Finite Group Databasebeta

← C22×C3×(C2×D3)⋊Dic3 #13175075

Subgroups — C22×C3×(C2×D3)⋊Dic3 #13175075

Order 1728.

4050 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C32×D3C211yesmaximal, characteristic
2C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C23×C32×Dic3C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D3)⋊Dic3C2112yesmaximal, direct product factor
5C22×C3×#217—31nomaximal
6C22×C3×(C2×D3)⋊C4—31nomaximal
7C22×(C2×D3)⋊Dic3C311yesmaximal, characteristic, direct product factor
8C24×C33C2211yescharacteristic
9C23×C32×D3—24no—
10C23×C32×D3—24no—
11C23×C32×D3C418yescomplemented
12C23×C32×D3C2216yes—
13C22×C3×C3⋊Dic3—24no—
14C22×C3×C3⋊Dic3C2216yescomplemented
15C22×C32×Dic3—24no—
16C22×C32×Dic3C2216yescomplemented
17C3×(C2×D3)⋊Dic3C22116yesdirect product factor
18C25×C32—31no—
19C24×C3×D3—21no—
20C24×C3×D3D311yescharacteristic
21C24×C3×D3C2×C311yescharacteristic
22C23×C32×C4—31no—
23C23×C3⋊Dic3C2×C311yescharacteristic, complemented
24C23×C3×Dic3—21no—
25C23×C3×Dic3—31no—
26C23×C3×Dic3—31no—
27C23×C3×Dic3—61no—
28C23×C3×Dic3D311yescharacteristic, complemented
29C23×C3×Dic3C2×C311yescharacteristic, complemented
30C2×C3×#217—312no—
31C2×C3×(C2×D3)⋊C4—312no—
32C2×(C2×D3)⋊Dic3C2×C3112yesdirect product factor
33C23×C33C2×C414yes—
34C23×C33D414yes—
35C23×C33D414yes—
36C23×C33C2313yes—
37C22×C32×D3—232no—
38C22×C32×D3—212no—
39C22×C32×D3—212no—
40C22×C32×D3C2×C4124yescomplemented
41C22×C32×D3C2314yes—
42C2×C3×C3⋊Dic3—212no—
43C2×C3×C3⋊Dic3C2314yescomplemented
44C2×C32×Dic3—212no—
45C2×C32×Dic3C2314yescomplemented
46C22×C3×C22⋊C4—91no—
47C22×#217—31no—
48C22×(C2×D3)⋊C4—31no—
49C24×C32—21no—
50C24×C32—21no—
51C24×C32—21no—
52C24×C32—38no—
53C24×C32—36no—
54C24×C32—41no—
55C24×C32—64no—
56C24×C32—64no—
57C24×C32C2×D311yescharacteristic
58C24×C32C2×D311yescharacteristic
59C24×C32C22×C311yescharacteristic
60C23×C3×D3—28no—
61C23×C3×D3—28no—
62C23×C3×D3—28no—
63C23×C3×D3—26no—
64C23×C3×D3—24no—
65C23×C3×D3—24no—
66C23×C3×D3—24no—
67C23×C3×D3—24no—
68C23×C3×D3Dic318yescomplemented
69C23×C3×D3C3×C418yescomplemented
70C23×C3×D3C2×D316yes—
71C23×C3×D3C22×C316yes—
72C22×C32×C4—36no—
73C22×C32×C4—64no—
74C22×C3⋊Dic3—24no—
75C22×C3⋊Dic3C22×C316yescomplemented
76C22×C3×Dic3—28no—
77C22×C3×Dic3—26no—
78C22×C3×Dic3—24no—
79C22×C3×Dic3—24no—
80C22×C3×Dic3—36no—
81C22×C3×Dic3—36no—
82C22×C3×Dic3—66no—
83C22×C3×Dic3—68no—
84C22×C3×Dic3—64no—
85C22×C3×Dic3—64no—
86C22×C3×Dic3C2×D316yescomplemented
87C22×C3×Dic3C22×C316yescomplemented
88C3×#217—316no—
89C3×(C2×D3)⋊C4—316no—
90(C2×D3)⋊Dic3C22×C3116yesdirect product factor
91C22×C33C22⋊C4116yescomplemented
92C22×C33C22×C416yes—
93C22×C33C2×D416yes—
94C22×C33C2×D416yes—
95C22×C33C2411yescharacteristic
96C2×C32×D3—248no—
97C2×C32×D3—24no—
98C2×C32×D3—24no—
99C2×C32×D3C22×C418yescomplemented
100C3×C3⋊Dic3—24no—
