Finite Group Databasebeta

← C22×D3×C3⋊S4 #13181832

Subgroups — C22×D3×C3⋊S4 #13181832

Order 1728.

3663 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 3663 conjugacy classes of subgroups — too many to draw. Only its 230 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⋊S4)C211yesmaximal, characteristic, complemented
3C22×C3×C3⋊S4C211yesmaximal, characteristic, complemented
4C2×D3×C3⋊S4C2112yesmaximal, direct product factor
5C23×C3⋊S4—31nomaximal
6C22×D3×S4—33nomaximal
7C22×D3×#67—31nomaximal
8C22×C32×A4C2211yescharacteristic, complemented
9C22×D3×C3⋊D3—41nomaximal
10C2×C3×D3×A4C2216yescomplemented
11C2×C3⋊(C3⋊S4)C2216yescomplemented
12C2×C3×C3⋊S4C2216yescomplemented
13D3×C3⋊S4C22116yesdirect product factor
14C24×C3×D3D311yescharacteristic, complemented
15C23×C3×A4—31no—
16C23×D32—31no—
17C22×D3×A4D313yescomplemented
18C22×C3⋊S4—31no—
19C22×C3⋊S4—33no—
20C22×C3⋊S4—312no—
21C22×C3⋊S4—61no—
22C22×C3⋊S4—63no—
23C22×C3⋊S4D311yescharacteristic, direct product factor
24C22×C3×S4—33no—
25C2×D3×S4—336no—
26C22×#638—31no—
27C22×C3×#67—31no—
28C2×D3×#67—312no—
29C2×D3×#67—312no—
30C22×#626—31no—
31C22×#625—31no—
32C22×D3×Dic3—31no—
33C22×C3⋊(C3⋊D3)—41no—
34C22×C3×C3⋊D3—41no—
35C22×C32×D3—41no—
36C2×C32×A4C2313yescomplemented
37C2×D3×C3⋊D3—412no—
38C3×D3×A4C2314yescomplemented
39C3⋊(C3⋊S4)C2314yescomplemented
40C3×C3⋊S4C2314yescomplemented
41C23×S4—93no—
42C23×#67—91no—
43C22×D3×D4—91no—
44C24×C32C2×D311yescharacteristic, complemented
45C23×C3⋊D3—31no—
46C23×C3⋊D3—121no—
47C23×C3×D3—31no—
48C23×C3×D3—31no—
49C23×C3×D3—36no—
50C23×C3×D3—31no—
51C23×C3×D3C2×D316yescomplemented
52C22×C3×A4—21no—
53C22×C3×A4—23no—
54C22×C3×A4—36no—
55C22×C3×A4C2×D313yescomplemented
56C22×C3×A4C2×D311yescharacteristic, complemented
57C22×D32—312no—
58C22×D32—61no—
59C22×D32—66no—
60C22×D32—61no—
61C22×D32—123no—
62C2×D3×A4C2×D3118yescomplemented
63C2×C3⋊S4—36no—
64C2×C3⋊S4—318no—
65C2×C3⋊S4—316no—
66C2×C3⋊S4—66no—
67C2×C3⋊S4—618no—
68C2×C3⋊S4C2×D316yesdirect product factor
69C2×C3×S4—318no—
70D3×S4—348no—
71C2×#638—36no—
72C2×#638—36no—
73C22×C3⋊Dic3—31no—
74C2×C3×#67—36no—
75C2×C3×#67—36no—
76C22×C3×Dic3—31no—
77D3×#67—348no—
78D3×#67—316no—
79C2×#626—312no—
80C2×#625—312no—
81C2×D3×Dic3—312no—
82C22×C33—41no—
83C2×C3⋊(C3⋊D3)—46no—
84C2×C3×C3⋊D3—46no—
85C2×C32×D3—46no—
86C32×A4C2411yescharacteristic, complemented
87D3×C3⋊D3—416no—
88C25×C3—31no—
89C24×D3—91no—
90C24×D3—91no—
91C24×D3C3⋊D311yescharacteristic, complemented
92C23×A4—33no—
93C22×S4—33no—
94C22×S4—936no—
95C22×S4—93no—
96C22×S4—183no—
97C22×C3×D4—91no—
98C22×#67—31no—
99C22×#67—91no—
100C22×#67—912no—
101C22×#67—912no—
