Finite Group Databasebeta

← C4×#11724052 #72269096

Subgroups — C4×#11724052 #72269096

Order 3456.

1959 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 1959 conjugacy classes of subgroups — too many to draw. Only its 61 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×#11724052C211yesmaximal, characteristic
2C4×C32⋊(C22⋊A4)C211yesmaximal, characteristic, complemented
3#13164539C211yesmaximal, characteristic, complemented
4C4×C3⋊#1020C311yesmaximal, characteristic, complemented
5C3×C4×#953—31nomaximal
6C2×C32⋊(C22⋊A4)C2211yescharacteristic
7#11724052C412yesdirect product factor
8C4×(C3⋊S4)⋊C3—43nomaximal
9C2×C3⋊#1020C2×C311yescharacteristic
10C24×C32×C4D311yescharacteristic, complemented
11C3×C4×C22⋊A4—21no—
12C3×C4×C22⋊A4C2×C311yescharacteristic, complemented
13C3⋊#5936C2×C311yescharacteristic, complemented
14C2×C3×#953—31no—
15C2×C3×C4×#67—33no—
16C3×#5247—31no—
17C3×#5213—33no—
18C3×#5210—33no—
19C3×C4×#217—33no—
20C32⋊(C22⋊A4)C2×C411yescharacteristic, complemented
21C2×(C3⋊S4)⋊C3—43no—
22C4×C22⋊H(3,𝔽3)—43no—
23C4×C22⋊H(3,𝔽3)—81no—
24(C3⋊SL(2,ℤ/4))⋊C3—43no—
25C4×#1020—91no—
26C3×C4×#120—91no—
27C4×#953—31no—
28C25×C32C2×D311yescharacteristic
29C2×C3×C22⋊A4—21no—
30C2×C3×C22⋊A4C22×C311yescharacteristic
31C3⋊#1020C3×C412yescomplemented
32C23×C32×C4—33no—
33C23×C32×C4—33no—
34C23×C32×C4—61no—
35C23×C32×C4—61no—
36C22×C3×#67—33no—
37C22×C3×C4×D3—31no—
38C4×C3⋊S4—43no—
39C3×#953—36no—
40C3×#953—32no—
41C2×C3×#217—33no—
42C2×C3×#217—33no—
43C2×C3×#217—33no—
44C3×C4×#67—63no—
45C3×C4×#67—63no—
46C3×C4×#67—63no—
47C3×C4×#67—63no—
48C2×C3×(C2×D3)⋊C4—33no—
49C2×C3×Dic3⋊C4—33no—
50C2×C3×C4×Dic3—33no—
51C3×C3⋊#95—36no—
52C2×C22⋊H(3,𝔽3)—43no—
53C2×C22⋊H(3,𝔽3)—81no—
54(C3⋊S4)⋊C3—46no—
55C4×(C3⋊D3)⋊C3—161no—
56C2×#1020—91no—
57C24×C3×C4—31no—
58C24×C3×C4C3×D311yescharacteristic
59C4×C22⋊A4—31no—
60C4×C22⋊A4—61no—
61#5936—91no—
62C2×C3×#120—91no—
63C2×C3×C4×D4—93no—
64C2×#953—31no—
65C2×C4×#67—33no—
66C3×#386—93no—
67C3×#385—93no—
68C3×#379—91no—
69C3×C4×C22⋊C4—93no—
70#5247—31no—
71#5213—33no—
72#5210—33no—
73C4×#217—33no—
74C24×C32—33no—
75C24×C32—33no—
76C24×C32—61no—
77C24×C32—61no—
78C24×C32C4×D311yescharacteristic, complemented
79C23×C3×D3—31no—
80C3×C22⋊A4—21no—
81C3×C22⋊A4C2×C3×C411yescharacteristic, complemented
82C2×C3⋊S4—43no—
83C22×C32×C4—21no—
84C22×C32×C4—31no—
