Finite Group Databasebeta

← C2×#11724202 #13180835

Subgroups — C2×#11724202 #13180835

Order 1728.

1471 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 1471 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⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
2C22×C3×C3⋊S4C211yesmaximal, characteristic, complemented
3C2×C3×Dic3×A4C211yesmaximal, characteristic, complemented
4#11724202C214yesmaximal, direct product factor
5C2×C4⋊(C3⋊S4)—31nomaximal
6C2×#64207—33nomaximal
7C2×#63978—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×C3⋊(C3⋊S4)—22no—
10C2×C3⋊(C3⋊S4)C2212yescomplemented
11C2×C3×C3⋊S4—22no—
12C2×C3×C3⋊S4C2212yescomplemented
13C2×#6292—41nomaximal
14C3×Dic3×A4C2212yescomplemented
15C22×C3⋊S4—31no—
16C22×C3⋊S4—33no—
17C22×C3⋊S4—61no—
18C22×C3⋊S4—63no—
19C22×C3⋊S4D311yescharacteristic, complemented
20C22×C3×S4—33no—
21C22×#638—31no—
22C22×C3×#67—31no—
23C23×C3×Dic3D311yescharacteristic, complemented
24C2×C3×C4×A4—31no—
25C22×#626—31no—
26C2×Dic3×A4D313yescomplemented
27C4⋊(C3⋊S4)—34no—
28#64207—312no—
29#63978—34no—
30#63978—34no—
31C2×Dic3⋊Dic3—31no—
32C2×#3722—31no—
33C2×(C2×D3)⋊Dic3—31no—
34C22×C3⋊(C3⋊D3)—41no—
35C22×C3×C3⋊D3—41no—
36C2×C32×A4D412yescomplemented
37C2×C32×A4C2311yescharacteristic
38C3⋊(C3⋊S4)—22no—
39C3×C3⋊S4—22no—
40C2×C32×Dic3—41no—
41#6292—44no—
42C2×C4⋊S4—93no—
43C2×#939—91no—
44C2×#930—91no—
45C24×C32C2×D311yescharacteristic
46C23×C3⋊D3—31no—
47C23×C3×D3—31no—
48C22×C3×A4—21no—
49C22×C3×A4—23no—
50C22×C3×A4C2×D313yes—
51C22×C3×A4C2×D311yescharacteristic
52C2×C3⋊S4—22no—
53C2×C3⋊S4—32no—
54C2×C3⋊S4—36no—
55C2×C3⋊S4—62no—
56C2×C3⋊S4—66no—
57C2×C3⋊S4—64no—
58C2×C3⋊S4—612no—
59C2×C3⋊S4—66no—
60C2×C3⋊S4—62no—
61C2×C3⋊S4C2×D312yescomplemented
62C2×C3×S4—36no—
63C2×C3×S4—66no—
64C2×#638—32no—
65C2×#638—32no—
66C2×#638—62no—
67C2×#638—62no—
68C22×C3⋊Dic3—31no—
69C2×C4⋊(C3⋊D3)—121no—
70C2×C3×#67—32no—
71C2×C3×#67—32no—
72C2×C3×#67—62no—
73C2×C3×#67—62no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—32no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3C2×D312yescomplemented
79C3×C4×A4—32no—
80C2×#626—34no—
81C2×#626—61no—
82C2×#626—62no—
83C2×#626—61no—
84C2×#626—123no—
85Dic3×A4C2×D316yescomplemented
86Dic3⋊Dic3—34no—
87#3722—34no—
88(C2×D3)⋊Dic3—34no—
89C22×C33—41no—
90C2×C3⋊(C3⋊D3)—42no—
91C2×C3⋊(C3⋊D3)—82no—
92C2×C3×C3⋊D3—42no—
93C2×C3×C3⋊D3—82no—
94C32×A4C2×D411yescharacteristic, complemented
95C32×Dic3—42no—
96C22×S4—33no—
97C22×S4—93no—
98C22×S4—183no—
99C22×C3×D4—91no—
100C23×C3×C4—31no—
101C22×#67—31no—
102C22×#67—91no—
103C22×#67—91no—
104C22×#67—91no—
105C22×#67—181no—