102C22×#67—91no—
103C22×#67—91no—
104C22×#67—91no—
105C22×#67—91no—
106C22×#67—181no—
107C23×Dic3—91no—
108C2×D3×D4—912no—
109C2×D3×D4—912no—
110C22×D12—91no—
111C22×C4×D3—91no—
112C23×C32—31no—
113C23×C32—33no—
114C23×C32—121no—
115C23×C32C22×D313yescomplemented
116C22×C3⋊D3—36no—
117C22×C3⋊D3—41no—
118C22×C3⋊D3—63no—
119C22×C3⋊D3—61no—
120C22×C3⋊D3—1212no—
121C22×C3⋊D3—121no—
122C22×C3⋊D3—123no—
123C22×C3⋊D3—241no—
124C22×C3⋊D3—243no—
125C22×C3×D3—31no—
126C22×C3×D3—36no—
127C22×C3×D3—36no—
128C22×C3×D3—36no—
129C22×C3×D3—312no—
130C22×C3×D3—36no—
131C22×C3×D3—43no—
132C22×C3×D3—64no—
133C22×C3×D3—63no—
134C22×C3×D3—63no—
135C22×C3×D3—61no—
136C22×C3×D3—123no—
137C22×C3×D3S411yescharacteristic, complemented
138C22×C3×D3C22×D314yescomplemented
139C2×C3×A4—23no—
140C2×C3×A4—29no—
141C2×C3×A4—34no—
142C2×C3×A4C22×D319yescomplemented
143C2×C3×A4C22×D313yescomplemented
144C2×D32—316no—
145C2×D32—612no—
146C2×D32—624no—
147C2×D32—612no—
148C2×D32—1236no—
149D3×A4C22×D3112yescomplemented
150C3⋊S4—34no—
151C3⋊S4—312no—
152C3⋊S4—64no—
153C3⋊S4—612no—
154C3⋊S4C22×D314yesdirect product factor
155C3×S4—312no—
156#638—312no—
157#638—34no—
158C2×C3⋊Dic3—36no—
159C3×#67—312no—
160C3×#67—34no—
161C2×C3×Dic3—36no—
162#626—316no—
163#625—316no—
164D3×Dic3—316no—
165C23×D4—271noSylow
166C2×C33—43no—
167C3⋊(C3⋊D3)—44no—
168C3×C3⋊D3—44no—
169C32×D3—44no—
170C24×C3—21no—
171C24×C3—36no—
172C24×C3—91no—
173C24×C3—91no—
174C24×C3—96no—
175C24×C3—91no—
176C24×C3D3211yescharacteristic, complemented
177C24×C3C2×C3⋊D311yescharacteristic, complemented
178C23×D3—31no—
179C23×D3—31no—
180C23×D3—36no—
181C23×D3—31no—
182C23×D3—912no—
183C23×D3—912no—
184C23×D3—91no—
185C23×D3—91no—
186C23×D3—181no—
187C23×D3—181no—
188C23×D3—186no—
189C23×D3—181no—
190C23×D3—186no—
191C23×D3—181no—
192C23×D3—181no—
193C23×D3—363no—
194C23×D3C2×C3⋊D316yescomplemented
195C22×A4—23no—
196C22×A4—318no—
197C22×A4D3213yescomplemented
198C2×S4—318no—
199C2×S4—948no—
200C2×S4—918no—
201C2×S4—1818no—
202C2×C3×D4—96no—
203C2×C3×D4—96no—
204C22×C3×C4—91no—
205C2×#67—36no—
206C2×#67—36no—
207C2×#67—912no—
208C2×#67—916no—
209C2×#67—948no—
210C2×#67—912no—
211C2×#67—912no—
212C2×#67—96no—
213C2×#67—96no—
214C2×#67—96no—
215C2×#67—96no—
216C2×#67—186no—
217C2×#67—186no—
218C22×Dic3—31no—
219C22×Dic3—912no—
220C22×Dic3—91no—
221C22×Dic3—91no—
222C22×Dic3—181no—
223D3×D4—948no—
224D3×D4—916no—
225C2×D12—912no—
226C2×C4×D3—912no—
227C22×C32—33no—
228C22×C32—33no—
229C22×C32—33no—
230C22×C32—41no—
231C22×C32—43no—
232C22×C32—61no—
233C22×C32—83no—
234C22×C32—81no—
235C22×C32—126no—
236C22×C32C2×S411yescharacteristic, complemented
