Finite Group Databasebeta

← C2×C3×C4⋊SL(2,ℤ/4) #11922765

Subgroups — C2×C3×C4⋊SL(2,ℤ/4) #11922765

Order 1152.

1052 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C22×C3×C4×A4C211yesmaximal, characteristic
2C22×C3×SL(2,ℤ/4)C212yesmaximal
3C3×C4⋊SL(2,ℤ/4)C214yesmaximal, direct product factor
4C2×C4⋊SL(2,ℤ/4)C311yesmaximal, characteristic, direct product factor
5C2×C3×#380—31nomaximal
6C23×C3×A4C2211yescharacteristic
7C2×C3×C4×A4C414yescomplemented
8C2×C3×C4×A4C2212yes—
9C2×C3×SL(2,ℤ/4)—24no—
10C2×C3×SL(2,ℤ/4)C2214yes—
11C2×C3×C4⋊Dic3—41nomaximal
12C24×C3×C4D311yescharacteristic
13C22×C4×A4—21no—
14C22×C4×A4C2×C311yescharacteristic
15C22×SL(2,ℤ/4)C2×C312yes—
16C22×C3×C4⋊C4—31no—
17C22×C3×C4⋊C4—31no—
18C22×C3×C22⋊C4—32no—
19C4⋊SL(2,ℤ/4)C2×C314yesdirect product factor
20C3×#380—34no—
21C3×#380—34no—
22C2×C3×#95—32no—
23C22×C3×A4C2×C412yes—
24C22×C3×A4D412yes—
25C22×C3×A4Q812yes—
26C22×C3×A4C2311yescharacteristic
27C22×C32×C4—41no—
28C22×C3×Dic3—42no—
29C3×C4×A4C2×C414yescomplemented
30C3×SL(2,ℤ/4)—24no—
31C3×C4⋊Dic3—44no—
32C2×#380—31noSylow
33C25×C3C2×D311yescharacteristic
34C23×A4—21no—
35C23×A4C22×C311yescharacteristic
36C23×C3×C4—31no—
37C23×C3×C4—34no—
38C23×C3×C4—31no—
39C23×C3×C4—32no—
40C23×C3×C4—32no—
41C23×C3×C4—32no—
42C23×C3×C4Dic314yescomplemented
43C23×C3×C4C2×D312yes—
44C2×C4×A4—24no—
45C2×C4×A4—22no—
46C2×C4×A4C3×C414yescomplemented
47C2×C4×A4C22×C312yes—
48C2×SL(2,ℤ/4)—24no—
49C2×SL(2,ℤ/4)C22×C314yes—
50C2×C3×C4⋊C4—34no—
51C2×C3×C4⋊C4—34no—
52C2×C3×C4⋊C4—61no—
53C2×C3×C4⋊C4—64no—
54C2×C3×C4⋊C4—62no—
55C2×C3×C4⋊C4—61no—
56C2×C3×C22⋊C4—34no—
57C2×C3×C22⋊C4—34no—
58C2×C3×C22⋊C4—64no—
59C2×C3×C22⋊C4—64no—
60C2×C4⋊Dic3—41no—
61C3×#95—38no—
62C23×C32—41no—
63C2×C3×A4C4⋊C414yescomplemented
64C2×C3×A4C22×C411yescharacteristic
65C2×C3×A4C2×D411yescharacteristic
66C2×C3×A4C2×Q811yescharacteristic
67C2×C32×C4—44no—
68C2×C32×C4—42no—
69C2×C3×Dic3—44no—
70C2×C3×Dic3—84no—
71C24×C4C3×D311yescharacteristic
72C22×C4⋊C4—31no—
73C22×C4⋊C4—31no—
74C22×C22⋊C4—32no—
75#380—34no—
76#380—34no—
77C2×#95—32no—
78C24×C3—31no—
79C24×C3—32no—
80C24×C3—31no—
81C24×C3—32no—
82C24×C3—32no—
83C24×C3Dic612yes—
84C24×C3D1212yes—
85C24×C3C2×Dic312yes—
86C24×C3C22×D311yescharacteristic
87C22×A4—22no—
88C22×A4—22no—
89C22×A4—22no—
90C22×A4—21no—
91C22×A4C2×C3×C412yes—
92C22×A4C3×D412yes—
93C22×A4C3×Q812yes—
94C22×A4C23×C311yescharacteristic
95C22×C3×C4—31no—
