Finite Group Databasebeta

← C3×C4×Dic3⋊Dic3 #13158251

Subgroups — C3×C4×Dic3⋊Dic3 #13158251

Order 1728.

1154 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×C4×Dic3C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4C2×C3×Dic3⋊Dic3C212yesmaximal
5C3×#63579C212yesmaximal
6C3×C4×C4⋊Dic3—31nomaximal
7C3×C4×Dic3⋊C4—31nomaximal
8C4×Dic3⋊Dic3C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2211yescharacteristic
14C3×C4×C3⋊Dic3—22no—
15C32×C4×Dic3—22no—
16C32×C4×Dic3C414yescomplemented
17C3×Dic3⋊Dic3C418yesdirect product factor
18C2×C32×C42—31no—
19C2×C32×C42—31no—
20C2×C4×C3⋊Dic3C2×C311yescharacteristic
21C2×C3×C4⋊Dic3—32no—
22C2×C3×Dic3⋊C4—32no—
23C2×C3×C4×Dic3—21no—
24C2×C3×C4×Dic3—21no—
25C2×C3×C4×Dic3—31no—
26C2×C3×C4×Dic3—31no—
27C2×C3×C4×Dic3—61no—
28C2×C3×C4×Dic3D311yescharacteristic
29C2×C3×C4×Dic3D311yescharacteristic
30C2×C3×C4×Dic3C2×C311yescharacteristic
31C2×C3×C4×Dic3C2×C311yescharacteristic
32C2×Dic3⋊Dic3C2×C312yes—
33C3×C3⋊#95—32no—
34C3×C3⋊#95—32no—
35#63579C2×C312yes—
36C23×C33C2311yescharacteristic
37C2×C33×C4C2×C412yes—
38C2×C33×C4D412yes—
39C2×C33×C4Q812yes—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3C2×C414yes—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3C2×C414yes—
47C2×C32×Dic3C2×C412yes—
48C2×C32×Dic3C2×C414yes—
49C2×C32×Dic3C2×C412yes—
50C3×C4×C4⋊C4—91no—
51C4×C4⋊Dic3—31no—
52C4×Dic3⋊C4—31no—
53C22×C32×C4—21no—
54C22×C32×C4—21no—
55C22×C32×C4—21no—
56C22×C32×C4—31no—
57C22×C32×C4—32no—
58C22×C32×C4—31no—
59C22×C32×C4—41no—
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C22×C311yescharacteristic
63C22×C3⋊Dic3C22×C312yes—
64C22×C3×Dic3—22no—
65C22×C3×Dic3—21no—
66C22×C3×Dic3—21no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—62no—
70C22×C3×Dic3C2×D312yes—
71C22×C3×Dic3C2×D311yescharacteristic
72C22×C3×Dic3C2×D311yescharacteristic
73C22×C3×Dic3C22×C311yescharacteristic
74C22×C3×Dic3C22×C312yes—
75C22×C3×Dic3C22×C311yescharacteristic
76C32×C42—34no—
77C32×C42—62no—
78C4×C3⋊Dic3—22no—
79C3×C4⋊Dic3—38no—
80C3×Dic3⋊C4—38no—
81C3×C4×Dic3—24no—
82C3×C4×Dic3—24no—
83C3×C4×Dic3—22no—
84C3×C4×Dic3—22no—
85C3×C4×Dic3—64no—
86C3×C4×Dic3—62no—
87C3×C4×Dic3—62no—
88C3×C4×Dic3Dic314yescomplemented
89C3×C4×Dic3C3×C414yescomplemented
90Dic3⋊Dic3C3×C418yesdirect product factor
91C22×C33C22×C411yescharacteristic
92C22×C33C22×C412yes—
93C22×C33C2×D411yescharacteristic
94C22×C33C2×Q811yescharacteristic
95C22×C33#4112yes—
96C33×C4C4⋊C414yescomplemented
97C3×C3⋊Dic3—24no—
98C32×Dic3—22no—
99C32×Dic3—24no—
100C32×Dic3C4214yescomplemented
101C2×C3×C4⋊C4—92no—
102C2×C3×C42—31no—
103C2×C3×C42—31no—
104C2×C3×C42—31no—
105C2×C3×C42—31no—
106C2×C3×C42—61no—
107C2×C3×C42—61no—
108C2×C3×C42—91no—
109C2×C4⋊Dic3—32no—
110C2×Dic3⋊C4—32no—
111C2×C4×Dic3—31no—
112C2×C4×Dic3—31no—
113C2×C4×Dic3—61no—
114C2×C4×Dic3C3×D311yescharacteristic
115C2×C4×Dic3C3×D311yescharacteristic
116C3×#95—92no—
117C3⋊#95—32no—
118C3⋊#95—32no—
119C23×C32—21no—
120C23×C32—21no—
121C23×C32—21no—
122C23×C32—41no—
123C23×C32C22×D311yescharacteristic
