Finite Group Databasebeta

← A4×Dic3⋊Dic3 #13166167

Subgroups — A4×Dic3⋊Dic3 #13166167

Order 1728.

602 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×A4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×Dic3×A4C211yesmaximal, characteristic
3C2×C3×Dic3×A4C211yesmaximal, characteristic
4C22×Dic3⋊Dic3C311yesmaximal, characteristic, complemented
5A4×C4⋊Dic3—31nomaximal
6A4×Dic3⋊C4—31nomaximal
7C22×C32×A4C2211yescharacteristic
8A4×C3⋊Dic3—21no—
9C3×Dic3×A4—21no—
10C3×Dic3×A4C412yescomplemented
11C3×Dic3⋊Dic3—41nomaximal
12C23×C3⋊Dic3C2×C311yescharacteristic
13C23×C3×Dic3C2×C311yescharacteristic
14C23×C3×Dic3C2×C311yescharacteristic
15C2×C3×C4×A4—31no—
16C2×C3×C4×A4—31no—
17C2×Dic3×A4—21no—
18C2×Dic3×A4—21no—
19C2×Dic3×A4—31no—
20C2×Dic3×A4—31no—
21C2×Dic3×A4—61no—
22C2×Dic3×A4D311yescharacteristic
23C2×Dic3×A4D311yescharacteristic
24C2×Dic3⋊Dic3—31no—
25C2×Dic3⋊Dic3—31no—
26C2×Dic3⋊Dic3—31no—
27C2×Dic3⋊Dic3—31no—
28C2×C32×A4C2×C411yescharacteristic
29C2×C32×A4D411yescharacteristic
30C2×C32×A4Q811yescharacteristic
31C2×C3×C3⋊Dic3—41no—
32C2×C32×Dic3—41no—
33C2×C32×Dic3—41no—
34C22×C4⋊Dic3—31no—
35C22×Dic3⋊C4—31no—
36A4×C4⋊C4—91no—
37C24×C32C22×C311yescharacteristic
38C22×C3×A4—21no—
39C22×C3×A4—21no—
40C22×C3×A4—21no—
41C22×C3×A4—41no—
42C22×C3×A4C2×D311yescharacteristic
43C22×C3×A4C2×D311yescharacteristic
44C22×C3⋊Dic3—21no—
45C22×C3⋊Dic3—31no—
46C22×C3⋊Dic3—31no—
47C22×C3⋊Dic3—61no—
48C22×C3×Dic3—21no—
49C22×C3×Dic3—31no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—31no—
52C22×C3×Dic3—31no—
53C22×C3×Dic3—31no—
54C22×C3×Dic3—61no—
55C22×C3×Dic3C3×C412yescomplemented
56C3×C4×A4—32no—
57C3×C4×A4—61no—
58Dic3×A4—22no—
59Dic3×A4—22no—
60Dic3×A4—21no—
61Dic3×A4—62no—
62Dic3×A4—61no—
63Dic3×A4—61no—
64Dic3×A4Dic312yescomplemented
65C3×C4⋊Dic3—121no—
66C3×Dic3⋊C4—121no—
67Dic3⋊Dic3—32no—
68Dic3⋊Dic3—31no—
69Dic3⋊Dic3—31no—
70Dic3⋊Dic3—31no—
71Dic3⋊Dic3A411yescharacteristic, direct product factor
72C22×C33—41no—
73C32×A4C4⋊C411yescharacteristic, complemented
74C3×C3⋊Dic3—81no—
75C32×Dic3—42no—
76C32×Dic3—81no—
77C23×C3×C4—31no—
78C23×C3×C4—31no—
79C23×Dic3—31no—
80C23×Dic3—31no—
81C23×Dic3—61no—
82C23×Dic3C3×D311yescharacteristic
83C23×Dic3C3×D311yescharacteristic
84C2×C4×A4—31no—
85C2×C4×A4—31no—
86C2×C4×A4—61no—
87C2×C4×A4—61no—
88C2×C4×A4—91no—
89C2×C4⋊Dic3—91no—
90C2×C4⋊Dic3—91no—
91C2×C4⋊Dic3—91no—
92C2×C4⋊Dic3—91no—
93C2×Dic3⋊C4—91no—
94C2×Dic3⋊C4—91no—
95C2×Dic3⋊C4—91no—
96C2×Dic3⋊C4—91no—
97C23×C32—31no—
98C23×C32—31no—
99C23×C32—31no—
100C23×C32—31no—
101C23×C32C2×C3×C411yescharacteristic
102C23×C32C3×D411yescharacteristic
103C23×C32C3×Q811yescharacteristic
104C2×C3×A4—21no—
105C2×C3×A4—21no—
106C2×C3×A4—21no—
107C2×C3×A4—21no—
108C2×C3×A4—21no—
109C2×C3×A4—21no—
110C2×C3×A4—21no—
111C2×C3×A4—21no—
112C2×C3×A4—21no—
113C2×C3×A4—41no—
