Finite Group Databasebeta

← C3×#32941 #10651548

Subgroups — C3×#32941 #10651548

Order 768.

1342 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#131C211yesmaximal, characteristic, complemented
2C2×C3×#471C212yesmaximal, complemented
3C2×C3×#465C211yesmaximal, characteristic, complemented
4C3×#3007C211yesmaximal, characteristic, complemented
5C3×#3006C211yesmaximal, characteristic
6C3×#1908C212yesmaximal, complemented
7C3×#1907C212yesmaximal, complemented
8C3×#1901C211yesmaximal, characteristic, complemented
9C3×#1900C211yesmaximal, characteristic, complemented
10C3×#1815C212yesmaximal, complemented
11C3×#1810C211yesmaximal, characteristic, complemented
12#32941C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#137C2212yes—
15C2×C3×AGL(1,ℤ/8)C2212yes—
16C22×C3×SD16C2211yescharacteristic, complemented
17C2×C3×#131—28no—
18C2×C3×#131—21no—
19C2×C3×#131—21no—
20C2×C3×#131C2212yes—
21C2×C3×#131C2212yes—
22C22×C3×M16—21no—
23C22×C3×M16C2211yescharacteristic
24C23×C3×C8—21no—
25C23×C3×C8C2211yescharacteristic
26C2×C3×C22⋊Q8C2211yescharacteristic
27C2×C3×#121C2211yescharacteristic
28C3×#471—28no—
29C3×#465—21no—
30C3×#465—21no—
31C3×#465—21no—
32C3×#465—21no—
33C3×#452—22no—
34C3×#451—21no—
35C3×#450—21no—
36C3×#449—22no—
37C3×#448—21no—
38C3×#447—21no—
39C2×C3×#108C2212yescomplemented
40C2×C3×#106C2211yescharacteristic, complemented
41C3×#417C2212yes—
42C3×#417C2212yes—
43C2×C3×#103C2211yescharacteristic, complemented
44C2×C3×#102C2211yescharacteristic, complemented
45C2×C3×#101C2212yescomplemented
46C2×C3×#100C2212yescomplemented
47C3×#407C2211yescharacteristic
48C3×#407C2211yescharacteristic
49C3×#343C2212yescomplemented
50C3×#343C2212yescomplemented
51C3×#341C2212yescomplemented
52C22×#131C2×C311yescharacteristic, complemented
53C2×#471C2×C312yescomplemented
54C2×#465C2×C311yescharacteristic, complemented
55#3007C2×C311yescharacteristic, complemented
56#3006C2×C311yescharacteristic
57#1908C2×C312yescomplemented
58#1907C2×C312yescomplemented
59#1901C2×C311yescharacteristic, complemented
60#1900C2×C311yescharacteristic, complemented
61#1815C2×C312yescomplemented
62#1810C2×C311yescharacteristic, complemented
63C2×C3×#41—24no—
64C2×C3×#41—24no—
65C2×C3×#41—21no—
66C2×C3×#41—21no—
67C2×C3×#41D412yes—
68C2×C3×#41C2312yes—
69C22×C3×Q8D411yescharacteristic
70C22×C3×Q8C2311yescharacteristic
71C22×C3×D4—21no—
72C22×C3×D4—21no—
73C22×C3×D4D411yescharacteristic
74C22×C3×D4C2311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C3×#137—24no—
78C3×#137—24no—
79C3×AGL(1,ℤ/8)—24no—
80C3×AGL(1,ℤ/8)—24no—
81C2×C3×Q16—22no—
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—21no—
85C2×C3×SD16—21no—
86C2×C3×SD16—24no—
87C2×C3×D8—22no—
88C3×#131—28no—
89C3×#131—28no—
90C3×#131—24no—
91C3×#131—24no—
92C3×#131—24no—
93C3×#131D418yescomplemented
94C2×C3×M16—28no—
95C2×C3×M16—21no—
96C2×C3×M16—24no—
97C2×C3×M16—22no—
98C2×C3×M16—22no—
99C2×C3×M16C2312yes—
100C2×C3×M16C2312yes—
101C2×C3×M16C2312yes—
102C22×C3×C8—28no—
103C22×C3×C8—21no—
104C22×C3×C8—22no—
105C22×C3×C8—22no—
106C22×C3×C8—24no—
107C22×C3×C8D411yescharacteristic
108C22×C3×C8D411yescharacteristic
109C22×C3×C8D411yescharacteristic
110C22×C3×C8D411yescharacteristic
111C22×C3×C8C2311yescharacteristic
112C22×C3×C8C2311yescharacteristic
113C3×C22⋊Q8—21no—
114C3×C22⋊Q8—21no—
115C3×C22⋊Q8—21no—
116C3×C22⋊Q8—21no—
117C3×#121—21no—
118C3×#121—21no—
119C3×#121—21no—
120C3×#121—21no—
