
47C23×Dic3—31no—
48C23×Dic3—31no—
49C2×C4×A4—91no—
50C2×C4×A4—91no—
51C2×SL(2,ℤ/4)—93no—
52C2×SL(2,ℤ/4)—93no—
53C2×#217—91no—
54C2×#217—91no—
55C2×#217—91no—
56C2×#217—91no—
57C23×C32—41no—
58C23×C32—41no—
59C2×C3×A4—31no—
60C2×C3×A4—31no—
61C2×C3×A4—31no—
62C2×C3×A4—31no—
63C2×C3×A4—43no—
64C2×C3×A4—43no—
65C2×C3×A4—83no—
66C2×C3×A4—81no—
67C2×C3×A4—81no—
68C2×C3×A4—83no—
69C2×C3×A4—241no—
70C2×C3×A4S411yescharacteristic
71C2×C3×A4C2×A411yescharacteristic
72C2×C3⋊Dic3—121no—
73C2×C3⋊Dic3—121no—
74C2×C3×Dic3—121no—
75C2×C3×Dic3—121no—
76C22×C22⋊C4—91noSylow
77C2×C33—161no—
78C24×C3—31no—
79C24×C3—31no—
80C24×C3—31no—
81C24×C3—31no—
82C24×C3—91no—
83C24×C3—91no—
84C24×C3C3×Dic311yescharacteristic, complemented
85C22⋊A4—23no—
86C22×A4—21no—
87C22×A4—33no—
88C22×A4—31no—
89C22×A4—31no—
90C22×A4—33no—
91C22×A4—61no—
92C22×A4—61no—
93C22×A4C3×Dic313yescomplemented
94C22×A4C3⋊Dic311yescharacteristic, complemented
95C22×Dic3—61no—
96C22×Dic3—61no—
97C22×Dic3—91no—
98C22×Dic3—91no—
99C22×Dic3—91no—
100C22×Dic3—91no—
101C22×Dic3—123no—
102C22×Dic3—181no—
103C22×Dic3—181no—
104C4×A4—181no—
105C4×A4—181no—
106SL(2,ℤ/4)—33no—
107SL(2,ℤ/4)—93no—
108C3×C22⋊C4—361no—
109#217—31no—
110#217—91no—
111#217—91no—
112#217—92no—
113#217—91no—
114C22×C32—41no—
115C22×C32—41no—
116C22×C32—121no—
117C22×C32—121no—
118C22×C32—121no—
119C22×C32—121no—
120C3×A4—43no—
121C3×A4—43no—
122C3×A4—83no—
123C3×A4—81no—
124C3×A4—81no—
125C3×A4—83no—
126C3×A4—241no—
127C3×A4SL(2,ℤ/4)11yescharacteristic, complemented
128C3×A4C4×A411yescharacteristic, complemented
129C3⋊Dic3—41no—
130C3⋊Dic3—121no—
131C3×Dic3—241no—
132C3×Dic3—241no—
133C3×Dic3—483no—
134C25C3×C3⋊D311yescharacteristic
135C23×C4—91no—
136C23×C4—91no—
137C2×C22⋊C4—271no—
138C2×C22⋊C4—271no—
139C2×C22⋊C4—271no—
140C2×C22⋊C4—271no—
141C33—161noSylow
142C23×C3—31no—
143C23×C3—31no—
144C23×C3—31no—
145C23×C3—31no—
146C23×C3—43no—
147C23×C3—41no—
148C23×C3—61no—
149C23×C3—81no—
150C23×C3—91no—
151C23×C3—91no—
152C23×C3—91no—
153C23×C3—91no—
154C23×C3—91no—
155C23×C3—91no—
156C23×C3—91no—
157C23×C3—91no—
158C23×C3—92no—
159C23×C3—91no—
160C23×C3—181no—
161C23×C3—181no—
162C23×C3C3×S411yescharacteristic
163C23×C3D3×A411yescharacteristic
164C2×A4—21no—
165C2×A4—31no—
166C2×A4—31no—
167C2×A4—33no—
168C2×A4—33no—
169C2×A4—61no—
170C2×A4—61no—
171C2×A4—83no—
172C2×A4—83no—
173C2×A4—243no—
174C2×A4C3⋊S411yescharacteristic
175C2×A4D3×A413yes—
176C2×C3×C4—361no—
177C2×C3×C4—361no—
178C2×Dic3—31no—
179C2×Dic3—31no—
180C2×Dic3—91no—
181C2×Dic3—91no—
182C2×Dic3—181no—
183C2×Dic3—181no—
184C2×Dic3—181no—
185C2×Dic3—181no—
186C2×Dic3—181no—
187C2×Dic3—181no—
188C2×Dic3—181no—
189C2×Dic3—181no—
190C2×Dic3—363no—
191C2×Dic3—363no—
192C2×C32—41no—
193C2×C32—41no—
194C2×C32—121no—
195C2×C32—121no—
196C2×C32—121no—
197C2×C32—121no—
198C2×C32—163no—
199C2×C32—323no—
200C2×C32—321no—
201C24—31no—
202C24—31no—
203C24—31no—
204C24—31no—
205C24—91no—
206C24—91no—
207C24C3×C3⋊Dic311yescharacteristic, complemented
208C22×C4—181no—
209C22×C4—181no—
210C22×C4—271no—
211C22×C4—271no—
212C22×C4—271no—
213C22×C4—271no—
214C22×C4—541no—
215C22×C4—541no—
216C22⋊C4—91no—
