Finite Group Databasebeta

← C3×D3×A4⋊Q8 #13180682

Subgroups — C3×D3×A4⋊Q8 #13180682

Order 1728.

770 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C3×C4×D3×A4C211yesmaximal, characteristic
2C3×#64259C211yesmaximal, characteristic, complemented
3C32×A4⋊Q8C211yesmaximal, characteristic, complemented
4C3×D3×SL(2,ℤ/4)C212yesmaximal
5C3×#64204C212yesmaximal, complemented
6C2×C3×A4⋊Q8—31nomaximal
7D3×A4⋊Q8C311yesmaximal, characteristic, direct product factor
8C3×D3×C22⋊Q8—31nomaximal
9C2×C3×D3×A4C2211yescharacteristic
10C32×C4×A4C2211yescharacteristic
11C3×D3×Dic6—41nomaximal
12C3×Dic3×A4C2211yescharacteristic
13C3×C3⋊SL(2,ℤ/4)C2212yes—
14C32×SL(2,ℤ/4)C2212yes—
15C2×C3×D3×Q8—31no—
16C22×C3×C4×D3D311yescharacteristic
17C2×C3×C4×A4—31no—
18C4×D3×A4—21no—
19C4×D3×A4C2×C311yescharacteristic
20#64259C2×C311yescharacteristic, complemented
21C2×C3×SL(2,ℤ/4)—32no—
22C3×A4⋊Q8—21no—
23C3×A4⋊Q8—31no—
24C3×A4⋊Q8—32no—
25C3×A4⋊Q8—61no—
26C3×A4⋊Q8D311yescharacteristic, direct product factor
27C3×A4⋊Q8C2×C311yescharacteristic, complemented
28D3×SL(2,ℤ/4)C2×C312yes—
29#64204C2×C312yescomplemented
30C32×C22⋊Q8—31no—
31C3×#946—31no—
32C3×#924—31no—
33C3×#897—31no—
34C3×#896—32no—
35C3×D3×C4⋊C4—31no—
36C3×D3×C4⋊C4—32no—
37C3×D3×C22⋊C4—32no—
38C3×#878—32no—
39C2×C32×A4C2311yescharacteristic
40C3×D3×A4Q812yescomplemented
41C3×#634—41no—
42C32×C4×D3—41no—
43C32×Dic6—41no—
44C3×#627—42no—
45C3×D3×Dic3—42no—
46C2×A4⋊Q8—31no—
47C2×C3×C22⋊Q8—91no—
48D3×C22⋊Q8—31no—
49C23×C3×D3C2×D311yescharacteristic
50C22×C3×A4—31no—
51C2×D3×A4—21no—
52C2×D3×A4C22×C311yescharacteristic
53C2×C32×Q8—31no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C3×Dic3C2×D311yescharacteristic
56C3×D3×Q8—61no—
57C3×D3×Q8—61no—
58C3×D3×Q8—61no—
59C3×D3×Q8—61no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—32no—
65C2×C3×C4×D3—32no—
66C2×C3×Dic6—32no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—121no—
69C3×C4×A4—21no—
70C3×C4×A4—21no—
71C3×C4×A4—21no—
72C3×C4×A4—31no—
73C3×C4×A4—41no—
74C3×C4×A4C2×D311yescharacteristic
75C3×C4×A4C22×C311yescharacteristic
76D3×Dic6—41no—
77Dic3×A4—21no—
78Dic3×A4C22×C311yescharacteristic
79C3⋊SL(2,ℤ/4)C22×C312yes—
80C3×SL(2,ℤ/4)—22no—
81C3×SL(2,ℤ/4)—32no—
82C3×SL(2,ℤ/4)—62no—
83C3×SL(2,ℤ/4)C2×D312yes—
84C3×SL(2,ℤ/4)C22×C312yes—
85C32×C4⋊C4—32no—
86C32×C4⋊C4—31no—
87C32×C22⋊C4—32no—
88C3×#217—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—31no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—32no—
96C2×C32×D3—41no—
97C32×A4C2×Q811yescharacteristic, complemented
98C33×C4—41no—
99C3×C3⋊Dic3—42no—
100C32×Dic3—42no—
101C32×Dic3—41no—
102C22×C3×Q8—91no—
103C23×C3×C4—31no—
104C2×D3×Q8—31no—
105C22×C4×D3C3×D311yescharacteristic
106C2×C4×A4—31no—
107C2×C4×A4—61no—
108C2×SL(2,ℤ/4)—32no—
109A4⋊Q8—31no—
110A4⋊Q8—32no—
111A4⋊Q8—61no—
112A4⋊Q8C3×D311yescharacteristic, direct product factor
113C3×C22⋊Q8—31no—
114C3×C22⋊Q8—31no—
115C3×C22⋊Q8—61no—
116C3×C22⋊Q8—92no—
117C3×C22⋊Q8—92no—
118C3×C22⋊Q8—91no—
119C3×C22⋊Q8—91no—
120C3×C22⋊Q8—91no—
121C2×C3×C4⋊C4—91no—