114C2×C3×A4—41no—
115C2×C3×A4—41no—
116C2×C3×A4Dic611yescharacteristic
117C2×C3×A4Dic611yescharacteristic
118C2×C3×A4C4×D311yescharacteristic
119C2×C3×A4D1211yescharacteristic
120C2×C3×A4C2×Dic311yescharacteristic
121C2×C3×A4#6711yescharacteristic
122C2×C32×C4—121no—
123C2×C32×C4—121no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—61no—
129C2×C3⋊Dic3C2×A411yescharacteristic
130C2×C3×Dic3—32no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—81no—
143C2×C3×Dic3—81no—
144C2×C3×Dic3—121no—
145C2×C3×Dic3—121no—
146C2×C3×Dic3—241no—
147C2×C3×Dic3C2×A411yescharacteristic
148C2×C3×Dic3C2×A411yescharacteristic
149C22×C4⋊C4—91noSylow
150C2×C33—41no—
151C2×C33—41no—
152C2×C33—41no—
153C24×C3—21no—
154C24×C3C2×C3×D311yescharacteristic
155C24×C3C2×C3×D311yescharacteristic
156C22×A4—21no—
157C22×A4—21no—
158C22×A4—41no—
159C22×A4D3211yescharacteristic
160C22×C3×C4—32no—
161C22×C3×C4—61no—
162C22×C3×C4—91no—
163C22×C3×C4—91no—
164C22×C3×C4—91no—
165C22×C3×C4—92no—
166C22×C3×C4—91no—
167C22×C3×C4—181no—
168C22×Dic3—21no—
169C22×Dic3—31no—
170C22×Dic3—31no—
171C22×Dic3—32no—
172C22×Dic3—31no—
173C22×Dic3—31no—
174C22×Dic3—61no—
175C22×Dic3—62no—
176C22×Dic3—61no—
177C22×Dic3—61no—
178C22×Dic3—91no—
179C22×Dic3—91no—
180C22×Dic3—91no—
181C22×Dic3—91no—
182C22×Dic3—181no—
183C22×Dic3—181no—
184C22×Dic3—182no—
185C22×Dic3—181no—
186C22×Dic3—181no—
187C22×Dic3C3×Dic312yescomplemented
188C4×A4—32no—
189C4×A4—62no—
190C4×A4—62no—
191C4×A4—61no—
192C4×A4—181no—
193C3×C4⋊C4—361no—
194C4⋊Dic3—31no—
195C4⋊Dic3—91no—
196C4⋊Dic3—91no—
197C4⋊Dic3—92no—
198C4⋊Dic3—91no—
199Dic3⋊C4—31no—
200Dic3⋊C4—91no—
201Dic3⋊C4—91no—
202Dic3⋊C4—92no—
203Dic3⋊C4—91no—
204C22×C32—31no—
205C22×C32—31no—
206C22×C32—31no—
207C22×C32—32no—
208C22×C32—31no—
209C22×C32—31no—
210C22×C32—31no—
211C22×C32—31no—
212C22×C32—31no—
213C22×C32—31no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—81no—
217C22×C32—81no—
218C22×C32—81no—
219C22×C32—161no—
220C22×C32C3×C4⋊C411yescharacteristic, complemented
221C22×C32C22×A411yescharacteristic
222C3×A4—21no—
223C3×A4—21no—
224C3×A4—21no—
225C3×A4—41no—
226C3×A4Dic3⋊C411yescharacteristic, complemented
227C3×A4C4⋊Dic311yescharacteristic, complemented
228C32×C4—122no—
229C32×C4—241no—
230C3⋊Dic3—21no—
231C3⋊Dic3—61no—
232C3×Dic3—21no—
233C3×Dic3—32no—
234C3×Dic3—42no—
235C3×Dic3—61no—
236C3×Dic3—81no—
237C3×Dic3—82no—
238C3×Dic3—82no—
239C3×Dic3—241no—
240C3×Dic3—241no—
241C3×Dic3—242no—
242C3×Dic3C4×A412yescomplemented
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—91no—
246C2×C4⋊C4—271no—
247C2×C4⋊C4—271no—
248C2×C4⋊C4—271no—
249C2×C4⋊C4—271no—
250C33—41noSylow
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—31no—
255C23×C3—31no—
256C23×C3—31no—
257C23×C3—31no—
258C23×C3—31no—
259C23×C3—31no—
260C23×C3—31no—
261C23×C3—31no—
262C23×C3—61no—
