Finite Group Databasebeta

← Dic3×C4⋊S4 #11921973

Subgroups — Dic3×C4⋊S4 #11921973

Order 1152.

892 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 93 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⋊S4C211yesmaximal, characteristic
2C2×Dic3×S4C212yesmaximal, complemented
3C4×Dic3×A4C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
5(C2×S4)⋊Dic3C212yesmaximal, complemented
6C4×C4⋊S4—31nomaximal
7Dic3×#121—31nomaximal
8C22×C3×S4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2211yescharacteristic
11C2×Dic3×A4C2211yescharacteristic
12C2×C3⋊SL(2,ℤ/4)C2212yes—
13C3×C4⋊S4C414yescomplemented
14Dic3×S4—24no—
15Dic3×D12—41nomaximal
16C2×C4⋊S4D311yescharacteristic
17C2×C4×S4—32no—
18C2×C3×#121—31no—
19C2×D4×Dic3—31no—
20C2×D4×Dic3—32no—
21C22×C4×Dic3D311yescharacteristic, complemented
22C42×A4—31no—
23(C2×S4)⋊C4—32no—
24C4⋊SL(2,ℤ/4)—31no—
25C4⋊SL(2,ℤ/4)—61no—
26#5221—31no—
27#5220—32no—
28#5212—31no—
29#4984—31no—
30Dic3×C4⋊C4—31no—
31#4951—32no—
32Dic3×C22⋊C4—32no—
33C22×C3×A4C2311yescharacteristic
34C2×C3×S4—22no—
35C2×C3×S4—22no—
36C2×C3×S4C2×C414yescomplemented
37C2×C3×D12—41no—
38C3×C4×A4C2×C412yescomplemented
39C2×D3×Dic3—42no—
40Dic3×A4—21no—
41Dic3×A4D412yescomplemented
42C3⋊SL(2,ℤ/4)—22no—
43C4⋊(C3⋊Dic3)—41no—
44C3×C4×Dic3—41no—
45(C2×D3)⋊Dic3—42no—
46C4×#121—91noSylow
47C22×S4C2×D312yes—
48C22×C3×D4—31no—
49C22×C3×D4—32no—
50C23×C3×C4C2×D311yescharacteristic
51C23×Dic3—32no—
52C23×Dic3C2×D311yescharacteristic
53C23×Dic3C2×D311yescharacteristic
54C2×C4×A4—21no—
55C2×C4×A4—31no—
56C2×C4×A4—31no—
57C2×C4×A4C2×D311yescharacteristic
58C2×SL(2,ℤ/4)—32no—
59C2×SL(2,ℤ/4)—62no—
60C4⋊S4Dic314yesdirect product factor
61C4×S4—64no—
62C3×#121—34no—
63C3×#121—34no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C22⋊C4—32no—
66C2×#217—32no—
67C2×#217—32no—
68C2×#217—32no—
69C2×#217—32no—
70D4×Dic3—61no—
71D4×Dic3—61no—
72D4×Dic3—61no—
73D4×Dic3—64no—
74D4×Dic3—64no—
75D4×Dic3—61no—
76C2×C4⋊Dic3—32no—
77C2×C4⋊Dic3—31no—
78C2×C4⋊Dic3—31no—
79C2×C4×Dic3—31no—
80C2×C4×Dic3—31no—
81C2×C4×Dic3—31no—
82C2×C4×Dic3—31no—
83C2×C4×Dic3—32no—
84C4×D12—121no—
85C3⋊#95—32no—
86C3⋊#95—32no—
87C3⋊#95—32no—
88C22×C3×D3—42no—
89C2×C3×A4C22×C411yescharacteristic
90C2×C3×A4C2×D411yescharacteristic
91C2×C3×A4#4111yescharacteristic
92C3×S4—24no—
93C2×C32×C4—41no—
94C2×C3⋊Dic3—42no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—41no—
97C3×D12—44no—
98D3×Dic3—84no—
99C2×#121—31no—
100C2×C4×D4—92no—
101C2×C4×D4—91no—
102C22×C42—31no—
103#390—91no—
104#389—91no—
105#388—92no—
106#386—92no—
107#380—91no—
108C4×C4⋊C4—91no—
109C4×C22⋊C4—92no—
110C24×C3—32no—
111C24×C3C22×D311yescharacteristic
112C22×A4—21no—
113C22×A4C22×D311yescharacteristic
114C2×S4—22no—
115C2×S4—22no—
116C2×S4C2×Dic314yescomplemented
117C2×C3×D4—34no—
118C2×C3×D4—34no—
119C2×C3×D4—34no—
120C2×C3×D4—61no—
121C2×C3×D4—62no—
122C2×C3×D4—62no—
123C2×C3×D4—62no—
124C2×C3×D4—62no—
125C2×C3×D4—62no—
126C2×C3×D4—61no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—32no—
130C22×C3×C4—32no—
131C22×C3×C4C4×D312yescomplemented
132C22×Dic3—21no—
133C22×Dic3—31no—
134C22×Dic3—31no—
135C22×Dic3—31no—
136C22×Dic3—32no—
137C22×Dic3—31no—
138C22×Dic3—32no—
139C22×Dic3—32no—
140C22×Dic3—61no—
141C22×Dic3—62no—
142C22×Dic3—64no—
143C22×Dic3—64no—
144C22×Dic3—62no—
145C22×Dic3D1212yescomplemented