122C2×C3×C4⋊C4—92no—
123C2×C3×C22⋊C4—92no—
124#946—31no—
125#924—31no—
126#897—31no—
127#896—32no—
128D3×C4⋊C4—31no—
129D3×C4⋊C4—32no—
130D3×C22⋊C4—32no—
131#878—32no—
132C23×C32C22×D311yescharacteristic
133C22×C3×D3—31no—
134C22×C3×D3—32no—
135C22×C3×D3—31no—
136C22×C3×D3Dic612yescomplemented
137C2×C3×A4—21no—
138C2×C3×A4—21no—
139C2×C3×A4—21no—
140C2×C3×A4—32no—
141C2×C3×A4—41no—
142C2×C3×A4C22×D311yescharacteristic
143C2×C3×A4C23×C311yescharacteristic
144D3×A4—22no—
145D3×A4C3×Q812yescomplemented
146C32×Q8—61no—
147C32×Q8—61no—
148C2×C32×C4—31no—
149C2×C32×C4—31no—
150C2×C32×C4—32no—
151C2×C32×C4—32no—
152C2×C32×C4—121no—
153#634—41no—
154C2×C3×Dic3—31no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—32no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—122no—
159C3×C4×D3—31no—
160C3×C4×D3—31no—
161C3×C4×D3—31no—
162C3×C4×D3—41no—
163C3×C4×D3—61no—
164C3×C4×D3—64no—
165C3×C4×D3—62no—
166C3×C4×D3—62no—
167C3×C4×D3—81no—
168C3×C4×D3S411yescharacteristic
169C3×Dic6—41no—
170C3×Dic6—41no—
171C3×Dic6—62no—
172C3×Dic6—61no—
173C3×Dic6—61no—
174C3×Dic6—62no—
175C3×Dic6—81no—
176C3×Dic6—122no—
177C3×Dic6—121no—
178C3×Dic6—241no—
179#627—42no—
180D3×Dic3—42no—
181C2×C22⋊Q8—91noSylow
182C2×C33—41no—
183C32×D3—42no—
184C24×C3—31no—
185C23×D3C2×C3×D311yescharacteristic
186C22×A4—31no—
187C22×A4—61no—
188C2×C3×Q8—31no—
189C2×C3×Q8—31no—
190C2×C3×Q8—61no—
191C2×C3×Q8—91no—
192C2×C3×Q8—92no—
193C2×C3×Q8—181no—
194C2×C3×Q8—181no—
195C2×C3×Q8—181no—
196C2×C3×Q8—181no—
197C22×C3×C4—21no—
198C22×C3×C4—31no—
199C22×C3×C4—91no—
200C22×C3×C4—91no—
201C22×C3×C4—91no—
202C22×C3×C4—91no—
203C22×C3×C4—92no—
204C22×C3×C4—92no—
205C22×C3×C4D3211yescharacteristic
206C22×C3×C4C2×C3×D311yescharacteristic
207C22×Dic3C2×C3×D311yescharacteristic
208D3×Q8—61no—
209D3×Q8—61no—
210D3×Q8—61no—
211D3×Q8—61no—
212C2×C4×D3—31no—
213C2×C4×D3—31no—
214C2×C4×D3—31no—
215C2×C4×D3—31no—
216C2×C4×D3—32no—
217C2×C4×D3—32no—
218C2×Dic6—32no—
219C2×Dic6—31no—
220C2×Dic6—121no—
221C4×A4—21no—
222C4×A4—21no—
223C4×A4—31no—
224C4×A4—41no—
225C4×A4—61no—
226C4×A4C2×C3×D311yescharacteristic
227SL(2,ℤ/4)—32no—
228SL(2,ℤ/4)—62no—
229SL(2,ℤ/4)C2×C3×D312yes—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4—62no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—92no—
241C3×C22⋊C4—32no—
242C3×C22⋊C4—32no—
243C3×C22⋊C4—62no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247#217—32no—
248(C2×D3)⋊C4—32no—
249(C2×D3)⋊C4—32no—
250C4⋊Dic3—32no—
251C4⋊Dic3—31no—
252Dic3⋊C4—32no—
253Dic3⋊C4—32no—
254Dic3⋊C4—32no—
255C22×C32—31no—
256C22×C32—31no—
257C22×C32—121no—
258C22×C32C2×Dic611yescharacteristic, complemented
259C2×C3×D3—31no—
260C2×C3×D3—32no—
261C2×C3×D3—32no—
262C2×C3×D3—32no—
263C2×C3×D3—41no—
264C2×C3×D3—61no—
265C2×C3×D3—81no—
266C2×C3×D3C2×S411yescharacteristic
267C3×A4—21no—
268C3×A4—21no—
269C3×A4—21no—
270C3×A4—41no—
271C3×A4D3×Q811yescharacteristic, complemented
272C3×A4C2×C3×Q811yescharacteristic, complemented
273C32×C4—31no—
274C32×C4—41no—
275C32×C4—41no—
276C32×C4—62no—
