Finite Group Databasebeta

← C3×#32998 #10651605

Subgroups — C3×#32998 #10651605

Order 768.

1284 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3428C211yesmaximal, characteristic, complemented
2C3×#3094C211yesmaximal, characteristic, complemented
3C3×#3006C211yesmaximal, characteristic
4C3×#2500C211yesmaximal, characteristic, complemented
5C3×#1883C212yesmaximal, complemented
6C3×#1883C212yesmaximal, complemented
7C3×#1860C212yesmaximal, complemented
8C3×#1859C212yesmaximal, complemented
9C3×#1826C211yesmaximal, characteristic, complemented
10C3×#1822C211yesmaximal, characteristic, complemented
11C3×#1789C211yesmaximal, characteristic, complemented
12#32998C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×AGL(1,ℤ/8)C2212yes—
15C2×C3×#121C2211yescharacteristic
16C3×#520—21no—
17C3×#520—21no—
18C3×#520C2212yes—
19C3×#520C2212yes—
20C3×#519—21no—
21C3×#518—21no—
22C3×#518—22no—
23C3×C4×#41—22no—
24C3×C4×#41—22no—
25C2×C3×#117—22no—
26C2×C3×#117C2211yescharacteristic
27C2×C3×#117C2211yescharacteristic
28C3×#482—22no—
29C3×#481—21no—
30C3×#480—21no—
31C3×#449—22no—
32C3×#448—21no—
33C3×#447—21no—
34C3×#428C2212yescomplemented
35C2×C3×#104C2212yes—
36C2×C3×#104C2212yes—
37C3×#418C2212yescomplemented
38C3×#417C2212yes—
39C3×#407C2211yescharacteristic
40C3×#407C2211yescharacteristic
41C3×#394—22no—
42C3×#390C2211yescharacteristic, complemented
43C3×#390C2211yescharacteristic, complemented
44C3×#388—22no—
45C3×#384C2211yescharacteristic, complemented
46C3×#382C2211yescharacteristic, complemented
47C3×C4×C22⋊C4C2211yescharacteristic
48C3×#343C2212yescomplemented
49C3×#343C2212yescomplemented
50C3×#339C2212yescomplemented
51C3×#339C2212yescomplemented
52#3428C2×C311yescharacteristic, complemented
53#3094C2×C311yescharacteristic, complemented
54#3006C2×C311yescharacteristic
55#2500C2×C311yescharacteristic, complemented
56#1883C2×C312yescomplemented
57#1883C2×C312yescomplemented
58#1860C2×C312yescomplemented
59#1859C2×C312yescomplemented
60#1826C2×C311yescharacteristic, complemented
61#1822C2×C311yescharacteristic, complemented
62#1789C2×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×D4—21no—
71C22×C3×D4—21no—
72C22×C3×D4D411yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C23×C3×C4—21no—
75C23×C3×C4C2311yescharacteristic
76C3×AGL(1,ℤ/8)—24no—
77C3×AGL(1,ℤ/8)—24no—
78C2×C3×SD16—22no—
79C2×C3×D8—22no—
80C2×C3×M16C2312yes—
81C2×C3×M16C2312yes—
82C3×#121—21no—
83C3×#121—21no—
84C3×#121—21no—
85C3×#121—21no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×D4—24no—
91C3×C4×D4—24no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—24no—
97C3×C4×D4—24no—
98C3×#117—24no—
99C3×#117—24no—
100C3×#117—21no—
101C3×#117—21no—
102C3×#117—21no—
103C3×#117—21no—
104C3×#117—21no—
105C3×#117—21no—
106C3×#117—22no—
107C3×#117—22no—
108C3×#117D412yes—
109C3×#117C2312yes—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—22no—
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4D411yescharacteristic
114C2×C3×C4⋊C4D411yescharacteristic
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C22⋊C4—24no—
119C2×C3×C22⋊C4—21no—
120C2×C3×C22⋊C4—21no—
