Finite Group Databasebeta

← C2×#11724204 #13180838

Subgroups — C2×#11724204 #13180838

Order 1728.

1459 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 1459 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C22×C3×D3×A4C211yesmaximal, characteristic, complemented
2C22×C3×C3⋊S4C211yesmaximal, characteristic, complemented
3C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4#11724204C214yesmaximal, direct product factor
5C2×#64269—31nomaximal
6C2×#64209—33nomaximal
7C2×#63976—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×C3×D3×A4—22no—
10C2×C3×D3×A4C2212yescomplemented
11C2×C3×C3⋊S4—22no—
12C2×C3×C3⋊S4C2212yescomplemented
13C2×#6290—41nomaximal
14C3⋊(C3⋊SL(2,ℤ/4))C2212yescomplemented
15C24×C3×D3D311yescharacteristic, complemented
16C23×C3×A4—31no—
17C22×D3×A4D313yescomplemented
18C22×C3⋊S4D311yescharacteristic, complemented
19C22×C3×S4—33no—
20C22×C3×#67—31no—
21C22×#625—31no—
22C22×#625—31no—
23#64269—34no—
24C2×C3⋊SL(2,ℤ/4)—31no—
25C2×C3⋊SL(2,ℤ/4)—33no—
26C2×C3⋊SL(2,ℤ/4)—61no—
27C2×C3⋊SL(2,ℤ/4)—63no—
28#64209—312no—
29C2×#3757—31no—
30#63976—34no—
31#63976—34no—
32C2×(C2×D3)⋊Dic3—31no—
33C2×(C2×D3)⋊Dic3—31no—
34C22×C3×C3⋊D3—41no—
35C22×C32×D3—41no—
36C2×C32×A4D412yescomplemented
37C2×C32×A4C2311yescharacteristic
38C3×D3×A4—22no—
39C3×C3⋊S4—22no—
40C2×C3⋊(C3⋊Dic3)—41no—
41#6290—44no—
42C2×GL(2,ℤ/4)—93no—
43C2×#953—91no—
44C2×#937—91no—
45C24×C32C2×D311yescharacteristic
46C23×C3×D3—22no—
47C23×C3×D3—31no—
48C23×C3×D3—31no—
49C23×C3×D3—32no—
50C23×C3×D3—31no—
51C23×C3×D3—62no—
52C23×C3×D3C2×D312yescomplemented
53C22×C3×A4—21no—
54C22×C3×A4—23no—
55C22×C3×A4—32no—
56C22×C3×A4—62no—
57C22×C3×A4C2×D313yes—
58C22×C3×A4C2×D311yescharacteristic
59C2×D3×A4—26no—
60C2×D3×A4C2×D316yescomplemented
61C2×C3⋊S4—22no—
62C2×C3⋊S4C2×D312yescomplemented
63C2×C3×S4—36no—
64C2×C3×S4—66no—
65C2×#638—121no—
66C22×C3⋊Dic3—31no—
67C22×C3⋊Dic3—31no—
68C2×C3×#67—32no—
69C2×C3×#67—32no—
70C2×C3×#67—62no—
71C2×C3×#67—62no—
72C22×C3×Dic3—31no—
73C2×#625—34no—
74C2×#625—34no—
75C2×#625—64no—
76C2×#625—61no—
77C2×#625—62no—
78C2×#625—61no—
79C2×#625—123no—
80C3⋊SL(2,ℤ/4)—32no—
81C3⋊SL(2,ℤ/4)—36no—
82C3⋊SL(2,ℤ/4)—62no—
83C3⋊SL(2,ℤ/4)—66no—
84#3757—32no—
85#3757—32no—
86(C2×D3)⋊Dic3—34no—
87(C2×D3)⋊Dic3—34no—
88C22×C33—41no—
89C2×C3×C3⋊D3—42no—
90C2×C3×C3⋊D3—82no—
91C2×C32×D3—42no—
92C2×C32×D3—82no—
93C32×A4C2×D411yescharacteristic, complemented
94C3⋊(C3⋊Dic3)—42no—
95C25×C3—31no—
96C24×D3C3⋊D311yescharacteristic, complemented
97C23×A4—33no—
98C22×S4—33no—
99C22×C3×D4—91no—
100C22×#67—31no—
101C22×#67—91no—
102C22×#67—91no—
103C22×#67—91no—
104C22×#67—91no—
105GL(2,ℤ/4)—912no—
106C2×SL(2,ℤ/4)—93no—
107C2×SL(2,ℤ/4)—183no—
108#953—94no—
109#953—94no—