277C32×C4—62no—
278C32×C4—81no—
279C32×C4—81no—
280C32×C4—81no—
281C32×C4—121no—
282C32×C4—161no—
283C32×C4C2×S411yescharacteristic
284C3⋊Dic3—42no—
285C3×Dic3—31no—
286C3×Dic3—42no—
287C3×Dic3—42no—
288C3×Dic3—41no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C3×Dic3—82no—
292C3×Dic3—81no—
293C3×Dic3—122no—
294C3×Dic3—242no—
295C3×Dic3C2×S411yescharacteristic
296C22×Q8—91no—
297C23×C4—31no—
298C22⋊Q8—31no—
299C22⋊Q8—92no—
300C22⋊Q8—92no—
301C22⋊Q8—91no—
302C22⋊Q8—91no—
303C22⋊Q8—91no—
304C2×C4⋊C4—91no—
305C2×C4⋊C4—92no—
306C2×C22⋊C4—92no—
307C33—41noSylow
308C23×C3—21no—
309C23×C3—32no—
310C23×C3—91no—
311C23×C3—91no—
312C23×C3—92no—
313C23×C3C2×D3211yescharacteristic
314C23×C3C22×C3×D311yescharacteristic
315C22×D3—31no—
316C22×D3—31no—
317C22×D3—32no—
318C22×D3C3×Dic612yescomplemented
319C2×A4—21no—
320C2×A4—21no—
321C2×A4—32no—
322C2×A4—41no—
323C2×A4—62no—
324C2×A4C22×C3×D311yescharacteristic
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—61no—
328C3×Q8—61no—
329C3×Q8—121no—
330C3×Q8—121no—
331C3×Q8—182no—
332C3×Q8—181no—
333C3×Q8—182no—
334C3×Q8—181no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—32no—
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—91no—
349C2×C3×C4—91no—
350C2×C3×C4—91no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4—121no—
356C2×C3×C4—181no—
357C2×C3×C4—182no—
358C2×C3×C4—184no—
359C2×C3×C4—182no—
360C2×C3×C4—241no—
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—122no—
366C4×D3—31no—
367C4×D3—31no—
368C4×D3—31no—
369C4×D3—61no—
370C4×D3—64no—
371C4×D3—62no—
372C4×D3—62no—
373C4×D3C3×S411yescharacteristic
374Dic6—41no—
375Dic6—62no—
376Dic6—61no—
377Dic6—61no—
378Dic6—62no—
379Dic6—122no—
380Dic6—121no—
381Dic6—241no—
382C2×C32—31no—
383C2×C32—31no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—81no—
387C2×C32—81no—
388C2×C32—81no—
389C2×C32—122no—
390C2×C32—161no—
391C2×C32C22×S411yescharacteristic
392C3×D3—32no—
393C3×D3—42no—
394C3×D3—82no—
395C3×D3A4⋊Q812yesdirect product factor
396C24—31no—
397C2×Q8—31no—
398C2×Q8—91no—
399C2×Q8—92no—
400C2×Q8—181no—
401C2×Q8—181no—
402C2×Q8—181no—
403C2×Q8—181no—
404C22×C4—31no—
405C22×C4—91no—
406C22×C4—91no—
407C22×C4—91no—
408C22×C4—91no—
409C22×C4—92no—
410C22×C4—92no—
411C22×C4C3×D3211yescharacteristic
412C4⋊C4—31no—
413C4⋊C4—32no—
414C4⋊C4—92no—
415C4⋊C4—92no—
416C4⋊C4—92no—
417C4⋊C4—91no—
418C4⋊C4—92no—
419C22⋊C4—32no—
420C22⋊C4—92no—
421C22⋊C4—92no—
422C22⋊C4—92no—
423C22×C3—21no—
424C22×C3—31no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—61no—
430C22×C3—61no—
431C22×C3—92no—
432C22×C3—92no—
433C22×C3—92no—
434C22×C3—91no—
435C22×C3—121no—
436C22×C3—181no—
437C22×C3—241no—
438C22×C3D3×Dic611yescharacteristic, complemented
439C22×C3C2×C3×Dic611yescharacteristic, complemented
440C2×D3—31no—
441C2×D3—32no—
442C2×D3—32no—
443C2×D3—32no—
444C2×D3—61no—
445C2×D3C2×C3×S411yescharacteristic
446A4—21no—
447A4—21no—
448A4—41no—
449A4C3×D3×Q811yescharacteristic, complemented
450C3×C4—21no—
451C3×C4—31no—
452C3×C4—31no—
453C3×C4—31no—
