Finite Group Databasebeta

← C3×#32925 #10651532

Subgroups — C3×#32925 #10651532

Order 768.

1352 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×2−1+4C211yesmaximal, characteristic, complemented
2C3×#3040C212yesmaximal, complemented
3C2×C3×#456C211yesmaximal, characteristic, complemented
4C2×C3×#454C211yesmaximal, characteristic
5C3×#2545C211yesmaximal, characteristic, complemented
6C3×#1888C211yesmaximal, characteristic, complemented
7C3×#1886C211yesmaximal, characteristic, complemented
8C3×#1883C212yesmaximal, complemented
9C3×#1882C212yesmaximal, complemented
10C3×#1806C212yesmaximal, complemented
11C3×#1801C211yesmaximal, characteristic, complemented
12#32925C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4C2211yescharacteristic
14C2×C3×#137C2212yes—
15C2×C3×AGL(1,ℤ/8)C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#124C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C3×#520—22no—
20C3×#520—22no—
21C3×#520C2212yes—
22C3×#519—22no—
23C3×#519C2211yescharacteristic
24C3×C4×#41—22no—
25C3×C4×#41—22no—
26C3×C4×#41C2212yes—
27C2×C3×C4×Q8—22no—
28C2×C3×C4×Q8C2211yescharacteristic
29C3×#464—22no—
30C3×#464—22no—
31C3×#463—22no—
32C3×#463—22no—
33C3×#456—24no—
34C3×#454—24no—
35C3×#423C2212yes—
36C2×C3×#104C2212yes—
37C2×C3×#104C2212yes—
38C3×#419C2212yes—
39C3×#418C2212yes—
40C2×C3×#101C2211yescharacteristic
41C2×C3×#101C2211yescharacteristic
42C3×#396—22no—
43C3×#391C2211yescharacteristic
44C3×#390C2211yescharacteristic
45C3×#388—22no—
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×#343C2212yes—
48C3×#339C2212yes—
49C3×#339C2212yes—
50C4×2−1+4C2×C311yescharacteristic, complemented
51#3040C2×C312yescomplemented
52C2×#456C2×C311yescharacteristic, complemented
53C2×#454C2×C311yescharacteristic
54#2545C2×C311yescharacteristic, complemented
55#1888C2×C311yescharacteristic, complemented
56#1886C2×C311yescharacteristic, complemented
57#1883C2×C312yescomplemented
58#1882C2×C312yescomplemented
59#1806C2×C312yescomplemented
60#1801C2×C311yescharacteristic, complemented
61C3×2−1+4—24no—
62C3×2−1+4—22no—
63C3×2−1+4D414yes—
64C2×C3×#41—22no—
65C2×C3×#41—22no—
66C2×C3×#41C2312yes—
67C22×C3×Q8—22no—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4C2311yescharacteristic
71C3×#137—24no—
72C3×#137—24no—
73C3×AGL(1,ℤ/8)—24no—
74C3×AGL(1,ℤ/8)—24no—
75C2×C3×Q16—22no—
76C2×C3×SD16—22no—
77C2×C3×SD16—22no—
78C2×C3×D8—22no—
79C2×C3×M16C2312yes—
80C2×C3×M16C2312yes—
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8—22no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124—22no—
86C3×#124—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—24no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8D412yes—
96C3×C4×Q8D412yes—
97C3×C4×D4—22no—
98C3×C4×D4—24no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×C4×D4—24no—
103C3×C4×D4—22no—
104C3×C4×D4D412yes—
105C3×C4×D4D412yes—
106C3×#117—24no—
107C3×#117—24no—
108C3×#117—22no—
109C3×#117—22no—
110C3×#117—22no—
111C3×#117C2312yes—
112C2×C3×C4⋊C4—24no—
113C2×C3×C4⋊C4—22no—
114C2×C3×C4⋊C4—21no—
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4D412yes—
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C22⋊C4—21no—
119C2×C3×C22⋊C4—22no—
120C2×C3×C22⋊C4—22no—
121C2×C3×C22⋊C4—21no—
122C2×C3×C42—24no—
123C2×C3×C42—22no—
124C2×C3×C42D412yes—
125C2×C3×C42C2311yescharacteristic
126C2×C3×C42C2311yescharacteristic
127C2×C3×C42C2311yescharacteristic
128C3×C4⋊C8—22no—
129C3×C4⋊C8—22no—
130C3×#104—24no—
131C3×#104—24no—
132C3×#103—22no—
133C3×#103—22no—
134C3×#102—22no—
135C3×#102—22no—
136C3×#101—22no—
137C3×#101—22no—
138C3×#95—21no—
139C3×#95—22no—
140C2×2−1+4C22×C311yescharacteristic
141C2×#137C22×C312yes—