110C2×#217—91no—
111C2×#217—91no—
112C2×#217—91no—
113C2×#217—91no—
114C2×#217—181no—
115#937—94no—
116#937—94no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C23×C32—31no—
120C23×C32—31no—
121C23×C32—32no—
122C23×C32—121no—
123C23×C32#6712yescomplemented
124C23×C32C22×D311yescharacteristic
125C22×C3⋊D3—41no—
126C22×C3×D3—22no—
127C22×C3×D3—32no—
128C22×C3×D3—34no—
129C22×C3×D3—31no—
130C22×C3×D3—32no—
131C22×C3×D3—32no—
132C22×C3×D3—32no—
133C22×C3×D3—43no—
134C22×C3×D3—62no—
135C22×C3×D3—62no—
136C22×C3×D3—61no—
137C22×C3×D3—62no—
138C22×C3×D3—64no—
139C22×C3×D3—64no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C22×C3×D3—61no—
143C22×C3×D3—62no—
144C22×C3×D3—61no—
145C22×C3×D3—123no—
146C22×C3×D3S411yescharacteristic, complemented
147C2×C3×A4—22no—
148C2×C3×A4—21no—
149C2×C3×A4—26no—
150C2×C3×A4—23no—
151C2×C3×A4—62no—
152C2×C3×A4#6716yescomplemented
153C2×C3×A4#6712yescomplemented
154C2×C3×A4C22×D313yes—
155C2×C3×A4C22×D311yescharacteristic
156D3×A4—26no—
157C3⋊S4—22no—
158C3×S4—66no—
159#638—124no—
160C2×C3⋊Dic3—32no—
161C2×C3⋊Dic3—32no—
162C2×C3⋊Dic3—62no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—121no—
166C2×C3⋊Dic3—123no—
167C2×C3⋊Dic3—241no—
168C2×C3⋊Dic3—243no—
169C3×#67—64no—
170C3×#67—62no—
171C3×#67—62no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—62no—
174#625—68no—
175#625—64no—
176#625—64no—
177#625—1212no—
178C2×#120—271noSylow
179C2×C33—42no—
180C2×C33—41no—
181C3×C3⋊D3—82no—
182C32×D3—82no—
183C24×C3—21no—
184C24×C3—32no—
185C24×C3—62no—
186C24×C3—91no—
187C24×C3—91no—
188C24×C3—92no—
189C24×C3—91no—
190C24×C3—182no—
191C24×C3D3211yescharacteristic
192C24×C3C2×C3⋊D311yescharacteristic
193C23×D3—22no—
194C23×D3—31no—
195C23×D3—31no—
196C23×D3—32no—
197C23×D3—31no—
198C23×D3—62no—
199C23×D3C2×C3⋊D312yescomplemented
200C22×A4—23no—
201C22×A4—36no—
202C22×A4—66no—
203C22×A4D3213yes—
204C2×S4—36no—
205C2×S4—66no—
206C2×C3×D4—92no—
207C2×C3×D4—92no—
208C2×C3×D4—182no—
209C2×C3×D4—182no—
210C22×C3×C4—91no—
211C2×#67—32no—
212C2×#67—32no—
213C2×#67—62no—
214C2×#67—62no—
215C2×#67—94no—
216C2×#67—94no—
217C2×#67—94no—
218C2×#67—94no—
219C2×#67—184no—
220C2×#67—182no—
221C2×#67—181no—
222C2×#67—181no—
223C2×#67—182no—
224C2×#67—184no—
225C2×#67—181no—
226C2×#67—181no—
227C2×#67—363no—
228C22×Dic3—31no—
229C22×Dic3—91no—
230C22×Dic3—91no—
231C22×Dic3—91no—
232C22×Dic3—91no—
233C22×Dic3—181no—
234C22×Dic3—181no—
235SL(2,ℤ/4)—96no—
236SL(2,ℤ/4)—186no—
237#217—94no—
238#217—94no—
239#217—92no—
240#217—92no—
241#217—92no—
242#217—92no—
243#217—182no—
244#217—182no—
245(C2×D3)⋊C4—94no—
246(C2×D3)⋊C4—94no—
247C22×C32—32no—
248C22×C32—31no—
249C22×C32—32no—
250C22×C32—32no—
251C22×C32—31no—
252C22×C32—31no—
253C22×C32—41no—
