Finite Group Databasebeta

← C4×(C3⋊S4)⋊C4 #11921967

Subgroups — C4×(C3⋊S4)⋊C4 #11921967

Order 1152.

1048 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 115 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×C4×C3⋊S4C211yesmaximal, characteristic
2C2×(C3⋊S4)⋊C4C212yesmaximal
3C4×Dic3×A4C211yesmaximal, characteristic, complemented
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5Dic3×SL(2,ℤ/4)C212yesmaximal, complemented
6C42×S4—31nomaximal
7C4×#67⋊C4—31nomaximal
8C22×C3⋊S4C2211yescharacteristic
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2212yes—
11C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
12C4×C3⋊S4C414yescomplemented
13C2×C3×SL(2,ℤ/4)C2212yes—
14(C3⋊S4)⋊C4C418yesdirect product factor
15C4×#624—41nomaximal
16C2×C4×S4—31no—
17C2×C4×S4—32no—
18C2×C4×S4—61no—
19C2×C4×#67—31no—
20C22×C4×Dic3D311yescharacteristic, complemented
21C2×#67⋊C4—32no—
22C2×C42×D3—31no—
23C42×A4—31no—
24C4×SL(2,ℤ/4)—32no—
25C4×SL(2,ℤ/4)D311yescharacteristic, complemented
26C3×C4×C22⋊C4—31no—
27Dic3×C22⋊C4—32no—
28C4×(C2×D3)⋊C4—31no—
29C4×Dic3⋊C4—31no—
30C42×Dic3—31no—
31#4666—32no—
32#4649—32no—
33C22×C3×A4C2311yescharacteristic
34C2×C3⋊S4C2×C412yes—
35C2×C3⋊S4C2×C414yes—
36C2×C4×C3⋊D3—41no—
37C3×C4×A4C2×C412yescomplemented
38C2×#624—42no—
39Dic3×A4C2×C414yescomplemented
40C3⋊SL(2,ℤ/4)C2×C412yescomplemented
41C3×SL(2,ℤ/4)C2×C414yescomplemented
42C3×C4×Dic3—41no—
43C3×C4×Dic3—41no—
44Dic32—42no—
45C42×D4—91noSylow
46C22×S4—31no—
47C22×S4—61no—
48C23×C3×C4C2×D311yescharacteristic
49C22×#67—31no—
50C23×Dic3C2×D312yes—
51C22×C4×D3—32no—
52C22×C4×D3—31no—
53C2×C4×A4—21no—
54C2×C4×A4—32no—
55C2×C4×A4C2×D311yescharacteristic
56C2×SL(2,ℤ/4)—31no—
57C2×SL(2,ℤ/4)—61no—
58C2×SL(2,ℤ/4)C2×D312yes—
59C4×S4—34no—
60C4×S4—38no—
61C4×S4—64no—
62C2×C3×C22⋊C4—32no—
63C2×C3×C42—31no—
64C2×C3×C42—31no—
65C2×#217—31no—
66C4×#67—34no—
67C4×#67—34no—
68C2×(C2×D3)⋊C4—31no—
69C2×(C2×D3)⋊C4—32no—
70C2×Dic3⋊C4—31no—
71C2×Dic3⋊C4—32no—
72C2×C4×Dic3—32no—
73C2×C4×Dic3—31no—
74C2×C4×Dic3—31no—
75C2×C4×Dic3—32no—
76C2×C4×Dic3—31no—
77C2×C4×Dic3—32no—
78C2×C4×Dic3—32no—
79#67⋊C4—38no—
80#67⋊C4—38no—
81C42×D3—61no—
82C42×D3—62no—
83C42×D3—61no—
84C42×D3—121no—
85C3×#95—32no—
86C3⋊#95—32no—
87C3⋊#95—32no—
88C22×C3⋊D3—41no—
89C2×C3×A4C22×C412yes—
90C2×C3×A4C22×C411yescharacteristic
91C3⋊S4C4214yescomplemented
92C2×C32×C4—41no—
93C2×C3⋊Dic3—41no—
94C4×C3⋊D3—44no—
95C2×C3×Dic3—42no—
96C2×C3×Dic3—42no—
97#624—48no—
98C2×C4×D4—92no—
99C2×C4×D4—91no—
100C22×C42—31no—
101C22×C42—91no—
102C4×C4⋊C4—92no—
103C4×C4⋊C4—91no—
104C4×C22⋊C4—31no—
105C4×C22⋊C4—92no—
106C4×C22⋊C4—91no—
107C4×C22⋊C4—92no—
108C43—91no—
109C24×C3C22×D311yescharacteristic
110C23×D3—31no—
111C22×A4—21no—
112C22×A4C22×D311yescharacteristic
113C2×S4—34no—
114C2×S4—32no—
115C2×S4—62no—
116C2×S4—64no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—32no—
120C22×C3×C4—32no—
121C22×C3×C4—32no—
122C22×C3×C4C4×D312yescomplemented
123C2×#67—34no—
124C2×#67—32no—
125C2×#67—34no—
126C2×#67—32no—
127C22×Dic3—31no—
128C22×Dic3—32no—
129C22×Dic3—32no—
130C22×Dic3—34no—
131C22×Dic3—31no—
132C22×Dic3—31no—
133C22×Dic3C4×D314yescomplemented
134C2×C4×D3—38no—
135C2×C4×D3—34no—
136C2×C4×D3—62no—
137C2×C4×D3—64no—