146C2×D12—41no—
147C2×C4×D3—122no—
148C4×A4—22no—
149C4×A4—32no—
150C4×A4—61no—
151C4×A4C2×Dic312yescomplemented
152SL(2,ℤ/4)—62no—
153SL(2,ℤ/4)—122no—
154C3×C4⋊C4—34no—
155C3×C22⋊C4—38no—
156C3×C42—121no—
157#217—64no—
158#217—62no—
159#217—64no—
160#217—62no—
161#217—62no—
162#217—62no—
163(C2×D3)⋊C4—122no—
164C4⋊Dic3—64no—
165C4⋊Dic3—62no—
166C4⋊Dic3—61no—
167C4⋊Dic3—61no—
168C4⋊Dic3—121no—
169C4⋊Dic3—241no—
170C4×Dic3—31no—
171C4×Dic3—31no—
172C4×Dic3—31no—
173C4×Dic3—61no—
174C4×Dic3—64no—
175C4×Dic3S411yescharacteristic, complemented
176C22×C32—41no—
177C2×C3×D3—44no—
178C2×C3×D3—82no—
179C2×C3×D3—82no—
180C3×A4C4×D411yescharacteristic, complemented
181C32×C4—42no—
182C3⋊Dic3—82no—
183C3×Dic3—42no—
184C3×Dic3—81no—
185C22×D4—31no—
186C22×D4—32no—
187C23×C4—31no—
188C23×C4—31no—
189C23×C4—92no—
190C23×C4D3211yescharacteristic
191#121—34no—
192#121—34no—
193C4×D4—181no—
194C4×D4—181no—
195C4×D4—181no—
196C4×D4—181no—
197C4×D4—184no—
198C4×D4—184no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—91no—
201C2×C4⋊C4—92no—
202C2×C4⋊C4—91no—
203C2×C22⋊C4—32no—
204C2×C22⋊C4—92no—
205C2×C22⋊C4—92no—
206C2×C22⋊C4—92no—
207C2×C22⋊C4—92no—
208C2×C42—91no—
209C2×C42—91no—
210C2×C42—91no—
211C2×C42—91no—
212C2×C42—92no—
213#95—92no—
214#95—92no—
215#95—92no—
216C23×C3—31no—
217C23×C3—31no—
218C23×C3—31no—
219C23×C3—31no—
220C23×C3—34no—
221C23×C3—62no—
222C23×C3—62no—
223C23×C3—62no—
224C23×C3—62no—
225C23×C3—62no—
226C23×C3—62no—
227C23×C3C2×C4×D311yescharacteristic
228C23×C3C2×D1211yescharacteristic
229C23×C3#23511yescharacteristic
230C22×D3—42no—
231C2×A4—21no—
232C2×A4—21no—
233C2×A4—21no—
234C2×A4D3×D411yescharacteristic
235C2×A4#23711yescharacteristic
236C2×A4C22×Dic311yescharacteristic
237S4—24no—
238C3×D4—64no—
239C3×D4—64no—
240C3×D4—64no—
241C3×D4—64no—
242C3×D4—64no—
243C3×D4—64no—
244C2×C3×C4—31no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—34no—
249C2×C3×C4—41no—
250C2×C3×C4—61no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—81no—
254C2×C3×C4—121no—
255C2×C3×C4—121no—
256C2×C3×C4C2×S411yescharacteristic
257C2×Dic3—31no—
258C2×Dic3—32no—
259C2×Dic3—32no—
260C2×Dic3—32no—
261C2×Dic3—32no—
262C2×Dic3—31no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267C2×Dic3—62no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—62no—
273C2×Dic3—122no—
274C2×Dic3—124no—
275C2×Dic3—124no—
276C2×Dic3—242no—
277C2×Dic3C2×S411yescharacteristic
278C2×Dic3C2×S411yescharacteristic
279D12—44no—
280C4×D3—244no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32—41no—
284C3×D3—84no—
285C24—32no—
286C24C2×D3211yescharacteristic
287C2×D4—34no—
288C2×D4—34no—
289C2×D4—34no—
290C2×D4—61no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—61no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—61no—
303C22×C4—91no—
304C22×C4—91no—
305C22×C4—91no—
306C22×C4—91no—
307C22×C4—92no—
308C22×C4—92no—
309C22×C4—92no—
310C22×C4—182no—
311C22×C4—181no—
312C22×C4—184no—
313C22×C4—182no—
314C22×C4—184no—
315C22×C4D3×Dic312yescomplemented
316C4⋊C4—34no—
317C4⋊C4—181no—
318C4⋊C4—182no—
319C4⋊C4—184no—
320C4⋊C4—181no—
321C22⋊C4—38no—
322C22⋊C4—182no—
323C22⋊C4—184no—
324C22⋊C4—182no—
325C22⋊C4—184no—
326C22⋊C4—182no—
327C22⋊C4—182no—
328C42—31no—
329C42—91no—
330C42—91no—
331C42—91no—
332C42—181no—