237C22×C32C23×D311yescharacteristic, complemented
238C2×C3⋊D3—34no—
239C2×C3⋊D3—46no—
240C2×C3⋊D3—66no—
241C2×C3⋊D3—66no—
242C2×C3⋊D3—1216no—
243C2×C3⋊D3—126no—
244C2×C3⋊D3—1218no—
245C2×C3⋊D3—246no—
246C2×C3⋊D3—2418no—
247C2×C3×D3—36no—
248C2×C3×D3—34no—
249C2×C3×D3—312no—
250C2×C3×D3—34no—
251C2×C3×D3—34no—
252C2×C3×D3—418no—
253C2×C3×D3—66no—
254C2×C3×D3—66no—
255C2×C3×D3—66no—
256C2×C3×D3—1218no—
257C2×C3×D3C2×S416yescomplemented
258C3×A4—21no—
259C3×A4—23no—
260C3×A4C23×D313yescomplemented
261C3×A4C23×D311yescharacteristic, complemented
262D32—616no—
263D32—616no—
264D32—1248no—
265C3⋊Dic3—34no—
266C3×Dic3—34no—
267C25—31no—
268C25—271no—
269C22×D4—91no—
270C22×D4—271no—
271C22×D4—2712no—
272C22×D4—2712no—
273C22×D4—271no—
274C22×D4—271no—
275C23×C4—271no—
276C33—41noSylow
277C23×C3—23no—
278C23×C3—31no—
279C23×C3—31no—
280C23×C3—31no—
281C23×C3—33no—
282C23×C3—33no—
283C23×C3—34no—
284C23×C3—61no—
285C23×C3—63no—
286C23×C3—96no—
287C23×C3—96no—
288C23×C3—912no—
289C23×C3—96no—
290C23×C3—96no—
291C23×C3—91no—
292C23×C3—123no—
293C23×C3—183no—
294C23×C3—183no—
295C23×C3—184no—
296C23×C3—181no—
297C23×C3C2×D3213yescomplemented
298C23×C3C22×C3⋊D313yescomplemented
299C22×D3—31no—
300C22×D3—36no—
301C22×D3—36no—
302C22×D3—36no—
303C22×D3—312no—
304C22×D3—36no—
305C22×D3—64no—
306C22×D3—63no—
307C22×D3—63no—
308C22×D3—61no—
309C22×D3—916no—
310C22×D3—916no—
311C22×D3—96no—
312C22×D3—96no—
313C22×D3—123no—
314C22×D3—186no—
315C22×D3—1812no—
316C22×D3—1824no—
317C22×D3—1812no—
318C22×D3—1824no—
319C22×D3—1812no—
320C22×D3—183no—
321C22×D3—183no—
322C22×D3—181no—
323C22×D3—181no—
324C22×D3—1812no—
325C22×D3—3636no—
326C22×D3—361no—
327C22×D3—363no—
328C22×D3—363no—
329C22×D3—723no—
330C22×D3C3⋊S411yescharacteristic, direct product factor
331C22×D3C22×C3⋊D314yescomplemented
332C2×A4—29no—
333C2×A4—312no—
334C2×A4C2×D3219yescomplemented
335S4—312no—
336S4—912no—
337S4—1812no—
338C3×D4—912no—
339C3×D4—94no—
340C2×C3×C4—96no—
341#67—312no—
342#67—34no—
343#67—916no—
344#67—916no—
345#67—916no—
346#67—912no—
347#67—912no—
348#67—94no—
349#67—94no—
350#67—184no—
351#67—1812no—
352C2×Dic3—36no—
353C2×Dic3—916no—
354C2×Dic3—96no—
355C2×Dic3—96no—
356C2×Dic3—186no—
357D12—916no—
358C4×D3—916no—
359C2×C32—31no—
360C2×C32—33no—
361C2×C32—43no—
362C2×C32—49no—
363C2×C32—89no—
364C2×C32—83no—
365C2×C32—124no—
366C2×C32C22×S413yescomplemented
367C3⋊D3—44no—
368C3⋊D3—64no—
369C3⋊D3—124no—
370C3⋊D3—1212no—
371C3⋊D3—244no—
372C3⋊D3—2412no—
373C3×D3—34no—
374C3×D3—412no—
375C3×D3—64no—
376C3×D3—1212no—
377C3×D3C22×S414yescomplemented
378C24—36no—
379C24—91no—
380C24—91no—