138C2×C4×D3—61no—
139C2×C4×D3—62no—
140C2×C4×D3—61no—
141C2×C4×D3—62no—
142C2×C4×D3—122no—
143C2×C4×D3—121no—
144C2×C4×D3—241no—
145C4×A4—22no—
146C4×A4—34no—
147C4×A4C4×D312yescomplemented
148SL(2,ℤ/4)—32no—
149SL(2,ℤ/4)—62no—
150SL(2,ℤ/4)C4×D314yescomplemented
151C3×C22⋊C4—34no—
152C3×C22⋊C4—34no—
153C3×C42—61no—
154C3×C42—62no—
155C3×C42—61no—
156C3×C42—121no—
157#217—32no—
158#217—32no—
159(C2×D3)⋊C4—38no—
160(C2×D3)⋊C4—34no—
161Dic3⋊C4—38no—
162Dic3⋊C4—34no—
163C4×Dic3—38no—
164C4×Dic3—34no—
165C4×Dic3—31no—
166C4×Dic3—32no—
167C4×Dic3—41no—
168C4×Dic3—61no—
169C4×Dic3—62no—
170C4×Dic3—64no—
171C4×Dic3—62no—
172C4×Dic3—122no—
173C4×Dic3S411yescharacteristic, complemented
174C22×C32—41no—
175C2×C3⋊D3—44no—
176C2×C3⋊D3—42no—
177C3×A4C2×C4211yescharacteristic, complemented
178C32×C4—42no—
179C3⋊Dic3—42no—
180C3×Dic3—44no—
181C3×Dic3—44no—
182C22×D4—91no—
183C23×C4—32no—
184C23×C4—91no—
185C23×C4—92no—
186C23×C4D3211yescharacteristic
187C4×D4—98no—
188C4×D4—98no—
189C4×D4—94no—
190C4×D4—94no—
191C2×C4⋊C4—91no—
192C2×C4⋊C4—92no—
193C2×C22⋊C4—32no—
194C2×C22⋊C4—91no—
195C2×C22⋊C4—92no—
196C2×C22⋊C4—91no—
197C2×C42—31no—
198C2×C42—31no—
199C2×C42—91no—
200C2×C42—91no—
201C2×C42—92no—
202C2×C42—92no—
203C2×C42—91no—
204C2×C42—92no—
205C2×C42—92no—
206C2×C42—181no—
207C2×C42—182no—
208C2×C42—181no—
209#95—32no—
210#95—92no—
211#95—92no—
212C23×C3—31no—
213C23×C3—31no—
214C23×C3—32no—
215C23×C3C2×C4×D312yes—
216C23×C3C2×C4×D311yescharacteristic
217C22×D3—34no—
218C22×D3—32no—
219C22×D3—62no—
220C22×D3—61no—
221C22×D3—61no—
222C22×D3—121no—
223C22×D3—241no—
224C2×A4—22no—
225C2×A4—21no—
226C2×A4C2×C4×D312yes—
227C2×A4C2×C4×D311yescharacteristic
228S4—34no—
229S4—64no—
230C2×C3×C4—31no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—34no—
235C2×C3×C4—34no—
236C2×C3×C4—41no—
237C2×C3×C4—61no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—81no—
243C2×C3×C4—122no—
244C2×C3×C4C2×S411yescharacteristic
245#67—34no—
246#67—38no—
247#67—34no—
248C2×Dic3—34no—
249C2×Dic3—32no—
250C2×Dic3—32no—
251C2×Dic3—34no—
252C2×Dic3—34no—
253C2×Dic3—34no—
254C2×Dic3—32no—
255C2×Dic3—32no—
256C2×Dic3—42no—
257C2×Dic3—62no—
258C2×Dic3—62no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—121no—
262C2×Dic3—241no—
263C2×Dic3C2×S412yes—
264C4×D3—68no—
265C4×D3—64no—
266C4×D3—64no—
267C4×D3—68no—
268C4×D3—128no—
269C4×D3—124no—
270C4×D3—244no—
271C2×C32—42no—
272C2×C32—41no—
273C3⋊D3—44no—
274C24—91no—
275C24C2×D3211yescharacteristic
276C2×D4—94no—
277C2×D4—92no—
278C2×D4—94no—
279C2×D4—92no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—32no—
283C22×C4—32no—
284C22×C4—32no—
285C22×C4—34no—
286C22×C4—92no—
287C22×C4—94no—
288C22×C4—98no—
289C22×C4—91no—
290C22×C4—92no—
291C22×C4—94no—
292C22×C4—91no—
293C22×C4—91no—
294C22×C4—182no—
295C22×C4—181no—
296C22×C4—182no—
297C22×C4—182no—
298C22×C4—184no—
299C22×C4—181no—
300C22×C4#62412yescomplemented
301C4⋊C4—94no—
302C4⋊C4—98no—
303C22⋊C4—34no—
304C22⋊C4—34no—
305C22⋊C4—94no—
306C22⋊C4—98no—
307C22⋊C4—92no—
308C22⋊C4—92no—
309C42—31no—
310C42—61no—
311C42—62no—
312C42—61no—
313C42—92no—
314C42—94no—
315C42—98no—
316C42—91no—
317C42—182no—
318C42—181no—
319C42—182no—
320C42—184no—
321C22×C3—32no—
322C22×C3—31no—