263C23×C3—61no—
264C23×C3—61no—
265C23×C3—61no—
266C23×C3C3×Dic611yescharacteristic
267C23×C3C3×Dic611yescharacteristic
268C23×C3C3×C4×D311yescharacteristic
269C23×C3C3×D1211yescharacteristic
270C23×C3C2×C3×Dic311yescharacteristic
271C23×C3C3×#6711yescharacteristic
272C2×A4—21no—
273C2×A4—21no—
274C2×A4—21no—
275C2×A4—21no—
276C2×A4—21no—
277C2×A4—21no—
278C2×A4—41no—
279C2×A4—41no—
280C2×A4—41no—
281C2×A4D3×Dic311yescharacteristic
282C2×A4#62611yescharacteristic
283C2×A4#62711yescharacteristic
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—92no—
287C2×C3×C4—92no—
288C2×C3×C4—92no—
289C2×C3×C4—91no—
290C2×C3×C4—92no—
291C2×C3×C4—91no—
292C2×C3×C4—121no—
293C2×C3×C4—121no—
294C2×C3×C4—181no—
295C2×C3×C4—181no—
296C2×C3×C4—181no—
297C2×C3×C4—181no—
298C2×C3×C4—241no—
299C2×C3×C4—241no—
300C2×C3×C4—361no—
301C2×Dic3—32no—
302C2×Dic3—31no—
303C2×Dic3—32no—
304C2×Dic3—32no—
305C2×Dic3—32no—
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—61no—
313C2×Dic3—61no—
314C2×Dic3—91no—
315C2×Dic3—91no—
316C2×Dic3—181no—
317C2×Dic3—181no—
318C2×Dic3—181no—
319C2×Dic3—181no—
320C2×Dic3—181no—
321C2×Dic3—182no—
322C2×Dic3—182no—
323C2×Dic3—182no—
324C2×Dic3—181no—
325C2×Dic3—181no—
326C2×Dic3—181no—
327C2×Dic3—181no—
328C2×Dic3—182no—
329C2×Dic3D3×A411yescharacteristic
330C2×Dic3D3×A411yescharacteristic
331C2×C32—31no—
332C2×C32—31no—
333C2×C32—31no—
334C2×C32—31no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32—81no—
342C2×C32—81no—
343C2×C32—81no—
344C2×C32—81no—
345C2×C32—81no—
346C2×C32—81no—
347C2×C32—81no—
348C2×C32—81no—
349C2×C32—81no—
350C2×C32—161no—
351C2×C32—161no—
352C2×C32—161no—
353C2×C32C2×C4×A411yescharacteristic
354C2×C32D4×A411yescharacteristic
355C2×C32Q8×A411yescharacteristic
356C24C3×D3211yescharacteristic
357C22×C4—32no—
358C22×C4—61no—
359C22×C4—91no—
360C22×C4—91no—
361C22×C4—91no—
362C22×C4—92no—
363C22×C4—91no—
364C22×C4—181no—
365C22×C4—181no—
366C22×C4—271no—
367C22×C4—271no—
368C22×C4—541no—
369C4⋊C4—91no—
370C4⋊C4—271no—
371C4⋊C4—271no—
372C4⋊C4—271no—
373C4⋊C4—272no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—31no—
381C22×C3—31no—
382C22×C3—32no—
383C22×C3—32no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—31no—
396C22×C3—41no—
397C22×C3—61no—
398C22×C3—61no—
399C22×C3—61no—
400C22×C3—61no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—61no—
404C22×C3—61no—
405C22×C3—61no—
406C22×C3—62no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—161no—
410C22×C3C3×Dic3⋊C411yescharacteristic, complemented
411C22×C3C3×C4⋊Dic311yescharacteristic, complemented
412C22×C3C2×D3×A411yescharacteristic
413C22×C3C2×D3×A411yescharacteristic
414A4—21no—
415A4—21no—
416A4—41no—
417A4Dic3⋊Dic311yescharacteristic, direct product factor
418C3×C4—32no—
419C3×C4—61no—
420C3×C4—92no—
421C3×C4—122no—
422C3×C4—181no—