454C3×C4—41no—
455C3×C4—61no—
456C3×C4—62no—
457C3×C4—62no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4—91no—
463C3×C4—121no—
464C3×C4—122no—
465C3×C4—122no—
466C3×C4—161no—
467C3×C4—182no—
468C3×C4—182no—
469C3×C4—241no—
470C3×C4D3×S411yescharacteristic
471C3×C4C2×C3×S411yescharacteristic
472Dic3—31no—
473Dic3—42no—
474Dic3—62no—
475Dic3—62no—
476Dic3—122no—
477Dic3—242no—
478Dic3C2×C3×S411yescharacteristic
479C32—41no—
480C32—41no—
481C32—81no—
482C32—81no—
483C32—81no—
484C32—161no—
485C32C2×A4⋊Q811yescharacteristic, complemented
486C23—32no—
487C23—91no—
488C23—91no—
489C23—92no—
490C23C2×C3×D3211yescharacteristic
491Q8—61no—
492Q8—61no—
493Q8—182no—
494Q8—181no—
495Q8—182no—
496Q8—181no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—31no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—91no—
503C2×C4—91no—
504C2×C4—91no—
505C2×C4—91no—
506C2×C4—91no—
507C2×C4—92no—
508C2×C4—92no—
509C2×C4—181no—
510C2×C4—182no—
511C2×C4—182no—
512C2×C4—184no—
513C2×C3—21no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—32no—
517C2×C3—31no—
518C2×C3—31no—
519C2×C3—41no—
520C2×C3—61no—
521C2×C3—61no—
522C2×C3—81no—
523C2×C3—81no—
524C2×C3—92no—
525C2×C3—122no—
526C2×C3—161no—
527C2×C3—242no—
528C2×C3C2×D3×S411yescharacteristic
529C2×C3C22×C3×S411yescharacteristic
530D3—32no—
531D3C3×A4⋊Q812yesdirect product factor
532C22—31no—
533C22—31no—
534C22—31no—
535C22—92no—
536C22—92no—
537C22—92no—
538C22—91no—
539C22—181no—
540C22C3×D3×Dic611yescharacteristic, complemented
541C4—31no—
542C4—31no—
543C4—62no—
544C4—62no—
545C4—91no—
546C4—182no—
547C4—182no—
548C4C3×D3×S411yescharacteristic
549C3—21no—
550C3—41no—
551C3—81no—
552C3—81no—
553C3—161no—
554C3D3×A4⋊Q811yescharacteristic, direct product factor
555C3C2×C3×A4⋊Q811yescharacteristic, complemented
556C2—31no—
557C2—31no—
558C2—32no—
559C2—92no—
560C2C2×C3×D3×S411yescharacteristic
subgroupG/H
centerC2×C3C2×D3×S4
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2C2×C3×D3×S4
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
prime radicalC2×C3×D3×A4C22
hypercenterC3×C4D3×S4
hypocenterC3×A4C2×C3×Q8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64259C2semidirect
3. C32×A4⋊Q8C2semidirect
5. C3×#64204C2semidirect
7. D3×A4⋊Q8C3direct (N × H)
20. #64259C2×C3semidirect
26. C3×A4⋊Q8D3direct (N × H)
27. C3×A4⋊Q8C2×C3semidirect
29. #64204C2×C3semidirect
40. C3×D3×A4Q8semidirect
97. C32×A4C2×Q8semidirect
112. A4⋊Q8C3×D3direct (N × H)
136. C22×C3×D3Dic6semidirect
145. D3×A4C3×Q8semidirect
258. C22×C32C2×Dic6semidirect
271. C3×A4D3×Q8semidirect
272. C3×A4C2×C3×Q8semidirect
318. C22×D3C3×Dic6semidirect
395. C3×D3A4⋊Q8direct (N × H)
438. C22×C3D3×Dic6semidirect
439. C22×C3C2×C3×Dic6semidirect
449. A4C3×D3×Q8semidirect
485. C32C2×A4⋊Q8semidirect
531. D3C3×A4⋊Q8direct (N × H)
540. C22C3×D3×Dic6semidirect
554. C3D3×A4⋊Q8direct (N × H)
555. C3C2×C3×A4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D3×A42
C3×#642592
C32×A4⋊Q82
C3×D3×SL(2,ℤ/4)2
C3×#642042
C2×C3×A4⋊Q83
D3×A4⋊Q83
C3×D3×C22⋊Q83
C3×D3×Dic64