101C32×Dic3—24no—
102C25×C3—31no—
103C25×C3—31no—
104C25×C3—61no—
105C24×D3C3×D311yescharacteristic
106C23×C3×C4—31no—
107C23×C3×C4—31no—
108C23×C3×C4—61no—
109C23×C3×C4—91no—
110C23×Dic3—31no—
111C23×Dic3—31no—
112C23×Dic3—61no—
113C23×Dic3C3×D311yescharacteristic, complemented
114C2×C3×C22⋊C4—912no—
115C2×#217—312no—
116C2×(C2×D3)⋊C4—312no—
117C23×C32—24no—
118C23×C32—24no—
119C23×C32—24no—
120C23×C32—24no—
121C23×C32—24no—
122C23×C32—24no—
123C23×C32—24no—
124C23×C32—24no—
125C23×C32—24no—
126C23×C32—23no—
127C23×C32—23no—
128C23×C32—23no—
129C23×C32—324no—
130C23×C32—34no—
131C23×C32—44no—
132C23×C32—44no—
133C23×C32—44no—
134C23×C32—43no—
135C23×C32—632no—
136C23×C32—612no—
137C23×C32—612no—
138C23×C32C4×D314yes—
139C23×C32D1214yes—
140C23×C32C2×Dic314yes—
141C23×C32#6714yes—
142C23×C32#6714yes—
143C23×C32#6714yes—
144C23×C32C2×C3×C414yes—
145C23×C32C3×D414yes—
146C23×C32C3×D414yes—
147C23×C32C22×D313yes—
148C23×C32C22×D313yes—
149C23×C32C23×C313yes—
150C22×C3×D3—264no—
151C22×C3×D3—224no—
152C22×C3×D3—224no—
153C22×C3×D3—232no—
154C22×C3×D3—232no—
155C22×C3×D3—212no—
156C22×C3×D3—212no—
157C22×C3×D3—212no—
158C22×C3×D3—212no—
159C22×C3×D3—224no—
160C22×C3×D3—24no—
161C22×C3×D3C2×Dic3124yescomplemented
162C22×C3×D3C2×C3×C4124yescomplemented
163C22×C3×D3C22×D314yes—
164C22×C3×D3C23×C314yes—
165C2×C32×C4—34no—
166C2×C32×C4—612no—
167C2×C3⋊Dic3—212no—
168C2×C3⋊Dic3C23×C314yescomplemented
169C2×C3×Dic3—224no—
170C2×C3×Dic3—212no—
171C2×C3×Dic3—212no—
172C2×C3×Dic3—24no—
173C2×C3×Dic3—34no—
174C2×C3×Dic3—34no—
175C2×C3×Dic3—64no—
176C2×C3×Dic3—624no—
177C2×C3×Dic3—612no—
178C2×C3×Dic3—612no—
179C2×C3×Dic3C22×D314yescomplemented
180C2×C3×Dic3C23×C314yescomplemented
181C22×C22⋊C4—91noSylow
182C2×C33C2×C22⋊C4112yescomplemented
183C2×C33C23×C411yescharacteristic
184C2×C33C22×D411yescharacteristic
185C2×C33C22×D411yescharacteristic
186C32×D3—28no—
187C24×C3—21no—
188C24×C3—21no—
189C24×C3—21no—
190C24×C3—38no—
191C24×C3—38no—
192C24×C3—36no—
193C24×C3—36no—
194C24×C3—41no—
195C24×C3—68no—
196C24×C3—66no—
197C24×C3—68no—
198C24×C3—68no—
199C24×C3—64no—
200C24×C3—64no—
201C24×C3—64no—
202C24×C3—64no—
203C24×C3D3211yescharacteristic
204C24×C3C2×C3×D311yescharacteristic
205C24×C3C2×C3×D311yescharacteristic
206C23×D3—24no—
207C23×D3—24no—
208C23×D3C3×Dic318yescomplemented
209C23×D3C2×C3×D316yes—
210C22×C3×C4—36no—
211C22×C3×C4—36no—
212C22×C3×C4—66no—
213C22×C3×C4—68no—
214C22×C3×C4—64no—
215C22×C3×C4—64no—
216C22×C3×C4—96no—
217C22×C3×C4—184no—
218C22×Dic3—24no—
219C22×Dic3—36no—
220C22×Dic3—36no—
221C22×Dic3—66no—
222C22×Dic3—68no—
223C22×Dic3—64no—
224C22×Dic3—64no—
225C22×Dic3C2×C3×D316yescomplemented
226C3×C22⋊C4—916no—
227#217—316no—
228(C2×D3)⋊C4—316no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—216no—
233C22×C32—216no—
234C22×C32—216no—