106C23×Dic3C3⋊D311yescharacteristic, complemented
107C22×D12—91no—
108C2×C4×A4—33no—
109C4⋊S4—912no—
110C2×#217—91no—
111#939—94no—
112#939—94no—
113#930—94no—
114#930—94no—
115C2×(C2×D3)⋊C4—91no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×C4⋊Dic3—91no—
119C2×Dic3⋊C4—91no—
120C23×C32—31no—
121C23×C32—31no—
122C23×C32—32no—
123C23×C32D1212yescomplemented
124C23×C32C22×D311yescharacteristic
125C22×C3⋊D3—32no—
126C22×C3⋊D3—41no—
127C22×C3⋊D3—62no—
128C22×C3⋊D3—62no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—61no—
131C22×C3⋊D3—121no—
132C22×C3⋊D3—123no—
133C22×C3⋊D3—241no—
134C22×C3⋊D3—243no—
135C22×C3×D3—32no—
136C22×C3×D3—62no—
137C22×C3×D3—61no—
138C22×C3×D3—62no—
139C22×C3×D3—61no—
140C22×C3×D3—123no—
141C2×C3×A4—22no—
142C2×C3×A4—21no—
143C2×C3×A4—26no—
144C2×C3×A4—23no—
145C2×C3×A4D1216yescomplemented
146C2×C3×A4#6712yescomplemented
147C2×C3×A4C22×D313yes—
148C2×C3×A4C22×D311yescharacteristic
149C3⋊S4—22no—
150C3⋊S4—612no—
151C3⋊S4—64no—
152C3⋊S4—66no—
153C3⋊S4—62no—
154C3×S4—66no—
155C2×C32×C4—121no—
156#638—64no—
157#638—62no—
158#638—62no—
159C2×C3⋊Dic3—32no—
160C2×C3⋊Dic3—62no—
161C4⋊(C3⋊D3)—124no—
162C3×#67—64no—
163C3×#67—62no—
164C3×#67—62no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—43no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3S411yescharacteristic, complemented
174#626—64no—
175#626—64no—
176#626—1212no—
177C2×#121—271noSylow
178C2×C33—42no—
179C2×C33—41no—
180C3⋊(C3⋊D3)—82no—
181C3×C3⋊D3—82no—
182C24×C3—21no—
183C24×C3—91no—
184C24×C3D3211yescharacteristic
185C24×C3C2×C3⋊D311yescharacteristic
186C23×D3—31no—
187C23×D3—91no—
188C23×D3—91no—
189C23×D3—181no—
190C22×A4—23no—
191C22×A4D3213yes—
192C2×S4—36no—
193C2×S4—66no—
194C2×S4—96no—
195C2×S4—186no—
196C2×S4—1812no—
197C2×S4—186no—
198C2×C3×D4—92no—
199C2×C3×D4—92no—
200C2×C3×D4—182no—
201C2×C3×D4—182no—
202C22×C3×C4—32no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—92no—
206C22×C3×C4—91no—
207C2×#67—32no—
208C2×#67—32no—
209C2×#67—62no—
210C2×#67—62no—
211C2×#67—94no—
212C2×#67—92no—
213C2×#67—92no—
214C2×#67—92no—
215C2×#67—92no—
216C2×#67—182no—
217C2×#67—182no—
218C2×#67—181no—
219C2×#67—182no—
220C2×#67—184no—
221C2×#67—182no—
222C2×#67—182no—
223C2×#67—184no—
224C2×#67—182no—
225C2×#67—182no—
226C2×#67—181no—
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—32no—
230C22×Dic3—31no—
231C22×Dic3—91no—
232C22×Dic3—91no—
233C22×Dic3—181no—
234C22×Dic3C2×C3⋊D312yescomplemented
235C2×D12—94no—
236C2×D12—181no—
237C2×D12—182no—
238C2×D12—181no—
239C2×D12—363no—
240C4×A4—36no—
241#217—94no—
242(C2×D3)⋊C4—94no—
243(C2×D3)⋊C4—94no—
244(C2×D3)⋊C4—94no—
245C4⋊Dic3—94no—
