Finite Group Databasebeta

← C3×#33001 #10651608

Subgroups — C3×#33001 #10651608

Order 768.

1242 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3428C211yesmaximal, characteristic
2C3×#3095C211yesmaximal, characteristic, complemented
3C3×#3007C211yesmaximal, characteristic
4C3×#2499C211yesmaximal, characteristic, complemented
5C3×#1882C212yesmaximal, complemented
6C3×#1882C212yesmaximal, complemented
7C3×#1860C212yesmaximal, complemented
8C3×#1859C212yesmaximal, complemented
9C3×#1826C211yesmaximal, characteristic, complemented
10C3×#1822C211yesmaximal, characteristic, complemented
11C3×#1789C211yesmaximal, characteristic
12#33001C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#137C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
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×#485—22no—
29C3×#484—21no—
30C3×#483—21no—
31C3×#452—22no—
32C3×#451—21no—
33C3×#450—21no—
34C3×#428C2212yescomplemented
35C2×C3×#104C2212yes—
36C2×C3×#104C2212yes—
37C3×#419C2212yescomplemented
38C3×#417C2212yes—
39C3×#407C2211yescharacteristic
40C3×#407C2211yescharacteristic
41C3×#397—22no—
42C3×#391C2211yescharacteristic, complemented
43C3×#391C2211yescharacteristic, complemented
44C3×#387—22no—
45C3×#384C2211yescharacteristic, complemented
46C3×#382C2211yescharacteristic, complemented
47C3×C4×C22⋊C4C2211yescharacteristic
48C3×#343C2212yescomplemented
49C3×#343C2212yescomplemented
50C3×#339C2212yescomplemented
51C3×#339C2212yescomplemented
52#3428C2×C311yescharacteristic
53#3095C2×C311yescharacteristic, complemented
54#3007C2×C311yescharacteristic
55#2499C2×C311yescharacteristic, complemented
56#1882C2×C312yescomplemented
57#1882C2×C312yescomplemented
58#1860C2×C312yescomplemented
59#1859C2×C312yescomplemented
60#1826C2×C311yescharacteristic, complemented
61#1822C2×C311yescharacteristic, complemented
62#1789C2×C311yescharacteristic
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×D4D411yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C3×#137—24no—
76C3×#137—24no—
77C2×C3×Q16—22no—
78C2×C3×SD16—22no—
79C2×C3×M16C2312yes—
80C2×C3×M16C2312yes—
81C3×C22⋊Q8—21no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—21no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×D4—24no—
90C3×C4×D4—24no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C3×C4×D4—24no—
96C3×C4×D4—24no—
97C3×#117—24no—
98C3×#117—24no—
99C3×#117—21no—
100C3×#117—21no—
101C3×#117—21no—
102C3×#117—21no—
103C3×#117—21no—
104C3×#117—21no—
105C3×#117—22no—
106C3×#117—22no—
107C3×#117D412yes—
108C3×#117C2312yes—
109C2×C3×C4⋊C4—22no—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—22no—
112C2×C3×C4⋊C4—21no—
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⋊C4D412yes—
123C2×C3×C42—24no—
124C2×C3×C42—21no—
125C2×C3×C42D412yes—
126C2×C3×C42C2311yescharacteristic
127C2×C3×C42C2311yescharacteristic
128C3×#107—22no—
129C3×#106—22no—
130C3×#104—24no—
131C3×#104—24no—
132C3×#103—22no—
133C3×#103—22no—
134C3×#103—22no—
135C3×#102—22no—
136C3×C22⋊C8—21no—
137C3×C22⋊C8—21no—
138C3×C22⋊C8—21no—
139C3×C22⋊C8—21no—
140C3×#95—21no—
141C3×#95—22no—
142C3×#95—22no—
143C3×#95—22no—
144C22×#41C22×C311yescharacteristic
145C2×#137C22×C312yes—
146C2×C22⋊Q8C22×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#485—22no—
160#484—21no—
161#483—21no—
162#452—22no—
163#451—21no—
164#450—21no—
165#428C22×C312yescomplemented
166C2×#104C22×C312yes—
167C2×#104C22×C312yes—
168#419C22×C312yescomplemented
169#417C22×C312yes—
170#407C22×C311yescharacteristic
171#407C22×C311yescharacteristic
172#397—22no—
173#391C22×C311yescharacteristic, complemented