235C22×C32—26no—
236C22×C32—26no—
237C22×C32—26no—
238C22×C32—26no—
239C22×C32—26no—
240C22×C32—26no—
241C22×C32—26no—
242C22×C32—26no—
243C22×C32—26no—
244C22×C32—38no—
245C22×C32—416no—
246C22×C32—46no—
247C22×C32—46no—
248C22×C32—46no—
249C22×C32—41no—
250C22×C32—648no—
251C22×C32—64no—
252C22×C32—64no—
253C22×C32(C2×D3)⋊C4116yescomplemented
254C22×C32#217116yescomplemented
255C22×C32C3×C22⋊C4116yescomplemented
256C22×C32C2×C4×D316yes—
257C22×C32C2×D1216yes—
258C22×C32C22×Dic316yes—
259C22×C32C2×#6716yes—
260C22×C32C2×#6716yes—
261C22×C32C2×#6716yes—
262C22×C32C22×C3×C416yes—
263C22×C32C2×C3×D416yes—
264C22×C32C2×C3×D416yes—
265C22×C32C23×D311yescharacteristic
266C22×C32C23×D311yescharacteristic
267C22×C32C24×C311yescharacteristic
268C2×C3×D3—296no—
269C2×C3×D3—28no—
270C2×C3×D3—28no—
271C2×C3×D3—248no—
272C2×C3×D3—248no—
273C2×C3×D3—24no—
274C2×C3×D3—24no—
275C2×C3×D3—24no—
276C2×C3×D3—24no—
277C2×C3×D3—28no—
278C2×C3×D3C22×Dic318yescomplemented
279C2×C3×D3C22×C3×C418yescomplemented
280C32×C4—64no—
281C3⋊Dic3—24no—
282C3×Dic3—28no—
283C3×Dic3—24no—
284C3×Dic3—24no—
285C3×Dic3—68no—
286C3×Dic3—64no—
287C3×Dic3—64no—
288C25—31no—
289C23×C4—31no—
290C23×C4—91no—
291C2×C22⋊C4—912no—
292C33C22×C22⋊C411yescharacteristic, complemented, Sylow
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—23no—
303C23×C3—23no—
304C23×C3—23no—
305C23×C3—34no—
306C23×C3—34no—
307C23×C3—324no—
308C23×C3—324no—
309C23×C3—44no—
310C23×C3—44no—
311C23×C3—44no—
312C23×C3—43no—
313C23×C3—64no—
314C23×C3—624no—
315C23×C3—624no—
316C23×C3—624no—
317C23×C3—664no—
318C23×C3—632no—
319C23×C3—632no—
320C23×C3—612no—
321C23×C3—612no—
322C23×C3—612no—
323C23×C3—612no—
324C23×C3D3×Dic314yes—
325C23×C3#62514yes—
326C23×C3#62614yes—
327C23×C3C3×C4×D314yes—
328C23×C3C3×D1214yes—
329C23×C3C2×C3×Dic314yes—
330C23×C3C3×#6714yes—
331C23×C3C3×#6714yes—
332C23×C3C3×#6714yes—
333C23×C3C2×D3213yes—
334C23×C3C22×C3×D313yes—
335C23×C3C22×C3×D313yes—
336C22×D3—232no—
337C22×D3—212no—
338C22×D3—212no—
339C22×D3C2×C3×Dic3124yescomplemented
340C22×D3C22×C3×D314yes—
341C2×C3×C4—34no—
342C2×C3×C4—34no—
343C2×C3×C4—64no—
344C2×C3×C4—624no—
345C2×C3×C4—612no—
346C2×C3×C4—612no—
347C2×C3×C4—94no—
348C2×C3×C4—1812no—
349C2×Dic3—212no—
350C2×Dic3—34no—
351C2×Dic3—34no—
352C2×Dic3—64no—
353C2×Dic3—624no—
354C2×Dic3—612no—
355C2×Dic3—612no—
356C2×Dic3C22×C3×D314yescomplemented
357C2×C32—212no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—212no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—212no—
369C2×C32—412no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—68no—
374C2×C32C2×(C2×D3)⋊C4112yescomplemented
375C2×C32C2×#217112yescomplemented
376C2×C32C2×C3×C22⋊C4112yescomplemented
377C2×C32C22×C4×D311yescharacteristic
378C2×C32C22×D1211yescharacteristic
379C2×C32C23×Dic311yescharacteristic