85C22×C32×C4—33no—
86C22×C32×C4—33no—
87C22×C32×C4—33no—
88C22×C32×C4—33no—
89C22×C32×C4—66no—
90C22×C32×C4—63no—
91C22×C32×C4—63no—
92C22×C32×C4—61no—
93C22×C32×C4—63no—
94C22×C32×C4S413yescomplemented
95C2×C3×#67—36no—
96C2×C3×#67—36no—
97C2×C3×#67—63no—
98C2×C3×#67—63no—
99C2×C3×#67—63no—
100C2×C3×#67—63no—
101C22×C3×Dic3—33no—
102C22×C3×Dic3—33no—
103C22×C3×Dic3—31no—
104C2×C3×C4×D3—63no—
105C2×C3×C4×D3—63no—
106C3×C4×A4—43no—
107C3×C4×A4—81no—
108C3×C4×A4—82no—
109C3×C4×A4—83no—
110C3⋊SL(2,ℤ/4)—43no—
111C3×#217—36no—
112C3×#217—36no—
113C3×#217—63no—
114C3×#217—63no—
115C3×#217—63no—
116C3×#217—63no—
117C3×(C2×D3)⋊C4—63no—
118C3×(C2×D3)⋊C4—63no—
119C3×Dic3⋊C4—63no—
120C3×Dic3⋊C4—63no—
121C3×C4×Dic3—63no—
122C3×C4×Dic3—63no—
123C4×#120—91noSylow
124C2×(C3⋊D3)⋊C3—161no—
125C22⋊H(3,𝔽3)—43no—
126C22⋊H(3,𝔽3)—81no—
127C4×H(3,𝔽3)—161no—
128(C3⋊Dic3)⋊C3—161no—
129C25×C3—31no—
130C25×C3C2×C3×D311yescharacteristic
131C2×C22⋊A4—31no—
132C2×C22⋊A4—61no—
133#1020—92no—
134C22×C3×D4—93no—
135C23×C3×C4—33no—
136C23×C3×C4—33no—
137C23×C3×C4—33no—
138C23×C3×C4—33no—
139C23×C3×C4—63no—
140C23×C3×C4—63no—
141C23×C3×C4—61no—
142C23×C3×C4—61no—
143C23×C3×C4—63no—
144C23×C3×C4—63no—
145C23×C3×C4—91no—
146C22×#67—33no—
147C22×C4×D3—31no—
148C4×S4—363no—
149C3×#120—96no—
150C3×#120—92no—
151C3×C4×D4—183no—
152C3×C4×D4—183no—
153C3×C4×D4—183no—
154C3×C4×D4—183no—
155C2×C3×C4⋊C4—93no—
156C2×C3×C22⋊C4—93no—
157C2×C3×C22⋊C4—93no—
158C2×C3×C22⋊C4—93no—
159C2×C3×C22⋊C4—93no—
160C2×C3×C42—93no—
161#953—36no—
162#953—32no—
163C2×#217—33no—
164C2×#217—33no—
165C2×#217—33no—
166C4×#67—63no—
167C4×#67—63no—
168C4×#67—63no—
169C4×#67—63no—
170C2×(C2×D3)⋊C4—33no—
171C2×Dic3⋊C4—33no—
172C2×C4×Dic3—33no—
173C3×#95—96no—
174C3⋊#95—36no—
175C23×C32—21no—
176C23×C32—31no—
177C23×C32—33no—
178C23×C32—33no—
179C23×C32—33no—
180C23×C32—33no—
181C23×C32—33no—
182C23×C32—61no—
183C23×C32—61no—
184C23×C32—66no—
185C23×C32—63no—
186C23×C32—63no—
187C23×C32—63no—
188C23×C32C2×S413yes—
189C22×C3×D3—32no—
190C22×C3×D3—63no—
191C22×C3×D3—63no—
192C2×C3×A4—43no—
193C2×C3×A4—81no—
194C2×C3×A4—82no—
195C2×C3×A4—83no—
196C3⋊S4—46no—
197C2×C32×C4—33no—
198C2×C32×C4—33no—
199C2×C32×C4—33no—
200C2×C32×C4—33no—
201C2×C32×C4—63no—
202C2×C32×C4—66no—
203C2×C32×C4—63no—
204C2×C32×C4—61no—
205C2×C32×C4—61no—
206C4×C3⋊D3—161no—