121C2×C3×C4⋊C4—21no—
122C2×C3×C4⋊C4C2311yescharacteristic
123C2×C3×C4⋊C4C2311yescharacteristic
124C2×C3×C22⋊C4—21no—
125C2×C3×C22⋊C4—21no—
126C3×#108—24no—
127C3×#106—21no—
128C3×#106—21no—
129C3×#103—22no—
130C3×#103—22no—
131C3×#103—21no—
132C3×#103—21no—
133C3×#102—22no—
134C3×#102—22no—
135C3×#102—21no—
136C3×#102—21no—
137C3×#101—24no—
138C3×#100—24no—
139C3×C22⋊C8—21no—
140C3×C22⋊C8—21no—
141C3×C22⋊C8—21no—
142C3×C22⋊C8—21no—
143C22×#41C22×C311yescharacteristic
144C2×#137C22×C312yes—
145C2×AGL(1,ℤ/8)C22×C312yes—
146C22×SD16C22×C311yescharacteristic, complemented
147C2×#131—28no—
148C2×#131—21no—
149C2×#131—21no—
150C2×#131C22×C312yes—
151C2×#131C22×C312yes—
152C22×M16—21no—
153C22×M16C22×C311yescharacteristic
154C23×C8—21no—
155C23×C8C22×C311yescharacteristic
156C2×C22⋊Q8C22×C311yescharacteristic
157C2×#121C22×C311yescharacteristic
158#471—28no—
159#465—21no—
160#465—21no—
161#465—21no—
162#465—21no—
163#452—22no—
164#451—21no—
165#450—21no—
166#449—22no—
167#448—21no—
168#447—21no—
169C2×#108C22×C312yescomplemented
170C2×#106C22×C311yescharacteristic, complemented
171#417C22×C312yes—
172#417C22×C312yes—
173C2×#103C22×C311yescharacteristic, complemented
174C2×#102C22×C311yescharacteristic, complemented
175C2×#101C22×C312yescomplemented
176C2×#100C22×C312yescomplemented
177#407C22×C311yescharacteristic
178#407C22×C311yescharacteristic
179#343C22×C312yescomplemented
180#343C22×C312yescomplemented
181#341C22×C312yescomplemented
182C24×C3—21no—
183C24×C3—21no—
184C24×C3#4111yescharacteristic
185C3×#41—24no—
186C3×#41—28no—
187C3×#41—24no—
188C3×#41—24no—
189C3×#41—24no—
190C3×#41—22no—
191C3×#41—22no—
192C3×#41C2×D414yes—
193C3×#41#4114yes—
194C2×C3×Q8—24no—
195C2×C3×Q8—21no—
196C2×C3×Q8—22no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—22no—
200C2×C3×Q8C2×D412yes—
201C2×C3×D4—28no—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—24no—
205C2×C3×D4—21no—
206C2×C3×D4—22no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—41no—
211C2×C3×D4—41no—
212C2×C3×D4—82no—
213C2×C3×D4C2×D412yes—
214C22×C3×C4—24no—
215C22×C3×C4—24no—
216C22×C3×C4—21no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4#4111yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C3×Q16—44no—
231C3×SD16—44no—
232C3×SD16—44no—
233C3×SD16—42no—
234C3×SD16—42no—
235C3×D8—44no—
236C3×M16—24no—
237C3×M16—28no—
238C3×M16—22no—
239C3×M16—28no—
240C3×M16—22no—
241C3×M16—22no—
242C3×M16—22no—
243C3×M16—22no—
244C3×M16C2×D414yes—
245C2×C3×C8—24no—
246C2×C3×C8—28no—
247C2×C3×C8—22no—
248C2×C3×C8—22no—
249C2×C3×C8—22no—
250C2×C3×C8—28no—
251C2×C3×C8—22no—
252C2×C3×C8—22no—
253C2×C3×C8—22no—
254C2×C3×C8—22no—
255C2×C3×C8—22no—
256C2×C3×C8C2×D411yescharacteristic
257C2×C3×C8C2×D411yescharacteristic
258C2×C3×C8C2×D411yescharacteristic
259C2×C3×C8C2×D411yescharacteristic
260C2×C3×C8C2×D414yes—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—41no—
266C3×C4⋊C4—41no—
267C3×C22⋊C4—41no—
268C3×C22⋊C4—41no—
269C3×C22⋊C4—41no—
270C3×C22⋊C4—41no—
271C2×#41—24no—
272C2×#41—24no—
273C2×#41—21no—
274C2×#41—21no—
275C2×#41C3×D412yes—
276C2×#41C23×C312yes—
277C22×Q8C3×D411yescharacteristic
278C22×Q8C23×C311yescharacteristic
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C3×D411yescharacteristic
282C22×D4C23×C311yescharacteristic
283C23×C4—21no—
284C23×C4C23×C311yescharacteristic
285#137—24no—
286#137—24no—
287AGL(1,ℤ/8)—24no—
288AGL(1,ℤ/8)—24no—