380C2×C32C22×#6711yescharacteristic
381C2×C32C22×#6711yescharacteristic
382C2×C32C22×#6711yescharacteristic
383C2×C32C23×C3×C411yescharacteristic
384C2×C32C22×C3×D411yescharacteristic
385C2×C32C22×C3×D411yescharacteristic
386C3×D3—216no—
387C3×D3—28no—
388C3×D3—28no—
389C24—38no—
390C24—36no—
391C24—64no—
392C24—64no—
393C24C3×D3211yescharacteristic
394C22×C4—36no—
395C22×C4—64no—
396C22×C4—96no—
397C22×C4—184no—
398C22⋊C4—916no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—26no—
403C22×C3—26no—
404C22×C3—26no—
405C22×C3—26no—
406C22×C3—26no—
407C22×C3—26no—
408C22×C3—26no—
409C22×C3—26no—
410C22×C3—26no—
411C22×C3—216no—
412C22×C3—216no—
413C22×C3—216no—
414C22×C3—38no—
415C22×C3—38no—
416C22×C3—41no—
417C22×C3—46no—
418C22×C3—46no—
419C22×C3—46no—
420C22×C3—416no—
421C22×C3—68no—
422C22×C3—68no—
423C22×C3—68no—
424C22×C3—696no—
425C22×C3—648no—
426C22×C3—64no—
427C22×C3—64no—
428C22×C3—64no—
429C22×C3—64no—
430C22×C3—648no—
431C22×C3(C2×D3)⋊Dic3116yesdirect product factor
432C22×C3C3×(C2×D3)⋊C4116yescomplemented
433C22×C3C3×#217116yescomplemented
434C22×C3C2×D3×Dic316yes—
435C22×C3C2×#62516yes—
436C22×C3C2×#62616yes—
437C22×C3C2×C3×C4×D316yes—
438C22×C3C2×C3×D1216yes—
439C22×C3C22×C3×Dic316yes—
440C22×C3C2×C3×#6716yes—
441C22×C3C2×C3×#6716yes—
442C22×C3C2×C3×#6716yes—
443C22×C3C22×D3211yescharacteristic
444C22×C3C23×C3×D311yescharacteristic
445C22×C3C23×C3×D311yescharacteristic
446C2×D3—248no—
447C2×D3—24no—
448C2×D3—24no—
449C2×D3C22×C3×Dic318yescomplemented
450C3×C4—68no—
451C3×C4—64no—
452C3×C4—64no—
453C3×C4—184no—
454Dic3—24no—
455Dic3—68no—
456Dic3—64no—
457Dic3—64no—
458C32—21no—
459C32—21no—
460C32—21no—
461C32—41no—
462C32C22×(C2×D3)⋊C411yescharacteristic, complemented
463C32C22×#21711yescharacteristic, complemented
464C32C22×C3×C22⋊C411yescharacteristic, complemented
465C23—34no—
466C23—324no—
467C23—612no—
468C23—612no—
469C23—632no—
470C23C3×D3×Dic314yes—
471C23C3×#62514yes—
472C23C3×#62614yes—
473C23C2×C3×D3213yes—
474C2×C4—34no—
475C2×C4—612no—
476C2×C4—94no—
477C2×C4—1812no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—212no—
488C2×C3—212no—
489C2×C3—212no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—412no—
494C2×C3—616no—
495C2×C3—68no—
496C2×C3—68no—
497C2×C3C2×(C2×D3)⋊Dic3112yesdirect product factor
498C2×C3C2×C3×(C2×D3)⋊C4112yescomplemented
499C2×C3C2×C3×#217112yescomplemented
500C2×C3C22×D3×Dic311yescharacteristic
501C2×C3C22×#62511yescharacteristic
502C2×C3C22×#62611yescharacteristic
503C2×C3C22×C3×C4×D311yescharacteristic
504C2×C3C22×C3×D1211yescharacteristic
505C2×C3C23×C3×Dic311yescharacteristic
506C2×C3C22×C3×#6711yescharacteristic
507C2×C3C22×C3×#6711yescharacteristic
508C2×C3C22×C3×#6711yescharacteristic
509D3—28no—
510C22—38no—
511C22—64no—
512C22—64no—
513C22—648no—