121C2×C3×C22⋊C4—21no—
122C2×C3×C22⋊C4—22no—
123C2×C3×C22⋊C4—22no—
124C2×C3×C22⋊C4D412yes—
125C2×C3×C42—24no—
126C2×C3×C42—21no—
127C2×C3×C42D412yes—
128C2×C3×C42C2311yescharacteristic
129C2×C3×C42C2311yescharacteristic
130C3×#107—22no—
131C3×#106—22no—
132C3×#104—24no—
133C3×#104—24no—
134C3×#103—22no—
135C3×#102—22no—
136C3×#102—22no—
137C3×#102—22no—
138C3×C22⋊C8—21no—
139C3×C22⋊C8—21no—
140C3×C22⋊C8—21no—
141C3×C22⋊C8—21no—
142C3×#95—21no—
143C3×#95—22no—
144C22×#41C22×C311yescharacteristic
145C2×AGL(1,ℤ/8)C22×C312yes—
146C2×#121C22×C311yescharacteristic
147#520—21no—
148#520—21no—
149#520C22×C312yes—
150#520C22×C312yes—
151#519—21no—
152#518—21no—
153#518—22no—
154C4×#41—22no—
155C4×#41—22no—
156C2×#117—22no—
157C2×#117C22×C311yescharacteristic
158C2×#117C22×C311yescharacteristic
159#482—22no—
160#481—21no—
161#480—21no—
162#449—22no—
163#448—21no—
164#447—21no—
165#428C22×C312yescomplemented
166C2×#104C22×C312yes—
167C2×#104C22×C312yes—
168#418C22×C312yescomplemented
169#417C22×C312yes—
170#407C22×C311yescharacteristic
171#407C22×C311yescharacteristic
172#394—22no—
173#390C22×C311yescharacteristic, complemented
174#390C22×C311yescharacteristic, complemented
175#388—22no—
176#384C22×C311yescharacteristic, complemented
177#382C22×C311yescharacteristic, complemented
178C4×C22⋊C4C22×C311yescharacteristic
179#343C22×C312yescomplemented
180#343C22×C312yescomplemented
181#339C22×C312yescomplemented
182#339C22×C312yescomplemented
183C24×C3—21no—
184C24×C3—21no—
185C24×C3C2×D411yescharacteristic
186C3×#41—24no—
187C3×#41—24no—
188C3×#41—28no—
189C3×#41—24no—
190C3×#41—24no—
191C3×#41—22no—
192C3×#41—22no—
193C3×#41#4114yes—
194C3×#41#4114yes—
195C2×C3×Q8—24no—
196C2×C3×Q8—21no—
197C2×C3×Q8C2×D412yes—
198C2×C3×D4—24no—
199C2×C3×D4—24no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—22no—
207C2×C3×D4—22no—
208C2×C3×D4—41no—
209C2×C3×D4—41no—
210C2×C3×D4—82no—
211C2×C3×D4C2×D412yes—
212C22×C3×C4—28no—
213C22×C3×C4—21no—
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—22no—
217C22×C3×C4—22no—
218C22×C3×C4—21no—
219C22×C3×C4—24no—
220C22×C3×C4—24no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D412yes—
231C22×C3×C4C2×D412yes—
232C22×C3×C4C2411yescharacteristic
233C3×SD16—44no—
234C3×D8—44no—
235C3×M16—22no—
236C3×M16—22no—
237C3×M16—22no—
238C3×M16—22no—
239C2×C3×C8—22no—
240C2×C3×C8—22no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—82no—
252C3×C4⋊C4C2×D412yes—
253C3×C4⋊C4C2×D412yes—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—44no—
261C3×C22⋊C4—42no—
262C3×C22⋊C4—41no—
263C3×C22⋊C4—41no—
264C3×C22⋊C4—82no—
265C3×C42—24no—
266C3×C42—24no—
267C3×C42—22no—
268C3×C42—22no—
269C3×C42—22no—
270C3×C42—22no—
271C3×C42—22no—
272C3×C42—22no—
273C2×#41—24no—
274C2×#41—24no—
275C2×#41—21no—
276C2×#41—21no—
277C2×#41C3×D412yes—
278C2×#41C23×C312yes—
279C22×Q8C3×D411yescharacteristic
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C3×D411yescharacteristic
283C22×D4C23×C311yescharacteristic
284C23×C4—21no—
285C23×C4C23×C311yescharacteristic
286AGL(1,ℤ/8)—24no—
287AGL(1,ℤ/8)—24no—
288C2×SD16—22no—
289C2×D8—22no—
290C2×M16C23×C312yes—