96C22×C3×C4—34no—
97C22×C3×C4—34no—
98C22×C3×C4—32no—
99C22×C3×C4—32no—
100C22×C3×C4—32no—
101C22×C3×C4—34no—
102C22×C3×C4—38no—
103C22×C3×C4—34no—
104C22×C3×C4—34no—
105C22×C3×C4—34no—
106C22×C3×C4—41no—
107C22×C3×C4—64no—
108C22×C3×C4—62no—
109C22×C3×C4—61no—
110C22×C3×C4—64no—
111C22×C3×C4—64no—
112C22×C3×C4—64no—
113C22×C3×C4—62no—
114C22×C3×C4—64no—
115C22×C3×C4—64no—
116C22×C3×C4—62no—
117C22×C3×C4—81no—
118C22×C3×C4C2×Dic314yescomplemented
119C22×C3×C4S411yescharacteristic
120C22×Dic3—42no—
121C4×A4—24no—
122C4×A4C2×C3×C414yescomplemented
123SL(2,ℤ/4)—24no—
124C3×C4⋊C4—64no—
125C3×C4⋊C4—68no—
126C3×C4⋊C4—64no—
127C3×C22⋊C4—68no—
128C3×C22⋊C4—64no—
129C3×C22⋊C4—64no—
130C4⋊Dic3—44no—
131C22×C32—42no—
132C22×C32—42no—
133C22×C32—42no—
134C22×C32—41no—
135C3×A4C2×C4⋊C411yescharacteristic, complemented
136C32×C4—44no—
137C3×Dic3—84no—
138C25C2×C3×D311yescharacteristic
139C23×C4—31no—
140C23×C4—31no—
141C23×C4—32no—
142C23×C4—34no—
143C23×C4—32no—
144C23×C4—32no—
145C23×C4C3×Dic314yescomplemented
146C23×C4C2×C3×D312yes—
147C2×C4⋊C4—34no—
148C2×C4⋊C4—34no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—64no—
151C2×C4⋊C4—62no—
152C2×C4⋊C4—61no—
153C2×C22⋊C4—34no—
154C2×C22⋊C4—34no—
155C2×C22⋊C4—64no—
156C2×C22⋊C4—64no—
157#95—38no—
158C23×C3—32no—
159C23×C3—32no—
160C23×C3—32no—
161C23×C3—32no—
162C23×C3—32no—
163C23×C3—32no—
164C23×C3—31no—
165C23×C3—31no—
166C23×C3—32no—
167C23×C3—31no—
168C23×C3—32no—
169C23×C3—31no—
170C23×C3—32no—
171C23×C3—34no—
172C23×C3—34no—
173C23×C3—34no—
174C23×C3—31no—
175C23×C3—41no—
176C23×C3—61no—
177C23×C3—61no—
178C23×C3—61no—
179C23×C3—64no—
180C23×C3—81no—
181C23×C3C4⋊Dic314yescomplemented
182C23×C3C2×Dic611yescharacteristic
183C23×C3C2×D1211yescharacteristic
184C23×C3C22×Dic311yescharacteristic
185C23×C3C2×S411yescharacteristic
186C2×A4—24no—
187C2×A4—21no—
188C2×A4—21no—
189C2×A4—21no—
190C2×A4C3×C4⋊C414yescomplemented
191C2×A4C22×C3×C411yescharacteristic
192C2×A4C2×C3×D411yescharacteristic
193C2×A4C2×C3×Q811yescharacteristic
194C2×C3×C4—34no—
195C2×C3×C4—32no—
196C2×C3×C4—34no—
197C2×C3×C4—34no—
198C2×C3×C4—38no—
199C2×C3×C4—34no—
200C2×C3×C4—42no—
201C2×C3×C4—44no—
202C2×C3×C4—62no—
203C2×C3×C4—64no—
204C2×C3×C4—64no—
205C2×C3×C4—64no—
206C2×C3×C4—64no—
207C2×C3×C4—64no—
208C2×C3×C4—64no—
209C2×C3×C4—64no—
210C2×C3×C4—82no—
211C2×C3×C4—84no—
212C2×C3×C4—128no—
213C2×C3×C4—124no—
214C2×C3×C4—124no—
