Finite Group Databasebeta

← C2×C3×(C2×S4)⋊C4 #11922767

Subgroups — C2×C3×(C2×S4)⋊C4 #11922767

Order 1152.

1476 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1476 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C23×C3×S4C211yesmaximal, characteristic
2C22×C3×C4×A4C211yesmaximal, characteristic, complemented
3C22×C3×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C3×(C2×S4)⋊C4C214yesmaximal, direct product factor
5C2×(C2×S4)⋊C4C311yesmaximal, characteristic, direct product factor
6C2×C3×#386—31nomaximal
7C23×C3×A4C2211yescharacteristic
8C22×C3×S4—22no—
9C22×C3×S4—22no—
10C22×C3×S4C414yescomplemented
11C22×C3×S4C2212yes—
12C2×C3×C4×A4—22no—
13C2×C3×C4×A4C2212yescomplemented
14C2×C3×SL(2,ℤ/4)—22no—
15C2×C3×SL(2,ℤ/4)C2212yescomplemented
16C2×C3×(C2×D3)⋊C4—41nomaximal
17C23×S4C2×C311yescharacteristic
18C23×C3×D4—31no—
19C24×C3×C4D311yescharacteristic, complemented
20C22×C4×A4—21no—
21C22×C4×A4C2×C311yescharacteristic, complemented
22C22×SL(2,ℤ/4)C2×C311yescharacteristic, complemented
23C22×C3×C22⋊C4—31no—
24C22×C3×C22⋊C4—31no—
25C22×C3×C22⋊C4—31no—
26(C2×S4)⋊C4C2×C314yesdirect product factor
27C3×#386—34no—
28C3×#386—34no—
29C2×C3×#95—31no—
30C2×C3×#95—31no—
31C23×C3×D3—41no—
32C22×C3×A4C2×C412yes—
33C22×C3×A4D412yes—
34C22×C3×A4D412yes—
35C22×C3×A4C2311yescharacteristic
36C2×C3×S4—28no—
37C2×C3×S4—22no—
38C2×C3×S4—22no—
39C2×C3×S4C2×C414yescomplemented
40C22×C32×C4—41no—
41C22×C3×Dic3—41no—
42C3×C4×A4—22no—
43C3×SL(2,ℤ/4)—22no—
44C3×(C2×D3)⋊C4—44no—
45C2×#386—31noSylow
46C25×C3—31no—
47C25×C3C2×D311yescharacteristic
48C23×A4—21no—
49C23×A4C22×C311yescharacteristic
50C22×S4—22no—
51C22×S4—22no—
52C22×S4C3×C414yescomplemented
53C22×S4C22×C312yes—
54C22×C3×D4—34no—
55C22×C3×D4—32no—
56C22×C3×D4—34no—
57C22×C3×D4—32no—
58C22×C3×D4—62no—
59C22×C3×D4—62no—
60C22×C3×D4—62no—
61C22×C3×D4—62no—
62C23×C3×C4—22no—
63C23×C3×C4—31no—
64C23×C3×C4—31no—
65C23×C3×C4—31no—
66C23×C3×C4—32no—
67C23×C3×C4—31no—
68C23×C3×C4—31no—
69C23×C3×C4—62no—
70C23×C3×C4C2×D312yescomplemented
71C2×C4×A4—24no—
72C2×C4×A4—22no—
73C2×C4×A4—22no—
74C2×C4×A4C22×C312yescomplemented
75C2×SL(2,ℤ/4)—22no—
76C2×SL(2,ℤ/4)C22×C312yescomplemented
77C2×C3×C22⋊C4—34no—
78C2×C3×C22⋊C4—34no—
79C2×C3×C22⋊C4—32no—
80C2×C3×C22⋊C4—32no—
81C2×C3×C22⋊C4—64no—
82C2×C3×C22⋊C4—61no—
83C2×C3×C22⋊C4—62no—
84C2×C3×C22⋊C4—62no—
85C2×C3×C22⋊C4—62no—
86C2×C3×C22⋊C4—61no—
87C2×(C2×D3)⋊C4—41no—
88C3×#95—34no—
89C3×#95—34no—
90C23×C32—41no—
91C22×C3×D3—44no—
92C22×C3×D3—42no—
93C22×C3×D3—82no—
94C22×C3×D3—82no—
95C2×C3×A4C22⋊C414yescomplemented