291C2×M16C23×C312yes—
292#121—21no—
293#121—21no—
294#121—21no—
295#121—21no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—24no—
307C4×D4—24no—
308#117—24no—
309#117—24no—
310#117—21no—
311#117—21no—
312#117—21no—
313#117—21no—
314#117—21no—
315#117—21no—
316#117—22no—
317#117—22no—
318#117C3×D412yes—
319#117C23×C312yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4C3×D411yescharacteristic
324C2×C4⋊C4C3×D411yescharacteristic
325C2×C4⋊C4C23×C311yescharacteristic
326C2×C4⋊C4C23×C311yescharacteristic
327C2×C4⋊C4C23×C311yescharacteristic
328C2×C22⋊C4—24no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C3×D412yes—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42C3×D412yes—
338C2×C42C23×C311yescharacteristic
339C2×C42C23×C311yescharacteristic
340#107—22no—
341#106—22no—
342#104—24no—
343#104—24no—
344#103—22no—
345#102—22no—
346#102—22no—
347#102—22no—
348C22⋊C8—21no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352#95—21no—
353#95—22no—
354C23×C3—24no—
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—42no—
361C23×C3—42no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—81no—
366C23×C3—81no—
367C23×C3#12011yescharacteristic
368C23×C3#12011yescharacteristic
369C23×C3C22×D411yescharacteristic
370C3×Q8—24no—
371C3×Q8—22no—
372C3×Q8#12314yes—
373C3×D4—28no—
374C3×D4—24no—
375C3×D4—24no—
376C3×D4—24no—
377C3×D4—22no—
378C3×D4—42no—
379C3×D4—42no—
380C3×D4—82no—
381C3×D4—82no—
382C3×D4#12314yes—
383C2×C3×C4—24no—
384C2×C3×C4—28no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—24no—
392C2×C3×C4—24no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—42no—
402C2×C3×C4—42no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—42no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—81no—
409C2×C3×C4—81no—
410C2×C3×C4—84no—
411C2×C3×C4—84no—
412C2×C3×C4#12011yescharacteristic
413C2×C3×C4#12011yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C2×#4112yes—
417C2×C3×C4C2×#4112yes—
418C3×C8—42no—
419C3×C8—42no—
420C24—21no—
421C24—21no—
422C24C2×C3×D411yescharacteristic
423#41—24no—
424#41—24no—
425#41—28no—
426#41—24no—
427#41—24no—
428#41—22no—
429#41—22no—
430#41C3×#4114yes—
431#41C3×#4114yes—
432C2×Q8—24no—
433C2×Q8—21no—
434C2×Q8C2×C3×D412yes—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—82no—
448C2×D4C2×C3×D412yes—
449C22×C4—28no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4C2×C3×D411yescharacteristic
463C22×C4C2×C3×D411yescharacteristic
464C22×C4C2×C3×D411yescharacteristic
465C22×C4C2×C3×D411yescharacteristic
466C22×C4C2×C3×D411yescharacteristic
467C22×C4C2×C3×D412yes—
468C22×C4C2×C3×D412yes—
469C22×C4C24×C311yescharacteristic
470SD16—44no—
471D8—44no—
472M16—22no—
473M16—22no—
474M16—22no—
475M16—22no—
476C2×C8—22no—
477C2×C8—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—82no—
489C4⋊C4C2×C3×D412yes—
490C4⋊C4C2×C3×D412yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—82no—
502C42—24no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C22×C3—21no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—24no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—82no—
524C22×C3—82no—