423C3×C4—242no—
424C3×C4—241no—
425C3×C4—242no—
426C3×C4—721no—
427Dic3—21no—
428Dic3—32no—
429Dic3—62no—
430Dic3—61no—
431Dic3—61no—
432Dic3—61no—
433Dic3—182no—
434Dic3—181no—
435Dic3—181no—
436Dic3Dic3×A412yescomplemented
437C32—41no—
438C32—41no—
439C32—81no—
440C32—81no—
441C32—81no—
442C32—161no—
443C32A4×C4⋊C411yescharacteristic, complemented
444C23—31no—
445C23—31no—
446C23—31no—
447C23—31no—
448C23C3×D3×Dic311yescharacteristic
449C23C3×#62611yescharacteristic
450C23C3×#62711yescharacteristic
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—92no—
454C2×C4—92no—
455C2×C4—92no—
456C2×C4—91no—
457C2×C4—92no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—181no—
463C2×C4—181no—
464C2×C4—271no—
465C2×C4—541no—
466C2×C4—541no—
467C2×C4—541no—
468C2×C4—541no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—31no—
477C2×C3—31no—
478C2×C3—31no—
479C2×C3—31no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—61no—
484C2×C3—61no—
485C2×C3—61no—
486C2×C3—61no—
487C2×C3—81no—
488C2×C3—81no—
489C2×C3—81no—
490C2×C3—81no—
491C2×C3—81no—
492C2×C3—81no—
493C2×C3—161no—
494C2×C3—161no—
495C2×C3—161no—
496C2×C3A4×Dic611yescharacteristic
497C2×C3A4×Dic611yescharacteristic
498C2×C3C4×D3×A411yescharacteristic
499C2×C3A4×D1211yescharacteristic
500C2×C3C2×Dic3×A411yescharacteristic
501C2×C3A4×#6711yescharacteristic
502C22—31no—
503C22—31no—
504C22—31no—
505C22—32no—
506C22—31no—
507C22—31no—
508C22—31no—
509C22—31no—
510C22—31no—
511C22—31no—
512C22C3×Dic3⋊Dic311yescharacteristic, complemented
513C22D32×A411yescharacteristic
514C4—32no—
515C4—61no—
516C4—92no—
517C4—181no—
518C4—181no—
519C4—541no—
520C3—21no—
521C3—41no—
522C3—81no—
523C3—81no—
524C3—161no—
525C3A4×Dic3⋊C411yescharacteristic, complemented
526C3A4×C4⋊Dic311yescharacteristic, complemented
527C2—31no—
528C2—31no—
529C2—31no—
530C2—31no—
531C2D3×Dic3×A411yescharacteristic
532C2A4×#62611yescharacteristic
533C2A4×#62711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutatorC23×C32C2×C3×C4
Fitting, socle, cosocleC24×C32C22×C3
prime radicalC22×C32×A4C22
hypocenterC22×C32C3×C4⋊C4
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. C22×Dic3⋊Dic3C3semidirect
10. C3×Dic3×A4C4semidirect
55. C22×C3×Dic3C3×C4semidirect
64. Dic3×A4Dic3semidirect
71. Dic3⋊Dic3A4direct (N × H)
73. C32×A4C4⋊C4semidirect
187. C22×Dic3C3×Dic3semidirect
220. C22×C32C3×C4⋊C4semidirect
226. C3×A4Dic3⋊C4semidirect
227. C3×A4C4⋊Dic3semidirect
242. C3×Dic3C4×A4semidirect
410. C22×C3C3×Dic3⋊C4semidirect
411. C22×C3C3×C4⋊Dic3semidirect
417. A4Dic3⋊Dic3direct (N × H)
436. Dic3Dic3×A4semidirect
443. C32A4×C4⋊C4semidirect
512. C22C3×Dic3⋊Dic3semidirect
525. C3A4×Dic3⋊C4semidirect
526. C3A4×C4⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×A4×C3⋊Dic32
C2×C3×Dic3×A42
C22×Dic3⋊Dic33
A4×C4⋊Dic33
A4×Dic3⋊C43
C3×Dic3⋊Dic34