254C22×C32—43no—
255C22×C32—61no—
256C22×C32—83no—
257C22×C32—81no—
258C22×C32—122no—
259C22×C32—242no—
260C22×C32C2×#6711yescharacteristic, complemented
261C22×C32C2×S411yescharacteristic
262C2×C3⋊D3—42no—
263C2×C3⋊D3—82no—
264C2×C3×D3—22no—
265C2×C3×D3—34no—
266C2×C3×D3—32no—
267C2×C3×D3—46no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—64no—
271C2×C3×D3—64no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—86no—
278C2×C3×D3—126no—
279C2×C3×D3—122no—
280C2×C3×D3—122no—
281C2×C3×D3—246no—
282C2×C3×D3C2×S412yescomplemented
283C3×A4—21no—
284C3×A4—23no—
285C3×A4C2×#6713yescomplemented
286C3×A4C2×#6711yescharacteristic, complemented
287C3⋊Dic3—62no—
288C3⋊Dic3—62no—
289C3⋊Dic3—122no—
290C3⋊Dic3—126no—
291C3⋊Dic3—242no—
292C3⋊Dic3—246no—
293C3×Dic3—62no—
294C25—31no—
295C22×D4—91no—
296C22×D4—271no—
297C22×D4—271no—
298#120—274no—
299#120—274no—
300C2×C22⋊C4—271no—
301C2×C22⋊C4—271no—
302C2×C22⋊C4—271no—
303C33—41noSylow
304C23×C3—22no—
305C23×C3—21no—
306C23×C3—31no—
307C23×C3—31no—
308C23×C3—31no—
309C23×C3—31no—
310C23×C3—32no—
311C23×C3—32no—
312C23×C3—31no—
313C23×C3—61no—
314C23×C3—61no—
315C23×C3—62no—
316C23×C3—62no—
317C23×C3—92no—
318C23×C3—92no—
319C23×C3—92no—
320C23×C3—94no—
321C23×C3—92no—
322C23×C3—91no—
323C23×C3—123no—
324C23×C3—182no—
325C23×C3—182no—
326C23×C3—182no—
327C23×C3—181no—
328C23×C3—184no—
329C23×C3—182no—
330C23×C3—182no—
331C23×C3—181no—
332C23×C3—182no—
333C23×C3—184no—
334C23×C3—181no—
335C23×C3#62512yescomplemented
336C23×C3#63812yescomplemented
337C23×C3C2×D3211yescharacteristic
338C23×C3C22×C3⋊D311yescharacteristic
339C22×D3—22no—
340C22×D3—32no—
341C22×D3—34no—
342C22×D3—31no—
343C22×D3—32no—
344C22×D3—32no—
345C22×D3—32no—
346C22×D3—62no—
347C22×D3—62no—
348C22×D3—61no—
349C22×D3—62no—
350C22×D3—64no—
351C22×D3—64no—
352C22×D3—62no—
353C22×D3—62no—
354C22×D3—62no—
355C22×D3—61no—
356C22×D3—61no—
357C22×D3—123no—
358C22×D3C3⋊S411yescharacteristic, complemented
359C2×A4—26no—
360C2×A4—23no—
361C2×A4—66no—
362C2×A4#62516yescomplemented
363C2×A4C2×D3213yes—
364S4—66no—
365C3×D4—184no—
366C3×D4—182no—
367C3×D4—182no—
368C2×C3×C4—92no—
369C2×C3×C4—182no—
370#67—64no—
371#67—62no—
372#67—62no—
373#67—188no—
374#67—184no—
375#67—184no—
376#67—188no—
377#67—184no—
378#67—184no—
379#67—3612no—
380C2×Dic3—32no—
381C2×Dic3—62no—
382C2×Dic3—92no—
383C2×Dic3—92no—
384C2×Dic3—92no—
385C2×Dic3—92no—
386C2×Dic3—182no—
387C2×Dic3—182no—
388C2×Dic3—182no—
389C2×Dic3—182no—
390C2×Dic3—181no—
391C2×Dic3—181no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—361no—
395C2×Dic3—363no—
396C2×Dic3—361no—
397C2×Dic3—362no—
398C2×Dic3—723no—
399C2×C32—31no—
400C2×C32—31no—
401C2×C32—32no—
402C2×C32—41no—
403C2×C32—42no—