525C22×C3—84no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3C2×#12011yescharacteristic
530C22×C3#52511yescharacteristic
531C22×C3#52511yescharacteristic
532C3×C4—21no—
533C3×C4—24no—
534C3×C4—21no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4—42no—
538C3×C4—42no—
539C3×C4—42no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4C2×#12312yes—
543C3×C4#52512yes—
544C23—24no—
545C23—22no—
546C23—21no—
547C23—21no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—81no—
556C23—81no—
557C23C3×#12011yescharacteristic
558C23C3×#12011yescharacteristic
559C23C22×C3×D411yescharacteristic
560Q8—24no—
561Q8—22no—
562Q8C3×#12314yes—
563D4—28no—
564D4—24no—
565D4—24no—
566D4—24no—
567D4—22no—
568D4—42no—
569D4—42no—
570D4—82no—
571D4—82no—
572D4C3×#12314yes—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4C3×#12011yescharacteristic
603C2×C4C3×#12011yescharacteristic
604C2×C4C22×C3×D411yescharacteristic
605C2×C4C22×C3×D411yescharacteristic
606C2×C4C2×C3×#4112yes—
607C2×C4C2×C3×#4112yes—
608C8—42no—
609C8—42no—
610C2×C3—21no—
611C2×C3—21no—
612C2×C3—41no—
613C2×C3—41no—
614C2×C3—44no—
615C2×C3—161no—
616C2×C3#284911yescharacteristic
617C2×C3#301811yescharacteristic
618C2×C3#302111yescharacteristic
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—81no—
631C22—81no—
632C22—82no—
633C22—82no—
634C22—84no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22C2×C3×#12011yescharacteristic
639C22C3×#52511yescharacteristic
640C22C3×#52511yescharacteristic
641C4—21no—
642C4—24no—
643C4—21no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4C2×C3×#12312yes—
652C4C3×#52512yes—
653C3#3299811yescharacteristic, direct product factor, Sylow
654C2—21no—
655C2—21no—
656C2—41no—
657C2—41no—
658C2—44no—
659C2—161no—
660C2C3×#284911yescharacteristic
661C2C3×#301811yescharacteristic
662C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
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. C3×#3428C2semidirect
2. C3×#3094C2semidirect
4. C3×#2500C2semidirect
5. C3×#1883C2semidirect
6. C3×#1883C2semidirect
7. C3×#1860C2semidirect
8. C3×#1859C2semidirect
9. C3×#1826C2semidirect
10. C3×#1822C2semidirect
11. C3×#1789C2semidirect
12. #32998C3direct (N × H)
34. C3×#428C22semidirect
37. C3×#418C22semidirect
42. C3×#390C22semidirect
43. C3×#390C22semidirect
45. C3×#384C22semidirect
46. C3×#382C22semidirect
48. C3×#343C22semidirect
49. C3×#343C22semidirect
50. C3×#339C22semidirect
51. C3×#339C22semidirect
52. #3428C2×C3semidirect
53. #3094C2×C3semidirect
55. #2500C2×C3semidirect
56. #1883C2×C3semidirect
57. #1883C2×C3semidirect
58. #1860C2×C3semidirect
59. #1859C2×C3semidirect
60. #1826C2×C3semidirect
61. #1822C2×C3semidirect
62. #1789C2×C3semidirect
165. #428C22×C3semidirect
168. #418C22×C3semidirect
173. #390C22×C3semidirect
174. #390C22×C3semidirect
176. #384C22×C3semidirect
177. #382C22×C3semidirect
179. #343C22×C3semidirect
180. #343C22×C3semidirect
181. #339C22×C3semidirect
182. #339C22×C3semidirect
653. C3#32998direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34282
C3×#30942
C3×#30062
C3×#25002
C3×#18832
C3×#18602
C3×#18592
C3×#18262
C3×#18222
C3×#17892
#329983