246Dic3⋊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×C32C2×D1211yescharacteristic, complemented
259C22×C32C2×S411yescharacteristic
260C2×C3⋊D3—42no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—62no—
263C2×C3⋊D3—62no—
264C2×C3⋊D3—82no—
265C2×C3⋊D3—122no—
266C2×C3⋊D3—126no—
267C2×C3⋊D3—122no—
268C2×C3⋊D3—122no—
269C2×C3⋊D3—242no—
270C2×C3⋊D3—246no—
271C2×C3⋊D3—246no—
272C2×C3⋊D3—242no—
273C2×C3⋊D3—2412no—
274C2×C3⋊D3—244no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—62no—
278C2×C3×D3—126no—
279C2×C3×D3—122no—
280C2×C3×D3—122no—
281C2×C3×D3—246no—
282C3×A4—21no—
283C3×A4—23no—
284C3×A4C2×D1213yescomplemented
285C3×A4C2×#6711yescharacteristic, complemented
286C32×C4—122no—
287C3⋊Dic3—62no—
288C3×Dic3—32no—
289C3×Dic3—46no—
290C3×Dic3—62no—
291C3×Dic3C2×S412yescomplemented
292C22×D4—91no—
293C22×D4—271no—
294C22×D4—271no—
295C23×C4—31no—
296#121—274no—
297#121—274no—
298C2×C4⋊C4—271no—
299C2×C22⋊C4—271no—
300C2×C22⋊C4—271no—
301C33—41noSylow
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—31no—
305C23×C3—31no—
306C23×C3—31no—
307C23×C3—31no—
308C23×C3—32no—
309C23×C3—32no—
310C23×C3—61no—
311C23×C3—61no—
312C23×C3—62no—
313C23×C3—92no—
314C23×C3—181no—
315C23×C3—182no—
316C23×C3—182no—
317C23×C3—181no—
318C23×C3#62612yescomplemented
319C23×C3C4⋊(C3⋊D3)12yescomplemented
320C23×C3C2×D3211yescharacteristic
321C23×C3C22×C3⋊D311yescharacteristic
322C22×D3—32no—
323C22×D3—61no—
324C22×D3—62no—
325C22×D3—62no—
326C22×D3—61no—
327C22×D3—92no—
328C22×D3—92no—
329C22×D3—123no—
330C22×D3—182no—
331C22×D3—181no—
332C22×D3—182no—
333C22×D3—182no—
334C22×D3—181no—
335C22×D3—182no—
336C22×D3—182no—
337C22×D3—184no—
338C22×D3—181no—
339C22×D3—181no—
340C22×D3—361no—
341C22×D3—361no—
342C22×D3—362no—
343C22×D3—363no—
344C22×D3—723no—
345C2×A4—26no—
346C2×A4—23no—
347C2×A4#62616yescomplemented
348C2×A4C2×D3213yes—
349S4—66no—
350S4—1812no—
351S4—186no—
352C3×D4—184no—
353C3×D4—182no—
354C3×D4—182no—
355C2×C3×C4—31no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—91no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4—123no—
362C2×C3×C4—181no—
363C2×C3×C4—182no—
364#67—64no—
365#67—62no—
366#67—62no—
367#67—184no—
368#67—184no—
369#67—184no—
370#67—182no—
371#67—184no—
372#67—182no—
373#67—184no—
374#67—188no—
375#67—182no—
376#67—182no—
377#67—184no—
378C2×Dic3—31no—
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—32no—
382C2×Dic3—32no—
383C2×Dic3—61no—
384C2×Dic3—62no—
385C2×Dic3—92no—
386C2×Dic3—92no—
387C2×Dic3—182no—
388C2×Dic3—182no—
389C2×Dic3—182no—
390C2×Dic3—184no—
391C2×Dic3C3⋊S411yescharacteristic, complemented