124C23×C32C22×D311yescharacteristic
125C23×C32C23×C311yescharacteristic
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—34no—
136C2×C32×C4—32no—
137C2×C32×C4—34no—
138C2×C32×C4—32no—
139C2×C32×C4—42no—
140C2×C32×C4—42no—
141C2×C32×C4—42no—
142C2×C32×C4—62no—
143C2×C32×C4—62no—
144C2×C32×C4—62no—
145C2×C32×C4Dic612yes—
146C2×C32×C4Dic612yes—
147C2×C32×C4C4×D312yes—
148C2×C32×C4D1212yes—
149C2×C32×C4C2×Dic312yes—
150C2×C32×C4#6712yes—
151C2×C32×C4C2×C3×C412yes—
152C2×C32×C4C3×D412yes—
153C2×C32×C4C3×Q812yes—
154C2×C3⋊Dic3—22no—
155C2×C3⋊Dic3—22no—
156C2×C3⋊Dic3C2×C3×C414yes—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—24no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—34no—
171C2×C3×Dic3—34no—
172C2×C3×Dic3—64no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—64no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3C4×D314yes—
180C2×C3×Dic3C4×D314yes—
181C2×C3×Dic3C2×Dic312yes—
182C2×C3×Dic3C2×Dic312yes—
183C2×C3×Dic3C2×C3×C414yes—
184C2×C3×Dic3C2×C3×C414yes—
185C2×C3×Dic3C2×C3×C412yes—
186C2×C3×Dic3C2×C3×C412yes—
187C4×C4⋊C4—91noSylow
188C2×C33C2×C4211yescharacteristic
189C2×C33C2×C4⋊C411yescharacteristic
190C2×C33#11711yescharacteristic
191C2×C33C4×D412yes—
192C2×C33C4×Q812yes—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—31no—
197C22×C3×C4—31no—
198C22×C3×C4—31no—
199C22×C3×C4—32no—
200C22×C3×C4—31no—
201C22×C3×C4—32no—
202C22×C3×C4—41no—
203C22×C3×C4—61no—
204C22×C3×C4—61no—
205C22×C3×C4—62no—
206C22×C3×C4—92no—
207C22×C3×C4D3211yescharacteristic
208C22×C3×C4C2×C3×D311yescharacteristic
209C22×C3×C4C2×C3×D311yescharacteristic
210C22×Dic3—32no—
211C22×Dic3—32no—
212C22×Dic3—62no—
213C22×Dic3C2×C3×D312yes—
214C22×Dic3C2×C3×D311yescharacteristic
215C22×Dic3C2×C3×D311yescharacteristic
216C3×C4⋊C4—98no—
217C3×C42—34no—
218C3×C42—34no—
219C3×C42—64no—
220C3×C42—64no—
221C3×C42—62no—
222C3×C42—62no—
223C3×C42—182no—
224C4⋊Dic3—38no—
225Dic3⋊C4—38no—
226C4×Dic3—22no—
227C4×Dic3—64no—
228C4×Dic3—62no—
229C4×Dic3—62no—
230C4×Dic3C3×Dic314yescomplemented
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—41no—
247C22×C32—42no—
248C22×C32—41no—
249C22×C32—41no—
250C22×C32—42no—
251C22×C32C2×Dic611yescharacteristic
252C22×C32C2×Dic611yescharacteristic
253C22×C32C2×C4×D312yes—
254C22×C32C2×C4×D312yes—
255C22×C32C2×C4×D311yescharacteristic
256C22×C32C2×D1211yescharacteristic
257C22×C32#23512yes—
258C22×C32#23512yes—
259C22×C32C22×Dic311yescharacteristic
260C22×C32C2×#6711yescharacteristic
261C22×C32C22×C3×C412yes—
262C22×C32C22×C3×C411yescharacteristic
263C22×C32C2×C3×D411yescharacteristic
264C22×C32C2×C3×Q811yescharacteristic
265C22×C32C3×#4112yes—
266C32×C4—24no—
267C32×C4—24no—
268C32×C4—24no—
269C32×C4—34no—
270C32×C4—44no—
271C32×C4—62no—
272C32×C4—64no—
273C32×C4Dic3⋊C414yescomplemented
274C32×C4C4⋊Dic314yescomplemented
275C32×C4C3×C4⋊C414yescomplemented
276C3⋊Dic3—24no—
277C3×Dic3—24no—
278C3×Dic3—28no—
279C3×Dic3—22no—
280C3×Dic3—24no—
281C3×Dic3—22no—
282C3×Dic3—24no—
283C3×Dic3—24no—
284C3×Dic3—68no—
285C3×Dic3—64no—
286C3×Dic3—64no—