96C2×C3×A4C22×C411yescharacteristic
97C2×C3×A4C2×D411yescharacteristic
98C2×C3×A4C2×D411yescharacteristic
99C3×S4—24no—
100C2×C32×C4—42no—
101C2×C32×C4—82no—
102C2×C3×Dic3—42no—
103C2×C3×Dic3—82no—
104C23×D4—31no—
105C24×C4C3×D311yescharacteristic, complemented
106C22×C22⋊C4—31no—
107C22×C22⋊C4—31no—
108C22×C22⋊C4—31no—
109#386—34no—
110#386—34no—
111C2×#95—31no—
112C2×#95—31no—
113C24×C3—31no—
114C24×C3—31no—
115C24×C3—32no—
116C24×C3—32no—
117C24×C3—32no—
118C24×C3—32no—
119C24×C3—34no—
120C24×C3—62no—
121C24×C3—62no—
122C24×C3—62no—
123C24×C3—61no—
124C24×C3—62no—
125C24×C3—62no—
126C24×C3—61no—
127C24×C3C4×D312yes—
128C24×C3D1212yes—
129C24×C3#6712yes—
130C24×C3C22×D311yescharacteristic
131C23×D3—41no—
132C22×A4—22no—
133C22×A4—22no—
134C22×A4—22no—
135C22×A4—21no—
136C22×A4C2×C3×C412yes—
137C22×A4C3×D412yes—
138C22×A4C3×D412yes—
139C22×A4C23×C311yescharacteristic
140C2×S4—28no—
141C2×S4—22no—
142C2×S4—22no—
143C2×S4C2×C3×C414yescomplemented
144C2×C3×D4—38no—
145C2×C3×D4—34no—
146C2×C3×D4—34no—
147C2×C3×D4—64no—
148C2×C3×D4—62no—
149C2×C3×D4—64no—
150C2×C3×D4—62no—
151C2×C3×D4—68no—
152C2×C3×D4—68no—
153C2×C3×D4—68no—
154C2×C3×D4—68no—
155C2×C3×D4—62no—
156C2×C3×D4—62no—
157C22×C3×C4—22no—
158C22×C3×C4—32no—
159C22×C3×C4—32no—
160C22×C3×C4—34no—
161C22×C3×C4—34no—
162C22×C3×C4—31no—
163C22×C3×C4—32no—
164C22×C3×C4—32no—
165C22×C3×C4—32no—
166C22×C3×C4—32no—
167C22×C3×C4—41no—
168C22×C3×C4—62no—
169C22×C3×C4—62no—
170C22×C3×C4—61no—
171C22×C3×C4—62no—
172C22×C3×C4—64no—
173C22×C3×C4—62no—
174C22×C3×C4—62no—
175C22×C3×C4—64no—
176C22×C3×C4—62no—
177C22×C3×C4—62no—
178C22×C3×C4—62no—
179C22×C3×C4—62no—
180C22×C3×C4—61no—
181C22×C3×C4—62no—
182C22×C3×C4—62no—
183C22×C3×C4—61no—
184C22×C3×C4—81no—
185C22×C3×C4S411yescharacteristic, complemented
186C22×Dic3—41no—
187C4×A4—24no—
188C4×A4—22no—
189SL(2,ℤ/4)—22no—
190C3×C22⋊C4—68no—
191C3×C22⋊C4—64no—
192C3×C22⋊C4—64no—
193C3×C22⋊C4—62no—
194C3×C22⋊C4—62no—
195C3×C22⋊C4—64no—
196(C2×D3)⋊C4—44no—
197C22×C32—42no—
198C22×C32—42no—
199C22×C32—42no—
200C22×C32—41no—
201C2×C3×D3—44no—
202C2×C3×D3—82no—
203C2×C3×D3—88no—
204C2×C3×D3—82no—
205C3×A4C2×C22⋊C411yescharacteristic, complemented
206C32×C4—82no—
207C3×Dic3—82no—
208C25—31no—
209C25C2×C3×D311yescharacteristic
210C22×D4—32no—
211C22×D4—34no—
212C22×D4—34no—
213C22×D4—32no—
214C22×D4—62no—
215C22×D4—62no—
216C22×D4—62no—
217C22×D4—62no—
218C23×C4—22no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—62no—
226C23×C4C2×C3×D312yescomplemented
227C2×C22⋊C4—34no—
228C2×C22⋊C4—34no—
