Finite Group Databasebeta

← C3×#25669 #10644276

Subgroups — C3×#25669 #10644276

Order 768.

992 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2C3×#3142C211yesmaximal, characteristic, complemented
3C3×#3080C211yesmaximal, characteristic, complemented
4C3×#2660C211yesmaximal, characteristic, complemented
5C3×#2040C212yesmaximal, complemented
6C3×#2039C212yesmaximal, complemented
7C3×#1997C211yesmaximal, characteristic, complemented
8C3×#1963C211yesmaximal, characteristic, complemented
9C3×#1860C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C3×#1759C211yesmaximal, characteristic, complemented
12#25669C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#556—21no—
14C3×#555—21no—
15C3×#554—21no—
16C3×#553—22no—
17C3×#548—22no—
18C3×#547—21no—
19C3×#542—22no—
20C3×#537C2212yes—
21C3×#533—21no—
22C3×#532—21no—
23C3×#532C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C2×C3×#125C2211yescharacteristic
26C3×#520C2212yes—
27C3×C4×#41C2211yescharacteristic
28C3×#491—21no—
29C3×#490—21no—
30C3×#488—22no—
31C3×#479—21no—
32C3×#479—21no—
33C3×#478—21no—
34C3×#478—21no—
35C3×#428C2211yescharacteristic
36C3×#428C2211yescharacteristic
37C3×#423C2211yescharacteristic
38C2×C3×#104C2212yes—
39C2×C3×#104C2212yes—
40C2×C3×#104C2212yes—
41C3×#417C2212yes—
42C3×#417C2212yes—
43C3×#417C2211yescharacteristic
44C3×#417C2211yescharacteristic
45C3×C4×M16C2211yescharacteristic
46C3×#398—22no—
47C3×#395—22no—
48C3×#384C2212yes—
49C3×#382C2212yes—
50C3×#376C2211yescharacteristic
51C3×#343C2212yes—
52C3×#343C2212yes—
53C3×#339C2212yes—
54#3511C2×C311yescharacteristic
55#3142C2×C311yescharacteristic, complemented
56#3080C2×C311yescharacteristic, complemented
57#2660C2×C311yescharacteristic, complemented
58#2040C2×C312yescomplemented
59#2039C2×C312yescomplemented
60#1997C2×C311yescharacteristic, complemented
61#1963C2×C311yescharacteristic, complemented
62#1860C2×C312yescomplemented
63#1859C2×C312yescomplemented
64#1759C2×C311yescharacteristic, complemented
65C2×C3×#41—22no—
66C2×C3×#41C2311yescharacteristic
67C2×C3×#41C2312yes—
68C22×C3×Q8D412yes—
69C2×C3×M16C2312yes—
70C2×C3×M16C2311yescharacteristic
71C2×C3×M16C2311yescharacteristic
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8D411yescharacteristic
77C3×#127D411yescharacteristic
78C3×#126—24no—
79C3×#125—22no—
80C3×#125—21no—
81C3×#125—21no—
82C3×#125—21no—
83C3×#125—21no—
84C3×#124—22no—
85C3×#124—24no—
86C3×#124D412yes—
87C3×#123—24no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—24no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×#121—22no—
93C3×#121—22no—
94C3×#121—22no—
95C3×#121—21no—
96C3×#121—21no—
97C3×#121—24no—
98C3×C4×Q8—22no—
99C3×C4×Q8—22no—
100C3×C4×Q8D411yescharacteristic
101C3×C4×Q8D411yescharacteristic
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×C4×D4—24no—
107C3×C4×D4—21no—
108C3×C4×D4—21no—
109C3×C4×D4D411yescharacteristic
110C3×C4×D4D411yescharacteristic
111C3×#117—22no—
112C3×#117—22no—
113C3×#117—21no—
114C3×#117C2311yescharacteristic
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—21no—
117C2×C3×C4⋊C4—21no—
118C2×C3×C4⋊C4—24no—
119C2×C3×C4⋊C4D412yes—
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C4⋊C4C2312yes—
123C2×C3×C42—21no—
124C2×C3×C42C2312yes—
125C2×C3×C42C2311yescharacteristic
126C3×#107—21no—
127C3×#107—21no—
128C3×#106—21no—
129C3×#106—21no—
130C3×C4⋊C8—21no—
131C3×C4⋊C8—21no—
132C3×#104—24no—
133C3×#104—24no—
134C3×#104—24no—
135C3×#103—21no—
136C3×#103—21no—
137C3×#103—22no—
138C3×#103—22no—
139C3×#102—21no—
140C3×#102—21no—
141C3×#102—22no—
142C3×#102—22no—
143C3×#97—21no—
144C3×C4×C8—21no—
145C3×#95—21no—
146C3×#95—21no—
147C3×#95—24no—
148#556—21no—
149#555—21no—
150#554—21no—
151#553—22no—
152#548—22no—
153#547—21no—
154#542—22no—
155#537C22×C312yes—
156#533—21no—
157#532—21no—
158#532C22×C311yescharacteristic
159C2×C4⋊Q8C22×C311yescharacteristic