287C3×Dic3C4×Dic314yescomplemented
288C3×Dic3C3×C4214yescomplemented
289C2×C4⋊C4—92no—
290C2×C42—31no—
291C2×C42—31no—
292C2×C42—91no—
293#95—92no—
294C33C4×C4⋊C411yescharacteristic, complemented, Sylow
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—41no—
299C23×C3C2×D3211yescharacteristic
300C23×C3C22×C3×D311yescharacteristic
301C23×C3C22×C3×D311yescharacteristic
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—34no—
312C2×C3×C4—32no—
313C2×C3×C4—34no—
314C2×C3×C4—32no—
315C2×C3×C4—34no—
316C2×C3×C4—32no—
317C2×C3×C4—34no—
318C2×C3×C4—32no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4—64no—
325C2×C3×C4—64no—
326C2×C3×C4—64no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—94no—
336C2×C3×C4—182no—
337C2×C3×C4—182no—
338C2×C3×C4D3×Dic312yes—
339C2×C3×C4#62612yes—
340C2×C3×C4#62712yes—
341C2×C3×C4C3×Dic612yes—
342C2×C3×C4C3×Dic612yes—
343C2×C3×C4C3×C4×D312yes—
344C2×C3×C4C3×D1212yes—
345C2×C3×C4C2×C3×Dic312yes—
346C2×C3×C4C3×#6712yes—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—34no—
351C2×Dic3—34no—
352C2×Dic3—64no—
353C2×Dic3—64no—
354C2×Dic3—64no—
355C2×Dic3—62no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—62no—
359C2×Dic3C3×C4×D314yes—
360C2×Dic3C3×C4×D314yes—
361C2×Dic3C2×C3×Dic312yes—
362C2×Dic3C2×C3×Dic312yes—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—22no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—22no—
371C2×C32—22no—
372C2×C32—21no—
373C2×C32—22no—
374C2×C32—22no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—22no—
378C2×C32—41no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—42no—
382C2×C32—42no—
383C2×C32C4×Dic612yes—
384C2×C32C4×Dic612yes—
385C2×C32C42×D311yescharacteristic
386C2×C32#87211yescharacteristic
387C2×C32C4×D1212yes—
388C2×C32C2×C4×Dic311yescharacteristic
389C2×C32C2×Dic3⋊C411yescharacteristic
390C2×C32C2×C4⋊Dic311yescharacteristic
391C2×C32#92611yescharacteristic
392C2×C32C4×#6712yes—
393C2×C32C2×C3×C4211yescharacteristic
394C2×C32C2×C3×C4⋊C411yescharacteristic
395C2×C32C3×#11711yescharacteristic
396C2×C32C3×C4×D412yes—
397C2×C32C3×C4×Q812yes—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—32no—
401C22×C4—92no—
402C22×C4C3×D3211yescharacteristic
403C4⋊C4—98no—
404C42—34no—
405C42—62no—
406C42—182no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3#379812yes—
428C22×C3C4×D3212yes—
429C22×C3C2×D3×Dic311yescharacteristic
430C22×C3C2×#62611yescharacteristic
431C22×C3C2×#62711yescharacteristic
432C22×C3C2×C3×Dic611yescharacteristic
433C22×C3C2×C3×Dic611yescharacteristic
434C22×C3C2×C3×C4×D312yes—
435C22×C3C2×C3×C4×D312yes—
436C22×C3C2×C3×C4×D311yescharacteristic
437C22×C3C2×C3×D1211yescharacteristic
438C22×C3C3×#23512yes—
439C22×C3C3×#23512yes—
440C22×C3C22×C3×Dic311yescharacteristic
441C22×C3C2×C3×#6711yescharacteristic
442C3×C4—24no—
443C3×C4—24no—
444C3×C4—24no—
445C3×C4—34no—
446C3×C4—34no—
447C3×C4—44no—
448C3×C4—64no—
449C3×C4—64no—
450C3×C4—68no—
451C3×C4—62no—
452C3×C4—64no—
453C3×C4—62no—
454C3×C4—64no—
455C3×C4—184no—
456C3×C4Dic3⋊Dic314yesdirect product factor