229C2×C22⋊C4—32no—
230C2×C22⋊C4—32no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—61no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—62no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—61no—
237#95—34no—
238#95—34no—
239C23×C3—32no—
240C23×C3—32no—
241C23×C3—32no—
242C23×C3—32no—
243C23×C3—32no—
244C23×C3—32no—
245C23×C3—31no—
246C23×C3—31no—
247C23×C3—32no—
248C23×C3—31no—
249C23×C3—32no—
250C23×C3—31no—
251C23×C3—32no—
252C23×C3—31no—
253C23×C3—34no—
254C23×C3—34no—
255C23×C3—34no—
256C23×C3—34no—
257C23×C3—41no—
258C23×C3—61no—
259C23×C3—61no—
260C23×C3—61no—
261C23×C3—64no—
262C23×C3—62no—
263C23×C3—62no—
264C23×C3—68no—
265C23×C3—62no—
266C23×C3—62no—
267C23×C3—64no—
268C23×C3—62no—
269C23×C3—68no—
270C23×C3—64no—
271C23×C3—62no—
272C23×C3—81no—
273C23×C3—122no—
274C23×C3—124no—
275C23×C3—124no—
276C23×C3—122no—
277C23×C3—122no—
278C23×C3—122no—
279C23×C3(C2×D3)⋊C414yescomplemented
280C23×C3C2×C4×D311yescharacteristic
281C23×C3C2×D1211yescharacteristic
282C23×C3C2×#6711yescharacteristic
283C23×C3C2×S411yescharacteristic
284C22×D3—42no—
285C22×D3—44no—
286C22×D3—82no—
287C22×D3—82no—
288C2×A4—24no—
289C2×A4—21no—
290C2×A4—21no—
291C2×A4—21no—
292C2×A4C3×C22⋊C414yescomplemented
293C2×A4C22×C3×C411yescharacteristic
294C2×A4C2×C3×D411yescharacteristic
295C2×A4C2×C3×D411yescharacteristic
296S4—24no—
297C3×D4—616no—
298C3×D4—64no—
299C3×D4—64no—
300C3×D4—64no—
301C3×D4—64no—
302C2×C3×C4—22no—
303C2×C3×C4—34no—
304C2×C3×C4—34no—
305C2×C3×C4—32no—
306C2×C3×C4—42no—
307C2×C3×C4—62no—
308C2×C3×C4—62no—
309C2×C3×C4—64no—
310C2×C3×C4—64no—
311C2×C3×C4—62no—
312C2×C3×C4—68no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—82no—
323C2×C3×C4—84no—
324C2×C3×C4—82no—
325C2×C3×C4—124no—
326C2×C3×C4—122no—
327C2×C3×C4—122no—
328C2×C3×C4C2×S412yescomplemented
329C2×Dic3—42no—
330C2×Dic3—82no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32—44no—
334C2×C32—41no—
335C3×D3—84no—
336C24—31no—
337C24—31no—
338C24—32no—
339C24—32no—
340C24—32no—
341C24—32no—
342C24—34no—
343C24—62no—
344C24—61no—
345C24—62no—
346C24—62no—
347C24—62no—
348C24—62no—
349C24—61no—
350C24C3×C4×D312yes—
351C24C3×D1212yes—
352C24C3×#6712yes—
353C24C22×C3×D311yescharacteristic
354C2×D4—38no—
355C2×D4—34no—
356C2×D4—34no—
357C2×D4—64no—
358C2×D4—62no—
359C2×D4—64no—
360C2×D4—62no—
361C2×D4—68no—
362C2×D4—68no—
363C2×D4—68no—
364C2×D4—68no—
365C2×D4—62no—
366C2×D4—62no—
367C22×C4—22no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—34no—
371C22×C4—34no—
372C22×C4—31no—
373C22×C4—32no—