142C2×AGL(1,ℤ/8)C22×C312yes—
143C2×C4⋊Q8C22×C311yescharacteristic
144C2×#124C22×C311yescharacteristic
145C2×#124C22×C311yescharacteristic
146#520—22no—
147#520—22no—
148#520C22×C312yes—
149#519—22no—
150#519C22×C311yescharacteristic
151C4×#41—22no—
152C4×#41—22no—
153C4×#41C22×C312yes—
154C2×C4×Q8—22no—
155C2×C4×Q8C22×C311yescharacteristic
156#464—22no—
157#464—22no—
158#463—22no—
159#463—22no—
160#456—24no—
161#454—24no—
162#423C22×C312yes—
163C2×#104C22×C312yes—
164C2×#104C22×C312yes—
165#419C22×C312yes—
166#418C22×C312yes—
167C2×#101C22×C311yescharacteristic
168C2×#101C22×C311yescharacteristic
169#396—22no—
170#391C22×C311yescharacteristic
171#390C22×C311yescharacteristic
172#388—22no—
173C4×C4⋊C4C22×C311yescharacteristic
174#343C22×C312yes—
175#339C22×C312yes—
176#339C22×C312yes—
177C24×C3—21no—
178C3×#41—28no—
179C3×#41—28no—
180C3×#41—28no—
181C3×#41—28no—
182C3×#41—22no—
183C3×#41—22no—
184C3×#41C2×D414yes—
185C3×#41#4114yes—
186C2×C3×Q8—22no—
187C2×C3×Q8—28no—
188C2×C3×Q8—28no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8C2×D412yes—
196C2×C3×Q8C2×D412yes—
197C2×C3×Q8#4112yes—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—22no—
204C2×C3×D4C2×D412yes—
205C22×C3×C4—24no—
206C22×C3×C4—22no—
207C22×C3×C4—21no—
208C22×C3×C4—24no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C3×Q16—44no—
219C3×SD16—44no—
220C3×SD16—44no—
221C3×D8—44no—
222C3×M16—22no—
223C3×M16—22no—
224C3×M16—22no—
225C3×M16—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—44no—
237C3×C4⋊C4C2×D412yes—
238C3×C4⋊C4C2×D412yes—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—44no—
244C3×C22⋊C4—41no—
245C3×C22⋊C4—41no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—44no—
248C3×C42—24no—
249C3×C42—24no—
250C3×C42—22no—
251C3×C42—22no—
252C3×C42—22no—
253C3×C42—22no—
254C3×C42—22no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42C2×D412yes—
260C3×C42C2×D412yes—
2612−1+4—24no—
2622−1+4—22no—
2632−1+4C3×D414yes—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41C23×C312yes—
267C22×Q8—22no—
268C22×Q8C23×C311yescharacteristic
269C22×Q8C23×C311yescharacteristic
270C22×D4C23×C311yescharacteristic
271#137—24no—
272#137—24no—
273AGL(1,ℤ/8)—24no—
274AGL(1,ℤ/8)—24no—
275C2×Q16—22no—
276C2×SD16—22no—
277C2×SD16—22no—
278C2×D8—22no—
279C2×M16C23×C312yes—
280C2×M16C23×C312yes—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283#124—22no—
284#124—22no—
285#124—22no—
286#124—22no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8C3×D412yes—
296C4×Q8C3×D412yes—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—24no—
303C4×D4—22no—
304C4×D4C3×D412yes—
305C4×D4C3×D412yes—
306#117—24no—
307#117—24no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117C23×C312yes—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4C3×D412yes—
317C2×C4⋊C4C23×C311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42C3×D412yes—
325C2×C42C23×C311yescharacteristic
326C2×C42C23×C311yescharacteristic
327C2×C42C23×C311yescharacteristic
328C4⋊C8—22no—
329C4⋊C8—22no—
330#104—24no—
331#104—24no—
332#103—22no—
333#103—22no—
334#102—22no—
335#102—22no—
336#101—22no—
337#101—22no—
338#95—21no—
339#95—22no—
340C23×C3—22no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3C22×D411yescharacteristic
347C3×Q8—28no—
348C3×Q8—24no—
349C3×Q8—22no—
350C3×Q8—24no—
351C3×Q8—42no—
352C3×Q8—42no—
353C3×Q8#12114yes—
354C3×D4—28no—
355C3×D4—24no—
356C3×D4—24no—
357C3×D4—22no—
358C3×D4—42no—
359C3×D4—42no—
360C3×D4#12114yes—
361C2×C3×C4—216no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—44no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—82no—
384C2×C3×C4—84no—
385C2×C3×C4#12011yescharacteristic