404C2×C32—46no—
405C2×C32—43no—
406C2×C32—83no—
407C2×C32—86no—
408C2×C32—81no—
409C2×C32—82no—
410C2×C32—242no—
411C2×C32GL(2,ℤ/4)12yescomplemented
412C2×C32C22×S411yescharacteristic
413C3⋊D3—82no—
414C3×D3—22no—
415C3×D3—62no—
416C3×D3—86no—
417C3×D3—122no—
418C3×D3—246no—
419C24—32no—
420C24—62no—
421C24—91no—
422C24—91no—
423C24—92no—
424C24—91no—
425C24—182no—
426C24D3×C3⋊D311yescharacteristic
427C2×D4—92no—
428C2×D4—92no—
429C2×D4—182no—
430C2×D4—182no—
431C2×D4—274no—
432C2×D4—274no—
433C2×D4—541no—
434C2×D4—542no—
435C2×D4—544no—
436C2×D4—541no—
437C22×C4—91no—
438C22×C4—271no—
439C22×C4—271no—
440C22⋊C4—274no—
441C22⋊C4—274no—
442C22⋊C4—272no—
443C22⋊C4—272no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—32no—
447C22×C3—31no—
448C22×C3—32no—
449C22×C3—31no—
450C22×C3—32no—
451C22×C3—32no—
452C22×C3—31no—
453C22×C3—31no—
454C22×C3—31no—
455C22×C3—31no—
456C22×C3—32no—
457C22×C3—32no—
458C22×C3—32no—
459C22×C3—43no—
460C22×C3—61no—
461C22×C3—62no—
462C22×C3—61no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—62no—
469C22×C3—83no—
470C22×C3—94no—
471C22×C3—92no—
472C22×C3—126no—
473C22×C3—121no—
474C22×C3—182no—
475C22×C3—182no—
476C22×C3—184no—
477C22×C3—182no—
478C22×C3—184no—
479C22×C3—182no—
480C22×C3—182no—
481C22×C3—182no—
482C22×C3—182no—
483C22×C3—246no—
484C22×C3—362no—
485C22×C3—362no—
486C22×C3C2×#62511yescharacteristic, complemented
487C22×C3C2×#63811yescharacteristic, complemented
488C22×C3D3×S411yescharacteristic
489C22×C3C2×C3⋊S411yescharacteristic
490C2×D3—22no—
491C2×D3—34no—
492C2×D3—32no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—64no—
496C2×D3—64no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—62no—
500C2×D3—62no—
501C2×D3—62no—
502C2×D3—126no—
503C2×D3—122no—
504C2×D3—122no—
505C2×D3—246no—
506C2×D3C2×C3⋊S412yescomplemented
507A4—23no—
508A4C2×#62513yescomplemented
509C3×C4—182no—
510Dic3—62no—
511Dic3—182no—
512Dic3—182no—
513Dic3—182no—
514Dic3—182no—
515Dic3—362no—
516Dic3—366no—
517Dic3—362no—
518Dic3—726no—
519C32—41no—
520C32—43no—
521C32—83no—
522C32—81no—
523C32C2×GL(2,ℤ/4)11yescharacteristic, complemented
524C23—31no—
525C23—31no—
526C23—32no—
527C23—31no—
528C23—62no—
529C23—92no—
530C23—92no—
531C23—92no—
532C23—94no—
533C23—92no—
534C23—91no—
535C23—182no—
536C23—182no—
537C23—182no—
538C23—181no—
539C23—184no—
540C23—182no—
541C23—182no—
542C23—182no—
543C23—181no—
544C23—184no—
545C23—181no—
546C23#629012yescomplemented
547C23C2×D3×C3⋊D311yescharacteristic
548D4—184no—
549D4—182no—
550D4—182no—
551D4—544no—
552D4—548no—
553D4—544no—
554C2×C4—92no—
555C2×C4—182no—
556C2×C4—272no—
557C2×C4—272no—
558C2×C4—542no—
559C2×C4—541no—
560C2×C4—541no—
561C2×C3—22no—
562C2×C3—21no—