392D12—184no—
393D12—184no—
394D12—3612no—
395C2×C32—31no—
396C2×C32—31no—
397C2×C32—32no—
398C2×C32—41no—
399C2×C32—42no—
400C2×C32—46no—
401C2×C32—43no—
402C2×C32—86no—
403C2×C32—83no—
404C2×C32—81no—
405C2×C32—82no—
406C2×C32C4⋊S412yescomplemented
407C2×C32C22×S411yescharacteristic
408C3⋊D3—82no—
409C3⋊D3—122no—
410C3⋊D3—242no—
411C3⋊D3—246no—
412C3⋊D3—2412no—
413C3⋊D3—244no—
414C3×D3—122no—
415C3×D3—246no—
416C24—91no—
417C24—271no—
418C24D3×C3⋊D311yescharacteristic
419C2×D4—92no—
420C2×D4—92no—
421C2×D4—182no—
422C2×D4—182no—
423C2×D4—274no—
424C2×D4—272no—
425C2×D4—272no—
426C2×D4—541no—
427C2×D4—542no—
428C2×D4—541no—
429C2×D4—542no—
430C2×D4—542no—
431C22×C4—32no—
432C22×C4—91no—
433C22×C4—91no—
434C22×C4—92no—
435C22×C4—91no—
436C22×C4—271no—
437C4⋊C4—274no—
438C22⋊C4—274no—
439C22⋊C4—274no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—32no—
443C22×C3—31no—
444C22×C3—32no—
445C22×C3—31no—
446C22×C3—32no—
447C22×C3—32no—
448C22×C3—31no—
449C22×C3—31no—
450C22×C3—31no—
451C22×C3—31no—
452C22×C3—32no—
453C22×C3—32no—
454C22×C3—43no—
455C22×C3—62no—
456C22×C3—61no—
457C22×C3—62no—
458C22×C3—61no—
459C22×C3—62no—
460C22×C3—61no—
461C22×C3—61no—
462C22×C3—61no—
463C22×C3—83no—
464C22×C3—121no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3—182no—
468C22×C3—362no—
469C22×C3—362no—
470C22×C3C2×#62611yescharacteristic, complemented
471C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic, complemented
472C22×C3D3×S411yescharacteristic
473C22×C3C2×C3⋊S411yescharacteristic
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—126no—
478C2×D3—122no—
479C2×D3—122no—
480C2×D3—182no—
481C2×D3—182no—
482C2×D3—182no—
483C2×D3—182no—
484C2×D3—184no—
485C2×D3—182no—
486C2×D3—182no—
487C2×D3—246no—
488C2×D3—362no—
489C2×D3—362no—
490C2×D3—366no—
491C2×D3—364no—
492C2×D3—362no—
493C2×D3—362no—
494C2×D3—364no—
495C2×D3—362no—
496C2×D3—362no—
497C2×D3—726no—
498C2×D3—726no—
499C2×D3—7212no—
500A4—23no—
501A4C2×#62613yescomplemented
502C3×C4—32no—
503C3×C4—92no—
504C3×C4—126no—
505C3×C4—182no—
506Dic3—32no—
507Dic3—62no—
508Dic3—182no—
509Dic3—182no—
510Dic3—184no—
511Dic3C2×C3⋊S412yescomplemented
512C32—41no—
513C32—43no—
514C32—83no—
515C32—81no—
516C32C2×C4⋊S411yescharacteristic, complemented
517C23—31no—
518C23—31no—
519C23—32no—
520C23—92no—
521C23—182no—
522C23—181no—
523C23—182no—
524C23—181no—
525C23—272no—
526C23—542no—
527C23—541no—
528C23—542no—
529C23—541no—
530C23#629212yescomplemented
531C23C2×D3×C3⋊D311yescharacteristic
532D4—184no—
533D4—182no—
534D4—182no—
535D4—544no—
536D4—544no—
537D4—544no—
538D4—542no—
539D4—542no—
540C2×C4—31no—