207C3×#67—66no—
208C3×#67—66no—
209C3×#67—66no—
210C3×#67—66no—
211C2×C3×Dic3—36no—
212C2×C3×Dic3—63no—
213C2×C3×Dic3—63no—
214C2×C3×Dic3—63no—
215C2×C3×Dic3—63no—
216C2×C3×Dic3—66no—
217C3×C4×D3—123no—
218C3×C4×D3—121no—
219C24×C4(C3⋊D3)⋊C311yescharacteristic, complemented
220C2×#120—91no—
221C2×C4×D4—93no—
222#386—93no—
223#385—93no—
224#379—91no—
225C4×C22⋊C4—93no—
226C2×H(3,𝔽3)—161no—
227(C3⋊D3)⋊C3—162no—
228C24×C3—33no—
229C24×C3—33no—
230C24×C3—33no—
231C24×C3—33no—
232C24×C3—31no—
233C24×C3—63no—
234C24×C3—63no—
235C24×C3—61no—
236C24×C3—61no—
237C24×C3—63no—
238C24×C3—63no—
239C24×C3—91no—
240C24×C3C3×C4×D311yescharacteristic
241C23×D3—31no—
242C22⋊A4—31no—
243C22⋊A4—61no—
244C2×S4—363no—
245C2×C3×D4—96no—
246C2×C3×D4—96no—
247C2×C3×D4—183no—
248C2×C3×D4—183no—
249C2×C3×D4—183no—
250C2×C3×D4—183no—
251C22×C3×C4—21no—
252C22×C3×C4—31no—
253C22×C3×C4—31no—
254C22×C3×C4—33no—
255C22×C3×C4—33no—
256C22×C3×C4—33no—
257C22×C3×C4—33no—
258C22×C3×C4—33no—
259C22×C3×C4—33no—
260C22×C3×C4—33no—
261C22×C3×C4—33no—
262C22×C3×C4—33no—
263C22×C3×C4—61no—
264C22×C3×C4—63no—
265C22×C3×C4—61no—
266C22×C3×C4—63no—
267C22×C3×C4—63no—
268C22×C3×C4—63no—
269C22×C3×C4—66no—
270C22×C3×C4—63no—
271C22×C3×C4—63no—
272C22×C3×C4—66no—
273C22×C3×C4—66no—
274C22×C3×C4—66no—
275C22×C3×C4—66no—
276C22×C3×C4—61no—
277C22×C3×C4—63no—
278C22×C3×C4—63no—
279C22×C3×C4—63no—
280C22×C3×C4—66no—
281C22×C3×C4—63no—
282C22×C3×C4—63no—
283C22×C3×C4—66no—
284C22×C3×C4—93no—
285C22×C3×C4—93no—
286C22×C3×C4—91no—
287C22×C3×C4—183no—
288C22×C3×C4—183no—
289C22×C3×C4C3×S413yes—
290C2×#67—36no—
291C2×#67—36no—
292C2×#67—63no—
293C2×#67—63no—
294C2×#67—63no—
295C2×#67—63no—
296C22×Dic3—33no—
297C22×Dic3—33no—
298C22×Dic3—31no—
299C2×C4×D3—63no—
300C2×C4×D3—63no—
301C4×A4—123no—
302C4×A4—241no—
303C4×A4—242no—
304C4×A4—243no—
305SL(2,ℤ/4)—363no—
306C3×C4⋊C4—183no—
307C3×C4⋊C4—183no—
308C3×C22⋊C4—96no—
309C3×C22⋊C4—96no—
310C3×C22⋊C4—183no—
311C3×C22⋊C4—183no—
312C3×C22⋊C4—183no—
313C3×C22⋊C4—183no—
314C3×C22⋊C4—183no—
315C3×C22⋊C4—183no—
316C3×C42—183no—
317C3×C42—183no—
318#217—36no—
319#217—36no—
320#217—63no—
321#217—63no—
322#217—63no—
323#217—63no—
324(C2×D3)⋊C4—63no—
325(C2×D3)⋊C4—63no—
326Dic3⋊C4—63no—
327Dic3⋊C4—63no—
328C4×Dic3—63no—
329C4×Dic3—63no—
330C22×C32—21no—
331C22×C32—33no—
332C22×C32—31no—
333C22×C32—33no—
