Finite Group Databasebeta

← C3×C4⋊C4×S4 #11922591

Subgroups — C3×C4⋊C4×S4 #11922591

Order 1152.

864 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 120 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×C4×S4C212yesmaximal
2C2×C3×C4×S4C211yesmaximal, characteristic
3C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C3×SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6C4⋊C4×S4C311yesmaximal, characteristic, direct product factor
7C3×D4×C4⋊C4—31nomaximal
8C22×C3×S4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×SL(2,ℤ/4)C2212yes—
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C3×C4×S4—24no—
14C3×C4×S4C414yescomplemented
15C3×D3×C4⋊C4—41nomaximal
16C2×C4×S4C2×C312yes—
17C2×C4×S4C2×C311yescharacteristic
18C2×C3×C4×D4—31no—
19C2×C3×C4×D4—32no—
20C22×C3×C4⋊C4—31no—
21C22×C3×C4⋊C4D311yescharacteristic, complemented
22A4×C4⋊C4—21no—
23A4×C4⋊C4C2×C311yescharacteristic, complemented
24C4⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
25SL(2,ℤ/4)⋊C4C2×C312yescomplemented
26C3×#389—32no—
27C3×#385—32no—
28C3×#385—32no—
29C3×#384—31no—
30C3×#380—31no—
31C3×#380—31no—
32C3×C4×C4⋊C4—31no—
33C22×C3×A4C2311yescharacteristic
34C2×C3×S4C2×C412yes—
35C2×C3×S4D412yes—
36C2×C3×S4Q812yes—
37C2×C3×C4×D3—41no—
38C2×C3×C4×D3—42no—
39C3×C4×A4—22no—
40C3×C4×A4C2×C412yescomplemented
41C3×SL(2,ℤ/4)—22no—
42C3×SL(2,ℤ/4)C2×C412yescomplemented
43C32×C4⋊C4—41no—
44C3×C4⋊Dic3—41no—
45C3×Dic3⋊C4—42no—
46D4×C4⋊C4—31noSylow
47C22×S4C22×C311yescharacteristic
48C22×C3×D4—31no—
49C23×C3×C4—31no—
50C23×C3×C4—32no—
51C23×C3×C4C2×D312yes—
52C23×C3×C4C2×D311yescharacteristic
53C2×C4×A4—21no—
54C2×C4×A4—22no—
55C2×C4×A4C22×C312yes—
56C2×C4×A4C22×C311yescharacteristic
57C2×SL(2,ℤ/4)C22×C312yes—
58C2×SL(2,ℤ/4)C22×C311yescharacteristic
59C4×S4—24no—
60C4×S4C3×C414yescomplemented
61C3×C4×D4—34no—
62C3×C4×D4—34no—
63C3×C4×D4—64no—
64C3×C4×D4—64no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C4⋊C4—31no—
67C2×C3×C4⋊C4—31no—
68C2×C3×C4⋊C4—32no—
69C2×C3×C4⋊C4—31no—
70C2×C3×C4⋊C4—31no—
71C2×C3×C4⋊C4—32no—
72C2×C3×C4⋊C4—32no—
73C2×C3×C4⋊C4—32no—
74C2×C3×C4⋊C4—62no—
75C2×C3×C4⋊C4—61no—
76C2×C3×C4⋊C4—61no—
77C2×C3×C22⋊C4—31no—
78C2×C3×C22⋊C4—31no—
79C2×C3×C22⋊C4—32no—
80C2×C3×C22⋊C4—32no—
81C2×C3×C42—31no—
82C2×C3×C42—32no—
83D3×C4⋊C4—41no—
84C3×#95—32no—
85C3×#95—32no—
86C3×#95—32no—
87C22×C3×D3—41no—
88C2×C3×A4C22×C411yescharacteristic
89C2×C3×A4C2×D411yescharacteristic
90C2×C3×A4C2×Q811yescharacteristic
91C3×S4C4⋊C414yesdirect product factor
92C2×C32×C4—41no—
93C2×C32×C4—42no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—42no—
96C3×C4×D3—44no—
97C3×C4×D3—84no—
98C2×C4×D4—31no—
99C2×C4×D4—32no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4C3×D311yescharacteristic, complemented
102#389—32no—
103#385—32no—
104#385—32no—
105#384—31no—
106#380—31no—
107#380—31no—
108C4×C4⋊C4—31no—
109C24×C3—31no—
110C24×C3C22×D311yescharacteristic
111C22×A4—21no—
112C22×A4C23×C311yescharacteristic
113C2×S4C2×C3×C412yes—
114C2×S4C3×D412yes—
115C2×S4C3×Q812yes—
116C2×C3×D4—32no—
117C2×C3×D4—32no—
118C2×C3×D4—32no—
119C2×C3×D4—32no—
120C2×C3×D4—32no—
121C2×C3×D4—32no—
122C22×C3×C4—22no—
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—32no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—34no—
130C22×C3×C4—32no—
131C22×C3×C4—32no—
132C22×C3×C4—32no—
133C22×C3×C4—32no—
134C22×C3×C4—61no—
135C22×C3×C4—62no—
136C22×C3×C4—62no—
137C22×C3×C4—61no—