514C22C3×(C2×D3)⋊Dic3116yesdirect product factor
515C22C2×C3×D3×Dic316yes—
516C22C2×C3×#62516yes—
517C22C2×C3×#62616yes—
518C22C22×C3×D3211yescharacteristic
519C4—64no—
520C4—184no—
521C3—21no—
522C3—21no—
523C3—21no—
524C3—41no—
525C3C22×(C2×D3)⋊Dic311yescharacteristic, direct product factor
526C3C22×C3×(C2×D3)⋊C411yescharacteristic, complemented
527C3C22×C3×#21711yescharacteristic, complemented
528C2—68no—
529C2C2×C3×(C2×D3)⋊Dic3112yesdirect product factor
530C2C22×C3×D3×Dic311yescharacteristic
531C2C22×C3×#62511yescharacteristic
532C2C22×C3×#62611yescharacteristic
subgroupG/H
center, hypercenterC24×C3D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
Fitting, socleC24×C33C22
prime radicalC24×C32×D3C2
cosocleC22×C32C24×C3
hypocenterC32C22×C3×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C23×C3×C3⋊Dic3C2semidirect
3. C23×C32×Dic3C2semidirect
4. C2×C3×(C2×D3)⋊Dic3C2direct (N × H)
7. C22×(C2×D3)⋊Dic3C3direct (N × H)
11. C23×C32×D3C4semidirect
14. C22×C3×C3⋊Dic3C22semidirect
16. C22×C32×Dic3C22semidirect
17. C3×(C2×D3)⋊Dic3C22direct (N × H)
23. C23×C3⋊Dic3C2×C3semidirect
28. C23×C3×Dic3D3semidirect
29. C23×C3×Dic3C2×C3semidirect
32. C2×(C2×D3)⋊Dic3C2×C3direct (N × H)
40. C22×C32×D3C2×C4semidirect
43. C2×C3×C3⋊Dic3C23semidirect
45. C2×C32×Dic3C23semidirect
68. C23×C3×D3Dic3semidirect
69. C23×C3×D3C3×C4semidirect
75. C22×C3⋊Dic3C22×C3semidirect
86. C22×C3×Dic3C2×D3semidirect
87. C22×C3×Dic3C22×C3semidirect
90. (C2×D3)⋊Dic3C22×C3direct (N × H)
91. C22×C33C22⋊C4semidirect
99. C2×C32×D3C22×C4semidirect
113. C23×Dic3C3×D3semidirect
161. C22×C3×D3C2×Dic3semidirect
162. C22×C3×D3C2×C3×C4semidirect
168. C2×C3⋊Dic3C23×C3semidirect
179. C2×C3×Dic3C22×D3semidirect
180. C2×C3×Dic3C23×C3semidirect
182. C2×C33C2×C22⋊C4semidirect
208. C23×D3C3×Dic3semidirect
225. C22×Dic3C2×C3×D3semidirect
253. C22×C32(C2×D3)⋊C4semidirect
254. C22×C32#217semidirect
255. C22×C32C3×C22⋊C4semidirect
278. C2×C3×D3C22×Dic3semidirect
279. C2×C3×D3C22×C3×C4semidirect
292. C33C22×C22⋊C4semidirect
339. C22×D3C2×C3×Dic3semidirect
356. C2×Dic3C22×C3×D3semidirect
374. C2×C32C2×(C2×D3)⋊C4semidirect
375. C2×C32C2×#217semidirect
376. C2×C32C2×C3×C22⋊C4semidirect
431. C22×C3(C2×D3)⋊Dic3direct (N × H)
432. C22×C3C3×(C2×D3)⋊C4semidirect
433. C22×C3C3×#217semidirect
449. C2×D3C22×C3×Dic3semidirect
462. C32C22×(C2×D3)⋊C4semidirect
463. C32C22×#217semidirect
464. C32C22×C3×C22⋊C4semidirect
497. C2×C3C2×(C2×D3)⋊Dic3direct (N × H)
498. C2×C3C2×C3×(C2×D3)⋊C4semidirect
499. C2×C3C2×C3×#217semidirect
514. C22C3×(C2×D3)⋊Dic3direct (N × H)
525. C3C22×(C2×D3)⋊Dic3direct (N × H)
526. C3C22×C3×(C2×D3)⋊C4semidirect
527. C3C22×C3×#217semidirect
529. C2C2×C3×(C2×D3)⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C32×D32
C23×C3×C3⋊Dic32
C23×C32×Dic32
C2×C3×(C2×D3)⋊Dic32
C22×C3×#2173
C22×C3×(C2×D3)⋊C43
C22×(C2×D3)⋊Dic33