563C2×C3—31no—
564C2×C3—31no—
565C2×C3—32no—
566C2×C3—31no—
567C2×C3—31no—
568C2×C3—32no—
569C2×C3—46no—
570C2×C3—43no—
571C2×C3—61no—
572C2×C3—61no—
573C2×C3—62no—
574C2×C3—62no—
575C2×C3—86no—
576C2×C3—83no—
577C2×C3—182no—
578C2×C3—246no—
579C2×C3—362no—
580C2×C3#6420912yescomplemented
581C2×C3#6426912yescomplemented
582C2×C3C2×D3×S411yescharacteristic
583C2×C3C22×C3⋊S411yescharacteristic
584D3—22no—
585D3—62no—
586D3—122no—
587D3—246no—
588C22—32no—
589C22—31no—
590C22—32no—
591C22—32no—
592C22—31no—
593C22—31no—
594C22—32no—
595C22—61no—
596C22—62no—
597C22—94no—
598C22—92no—
599C22—182no—
600C22—182no—
601C22—184no—
602C22—182no—
603C22—184no—
604C22—182no—
605C22—182no—
606C22—182no—
607C22—182no—
608C22—362no—
609C22—362no—
610C22C2×#629011yescharacteristic, complemented
611C22D3×C3⋊S411yescharacteristic
612C4—182no—
613C4—542no—
614C4—542no—
615C3—21no—
616C3—43no—
617C3—83no—
618C3C2×#6420911yescharacteristic, complemented
619C3C2×#6426911yescharacteristic, complemented
620C2—31no—
621C2—31no—
622C2—32no—
623C2—62no—
624C2—182no—
625C2—362no—
626C2#1172420412yesdirect product factor
627C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC22D3×C3⋊S4
commutator, cosocleC2×C32×A4C23
FrattiniC2C2×D3×C3⋊S4
Fitting, socleC24×C32C2×D3
hypocenterC32×A4C2×D4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×D3×A4C2semidirect
2. C22×C3×C3⋊S4C2semidirect
3. C2×C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
4. #11724204C2direct (N × H)
10. C2×C3×D3×A4C22semidirect
12. C2×C3×C3⋊S4C22semidirect
14. C3⋊(C3⋊SL(2,ℤ/4))C22semidirect
15. C24×C3×D3D3semidirect
17. C22×D3×A4D3semidirect
18. C22×C3⋊S4D3semidirect
36. C2×C32×A4D4semidirect
52. C23×C3×D3C2×D3semidirect
60. C2×D3×A4C2×D3semidirect
62. C2×C3⋊S4C2×D3semidirect
93. C32×A4C2×D4semidirect
96. C24×D3C3⋊D3semidirect
123. C23×C32#67semidirect
146. C22×C3×D3S4semidirect
152. C2×C3×A4#67semidirect
153. C2×C3×A4#67semidirect
199. C23×D3C2×C3⋊D3semidirect
260. C22×C32C2×#67semidirect
282. C2×C3×D3C2×S4semidirect
285. C3×A4C2×#67semidirect
286. C3×A4C2×#67semidirect
335. C23×C3#625semidirect
336. C23×C3#638semidirect
358. C22×D3C3⋊S4semidirect
362. C2×A4#625semidirect
411. C2×C32GL(2,ℤ/4)semidirect
486. C22×C3C2×#625semidirect
487. C22×C3C2×#638semidirect
506. C2×D3C2×C3⋊S4semidirect
508. A4C2×#625semidirect
523. C32C2×GL(2,ℤ/4)semidirect
546. C23#6290semidirect
580. C2×C3#64209semidirect
581. C2×C3#64269semidirect
610. C22C2×#6290semidirect
618. C3C2×#64209semidirect
619. C3C2×#64269semidirect
626. C2#11724204direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D3×A42
C22×C3×C3⋊S42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
#117242042
C2×#642693
C2×#642093
C2×#639763
C2×#62904