289C2×Q16—22no—
290C2×SD16—22no—
291C2×SD16—22no—
292C2×SD16—21no—
293C2×SD16—21no—
294C2×SD16—24no—
295C2×D8—22no—
296#131—28no—
297#131—28no—
298#131—24no—
299#131—24no—
300#131—24no—
301#131C3×D418yescomplemented
302C2×M16—28no—
303C2×M16—21no—
304C2×M16—24no—
305C2×M16—22no—
306C2×M16—22no—
307C2×M16C23×C312yes—
308C2×M16C23×C312yes—
309C2×M16C23×C312yes—
310C22×C8—28no—
311C22×C8—21no—
312C22×C8—22no—
313C22×C8—22no—
314C22×C8—24no—
315C22×C8C3×D411yescharacteristic
316C22×C8C3×D411yescharacteristic
317C22×C8C3×D411yescharacteristic
318C22×C8C3×D411yescharacteristic
319C22×C8C23×C311yescharacteristic
320C22×C8C23×C311yescharacteristic
321C22⋊Q8—21no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325#121—21no—
326#121—21no—
327#121—21no—
328#121—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C23×C311yescharacteristic
331C2×C4⋊C4C23×C311yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334#108—24no—
335#106—21no—
336#106—21no—
337#103—22no—
338#103—22no—
339#103—21no—
340#103—21no—
341#102—22no—
342#102—22no—
343#102—21no—
344#102—21no—
345#101—24no—
346#100—24no—
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C23×C3—24no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—41no—
356C23×C3—42no—
357C23×C3—42no—
358C23×C3—41no—
359C23×C3—42no—
360C23×C3—81no—
361C23×C3—81no—
362C23×C3#12111yescharacteristic
363C23×C3#12111yescharacteristic
364C23×C3C2×#4111yescharacteristic
365C3×Q8—24no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—42no—
369C3×Q8—41no—
370C3×Q8#12114yes—
371C3×D4—28no—
372C3×D4—24no—
373C3×D4—24no—
374C3×D4—24no—
375C3×D4—22no—
376C3×D4—41no—
377C3×D4—42no—
378C3×D4—41no—
379C3×D4—82no—
380C3×D4—82no—
381C3×D4#12114yes—
382C2×C3×C4—24no—
383C2×C3×C4—28no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—24no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—24no—
393C2×C3×C4—21no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—42no—
398C2×C3×C4—81no—
399C2×C3×C4—81no—
400C2×C3×C4#12111yescharacteristic
401C2×C3×C4#12111yescharacteristic
402C2×C3×C4C22×D412yes—
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4112yes—
406C3×C8—21no—
407C3×C8—24no—
408C3×C8—21no—
409C3×C8—42no—
410C3×C8—42no—
411C3×C8#12012yes—
412C3×C8#12012yes—
413C24—21no—
414C24—21no—
415C24C3×#4111yescharacteristic
416#41—24no—
417#41—28no—
418#41—24no—
419#41—24no—
420#41—24no—
421#41—22no—
422#41—22no—
423#41C2×C3×D414yes—
424#41C3×#4114yes—
425C2×Q8—24no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8C2×C3×D412yes—
432C2×D4—28no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4—82no—
444C2×D4C2×C3×D412yes—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4C2×C3×D411yescharacteristic
456C22×C4C2×C3×D411yescharacteristic
457C22×C4C2×C3×D411yescharacteristic
458C22×C4C2×C3×D411yescharacteristic
459C22×C4C3×#4111yescharacteristic
460C22×C4C24×C311yescharacteristic
461Q16—44no—
462SD16—44no—
463SD16—44no—
464SD16—42no—
465SD16—42no—
466D8—44no—
467M16—24no—
468M16—28no—
469M16—22no—
470M16—28no—
471M16—22no—
472M16—22no—
473M16—22no—
474M16—22no—
475M16C2×C3×D414yes—
476C2×C8—24no—
477C2×C8—28no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—28no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8C2×C3×D411yescharacteristic
488C2×C8C2×C3×D411yescharacteristic
489C2×C8C2×C3×D411yescharacteristic
490C2×C8C2×C3×D411yescharacteristic