374C22×C4—32no—
375C22×C4—32no—
376C22×C4—32no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—64no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—64no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—61no—
393C22×C4C3×S411yescharacteristic, complemented
394C22⋊C4—68no—
395C22⋊C4—64no—
396C22⋊C4—64no—
397C22⋊C4—62no—
398C22⋊C4—62no—
399C22⋊C4—64no—
400C22×C3—34no—
401C22×C3—31no—
402C22×C3—31no—
403C22×C3—31no—
404C22×C3—31no—
405C22×C3—31no—
406C22×C3—32no—
407C22×C3—31no—
408C22×C3—31no—
409C22×C3—32no—
410C22×C3—34no—
411C22×C3—34no—
412C22×C3—34no—
413C22×C3—34no—
414C22×C3—31no—
415C22×C3—31no—
416C22×C3—32no—
417C22×C3—42no—
418C22×C3—41no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—61no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3—64no—
426C22×C3—64no—
427C22×C3—64no—
428C22×C3—82no—
429C22×C3—81no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—122no—
433C22×C3—128no—
434C22×C3—122no—
435C22×C3—122no—
436C22×C3—128no—
437C22×C3—122no—
438C22×C3C2×(C2×D3)⋊C411yescharacteristic, complemented
439C22×C3C4×S412yes—
440C22×C3C4⋊S412yes—
441C22×C3GL(2,ℤ/4)12yes—
442C22×C3C22×S411yescharacteristic
443C2×D3—44no—
444C2×D3—88no—
445C2×D3—82no—
446C2×D3—82no—
447A4—21no—
448A4C2×C3×C22⋊C411yescharacteristic, complemented
449C3×C4—22no—
450C3×C4—64no—
451C3×C4—62no—
452C3×C4—82no—
453C3×C4—84no—
454C3×C4—122no—
455C3×C4—122no—
456Dic3—82no—
457C32—41noSylow
458C23—32no—
459C23—32no—
460C23—31no—
461C23—31no—
462C23—32no—
463C23—32no—
464C23—32no—
465C23—32no—
466C23—32no—
467C23—31no—
468C23—32no—
469C23—31no—
470C23—32no—
471C23—31no—
472C23—34no—
473C23—34no—
474C23—34no—
475C23—34no—
476C23—61no—
477C23—64no—
478C23—61no—
479C23—61no—
480C23—62no—
481C23—68no—
482C23—62no—
483C23—62no—
484C23—68no—
485C23—62no—
486C23—64no—
487C23—62no—
488C23—62no—
489C23—64no—
490C23—124no—
491C23—124no—
492C23—122no—
493C23—122no—
494C23—122no—
495C23—122no—
496C23C3×(C2×D3)⋊C414yescomplemented
497C23C2×C3×C4×D311yescharacteristic
498C23C2×C3×D1211yescharacteristic
499C23C2×C3×#6711yescharacteristic
500C23C2×C3×S411yescharacteristic
501D4—616no—
502D4—64no—
503D4—64no—
504D4—64no—
505D4—64no—
506C2×C4—22no—
507C2×C4—34no—
508C2×C4—34no—
509C2×C4—32no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—68no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—124no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4C2×C3×S412yescomplemented
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—34no—
533C2×C3—31no—
534C2×C3—44no—
535C2×C3—41no—