381C24—96no—
382C24—91no—
383C24—271no—
384C24—2712no—
385C24—541no—
386C24—546no—
387C24—541no—
388C24D3×C3⋊D311yescharacteristic, complemented
389C2×D4—96no—
390C2×D4—96no—
391C2×D4—2712no—
392C2×D4—2716no—
393C2×D4—2748no—
394C2×D4—276no—
395C2×D4—2712no—
396C2×D4—276no—
397C22×C4—91no—
398C22×C4—2712no—
399C22×C4—271no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—33no—
403C22×C3—33no—
404C22×C3—33no—
405C22×C3—33no—
406C22×C3—36no—
407C22×C3—33no—
408C22×C3—33no—
409C22×C3—43no—
410C22×C3—63no—
411C22×C3—63no—
412C22×C3—63no—
413C22×C3—61no—
414C22×C3—61no—
415C22×C3—83no—
416C22×C3—94no—
417C22×C3—912no—
418C22×C3—94no—
419C22×C3—94no—
420C22×C3—96no—
421C22×C3—1218no—
422C22×C3—121no—
423C22×C3—186no—
424C22×C3—186no—
425C22×C3—186no—
426C22×C3D3×S411yescharacteristic, complemented
427C22×C3C2×C3⋊S411yescharacteristic, complemented
428C22×C3C22×D3211yescharacteristic, complemented
429C22×C3C23×C3⋊D311yescharacteristic, complemented
430C2×D3—36no—
431C2×D3—34no—
432C2×D3—312no—
433C2×D3—34no—
434C2×D3—34no—
435C2×D3—66no—
436C2×D3—66no—
437C2×D3—66no—
438C2×D3—94no—
439C2×D3—94no—
440C2×D3—1218no—
441C2×D3—184no—
442C2×D3—1816no—
443C2×D3—1816no—
444C2×D3—1816no—
445C2×D3—186no—
446C2×D3—186no—
447C2×D3—186no—
448C2×D3—186no—
449C2×D3—1816no—
450C2×D3—3648no—
451C2×D3—366no—
452C2×D3—3618no—
453C2×D3—366no—
454C2×D3—7218no—
455C2×D3C2×C3⋊S416yesdirect product factor
456A4—23no—
457A4C22×D3213yescomplemented
458C3×C4—94no—
459Dic3—34no—
460Dic3—94no—
461Dic3—94no—
462Dic3—184no—
463C32—41no—
464C32—43no—
465C32—83no—
466C32—81no—
467C32C23×S411yescharacteristic, complemented
468C23—31no—
469C23—31no—
470C23—33no—
471C23—34no—
472C23—96no—
473C23—96no—
474C23—912no—
475C23—96no—
476C23—96no—
477C23—91no—
478C23—183no—
479C23—183no—
480C23—184no—
481C23—181no—
482C23—276no—
483C23—2716no—
484C23—5412no—
485C23—543no—
486C23—5424no—
487C23—5412no—
488C23—541no—
489C23C2×D3×C3⋊D313yescomplemented
490D4—912no—
491D4—94no—
492D4—2716no—
493D4—2712no—
494D4—2716no—
495D4—274no—
496C2×C4—96no—
497C2×C4—2716no—
498C2×C4—276no—
499C2×C3—23no—
500C2×C3—31no—
501C2×C3—33no—
502C2×C3—34no—
503C2×C3—31no—
504C2×C3—33no—
505C2×C3—49no—
506C2×C3—61no—
507C2×C3—63no—
508C2×C3—89no—
509C2×C3—94no—
510C2×C3—1212no—
511C2×C3—184no—
512C2×C3C2×D3×S413yescomplemented
513C2×C3C22×C3⋊S413yescomplemented
514D3—34no—
515D3—64no—
516D3—1212no—
517D3—184no—
518D3—184no—
519D3—364no—
520D3—3612no—
521D3—7212no—
522D3C22×C3⋊S414yesdirect product factor
523C22—33no—
524C22—33no—
525C22—33no—
526C22—36no—
527C22—61no—
528C22—94no—
529C22—912no—
530C22—94no—
531C22—94no—