334C22×C32—33no—
335C22×C32—33no—
336C22×C32—33no—
337C22×C32—63no—
338C22×C32—66no—
339C22×C32—63no—
340C22×C32—63no—
341C22×C32—61no—
342C22×C32—61no—
343C22×C32C4×S413yescomplemented
344C2×C3⋊D3—161no—
345C2×C3×D3—66no—
346C2×C3×D3—123no—
347C2×C3×D3—121no—
348C3×A4—43no—
349C3×A4—81no—
350C3×A4—82no—
351C3×A4—83no—
352C32×C4—33no—
353C32×C4—61no—
354C32×C4—161no—
355C32×C4—321no—
356C32×C4#102011yescharacteristic, complemented
357C3⋊Dic3—161no—
358C3×Dic3—66no—
359C3×Dic3—123no—
360C3×Dic3—121no—
361C25C2×(C3⋊D3)⋊C311yescharacteristic
362C22×D4—93no—
363C23×C4—33no—
364C23×C4—33no—
365C23×C4—61no—
366C23×C4—61no—
367C23×C4—91no—
368#120—96no—
369#120—92no—
370C4×D4—183no—
371C4×D4—183no—
372C4×D4—183no—
373C4×D4—183no—
374C2×C4⋊C4—93no—
375C2×C22⋊C4—93no—
376C2×C22⋊C4—93no—
377C2×C22⋊C4—93no—
378C2×C22⋊C4—93no—
379C2×C42—93no—
380#95—96no—
381H(3,𝔽3)—161noSylow
382C23×C3—21no—
383C23×C3—31no—
384C23×C3—31no—
385C23×C3—33no—
386C23×C3—33no—
387C23×C3—33no—
388C23×C3—33no—
389C23×C3—33no—
390C23×C3—33no—
391C23×C3—33no—
392C23×C3—33no—
393C23×C3—33no—
394C23×C3—33no—
395C23×C3—33no—
396C23×C3—61no—
397C23×C3—63no—
398C23×C3—63no—
399C23×C3—61no—
400C23×C3—63no—
401C23×C3—63no—
402C23×C3—63no—
403C23×C3—61no—
404C23×C3—61no—
405C23×C3—63no—
406C23×C3—66no—
407C23×C3—63no—
408C23×C3—66no—
409C23×C3—63no—
410C23×C3—66no—
411C23×C3—66no—
412C23×C3—66no—
413C23×C3—63no—
414C23×C3—63no—
415C23×C3—63no—
416C23×C3—63no—
417C23×C3—66no—
418C23×C3—63no—
419C23×C3—66no—
420C23×C3—92no—
421C23×C3—183no—
422C23×C3—183no—
423C23×C3C2×C3×S413yes—
424C22×D3—32no—
425C22×D3—63no—
426C22×D3—63no—
427C2×A4—123no—
428C2×A4—241no—
429C2×A4—242no—
430C2×A4—243no—
431S4—366no—
432C3×D4—186no—
433C3×D4—186no—
434C3×D4—186no—
435C3×D4—186no—
436C2×C3×C4—33no—
437C2×C3×C4—33no—
438C2×C3×C4—33no—
439C2×C3×C4—33no—
440C2×C3×C4—33no—
441C2×C3×C4—33no—
442C2×C3×C4—33no—
443C2×C3×C4—33no—
444C2×C3×C4—63no—
445C2×C3×C4—63no—
446C2×C3×C4—63no—
447C2×C3×C4—63no—
448C2×C3×C4—63no—
449C2×C3×C4—66no—
450C2×C3×C4—66no—
451C2×C3×C4—66no—
452C2×C3×C4—63no—
453C2×C3×C4—63no—
454C2×C3×C4—63no—
455C2×C3×C4—66no—
456C2×C3×C4—63no—
457C2×C3×C4—66no—
458C2×C3×C4—66no—
459C2×C3×C4—63no—
460C2×C3×C4—61no—
461C2×C3×C4—61no—
462C2×C3×C4—96no—
463C2×C3×C4—183no—
464C2×C3×C4—183no—
465C2×C3×C4—183no—
466C2×C3×C4—183no—