160C2×#125C22×C311yescharacteristic
161#520C22×C312yes—
162C4×#41C22×C311yescharacteristic
163#491—21no—
164#490—21no—
165#488—22no—
166#479—21no—
167#479—21no—
168#478—21no—
169#478—21no—
170#428C22×C311yescharacteristic
171#428C22×C311yescharacteristic
172#423C22×C311yescharacteristic
173C2×#104C22×C312yes—
174C2×#104C22×C312yes—
175C2×#104C22×C312yes—
176#417C22×C311yescharacteristic
177#417C22×C311yescharacteristic
178#417C22×C312yes—
179#417C22×C312yes—
180C4×M16C22×C311yescharacteristic
181#398—22no—
182#395—22no—
183#384C22×C312yes—
184#382C22×C312yes—
185#376C22×C311yescharacteristic
186#343C22×C312yes—
187#343C22×C312yes—
188#339C22×C312yes—
189C3×#41—22no—
190C3×#41—22no—
191C3×#41—22no—
192C3×#41—22no—
193C3×#41—22no—
194C3×#41—22no—
195C3×#41—84no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—82no—
199C2×C3×Q8C2×D412yes—
200C2×C3×Q8C2×D411yescharacteristic
201C2×C3×D4—21no—
202C2×C3×D4—22no—
203C2×C3×D4—24no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4C2×D412yes—
207C2×C3×D4C2×D411yescharacteristic
208C22×C3×C4—22no—
209C22×C3×C4—24no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—22no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D414yes—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C3×M16—22no—
221C3×M16—22no—
222C3×M16—22no—
223C3×M16—22no—
224C2×C3×C8—22no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—84no—
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C4⋊C4C2×D411yescharacteristic
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4—21no—
253C3×C22⋊C4—21no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C42—24no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—22no—
261C3×C42C2×D411yescharacteristic
262C3×C42C2×D411yescharacteristic
263C3×C42C2×D411yescharacteristic
264C3×C42C2×D411yescharacteristic
265C3×C42#4114yescomplemented
266C2×#41—22no—
267C2×#41C23×C311yescharacteristic
268C2×#41C23×C312yes—
269C22×Q8C3×D412yes—
270C2×M16C23×C312yes—
271C2×M16C23×C311yescharacteristic
272C2×M16C23×C311yescharacteristic
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8C3×D411yescharacteristic
278#127C3×D411yescharacteristic
279#126—24no—
280#125—22no—
281#125—21no—
282#125—21no—
283#125—21no—
284#125—21no—
285#124—22no—
286#124—24no—
287#124C3×D412yes—
288#123—24no—
289C22⋊Q8—22no—
290C22⋊Q8—24no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293#121—22no—
294#121—22no—
295#121—22no—
296#121—21no—
297#121—21no—
298#121—24no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8C3×D411yescharacteristic
302C4×Q8C3×D411yescharacteristic
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—21no—
306C4×D4—21no—
307C4×D4—24no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4C3×D411yescharacteristic
311C4×D4C3×D411yescharacteristic
312#117—22no—
313#117—22no—
314#117—21no—
315#117C23×C311yescharacteristic
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4C3×D412yes—
321C2×C4⋊C4C23×C311yescharacteristic
322C2×C4⋊C4C23×C311yescharacteristic
323C2×C4⋊C4C23×C312yes—
324C2×C42—21no—
325C2×C42C23×C312yes—
326C2×C42C23×C311yescharacteristic
327#107—21no—
328#107—21no—
329#106—21no—
330#106—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333#104—24no—
334#104—24no—
335#104—24no—
336#103—21no—
337#103—21no—
338#103—22no—
339#103—22no—
340#102—21no—
341#102—21no—
342#102—22no—
343#102—22no—
344#97—21no—
345C4×C8—21no—
346#95—21no—
347#95—21no—
348#95—24no—
349C23×C3—21no—
350C23×C3—22no—
351C23×C3C22×D411yescharacteristic
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—44no—
357C3×Q8—82no—
358C3×D4—21no—
359C3×D4—21no—
360C3×D4—22no—
361C3×D4—22no—
362C3×D4—42no—
363C3×D4—42no—
364C3×D4—42no—
365C3×D4—42no—
366C3×D4—42no—
367C3×D4—84no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—44no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—84no—
389C2×C3×C4#12011yescharacteristic