491C2×C8C2×C3×D414yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—24no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—44no—
511C22×C3—48no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—162no—
518C22×C3#47112yescomplemented
519C22×C3C2×#12111yescharacteristic
520C22×C3#52512yes—
521C3×C4—21no—
522C3×C4—24no—
523C3×C4—21no—
524C3×C4—81no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4C2×#12011yescharacteristic
528C3×C4C2×#12112yes—
529C3×C4#52511yescharacteristic
530C23—24no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—81no—
540C23—81no—
541C23C3×#12111yescharacteristic
542C23C3×#12111yescharacteristic
543C23C2×C3×#4111yescharacteristic
544Q8—24no—
545Q8—22no—
546Q8—41no—
547Q8—42no—
548Q8—41no—
549Q8C3×#12114yes—
550D4—28no—
551D4—24no—
552D4—24no—
553D4—24no—
554D4—22no—
555D4—41no—
556D4—42no—
557D4—41no—
558D4—82no—
559D4—82no—
560D4C3×#12114yes—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4C3×#12111yescharacteristic
580C2×C4C3×#12111yescharacteristic
581C2×C4C22×C3×D412yes—
582C2×C4C22×C3×D411yescharacteristic
583C2×C4C22×C3×D411yescharacteristic
584C2×C4C2×C3×#4112yes—
585C8—21no—
586C8—24no—
587C8—21no—
588C8—42no—
589C8—42no—
590C8C3×#12012yes—
591C8C3×#12012yes—
592C2×C3—22no—
593C2×C3—22no—
594C2×C3—41no—
595C2×C3—44no—
596C2×C3—161no—
597C2×C3#284711yescharacteristic
598C2×C3C2×#47111yescharacteristic
599C2×C3#306711yescharacteristic
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—24no—
608C22—44no—
609C22—48no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—162no—
616C22C3×#47112yescomplemented
617C22C2×C3×#12111yescharacteristic
618C22C3×#52512yes—
619C4—21no—
620C4—24no—
621C4—21no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4C2×C3×#12011yescharacteristic
626C4C2×C3×#12112yes—
627C4C3×#52511yescharacteristic
628C3#3294111yescharacteristic, direct product factor, Sylow
629C2—22no—
630C2—22no—
631C2—41no—
632C2—44no—
633C2—161no—
634C2C3×#284711yescharacteristic
635C2C2×C3×#47111yescharacteristic
636C2C3×#306711yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#131C2semidirect
2. C2×C3×#471C2semidirect
3. C2×C3×#465C2semidirect
4. C3×#3007C2semidirect
6. C3×#1908C2semidirect
7. C3×#1907C2semidirect
8. C3×#1901C2semidirect
9. C3×#1900C2semidirect
10. C3×#1815C2semidirect
11. C3×#1810C2semidirect
12. #32941C3direct (N × H)
16. C22×C3×SD16C22semidirect
39. C2×C3×#108C22semidirect
40. C2×C3×#106C22semidirect
43. C2×C3×#103C22semidirect
44. C2×C3×#102C22semidirect
45. C2×C3×#101C22semidirect
46. C2×C3×#100C22semidirect
49. C3×#343C22semidirect
50. C3×#343C22semidirect
51. C3×#341C22semidirect
52. C22×#131C2×C3semidirect
53. C2×#471C2×C3semidirect
54. C2×#465C2×C3semidirect
55. #3007C2×C3semidirect
57. #1908C2×C3semidirect
58. #1907C2×C3semidirect
59. #1901C2×C3semidirect
60. #1900C2×C3semidirect
61. #1815C2×C3semidirect
62. #1810C2×C3semidirect
93. C3×#131D4semidirect
146. C22×SD16C22×C3semidirect
169. C2×#108C22×C3semidirect
170. C2×#106C22×C3semidirect
173. C2×#103C22×C3semidirect
174. C2×#102C22×C3semidirect
175. C2×#101C22×C3semidirect
176. C2×#100C22×C3semidirect
179. #343C22×C3semidirect
180. #343C22×C3semidirect
181. #341C22×C3semidirect
301. #131C3×D4semidirect
518. C22×C3#471semidirect
616. C22C3×#471semidirect
628. C3#32941direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1312
C2×C3×#4712
C2×C3×#4652
C3×#30072
C3×#30062
C3×#19082
C3×#19072
C3×#19012
C3×#19002
C3×#18152
C3×#18102
#329413