541C2×C4—92no—
542C2×C4—92no—
543C2×C4—91no—
544C2×C4—92no—
545C2×C4—92no—
546C2×C4—181no—
547C2×C4—182no—
548C2×C4—272no—
549C2×C4—542no—
550C2×C3—22no—
551C2×C3—21no—
552C2×C3—31no—
553C2×C3—31no—
554C2×C3—32no—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—32no—
558C2×C3—46no—
559C2×C3—43no—
560C2×C3—61no—
561C2×C3—61no—
562C2×C3—62no—
563C2×C3—86no—
564C2×C3—83no—
565C2×C3—362no—
566C2×C3#6420712yescomplemented
567C2×C3C4⋊(C3⋊S4)12yescomplemented
568C2×C3C2×D3×S411yescharacteristic
569C2×C3C22×C3⋊S411yescharacteristic
570D3—122no—
571D3—246no—
572D3—364no—
573D3—362no—
574D3—362no—
575D3—726no—
576D3—7212no—
577C22—32no—
578C22—31no—
579C22—32no—
580C22—32no—
581C22—31no—
582C22—31no—
583C22—61no—
584C22—182no—
585C22—182no—
586C22—182no—
587C22—362no—
588C22—362no—
589C22—542no—
590C22—542no—
591C22—542no—
592C22—1082no—
593C22—1082no—
594C22C2×#629211yescharacteristic, complemented
595C22D3×C3⋊S411yescharacteristic
596C4—32no—
597C4—92no—
598C4—182no—
599C4—542no—
600C3—21no—
601C3—43no—
602C3—83no—
603C3C2×#6420711yescharacteristic, complemented
604C3C2×C4⋊(C3⋊S4)11yescharacteristic, complemented
605C2—31no—
606C2—31no—
607C2—32no—
608C2—362no—
609C2—1082no—
610C2#1172420212yesdirect product factor
611C2C2×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⋊(C3⋊S4)C2semidirect
2. C22×C3×C3⋊S4C2semidirect
3. C2×C3×Dic3×A4C2semidirect
4. #11724202C2direct (N × H)
10. C2×C3⋊(C3⋊S4)C22semidirect
12. C2×C3×C3⋊S4C22semidirect
14. C3×Dic3×A4C22semidirect
19. C22×C3⋊S4D3semidirect
23. C23×C3×Dic3D3semidirect
26. C2×Dic3×A4D3semidirect
36. C2×C32×A4D4semidirect
61. C2×C3⋊S4C2×D3semidirect
78. C22×C3×Dic3C2×D3semidirect
85. Dic3×A4C2×D3semidirect
94. C32×A4C2×D4semidirect
106. C23×Dic3C3⋊D3semidirect
123. C23×C32D12semidirect
145. C2×C3×A4D12semidirect
146. C2×C3×A4#67semidirect
173. C2×C3×Dic3S4semidirect
234. C22×Dic3C2×C3⋊D3semidirect
258. C22×C32C2×D12semidirect
284. C3×A4C2×D12semidirect
285. C3×A4C2×#67semidirect
291. C3×Dic3C2×S4semidirect
318. C23×C3#626semidirect
319. C23×C3C4⋊(C3⋊D3)semidirect
347. C2×A4#626semidirect
391. C2×Dic3C3⋊S4semidirect
406. C2×C32C4⋊S4semidirect
470. C22×C3C2×#626semidirect
471. C22×C3C2×C4⋊(C3⋊D3)semidirect
501. A4C2×#626semidirect
511. Dic3C2×C3⋊S4semidirect
516. C32C2×C4⋊S4semidirect
530. C23#6292semidirect
566. C2×C3#64207semidirect
567. C2×C3C4⋊(C3⋊S4)semidirect
594. C22C2×#6292semidirect
603. C3C2×#64207semidirect
604. C3C2×C4⋊(C3⋊S4)semidirect
610. C2#11724202direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊S4)2
C22×C3×C3⋊S42
C2×C3×Dic3×A42
#117242022
C2×C4⋊(C3⋊S4)3
C2×#642073
C2×#639783
C2×#62924