532C22—96no—
533C22—186no—
534C22—186no—
535C22—186no—
536C22—274no—
537C22—5416no—
538C22—546no—
539C22—5416no—
540C22—546no—
541C22D3×C3⋊S411yescharacteristic, direct product factor
542C22C22×D3×C3⋊D311yescharacteristic, complemented
543C4—94no—
544C4—274no—
545C3—21no—
546C3—43no—
547C3—83no—
548C3C22×D3×S411yescharacteristic, complemented
549C3C23×C3⋊S411yescharacteristic, complemented
550C2—31no—
551C2—33no—
552C2—34no—
553C2—94no—
554C2—184no—
555C2—544no—
556C2C2×D3×C3⋊S413yesdirect product factor
subgroupG/H
center, hypercenterC22D3×C3⋊S4
commutator, cosocle, hypocenterC32×A4C24
FrattiniC1G
Fitting, socleC24×C32C2×D3
LCSG ▷ 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⋊S4)C2semidirect
3. C22×C3×C3⋊S4C2semidirect
4. C2×D3×C3⋊S4C2direct (N × H)
8. C22×C32×A4C22semidirect
10. C2×C3×D3×A4C22semidirect
11. C2×C3⋊(C3⋊S4)C22semidirect
12. C2×C3×C3⋊S4C22semidirect
13. D3×C3⋊S4C22direct (N × H)
14. C24×C3×D3D3semidirect
17. C22×D3×A4D3semidirect
23. C22×C3⋊S4D3direct (N × H)
36. C2×C32×A4C23semidirect
38. C3×D3×A4C23semidirect
39. C3⋊(C3⋊S4)C23semidirect
40. C3×C3⋊S4C23semidirect
44. C24×C32C2×D3semidirect
51. C23×C3×D3C2×D3semidirect
55. C22×C3×A4C2×D3semidirect
56. C22×C3×A4C2×D3semidirect
62. C2×D3×A4C2×D3semidirect
68. C2×C3⋊S4C2×D3direct (N × H)
86. C32×A4C24semidirect
91. C24×D3C3⋊D3semidirect
115. C23×C32C22×D3semidirect
137. C22×C3×D3S4semidirect
138. C22×C3×D3C22×D3semidirect
142. C2×C3×A4C22×D3semidirect
143. C2×C3×A4C22×D3semidirect
149. D3×A4C22×D3semidirect
154. C3⋊S4C22×D3direct (N × H)
176. C24×C3D32semidirect
177. C24×C3C2×C3⋊D3semidirect
194. C23×D3C2×C3⋊D3semidirect
197. C22×A4D32semidirect
236. C22×C32C2×S4semidirect
237. C22×C32C23×D3semidirect
257. C2×C3×D3C2×S4semidirect
260. C3×A4C23×D3semidirect
261. C3×A4C23×D3semidirect
297. C23×C3C2×D32semidirect
298. C23×C3C22×C3⋊D3semidirect
330. C22×D3C3⋊S4direct (N × H)
331. C22×D3C22×C3⋊D3semidirect
334. C2×A4C2×D32semidirect
366. C2×C32C22×S4semidirect
377. C3×D3C22×S4semidirect
388. C24D3×C3⋊D3semidirect
426. C22×C3D3×S4semidirect
427. C22×C3C2×C3⋊S4semidirect
428. C22×C3C22×D32semidirect
429. C22×C3C23×C3⋊D3semidirect
455. C2×D3C2×C3⋊S4direct (N × H)
457. A4C22×D32semidirect
467. C32C23×S4semidirect
489. C23C2×D3×C3⋊D3semidirect
512. C2×C3C2×D3×S4semidirect
513. C2×C3C22×C3⋊S4semidirect
522. D3C22×C3⋊S4direct (N × H)
541. C22D3×C3⋊S4direct (N × H)
542. C22C22×D3×C3⋊D3semidirect
548. C3C22×D3×S4semidirect
549. C3C23×C3⋊S4semidirect
556. C2C2×D3×C3⋊S4direct (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⋊S4)2
C22×C3×C3⋊S42
C2×D3×C3⋊S42
C23×C3⋊S43
C22×D3×S43
C22×D3×#673
C22×D3×C3⋊D34