467C2×C3×C4—186no—
468C2×C3×C4—363no—
469C2×C3×C4—361no—
470#67—66no—
471#67—66no—
472#67—66no—
473#67—66no—
474C2×Dic3—36no—
475C2×Dic3—63no—
476C2×Dic3—63no—
477C2×Dic3—63no—
478C2×Dic3—63no—
479C2×Dic3—66no—
480C4×D3—123no—
481C4×D3—121no—
482C4×D3—1441no—
483C2×C32—33no—
484C2×C32—33no—
485C2×C32—61no—
486C2×C32—61no—
487C2×C32—161no—
488C2×C32—321no—
489C2×C32C2×#102011yescharacteristic
490C3⋊D3—162no—
491C3×D3—122no—
492C24—33no—
493C24—33no—
494C24—61no—
495C24—61no—
496C24—91no—
497C24C4×(C3⋊D3)⋊C311yescharacteristic, complemented
498C2×D4—96no—
499C2×D4—96no—
500C2×D4—183no—
501C2×D4—183no—
502C2×D4—183no—
503C2×D4—183no—
504C22×C4—21no—
505C22×C4—31no—
506C22×C4—33no—
507C22×C4—33no—
508C22×C4—33no—
509C22×C4—33no—
510C22×C4—66no—
511C22×C4—63no—
512C22×C4—63no—
513C22×C4—61no—
514C22×C4—63no—
515C22×C4—93no—
516C22×C4—93no—
517C22×C4—91no—
518C22×C4—183no—
519C22×C4—183no—
520C22×C4(C3⋊S4)⋊C313yescomplemented
521C4⋊C4—183no—
522C4⋊C4—183no—
523C22⋊C4—96no—
524C22⋊C4—96no—
525C22⋊C4—183no—
526C22⋊C4—183no—
527C22⋊C4—183no—
528C22⋊C4—183no—
529C22⋊C4—183no—
530C22⋊C4—183no—
531C42—183no—
532C42—183no—
533C22×C3—21no—
534C22×C3—33no—
535C22×C3—31no—
536C22×C3—33no—
537C22×C3—33no—
538C22×C3—31no—
539C22×C3—33no—
540C22×C3—33no—
541C22×C3—33no—
542C22×C3—33no—
543C22×C3—33no—
544C22×C3—33no—
545C22×C3—33no—
546C22×C3—33no—
547C22×C3—61no—
548C22×C3—63no—
549C22×C3—63no—
550C22×C3—61no—
551C22×C3—63no—
552C22×C3—63no—
553C22×C3—63no—
554C22×C3—63no—
555C22×C3—66no—
556C22×C3—63no—
557C22×C3—63no—
558C22×C3—66no—
559C22×C3—66no—
560C22×C3—66no—
561C22×C3—66no—
562C22×C3—63no—
563C22×C3—63no—
564C22×C3—61no—
565C22×C3—63no—
566C22×C3—66no—
567C22×C3—63no—
568C22×C3—66no—
569C22×C3—63no—
570C22×C3—61no—
571C22×C3—186no—
572C22×C3—361no—
573C22×C3—363no—
574C22×C3C3×C4×S413yes—
575C2×D3—66no—
576C2×D3—123no—
577C2×D3—121no—
578C2×D3—1441no—
579A4—123no—
580A4—241no—
581A4—242no—
582A4—243no—
583C3×C4—33no—
584C3×C4—33no—
585C3×C4—31no—
586C3×C4—63no—
587C3×C4—63no—
588C3×C4—61no—
589C3×C4—186no—
590C3×C4—361no—
591C3×C4—363no—
592C3×C4—481no—
593C3×C4—961no—
594C3×C4C3×#102011yescharacteristic
595Dic3—66no—
596Dic3—123no—
597Dic3—121no—
598Dic3—1441no—
599C32—161no—
600C32—321no—
601C32C4×#102011yescharacteristic, complemented
602C23—21no—
603C23—31no—
604C23—33no—