174#391C22×C311yescharacteristic, complemented
175#387—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×C3C2×D411yescharacteristic
185C3×#41—24no—
186C3×#41—24no—
187C3×#41—28no—
188C3×#41—24no—
189C3×#41—24no—
190C3×#41—22no—
191C3×#41—22no—
192C3×#41#4114yes—
193C3×#41#4114yes—
194C2×C3×Q8—24no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—22no—
199C2×C3×Q8—22no—
200C2×C3×Q8C2×D412yes—
201C2×C3×D4—24no—
202C2×C3×D4—24no—
203C2×C3×D4—22no—
204C2×C3×D4—22no—
205C2×C3×D4—24no—
206C2×C3×D4—21no—
207C2×C3×D4C2×D412yes—
208C22×C3×C4—28no—
209C22×C3×C4—21no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4—21no—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2411yescharacteristic
230C3×Q16—44no—
231C3×SD16—44no—
232C3×M16—22no—
233C3×M16—22no—
234C3×M16—22no—
235C3×M16—22no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—44no—
249C3×C4⋊C4—41no—
250C3×C4⋊C4—41no—
251C3×C4⋊C4C2×D412yes—
252C3×C4⋊C4C2×D412yes—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—42no—
260C3×C22⋊C4—41no—
261C3×C22⋊C4—41no—
262C3×C42—24no—
263C3×C42—24no—
264C3×C42—22no—
265C3×C42—22no—
266C3×C42—22no—
267C3×C42—22no—
268C3×C42—22no—
269C3×C42—22no—
270C2×#41—24no—
271C2×#41—24no—
272C2×#41—21no—
273C2×#41—21no—
274C2×#41C3×D412yes—
275C2×#41C23×C312yes—
276C22×Q8C3×D411yescharacteristic
277C22×Q8C23×C311yescharacteristic
278C22×D4—21no—
279C22×D4C3×D411yescharacteristic
280C23×C4—21no—
281C23×C4C23×C311yescharacteristic
282#137—24no—
283#137—24no—
284C2×Q16—22no—
285C2×SD16—22no—
286C2×M16C23×C312yes—
287C2×M16C23×C312yes—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—24no—
303C4×D4—24no—
304#117—24no—
305#117—24no—
306#117—21no—
307#117—21no—
308#117—21no—
309#117—21no—
310#117—21no—
311#117—21no—
312#117—22no—
313#117—22no—
314#117C3×D412yes—
315#117C23×C312yes—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4C3×D411yescharacteristic
321C2×C4⋊C4C3×D411yescharacteristic
322C2×C4⋊C4C23×C311yescharacteristic
323C2×C4⋊C4C23×C311yescharacteristic
324C2×C4⋊C4C23×C311yescharacteristic
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4C3×D412yes—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42C3×D412yes—
333C2×C42C23×C311yescharacteristic
334C2×C42C23×C311yescharacteristic
335#107—22no—
336#106—22no—
337#104—24no—
338#104—24no—
339#103—22no—
340#103—22no—
341#103—22no—
342#102—22no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—22no—
351C23×C3—24no—
352C23×C3—22no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—81no—
358C23×C3—81no—
359C23×C3#12011yescharacteristic
360C23×C3#12011yescharacteristic
361C23×C3C22×D411yescharacteristic
362C3×Q8—24no—
363C3×Q8—22no—
364C3×Q8—42no—
365C3×Q8—42no—
366C3×Q8#12314yes—
367C3×D4—28no—
368C3×D4—24no—
369C3×D4—24no—
370C3×D4—24no—
371C3×D4—22no—
372C3×D4#12314yes—
373C2×C3×C4—24no—
374C2×C3×C4—28no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—42no—
392C2×C3×C4—42no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—81no—
402C2×C3×C4—81no—
403C2×C3×C4—84no—
404C2×C3×C4—84no—
405C2×C3×C4#12011yescharacteristic
406C2×C3×C4#12011yescharacteristic
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4112yes—
410C2×C3×C4C2×#4112yes—
411C3×C8—42no—
412C3×C8—42no—
413C24—21no—
414C24C2×C3×D411yescharacteristic
415#41—24no—
416#41—24no—
417#41—28no—
418#41—24no—
419#41—24no—
420#41—22no—
421#41—22no—
422#41C3×#4114yes—
423#41C3×#4114yes—
424C2×Q8—24no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8C2×C3×D412yes—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4C2×C3×D412yes—