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—84no—
539C2×C3—81no—
540C2×C3—81no—
541C2×C3—81no—
542C2×C3—124no—
543C2×C3(C2×S4)⋊C414yesdirect product factor
544C2×C3C2×C4×S411yescharacteristic
545C2×C3C2×C4⋊S411yescharacteristic
546C2×C3C2×GL(2,ℤ/4)11yescharacteristic
547D3—84no—
548C22—34no—
549C22—31no—
550C22—31no—
551C22—31no—
552C22—34no—
553C22—34no—
554C22—34no—
555C22—34no—
556C22—31no—
557C22—31no—
558C22—32no—
559C22—31no—
560C22—31no—
561C22—32no—
562C22—31no—
563C22—31no—
564C22—32no—
565C22—61no—
566C22—62no—
567C22—62no—
568C22—62no—
569C22—64no—
570C22—64no—
571C22—64no—
572C22—122no—
573C22—128no—
574C22—122no—
575C22—128no—
576C22—122no—
577C22—122no—
578C22C2×C3×(C2×D3)⋊C411yescharacteristic, complemented
579C22C3×C4×S412yes—
580C22C3×C4⋊S412yes—
581C22C3×GL(2,ℤ/4)12yes—
582C22C22×C3×S411yescharacteristic
583C4—22no—
584C4—64no—
585C4—62no—
586C4—122no—
587C4—122no—
588C3—41no—
589C3—81no—
590C3C2×(C2×S4)⋊C411yescharacteristic, direct product factor
591C2—31no—
592C2—31no—
593C2—31no—
594C2—34no—
595C2—31no—
596C2—124no—
597C2C3×(C2×S4)⋊C414yesdirect product factor
598C2C2×C3×C4×S411yescharacteristic
599C2C2×C3×C4⋊S411yescharacteristic
600C2C2×C3×GL(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×S4C2
cosocleC22×A4C23×C3
hypercenterC22×C3×C4S4
hypocenterA4C2×C3×C22⋊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
2. C22×C3×C4×A4C2semidirect
3. C22×C3×SL(2,ℤ/4)C2semidirect
4. C3×(C2×S4)⋊C4C2direct (N × H)
5. C2×(C2×S4)⋊C4C3direct (N × H)
10. C22×C3×S4C4semidirect
13. C2×C3×C4×A4C22semidirect
15. C2×C3×SL(2,ℤ/4)C22semidirect
19. C24×C3×C4D3semidirect
21. C22×C4×A4C2×C3semidirect
22. C22×SL(2,ℤ/4)C2×C3semidirect
26. (C2×S4)⋊C4C2×C3direct (N × H)
39. C2×C3×S4C2×C4semidirect
52. C22×S4C3×C4semidirect
70. C23×C3×C4C2×D3semidirect
74. C2×C4×A4C22×C3semidirect
76. C2×SL(2,ℤ/4)C22×C3semidirect
95. C2×C3×A4C22⋊C4semidirect
105. C24×C4C3×D3semidirect
143. C2×S4C2×C3×C4semidirect
185. C22×C3×C4S4semidirect
205. C3×A4C2×C22⋊C4semidirect
226. C23×C4C2×C3×D3semidirect
279. C23×C3(C2×D3)⋊C4semidirect
292. C2×A4C3×C22⋊C4semidirect
328. C2×C3×C4C2×S4semidirect
393. C22×C4C3×S4semidirect
438. C22×C3C2×(C2×D3)⋊C4semidirect
448. A4C2×C3×C22⋊C4semidirect
496. C23C3×(C2×D3)⋊C4semidirect
528. C2×C4C2×C3×S4semidirect
543. C2×C3(C2×S4)⋊C4direct (N × H)
578. C22C2×C3×(C2×D3)⋊C4semidirect
590. C3C2×(C2×S4)⋊C4direct (N × H)
597. C2C3×(C2×S4)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×S42
C22×C3×C4×A42
C22×C3×SL(2,ℤ/4)2
C3×(C2×S4)⋊C42
C2×(C2×S4)⋊C43
C2×C3×#3863
C2×C3×(C2×D3)⋊C44