390C2×C3×C4#12011yescharacteristic
391C2×C3×C4#12011yescharacteristic
392C2×C3×C4#12011yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C2×#4112yes—
396C2×C3×C42−1+412yes—
397C3×C8—41no—
398C3×C8—41no—
399C3×C8—42no—
400#41—22no—
401#41—22no—
402#41—22no—
403#41—22no—
404#41—22no—
405#41—22no—
406#41—84no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—82no—
410C2×Q8C2×C3×D412yes—
411C2×Q8C2×C3×D411yescharacteristic
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4C2×C3×D412yes—
418C2×D4C2×C3×D411yescharacteristic
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4C2×C3×D414yes—
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C24×C311yescharacteristic
431M16—22no—
432M16—22no—
433M16—22no—
434M16—22no—
435C2×C8—22no—
436C2×C8—21no—
437C2×C8—21no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—84no—
457C4⋊C4C2×C3×D411yescharacteristic
458C4⋊C4C2×C3×D411yescharacteristic
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C42—24no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—22no—
472C42C2×C3×D411yescharacteristic
473C42C2×C3×D411yescharacteristic
474C42C2×C3×D411yescharacteristic
475C42C2×C3×D411yescharacteristic
476C42C3×#4114yescomplemented
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—42no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—42no—
483C22×C3—41no—
484C22×C3—42no—
485C22×C3C2×#12011yescharacteristic
486C22×C3#54812yes—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—44no—
492C3×C4—42no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—44no—
496C3×C4—41no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4#53211yescharacteristic
500C3×C4#53611yescharacteristic
501C3×C4#54712yes—
502C23—21no—
503C23—22no—
504C23C22×C3×D411yescharacteristic
505Q8—21no—
506Q8—21no—
507Q8—22no—
508Q8—22no—
509Q8—44no—
510Q8—82no—
511D4—21no—
512D4—21no—
513D4—22no—
514D4—22no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—84no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4C3×#12011yescharacteristic
543C2×C4C3×#12011yescharacteristic
544C2×C4C3×#12011yescharacteristic
545C2×C4C3×#12011yescharacteristic
546C2×C4C22×C3×D411yescharacteristic
547C2×C4C22×C3×D411yescharacteristic
548C2×C4C2×C3×#4112yes—
549C2×C4C3×2−1+412yes—
550C8—41no—
551C8—41no—
552C8—42no—
553C2×C3—22no—
554C2×C3—81no—
555C2×C3—82no—
556C2×C3#243211yescharacteristic
557C2×C3#301911yescharacteristic
558C2×C3#302011yescharacteristic
559C22—21no—
560C22—21no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22—41no—
566C22—42no—
567C22C2×C3×#12011yescharacteristic
568C22C3×#54812yes—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—44no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—44no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4C3×#53211yescharacteristic
582C4C3×#53611yescharacteristic
583C4C3×#54712yes—
584C3#2566911yescharacteristic, direct product factor, Sylow
585C2—22no—
586C2—81no—
587C2—82no—
588C2C3×#243211yescharacteristic
589C2C3×#301911yescharacteristic
590C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3511C2
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
2. C3×#3142C2semidirect
3. C3×#3080C2semidirect
4. C3×#2660C2semidirect
5. C3×#2040C2semidirect
6. C3×#2039C2semidirect
7. C3×#1997C2semidirect
8. C3×#1963C2semidirect
9. C3×#1860C2semidirect
10. C3×#1859C2semidirect
11. C3×#1759C2semidirect
12. #25669C3direct (N × H)
55. #3142C2×C3semidirect
56. #3080C2×C3semidirect
57. #2660C2×C3semidirect
58. #2040C2×C3semidirect
59. #2039C2×C3semidirect
60. #1997C2×C3semidirect
61. #1963C2×C3semidirect
62. #1860C2×C3semidirect
63. #1859C2×C3semidirect
64. #1759C2×C3semidirect
265. C3×C42#41semidirect
476. C42C3×#41semidirect
584. C3#25669direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35112
C3×#31422
C3×#30802
C3×#26602
C3×#20402
C3×#20392
C3×#19972
C3×#19632
C3×#18602
C3×#18592
C3×#17592
#256693