605C23—33no—
606C23—33no—
607C23—33no—
608C23—33no—
609C23—61no—
610C23—61no—
611C23—63no—
612C23—66no—
613C23—63no—
614C23—63no—
615C23—92no—
616C23—183no—
617C23—183no—
618C23C2×(C3⋊S4)⋊C313yes—
619D4—186no—
620D4—186no—
621D4—186no—
622D4—186no—
623C2×C4—33no—
624C2×C4—33no—
625C2×C4—33no—
626C2×C4—33no—
627C2×C4—63no—
628C2×C4—66no—
629C2×C4—63no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—96no—
633C2×C4—183no—
634C2×C4—183no—
635C2×C4—183no—
636C2×C4—183no—
637C2×C4—186no—
638C2×C4—363no—
639C2×C4—361no—
640C2×C3—33no—
641C2×C3—33no—
642C2×C3—33no—
643C2×C3—33no—
644C2×C3—31no—
645C2×C3—63no—
646C2×C3—63no—
647C2×C3—63no—
648C2×C3—63no—
649C2×C3—61no—
650C2×C3—61no—
651C2×C3—362no—
652C2×C3—481no—
653C2×C3—961no—
654C2×C3C2×C3×#102011yescharacteristic
655D3—122no—
656D3—1442no—
657C22—21no—
658C22—33no—
659C22—31no—
660C22—33no—
661C22—33no—
662C22—33no—
663C22—33no—
664C22—63no—
665C22—66no—
666C22—63no—
667C22—63no—
668C22—61no—
669C22—61no—
670C22—186no—
671C22—363no—
672C22—361no—
673C22C4×(C3⋊S4)⋊C313yescomplemented
674C4—33no—
675C4—61no—
676C4—186no—
677C4—363no—
678C4—361no—
679C4#1172405211yescharacteristic, direct product factor
680C3—31no—
681C3—481no—
682C3—961no—
683C3C3×C4×#102011yescharacteristic
684C2—33no—
685C2—33no—
686C2—61no—
687C2—61no—
688C2—362no—
689C2C2×#1172405211yescharacteristic
subgroupG/H
center, hypercenterC4#11724052
commutator, hypocenterC3×C22⋊A4C2×C3×C4
FrattiniC2×C3C2×C3×#1020
FittingC24×C32×C4D3
socleC25×C3C2×C3×D3
prime radicalC2×#11724052C2
cosocleC2×C3×C22⋊A4C22×C3
LCSG ▷ C3×C22⋊A4
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C4×C32⋊(C22⋊A4)C2semidirect
3. #13164539C2semidirect
4. C4×C3⋊#1020C3semidirect
7. #11724052C4direct (N × H)
10. C24×C32×C4D3semidirect
12. C3×C4×C22⋊A4C2×C3semidirect
13. C3⋊#5936C2×C3semidirect
20. C32⋊(C22⋊A4)C2×C4semidirect
31. C3⋊#1020C3×C4semidirect
78. C24×C32C4×D3semidirect
81. C3×C22⋊A4C2×C3×C4semidirect
94. C22×C32×C4S4semidirect
219. C24×C4(C3⋊D3)⋊C3semidirect
343. C22×C32C4×S4semidirect
356. C32×C4#1020semidirect
497. C24C4×(C3⋊D3)⋊C3semidirect
520. C22×C4(C3⋊S4)⋊C3semidirect
601. C32C4×#1020semidirect
673. C22C4×(C3⋊S4)⋊C3semidirect
679. C4#11724052direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117240522
C4×C32⋊(C22⋊A4)2
#131645392
C4×C3⋊#10203
C3×C4×#9533
C4×(C3⋊S4)⋊C34