438C22×C4—28no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4C2×C3×D411yescharacteristic
453C22×C4C2×C3×D411yescharacteristic
454C22×C4C2×C3×D411yescharacteristic
455C22×C4C2×C3×D411yescharacteristic
456C22×C4C2×C3×D411yescharacteristic
457C22×C4C2×C3×D412yes—
458C22×C4C2×C3×D412yes—
459C22×C4C24×C311yescharacteristic
460Q16—44no—
461SD16—44no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466C2×C8—22no—
467C2×C8—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—44no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4C2×C3×D412yes—
482C4⋊C4C2×C3×D412yes—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—42no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—24no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—84no—
514C22×C3C2×#12011yescharacteristic
515C22×C3#52511yescharacteristic
516C22×C3#52511yescharacteristic
517C3×C4—21no—
518C3×C4—24no—
519C3×C4—21no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—42no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4—81no—
528C3×C4C2×#12312yes—
529C3×C4#52512yes—
530C23—24no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—41no—
535C23—41no—
536C23—81no—
537C23—81no—
538C23C3×#12011yescharacteristic
539C23C3×#12011yescharacteristic
540C23C22×C3×D411yescharacteristic
541Q8—24no—
542Q8—22no—
543Q8—42no—
544Q8—42no—
545Q8C3×#12314yes—
546D4—28no—
547D4—24no—
548D4—24no—
549D4—24no—
550D4—22no—
551D4C3×#12314yes—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4C3×#12011yescharacteristic
585C2×C4C3×#12011yescharacteristic
586C2×C4C22×C3×D411yescharacteristic
587C2×C4C22×C3×D411yescharacteristic
588C2×C4C2×C3×#4112yes—
589C2×C4C2×C3×#4112yes—
590C8—42no—
591C8—42no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—41no—
595C2×C3—41no—
596C2×C3—44no—
597C2×C3#284911yescharacteristic
598C2×C3#301811yescharacteristic
599C2×C3#302211yescharacteristic
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—81no—
612C22—81no—
613C22—84no—
614C22C2×C3×#12011yescharacteristic
615C22C3×#52511yescharacteristic
616C22C3×#52511yescharacteristic
617C4—21no—
618C4—24no—
619C4—21no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4C2×C3×#12312yes—
629C4C3×#52512yes—
630C3#3300111yescharacteristic, direct product factor, Sylow
631C2—21no—
632C2—21no—
633C2—41no—
634C2—41no—
635C2—44no—
636C2C3×#284911yescharacteristic
637C2C3×#301811yescharacteristic
638C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
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
2. C3×#3095C2semidirect
4. C3×#2499C2semidirect
5. C3×#1882C2semidirect
6. C3×#1882C2semidirect
7. C3×#1860C2semidirect
8. C3×#1859C2semidirect
9. C3×#1826C2semidirect
10. C3×#1822C2semidirect
12. #33001C3direct (N × H)
34. C3×#428C22semidirect
37. C3×#419C22semidirect
42. C3×#391C22semidirect
43. C3×#391C22semidirect
45. C3×#384C22semidirect
46. C3×#382C22semidirect
48. C3×#343C22semidirect
49. C3×#343C22semidirect
50. C3×#339C22semidirect
51. C3×#339C22semidirect
53. #3095C2×C3semidirect
55. #2499C2×C3semidirect
56. #1882C2×C3semidirect
57. #1882C2×C3semidirect
58. #1860C2×C3semidirect
59. #1859C2×C3semidirect
60. #1826C2×C3semidirect
61. #1822C2×C3semidirect
165. #428C22×C3semidirect
168. #419C22×C3semidirect
173. #391C22×C3semidirect
174. #391C22×C3semidirect
176. #384C22×C3semidirect
177. #382C22×C3semidirect
179. #343C22×C3semidirect
180. #343C22×C3semidirect
181. #339C22×C3semidirect
182. #339C22×C3semidirect
630. C3#33001direct (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×#30952
C3×#30072
C3×#24992
C3×#18822
C3×#18602
C3×#18592
C3×#18262
C3×#18222
C3×#17892
#330013