333C42—184no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—41no—
344C22×C3—61no—
345C22×C3—64no—
346C22×C3—64no—
347C22×C3—64no—
348C22×C3—81no—
349C22×C3—122no—
350C22×C3—122no—
351C22×C3—122no—
352C22×C3—122no—
353C22×C3C4×D1211yescharacteristic, complemented
354C22×C3C22×S411yescharacteristic
355C2×D3—44no—
356C2×D3—82no—
357C2×D3—82no—
358A4—21no—
359A4D4×Dic311yescharacteristic, complemented
360C3×C4—32no—
361C3×C4—42no—
362C3×C4—64no—
363C3×C4—82no—
364C3×C4—122no—
365C3×C4—241no—
366C3×C4C4×S412yescomplemented
367Dic3—21no—
368Dic3—32no—
369Dic3—61no—
370Dic3—122no—
371Dic3—122no—
372Dic3—242no—
373Dic3—482no—
374Dic3C4⋊S412yesdirect product factor
375C32—41noSylow
376C23—31no—
377C23—31no—
378C23—31no—
379C23—31no—
380C23—34no—
381C23—62no—
382C23—62no—
383C23—62no—
384C23—62no—
385C23—62no—
386C23—62no—
387C23#379511yescharacteristic
388C23D3×D1211yescharacteristic
389C23C2×D3×Dic311yescharacteristic
390D4—64no—
391D4—64no—
392D4—64no—
393D4—64no—
394D4—64no—
395D4—64no—
396C2×C4—31no—
397C2×C4—31no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—34no—
402C2×C4—31no—
403C2×C4—61no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—92no—
407C2×C4—92no—
408C2×C4—91no—
409C2×C4—92no—
410C2×C4—92no—
411C2×C4—91no—
412C2×C4—181no—
413C2×C4—181no—
414C2×C4—181no—
415C2×C4—181no—
416C2×C4—182no—
417C2×C4—182no—
418C2×C4—182no—
419C2×C4—182no—
420C2×C4—182no—
421C2×C4—182no—
422C2×C4—364no—
423C2×C4—364no—
424C2×C4D3×S411yescharacteristic
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—81no—
433C2×C3—81no—
434C2×C3—81no—
435C2×C3—124no—
436C2×C3C2×C4×S411yescharacteristic
437C2×C3C2×C4⋊S411yescharacteristic
438C2×C3#591211yescharacteristic
439D3—84no—
440C22—31no—
441C22—31no—
442C22—31no—
443C22—31no—
444C22—31no—
445C22—31no—
446C22—31no—
447C22—31no—
448C22—31no—
449C22—61no—
450C22—64no—
451C22—64no—
452C22—64no—
453C22—122no—
454C22—122no—
455C22—122no—
456C22—122no—
457C22Dic3×D1211yescharacteristic, complemented
458C22C2×D3×S411yescharacteristic
459C4—32no—
460C4—32no—
461C4—64no—
462C4—61no—
463C4—92no—
464C4—181no—
465C4—362no—
466C4—362no—
467C4Dic3×S412yescomplemented
468C3—41no—
469C3—81no—
470C3C4×C4⋊S411yescharacteristic, complemented
471C2—31no—
472C2—31no—
473C2—31no—
474C2—31no—
475C2—124no—
476C2#1059642711yescharacteristic
477C2D3×C4⋊S411yescharacteristic
478C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC2×C3×C4⋊S4C2
cosocleC22×C3×A4C23
hypercenterC2×C4D3×S4
hypocenterC3×A4C4×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×S4C2semidirect
3. C4×Dic3×A4C2semidirect
4. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
5. (C2×S4)⋊Dic3C2semidirect
13. C3×C4⋊S4C4semidirect
21. C22×C4×Dic3D3semidirect
36. C2×C3×S4C2×C4semidirect
38. C3×C4×A4C2×C4semidirect
41. Dic3×A4D4semidirect
60. C4⋊S4Dic3direct (N × H)
116. C2×S4C2×Dic3semidirect
131. C22×C3×C4C4×D3semidirect
145. C22×Dic3D12semidirect
151. C4×A4C2×Dic3semidirect
175. C4×Dic3S4semidirect
180. C3×A4C4×D4semidirect
315. C22×C4D3×Dic3semidirect
353. C22×C3C4×D12semidirect
359. A4D4×Dic3semidirect
366. C3×C4C4×S4semidirect
374. Dic3C4⋊S4direct (N × H)
457. C22Dic3×D12semidirect
467. C4Dic3×S4semidirect
470. C3C4×C4⋊S4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊S42
C2×Dic3×S42
C4×Dic3×A42
C4⋊(C3⋊SL(2,ℤ/4))2
(C2×S4)⋊Dic32
C4×C4⋊S43
Dic3×#1213
Dic3×D124