215C2×C3×C4SL(2,ℤ/4)14yescomplemented
216C2×C3×C4C2×S412yes—
217C2×Dic3—44no—
218C2×Dic3—84no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—44no—
222C2×C32—41no—
223C24—31no—
224C24—31no—
225C24—32no—
226C24—32no—
227C24—32no—
228C24C3×Dic612yes—
229C24C3×D1212yes—
230C24C2×C3×Dic312yes—
231C24C22×C3×D311yescharacteristic
232C22×C4—31no—
233C22×C4—32no—
234C22×C4—32no—
235C22×C4—34no—
236C22×C4—34no—
237C22×C4—32no—
238C22×C4—34no—
239C22×C4—38no—
240C22×C4—34no—
241C22×C4—34no—
242C22×C4—34no—
243C22×C4—64no—
244C22×C4—62no—
245C22×C4—61no—
246C22×C4—64no—
247C22×C4—64no—
248C22×C4—64no—
249C22×C4—62no—
250C22×C4—64no—
251C22×C4—64no—
252C22×C4—62no—
253C22×C4C2×C3×Dic314yescomplemented
254C22×C4C3×S411yescharacteristic
255C4⋊C4—64no—
256C4⋊C4—68no—
257C4⋊C4—64no—
258C22⋊C4—68no—
259C22⋊C4—64no—
260C22⋊C4—64no—
261C22×C3—34no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—32no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—32no—
271C22×C3—34no—
272C22×C3—34no—
273C22×C3—34no—
274C22×C3—34no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—32no—
278C22×C3—42no—
279C22×C3—41no—
280C22×C3—42no—
281C22×C3—42no—
282C22×C3—62no—
283C22×C3—61no—
284C22×C3—62no—
285C22×C3—62no—
286C22×C3—82no—
287C22×C3—81no—
288C22×C3—82no—
289C22×C3—82no—
290C22×C3C2×C4⋊Dic311yescharacteristic, complemented
291C22×C3A4⋊Q812yes—
292C22×C3C4⋊S412yes—
293C22×C3C2×SL(2,ℤ/4)12yes—
294C22×C3C22×S411yescharacteristic
295A4—21no—
296A4C2×C3×C4⋊C411yescharacteristic, complemented
297C3×C4—34no—
298C3×C4—44no—
299C3×C4—84no—
300C3×C4—124no—
301C3×C4—124no—
302C3×C4C2×SL(2,ℤ/4)14yescomplemented
303Dic3—84no—
304C32—41noSylow
305C23—32no—
306C23—32no—
307C23—31no—
308C23—31no—
309C23—32no—
310C23—32no—
311C23—32no—
312C23—32no—
313C23—32no—
314C23—31no—
315C23—32no—
316C23—31no—
317C23—32no—
318C23—34no—
319C23—34no—
320C23—34no—
321C23—31no—
322C23—61no—
323C23—64no—
324C23—61no—
325C23—61no—
326C23C3×C4⋊Dic314yescomplemented
327C23C2×C3×Dic611yescharacteristic
328C23C2×C3×D1211yescharacteristic
329C23C22×C3×Dic311yescharacteristic
330C23C2×C3×S411yescharacteristic
331C2×C4—34no—
332C2×C4—32no—
333C2×C4—34no—
334C2×C4—34no—
335C2×C4—38no—
336C2×C4—34no—
337C2×C4—62no—
338C2×C4—64no—
339C2×C4—64no—
340C2×C4—64no—
341C2×C4—64no—
342C2×C4—64no—
343C2×C4—64no—
344C2×C4—64no—
345C2×C4—128no—
346C2×C4—124no—
347C2×C4—124no—
348C2×C4C3×SL(2,ℤ/4)14yescomplemented