138C22×C3×C4—62no—
139C22×C3×C4—64no—
140C22×C3×C4—62no—
141C22×C3×C4—122no—
142C22×C3×C4C4×D312yescomplemented
143C2×C4×D3—41no—
144C2×C4×D3—42no—
145C4×A4—22no—
146C4×A4—22no—
147C4×A4—42no—
148C4×A4C2×C3×C412yescomplemented
149SL(2,ℤ/4)—22no—
150SL(2,ℤ/4)C2×C3×C412yescomplemented
151C3×C4⋊C4—34no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—62no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4—62no—
159C3×C4⋊C4—64no—
160C3×C4⋊C4—62no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—81no—
163C3×C4⋊C4—122no—
164C3×C4⋊C4S411yescharacteristic, direct product factor
165C3×C22⋊C4—34no—
166C3×C22⋊C4—32no—
167C3×C22⋊C4—32no—
168C3×C22⋊C4—64no—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—62no—
171C3×C42—34no—
172C3×C42—64no—
173C4⋊Dic3—41no—
174Dic3⋊C4—42no—
175C22×C32—41no—
176C2×C3×D3—42no—
177C2×C3×D3—42no—
178C2×C3×D3—42no—
179C3×A4C2×C4⋊C411yescharacteristic, complemented
180C32×C4—42no—
181C32×C4—82no—
182C3×Dic3—42no—
183C3×Dic3—82no—
184C22×D4—31no—
185C23×C4—31no—
186C23×C4—32no—
187C23×C4C2×C3×D312yes—
188C23×C4C2×C3×D311yescharacteristic
189C4×D4—34no—
190C4×D4—34no—
191C4×D4—64no—
192C4×D4—64no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—32no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—61no—
205C2×C22⋊C4—31no—
206C2×C22⋊C4—31no—
207C2×C22⋊C4—32no—
208C2×C22⋊C4—32no—
209C2×C42—31no—
210C2×C42—32no—
211#95—32no—
212#95—32no—
213#95—32no—
214C23×C3—31no—
215C23×C3—31no—
216C23×C3—31no—
217C23×C3—31no—
218C23×C3—32no—
219C23×C3—32no—
220C23×C3—32no—
221C23×C3—61no—
222C23×C3—61no—
223C23×C3—61no—
224C23×C3—61no—
225C23×C3C2×C4×D311yescharacteristic
226C23×C3D3×D411yescharacteristic
227C23×C3D3×Q811yescharacteristic
228C22×D3—41no—
229C2×A4—21no—
230C2×A4—21no—
231C2×A4—21no—
232C2×A4C22×C3×C411yescharacteristic
233C2×A4C2×C3×D411yescharacteristic
234C2×A4C2×C3×Q811yescharacteristic
235S4C3×C4⋊C414yesdirect product factor
236C3×D4—34no—
237C3×D4—34no—
238C3×D4—34no—
239C3×D4—34no—
240C2×C3×C4—32no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—31no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—32no—
249C2×C3×C4—32no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—61no—
255C2×C3×C4—64no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—64no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—81no—
270C2×C3×C4—82no—
271C2×C3×C4—124no—
272C2×C3×C4—124no—
273C2×C3×C4C2×S412yes—
274C2×C3×C4C2×S411yescharacteristic
275C2×Dic3—41no—
276C2×Dic3—42no—
277C4×D3—44no—
278C4×D3—84no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—41no—
282C3×D3—44no—
283C24—31no—
284C24C22×C3×D311yescharacteristic
285C2×D4—32no—
286C2×D4—32no—
287C2×D4—32no—
288C2×D4—32no—
289C2×D4—32no—
290C2×D4—32no—
291C22×C4—22no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—32no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—34no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—61no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—61no—
307C22×C4—62no—
308C22×C4—64no—
309C22×C4—62no—
310C22×C4—122no—
311C22×C4C3×C4×D312yescomplemented
312C4⋊C4—34no—
313C4⋊C4—32no—
314C4⋊C4—31no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—62no—
318C4⋊C4—64no—
319C4⋊C4—62no—
320C4⋊C4—62no—
321C4⋊C4—61no—
322C4⋊C4—122no—
323C4⋊C4C3×S411yescharacteristic, direct product factor
324C22⋊C4—34no—
325C22⋊C4—32no—
326C22⋊C4—32no—
327C22⋊C4—64no—
328C22⋊C4—62no—
329C22⋊C4—62no—
330C42—34no—