457C3×C4C3×Dic3⋊C414yescomplemented
458C3×C4C3×C4⋊Dic314yescomplemented
459Dic3—22no—
460Dic3—24no—
461Dic3—68no—
462Dic3—64no—
463Dic3—64no—
464Dic3C3×C4×Dic314yescomplemented
465C32—21no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32C4×Dic3⋊C411yescharacteristic, complemented
470C32C4×C4⋊Dic311yescharacteristic, complemented
471C32C3×C4×C4⋊C411yescharacteristic, complemented
472C23C2×C3×D3211yescharacteristic
473C2×C4—34no—
474C2×C4—32no—
475C2×C4—34no—
476C2×C4—32no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—94no—
481C2×C4—182no—
482C2×C4—182no—
483C2×C4C3×D3×Dic312yes—
484C2×C4C3×#62612yes—
485C2×C4C3×#62712yes—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—22no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—22no—
499C2×C3—22no—
500C2×C3—22no—
501C2×C3—41no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—42no—
505C2×C3—42no—
506C2×C3#6385511yescharacteristic
507C2×C3C4×D3×Dic311yescharacteristic
508C2×C3C4×#62612yes—
509C2×C3C4×#62712yes—
510C2×C3C2×Dic3⋊Dic311yescharacteristic
511C2×C3C3×C4×Dic612yes—
512C2×C3C3×C4×Dic612yes—
513C2×C3C3×C42×D311yescharacteristic
514C2×C3C3×#87211yescharacteristic
515C2×C3C3×C4×D1212yes—
516C2×C3C2×C3×C4×Dic311yescharacteristic
517C2×C3C2×C3×Dic3⋊C411yescharacteristic
518C2×C3C2×C3×C4⋊Dic311yescharacteristic
519C2×C3C3×#92611yescharacteristic
520C2×C3C3×C4×#6712yes—
521C22C3×#379812yes—
522C22C3×C4×D3212yes—
523C22C2×C3×D3×Dic311yescharacteristic
524C22C2×C3×#62611yescharacteristic
525C22C2×C3×#62711yescharacteristic
526C4—34no—
527C4—62no—
528C4—64no—
529C4—184no—
530C4C3×Dic3⋊Dic314yesdirect product factor
531C3—21no—
532C3—21no—
533C3—21no—
534C3—41no—
535C3C4×Dic3⋊Dic311yescharacteristic, direct product factor
536C3C3×C4×Dic3⋊C411yescharacteristic, complemented
537C3C3×C4×C4⋊Dic311yescharacteristic, complemented
538C2C3×#6385511yescharacteristic
539C2C3×C4×D3×Dic311yescharacteristic
540C2C3×C4×#62612yes—
541C2C3×C4×#62712yes—
542C2C2×C3×Dic3⋊Dic311yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutatorC2×C32C2×C3×C42
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
8. C4×Dic3⋊Dic3C3direct (N × H)
16. C32×C4×Dic3C4semidirect
17. C3×Dic3⋊Dic3C4direct (N × H)
88. C3×C4×Dic3Dic3semidirect
89. C3×C4×Dic3C3×C4semidirect
90. Dic3⋊Dic3C3×C4direct (N × H)
96. C33×C4C4⋊C4semidirect
100. C32×Dic3C42semidirect
230. C4×Dic3C3×Dic3semidirect
273. C32×C4Dic3⋊C4semidirect
274. C32×C4C4⋊Dic3semidirect
275. C32×C4C3×C4⋊C4semidirect
287. C3×Dic3C4×Dic3semidirect
288. C3×Dic3C3×C42semidirect
294. C33C4×C4⋊C4semidirect
456. C3×C4Dic3⋊Dic3direct (N × H)
457. C3×C4C3×Dic3⋊C4semidirect
458. C3×C4C3×C4⋊Dic3semidirect
464. Dic3C3×C4×Dic3semidirect
469. C32C4×Dic3⋊C4semidirect
470. C32C4×C4⋊Dic3semidirect
471. C32C3×C4×C4⋊C4semidirect
530. C4C3×Dic3⋊Dic3direct (N × H)
535. C3C4×Dic3⋊Dic3direct (N × H)
536. C3C3×C4×Dic3⋊C4semidirect
537. C3C3×C4×C4⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C2×C32×C4×Dic32
C2×C3×Dic3⋊Dic32
C3×#635792
C3×C4×C4⋊Dic33
C3×C4×Dic3⋊C43
C4×Dic3⋊Dic33