217C22⋊C4—271no—
218C22⋊C4—271no—
219C22⋊C4—272no—
220C22⋊C4—271no—
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—31no—
225C22×C3—43no—
226C22×C3—41no—
227C22×C3—61no—
228C22×C3—81no—
229C22×C3—91no—
230C22×C3—91no—
231C22×C3—91no—
232C22×C3—91no—
233C22×C3—91no—
234C22×C3—91no—
235C22×C3—91no—
236C22×C3—91no—
237C22×C3—92no—
238C22×C3—91no—
239C22×C3—123no—
240C22×C3—121no—
241C22×C3—123no—
242C22×C3—121no—
243C22×C3—181no—
244C22×C3—181no—
245C22×C3—241no—
246C22×C3—241no—
247C22×C3C3×SL(2,ℤ/4)11yescharacteristic, complemented
248C22×C3Dic3×A411yescharacteristic, complemented
249A4—21no—
250A4—83no—
251A4—83no—
252A4—243no—
253A4C3⋊SL(2,ℤ/4)11yescharacteristic, direct product factor
254A4Dic3×A413yescomplemented
255C3×C4—721no—
256C3×C4—721no—
257Dic3—61no—
258Dic3—61no—
259Dic3—123no—
260Dic3—181no—
261Dic3—181no—
262Dic3—363no—
263C32—41no—
264C32—41no—
265C32—163no—
266C32—323no—
267C32—321no—
268C23—31no—
269C23—31no—
270C23—31no—
271C23—31no—
272C23—61no—
273C23—91no—
274C23—91no—
275C23—91no—
276C23—91no—
277C23—91no—
278C23—91no—
279C23—91no—
280C23—91no—
281C23—92no—
282C23—91no—
283C23—181no—
284C23—181no—
285C23C3×C3⋊S411yescharacteristic
286C23A4×C3⋊D311yescharacteristic
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—271no—
290C2×C4—271no—
291C2×C4—541no—
292C2×C4—541no—
293C2×C4—541no—
294C2×C4—541no—
295C2×C4—541no—
296C2×C4—541no—
297C2×C4—541no—
298C2×C4—541no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—43no—
304C2×C3—41no—
305C2×C3—81no—
306C2×C3—91no—
307C2×C3—91no—
308C2×C3—121no—
309C2×C3—123no—
310C2×C3—121no—
311C2×C3—123no—
312C2×C3—241no—
313C2×C3—241no—
314C2×C3—323no—
315C2×C3A4×S411yescharacteristic
316C22—31no—
317C22—31no—
318C22—31no—
319C22—31no—
320C22—61no—
321C22—91no—
322C22—91no—
323C22—91no—
324C22—91no—
325C22—91no—
326C22—91no—
327C22—91no—
328C22—91no—
329C22—92no—
330C22—91no—
331C22—181no—
332C22—181no—
333C22C3×C3⋊SL(2,ℤ/4)11yescharacteristic, complemented
334C22A4×C3⋊Dic311yescharacteristic, complemented
335C4—181no—
336C4—181no—
337C4—541no—
338C4—541no—
339C3—41no—
340C3—43no—
341C3—81no—
342C3—323no—
343C3A4×SL(2,ℤ/4)11yescharacteristic, complemented
344C2—31no—
345C2—31no—
346C2—31no—
347C2—31no—
348C2—91no—
349C2—91no—
350C2A4×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2A4×C3⋊S4
commutator, hypocenterC22×C3×A4C3×C4
Fitting, socleC25×C3C3×D3
prime radicalC2×C3×A42C2
cosocleC23×C3×A4C2×C3
LCSG ▷ C22×C3×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊SL(2,ℤ/4)C3semidirect
5. C3×A42C4semidirect
27. C22×C3×A4Dic3semidirect
28. C22×C3×A4C3×C4semidirect
30. A42Dic3semidirect
36. C3⋊SL(2,ℤ/4)A4direct (N × H)
84. C24×C3C3×Dic3semidirect
93. C22×A4C3×Dic3semidirect
94. C22×A4C3⋊Dic3semidirect
127. C3×A4SL(2,ℤ/4)semidirect
128. C3×A4C4×A4semidirect
207. C24C3×C3⋊Dic3semidirect
247. C22×C3C3×SL(2,ℤ/4)semidirect
248. C22×C3Dic3×A4semidirect
253. A4C3⋊SL(2,ℤ/4)direct (N × H)
254. A4Dic3×A4semidirect
333. C22C3×C3⋊SL(2,ℤ/4)semidirect
334. C22A4×C3⋊Dic3semidirect
343. C3A4×SL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×C3×A422
C22×C3⋊SL(2,ℤ/4)3
A4×SL(2,ℤ/4)3
A4×#2173
A4×C3⋊Dic34
C3×C3⋊SL(2,ℤ/4)4