331C42—64no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—34no—
342C22×C3—41no—
343C22×C3—61no—
344C22×C3—62no—
345C22×C3—62no—
346C22×C3—62no—
347C22×C3—62no—
348C22×C3—62no—
349C22×C3—62no—
350C22×C3—81no—
351C22×C3D3×C4⋊C411yescharacteristic, complemented
352C22×C3C22×S411yescharacteristic
353C2×D3—42no—
354C2×D3—42no—
355C2×D3—42no—
356A4—21no—
357A4C2×C3×C4⋊C411yescharacteristic, complemented
358C3×C4—22no—
359C3×C4—34no—
360C3×C4—32no—
361C3×C4—42no—
362C3×C4—62no—
363C3×C4—62no—
364C3×C4—62no—
365C3×C4—82no—
366C3×C4—82no—
367C3×C4—122no—
368C3×C4—122no—
369C3×C4—162no—
370C3×C4C4×S412yescomplemented
371Dic3—42no—
372Dic3—82no—
373C32—41noSylow
374C23—31no—
375C23—31no—
376C23—31no—
377C23—31no—
378C23—32no—
379C23—32no—
380C23—32no—
381C23—61no—
382C23—61no—
383C23—61no—
384C23—61no—
385C23C2×C3×C4×D311yescharacteristic
386C23C3×D3×D411yescharacteristic
387C23C3×D3×Q811yescharacteristic
388D4—34no—
389D4—34no—
390D4—34no—
391D4—34no—
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—32no—
395C2×C4—31no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—61no—
405C2×C4—64no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—64no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—124no—
420C2×C4—124no—
421C2×C4C2×C3×S412yes—
422C2×C4C2×C3×S411yescharacteristic
423C2×C3—31no—
424C2×C3—31no—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—64no—
431C2×C3—81no—
432C2×C3—81no—
433C2×C3—81no—
434C2×C3C2×C4×S411yescharacteristic
435C2×C3D4×S411yescharacteristic
436C2×C3Q8×S411yescharacteristic
437D3—44no—
438C22—31no—
439C22—31no—
440C22—31no—
441C22—31no—
442C22—31no—
443C22—31no—
444C22—31no—
445C22—31no—
446C22—31no—
447C22—34no—
448C22—61no—
449C22—62no—
450C22—62no—
451C22—62no—
452C22—62no—
453C22—62no—
454C22—62no—
455C22C3×D3×C4⋊C411yescharacteristic, complemented
456C22C22×C3×S411yescharacteristic
457C4—22no—
458C4—34no—
459C4—32no—
460C4—62no—
461C4—62no—
462C4—62no—
463C4—122no—
464C4—122no—
465C4C3×C4×S412yescomplemented
466C3—41no—
467C3—81no—
468C3C4⋊C4×S411yescharacteristic, direct product factor
469C2—31no—
470C2—31no—
471C2—31no—
472C2—31no—
473C2—64no—
474C2C2×C3×C4×S411yescharacteristic
475C2C3×D4×S411yescharacteristic
476C2C3×Q8×S411yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutatorC2×A4C22×C3×C4
FrattiniC22C22×C3×S4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
cosocleC22×A4C23×C3
hypercenterC3×C4⋊C4S4
hypocenterA4C2×C3×C4⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×A4×C4⋊C4C2semidirect
4. C3×C4⋊SL(2,ℤ/4)C2semidirect
5. C3×SL(2,ℤ/4)⋊C4C2semidirect
6. C4⋊C4×S4C3direct (N × H)
14. C3×C4×S4C4semidirect
21. C22×C3×C4⋊C4D3semidirect
23. A4×C4⋊C4C2×C3semidirect
24. C4⋊SL(2,ℤ/4)C2×C3semidirect
25. SL(2,ℤ/4)⋊C4C2×C3semidirect
40. C3×C4×A4C2×C4semidirect
42. C3×SL(2,ℤ/4)C2×C4semidirect
60. C4×S4C3×C4semidirect
91. C3×S4C4⋊C4direct (N × H)
101. C22×C4⋊C4C3×D3semidirect
142. C22×C3×C4C4×D3semidirect
148. C4×A4C2×C3×C4semidirect
150. SL(2,ℤ/4)C2×C3×C4semidirect
164. C3×C4⋊C4S4direct (N × H)
179. C3×A4C2×C4⋊C4semidirect
235. S4C3×C4⋊C4direct (N × H)
311. C22×C4C3×C4×D3semidirect
323. C4⋊C4C3×S4direct (N × H)
351. C22×C3D3×C4⋊C4semidirect
357. A4C2×C3×C4⋊C4semidirect
370. C3×C4C4×S4semidirect
455. C22C3×D3×C4⋊C4semidirect
465. C4C3×C4×S4semidirect
468. C3C4⋊C4×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×S42
C3×A4×C4⋊C42
C3×C4⋊SL(2,ℤ/4)2
C3×SL(2,ℤ/4)⋊C42
C4⋊C4×S43
C3×D4×C4⋊C43
C3×D3×C4⋊C44