323C22×C3—32no—
324C22×C3—31no—
325C22×C3—32no—
326C22×C3—31no—
327C22×C3—41no—
328C22×C3—61no—
329C22×C3—81no—
330C22×C3C42×D311yescharacteristic, complemented
331C22×C3C22×S411yescharacteristic
332C2×D3—34no—
333C2×D3—62no—
334C2×D3—64no—
335C2×D3—64no—
336C2×D3—62no—
337C2×D3—124no—
338C2×D3—122no—
339C2×D3—242no—
340C2×D3—244no—
341A4—21no—
342A4C42×D311yescharacteristic, complemented
343C3×C4—32no—
344C3×C4—42no—
345C3×C4—64no—
346C3×C4—64no—
347C3×C4—82no—
348C3×C4—124no—
349C3×C4C4×S412yescomplemented
350Dic3—34no—
351Dic3—34no—
352Dic3—44no—
353Dic3—62no—
354Dic3—62no—
355Dic3—122no—
356Dic3—242no—
357Dic3C4×S414yescomplemented
358C32—41noSylow
359C23—31no—
360C23—32no—
361C23—31no—
362C23—94no—
363C23—92no—
364C23—182no—
365C23—181no—
366C23—181no—
367C23C4×D3212yes—
368C23C2×#62411yescharacteristic
369D4—98no—
370D4—94no—
371D4—94no—
372C2×C4—32no—
373C2×C4—31no—
374C2×C4—32no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—34no—
378C2×C4—34no—
379C2×C4—61no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—94no—
385C2×C4—94no—
386C2×C4—94no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—94no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—188no—
393C2×C4—184no—
394C2×C4—188no—
395C2×C4—182no—
396C2×C4—182no—
397C2×C4—181no—
398C2×C4—181no—
399C2×C4—184no—
400C2×C4D3×S411yescharacteristic
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—32no—
404C2×C3—42no—
405C2×C3—41no—
406C2×C3—82no—
407C2×C3—81no—
408C2×C3C2×C4×S412yes—
409C2×C3C2×C4×S411yescharacteristic
410D3—64no—
411D3—124no—
412D3—244no—
413C22—32no—
414C22—31no—
415C22—32no—
416C22—32no—
417C22—31no—
418C22—31no—
419C22—61no—
420C22—94no—
421C22—184no—
422C22—182no—
423C22—184no—
424C22—182no—
425C22C4×#62411yescharacteristic, complemented
426C22C2×D3×S411yescharacteristic
427C4—34no—
428C4—32no—
429C4—64no—
430C4—64no—
431C4—94no—
432C4—94no—
433C4—182no—
434C4—182no—
435C4(C3⋊S4)⋊C412yesdirect product factor
436C3—41no—
437C3—81no—
438C3C42×S411yescharacteristic, complemented
439C2—32no—
440C2—31no—
441C2—31no—
442C2—184no—
443C2C4×D3×S412yes—
444C2C2×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×S4
commutator, hypocenterC3×A4C2×C42
FrattiniC22C2×D3×S4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC22×C3⋊S4C22
cosocleC22×C3×A4C23
LCSG ▷ C3×A4
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C4×Dic3×A4C2semidirect
4. C3×C4×SL(2,ℤ/4)C2semidirect
5. Dic3×SL(2,ℤ/4)C2semidirect
12. C4×C3⋊S4C4semidirect
14. (C3⋊S4)⋊C4C4direct (N × H)
20. C22×C4×Dic3D3semidirect
25. C4×SL(2,ℤ/4)D3semidirect
37. C3×C4×A4C2×C4semidirect
39. Dic3×A4C2×C4semidirect
40. C3⋊SL(2,ℤ/4)C2×C4semidirect
41. C3×SL(2,ℤ/4)C2×C4semidirect
91. C3⋊S4C42semidirect
122. C22×C3×C4C4×D3semidirect
133. C22×Dic3C4×D3semidirect
147. C4×A4C4×D3semidirect
150. SL(2,ℤ/4)C4×D3semidirect
173. C4×Dic3S4semidirect
177. C3×A4C2×C42semidirect
300. C22×C4#624semidirect
330. C22×C3C42×D3semidirect
342. A4C42×D3semidirect
349. C3×C4C4×S4semidirect
357. Dic3C4×S4semidirect
425. C22C4×#624semidirect
435. C4(C3⋊S4)⋊C4direct (N × H)
438. C3C42×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊S42
C2×(C3⋊S4)⋊C42
C4×Dic3×A42
C3×C4×SL(2,ℤ/4)2
Dic3×SL(2,ℤ/4)2
C42×S43
C4×#67⋊C43
C4×#6244