386C2×C3×C4#12011yescharacteristic
387C2×C3×C4#12012yes—
388C2×C3×C4#12114yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D412yes—
391C2×C3×C4C2×#4112yes—
392C2×C3×C4C2×#4111yescharacteristic
393C3×C8—42no—
394C3×C8—42no—
395C24—21no—
396#41—28no—
397#41—28no—
398#41—28no—
399#41—28no—
400#41—22no—
401#41—22no—
402#41C2×C3×D414yes—
403#41C3×#4114yes—
404C2×Q8—22no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8C2×C3×D412yes—
414C2×Q8C2×C3×D412yes—
415C2×Q8C3×#4112yes—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4C2×C3×D412yes—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4C2×C3×D412yes—
432C22×C4C2×C3×D412yes—
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C24×C311yescharacteristic
436Q16—44no—
437SD16—44no—
438SD16—44no—
439D8—44no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—22no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4C2×C3×D412yes—
456C4⋊C4C2×C3×D412yes—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—44no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C42—24no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42C2×C3×D412yes—
478C42C2×C3×D412yes—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—24no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3C2×#12011yescharacteristic
490C22×C3C2×#12111yescharacteristic
491C22×C3#52511yescharacteristic
492C3×C4—24no—
493C3×C4—24no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4C2×#12112yes—
504C3×C4#52512yes—
505C23—22no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C22×C3×D411yescharacteristic
512Q8—28no—
513Q8—24no—
514Q8—22no—
515Q8—24no—
516Q8—42no—
517Q8—42no—
518Q8C3×#12114yes—
519D4—28no—
520D4—24no—
521D4—24no—
522D4—22no—
523D4—42no—
524D4—42no—
525D4C3×#12114yes—
526C2×C4—216no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—82no—
549C2×C4—84no—
550C2×C4C3×#12011yescharacteristic
551C2×C4C3×#12011yescharacteristic
552C2×C4C3×#12012yes—
553C2×C4C3×#12114yes—
554C2×C4C22×C3×D411yescharacteristic
555C2×C4C22×C3×D412yes—
556C2×C4C2×C3×#4112yes—
557C2×C4C2×C3×#4111yescharacteristic
558C8—42no—
559C8—42no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—44no—
563C2×C3—161no—
564C2×C3#284711yescharacteristic
565C2×C3#301811yescharacteristic
566C2×C3#302011yescharacteristic
567C22—21no—
568C22—21no—
569C22—24no—
570C22—42no—
571C22—42no—
572C22—82no—
573C22—82no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C2×C3×#12011yescharacteristic
578C22C2×C3×#12111yescharacteristic
579C22C3×#52511yescharacteristic
580C4—22no—
581C4—21no—
582C4—21no—
583C4—24no—
584C4—24no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4C2×C3×#12112yes—
592C4C3×#52512yes—
593C3#3292511yescharacteristic, direct product factor, Sylow
594C2—21no—
595C2—21no—
596C2—44no—
597C2—161no—
598C2C3×#284711yescharacteristic
599C2C3×#301811yescharacteristic
600C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#454C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C4×2−1+4C2semidirect
2. C3×#3040C2semidirect
3. C2×C3×#456C2semidirect
5. C3×#2545C2semidirect
6. C3×#1888C2semidirect
7. C3×#1886C2semidirect
8. C3×#1883C2semidirect
9. C3×#1882C2semidirect
10. C3×#1806C2semidirect
11. C3×#1801C2semidirect
12. #32925C3direct (N × H)
50. C4×2−1+4C2×C3semidirect
51. #3040C2×C3semidirect
52. C2×#456C2×C3semidirect
54. #2545C2×C3semidirect
55. #1888C2×C3semidirect
56. #1886C2×C3semidirect
57. #1883C2×C3semidirect
58. #1882C2×C3semidirect
59. #1806C2×C3semidirect
60. #1801C2×C3semidirect
593. C3#32925direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×2−1+42
C3×#30402
C2×C3×#4562
C2×C3×#4542
C3×#25452
C3×#18882
C3×#18862
C3×#18832
C3×#18822
C3×#18062
C3×#18012
#329253