349C2×C4C2×C3×S412yes—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—34no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—44no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—84no—
360C2×C3—81no—
361C2×C3—81no—
362C2×C3—81no—
363C2×C3C4⋊SL(2,ℤ/4)14yesdirect product factor
364C2×C3C2×A4⋊Q811yescharacteristic
365C2×C3C2×C4⋊S411yescharacteristic
366C2×C3C22×SL(2,ℤ/4)11yescharacteristic
367C22—34no—
368C22—31no—
369C22—31no—
370C22—31no—
371C22—34no—
372C22—34no—
373C22—34no—
374C22—34no—
375C22—31no—
376C22—31no—
377C22—32no—
378C22—31no—
379C22—31no—
380C22—32no—
381C22—31no—
382C22—31no—
383C22—32no—
384C22—62no—
385C22—61no—
386C22—62no—
387C22—62no—
388C22C2×C3×C4⋊Dic311yescharacteristic, complemented
389C22C3×A4⋊Q812yes—
390C22C3×C4⋊S412yes—
391C22C2×C3×SL(2,ℤ/4)12yes—
392C22C22×C3×S411yescharacteristic
393C4—34no—
394C4—124no—
395C4—124no—
396C4C2×C3×SL(2,ℤ/4)14yescomplemented
397C3—41no—
398C3—81no—
399C3C2×C4⋊SL(2,ℤ/4)11yescharacteristic, direct product factor
400C2—31no—
401C2—31no—
402C2—34no—
403C2—31no—
404C2—31no—
405C2C3×C4⋊SL(2,ℤ/4)14yesdirect product factor
406C2C2×C3×A4⋊Q811yescharacteristic
407C2C2×C3×C4⋊S411yescharacteristic
408C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC23×C3C2×S4
commutatorC2×A4C22×C3×C4
FrattiniC22C22×C3×S4
FittingC24×C3×C4D3
socleC25×C3C2×D3
prime radicalC23×C3×A4C22
cosocleC22×A4C23×C3
hypercenterC22×C3×C4S4
hypocenterA4C2×C3×C4⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×C4⋊SL(2,ℤ/4)C2direct (N × H)
4. C2×C4⋊SL(2,ℤ/4)C3direct (N × H)
7. C2×C3×C4×A4C4semidirect
19. C4⋊SL(2,ℤ/4)C2×C3direct (N × H)
29. C3×C4×A4C2×C4semidirect
42. C23×C3×C4Dic3semidirect
46. C2×C4×A4C3×C4semidirect
63. C2×C3×A4C4⋊C4semidirect
118. C22×C3×C4C2×Dic3semidirect
122. C4×A4C2×C3×C4semidirect
135. C3×A4C2×C4⋊C4semidirect
145. C23×C4C3×Dic3semidirect
181. C23×C3C4⋊Dic3semidirect
190. C2×A4C3×C4⋊C4semidirect
215. C2×C3×C4SL(2,ℤ/4)semidirect
253. C22×C4C2×C3×Dic3semidirect
290. C22×C3C2×C4⋊Dic3semidirect
296. A4C2×C3×C4⋊C4semidirect
302. C3×C4C2×SL(2,ℤ/4)semidirect
326. C23C3×C4⋊Dic3semidirect
348. C2×C4C3×SL(2,ℤ/4)semidirect
363. C2×C3C4⋊SL(2,ℤ/4)direct (N × H)
388. C22C2×C3×C4⋊Dic3semidirect
396. C4C2×C3×SL(2,ℤ/4)semidirect
399. C3C2×C4⋊SL(2,ℤ/4)direct (N × H)
405. C2C3×C4⋊SL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×A42
C22×C3×SL(2,ℤ/4)2
C3×C4⋊SL(2,ℤ/4)2
C2×C4⋊SL(2,ℤ/4)3
C2×C3×#3803
C2×C3×C4⋊Dic34