Finite Group Databasebeta

← C3×#21014 #10639621

Subgroups — C3×#21014 #10639621

Order 768.

1748 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3449C211yesmaximal, characteristic
2C3×#3428C211yesmaximal, characteristic, complemented
3C2×C3×#410C212yesmaximal, complemented
4C2×C3×#410C211yesmaximal, characteristic, complemented
5C3×#1913C218yesmaximal, complemented
6C3×#1740C212yesmaximal, complemented
7#21014C311yesmaximal, characteristic, direct product factor, Sylow
8C22×C3×#41C2211yescharacteristic
9C22×C3×#41C2211yescharacteristic
10C3×#537—22no—
11C3×#536—21no—
12C3×#536C412yescomplemented
13C3×#535—21no—
14C3×#535C412yescomplemented
15C3×#534C2214yes—
16C3×#532—22no—
17C3×#525—22no—
18C3×#525C414yescomplemented
19C3×#520—24no—
20C3×#519—21no—
21C3×#518—21no—
22C3×#518C2214yes—
23C3×C4×#41—24no—
24C2×C3×#117—22no—
25C2×C3×#117C2212yes—
26C2×C3×#117C2211yescharacteristic
27C2×C3×#117C2212yes—
28C3×#410—22no—
29C3×#410—22no—
30C3×#410—24no—
31C3×#410—24no—
32C2×C3×#99C2214yes—
33C2×C3×#99C2218yes—
34C3×#342C2218yes—
35#3449C2×C311yescharacteristic
36#3428C2×C311yescharacteristic, complemented
37C2×#410C2×C311yescharacteristic, complemented
38C2×#410C2×C312yescomplemented
39#1913C2×C318yescomplemented
40#1740C2×C312yescomplemented
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—22no—
45C2×C3×#41—21no—
46C2×C3×#41—21no—
47C2×C3×#41—21no—
48C2×C3×#41—21no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41C2×C412yes—
54C2×C3×#41C2×C412yes—
55C2×C3×#41D412yes—
56C2×C3×#41D412yes—
57C22×C3×Q8C2×C411yescharacteristic, complemented
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4C2×C411yescharacteristic, complemented
60C22×C3×D4D411yescharacteristic
61C22×C3×D4C2312yes—
62C22×C3×D4C2312yes—
63C23×C3×C4C2×C412yescomplemented
64C23×C3×C4D412yes—
65C23×C3×C4C2311yescharacteristic
66C3×C4⋊Q8—22no—
67C3×#127—22no—
68C3×#124—22no—
69C3×#124—22no—
70C3×#123—24no—
71C3×#123C2×C418yescomplemented
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—24no—
74C3×#121—24no—
75C3×#121—28no—
76C3×#121—24no—
77C3×#120—24no—
78C3×#120C2×C418yescomplemented
79C3×C4×Q8—28no—
80C3×C4×D4—28no—
81C3×C4×D4—28no—
82C3×C4×D4—216no—
83C3×#117—24no—
84C3×#117—24no—
85C3×#117—22no—
86C3×#117—24no—
87C3×#117—22no—
88C3×#117—21no—
89C3×#117—21no—
90C3×#117—22no—
91C3×#117—22no—
92C3×#117—22no—
93C3×#117—22no—
94C3×#117C2×C412yescomplemented
95C3×#117C2×C412yescomplemented
96C3×#117D414yescomplemented
97C3×#117D414yescomplemented
98C2×C3×C4⋊C4—24no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C22⋊C4—24no—
103C2×C3×C22⋊C4C2314yes—
104C2×C3×C22⋊C4C2312yes—
105C2×C3×C22⋊C4C2314yes—
106C2×C3×C42—24no—
107C2×C3×C42—22no—
108C2×C3×C42—21no—
109C2×C3×C42—22no—
110C3×#99—28no—
111C3×#99—216no—
112C22×#41C22×C311yescharacteristic
113C22×#41C22×C311yescharacteristic
114#537—22no—
115#536—21no—
116#536C3×C412yescomplemented
117#535—21no—
118#535C3×C412yescomplemented
119#534C22×C314yes—
120#532—22no—
121#525—22no—
122#525C3×C414yescomplemented
123#520—24no—
124#519—21no—
125#518—21no—
126#518C22×C314yes—
127C4×#41—24no—
128C2×#117—22no—
129C2×#117C22×C312yes—
130C2×#117C22×C311yescharacteristic
131C2×#117C22×C312yes—
132#410—22no—
133#410—22no—
134#410—24no—
135#410—24no—
136C2×#99C22×C314yes—
137C2×#99C22×C318yes—
138#342C22×C318yes—
139C24×C3C22×C412yes—
140C24×C3C2×D412yes—
141C24×C3C2411yescharacteristic
142C3×#41—24no—
143C3×#41—24no—
144C3×#41—24no—
145C3×#41—24no—
146C3×#41—24no—
147C3×#41—24no—
148C3×#41—24no—
149C3×#41—24no—
150C3×#41—42no—
151C3×#41—41no—
152C3×#41—41no—
153C3×#41—42no—
154C3×#41—42no—
155C3×#41—42no—
156C3×#41—42no—
157C3×#41—41no—
158C3×#41—42no—
159C3×#41—41no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8C22⋊C412yes—
167C2×C3×Q8C22⋊C412yes—
168C2×C3×D4—24no—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—24no—
174C2×C3×D4—28no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4C22⋊C412yes—
182C2×C3×D4C22⋊C412yes—
183C2×C3×D4C22×C414yes—
184C2×C3×D4C2×D414yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—28no—
190C22×C3×C4—24no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4—24no—
197C22×C3×C4C22⋊C414yes—
198C22×C3×C4C22⋊C414yes—
199C22×C3×C4C22×C412yes—
200C22×C3×C4C22×C411yescharacteristic
201C22×C3×C4C22×C411yescharacteristic
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4#4112yes—
210C22×C3×C4#4112yes—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—42no—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—28no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4C22×C414yes—
227C3×C22⋊C4C2×D418yes—
228C3×C42—28no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—24no—
232C3×C42—24no—
233C3×C42—42no—
234C3×C42—42no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—21no—
238C2×#41—21no—
239C2×#41—21no—
240C2×#41—21no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41C2×C3×C412yes—
248C2×#41C2×C3×C412yes—
249C2×#41C3×D412yes—
250C2×#41C3×D412yes—
251C22×Q8C2×C3×C411yescharacteristic, complemented
252C22×Q8C3×D411yescharacteristic
253C22×D4C2×C3×C411yescharacteristic, complemented
254C22×D4C3×D411yescharacteristic
255C22×D4C23×C312yes—
256C22×D4C23×C312yes—
257C23×C4C2×C3×C412yescomplemented
258C23×C4C3×D412yes—
259C23×C4C23×C311yescharacteristic
260C4⋊Q8—22no—
261#127—22no—
262#124—22no—
263#124—22no—
264#123—24no—
265#123C2×C3×C418yescomplemented
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268#121—24no—
269#121—28no—
270#121—24no—
271#120—24no—
272#120C2×C3×C418yescomplemented
273C4×Q8—28no—
274C4×D4—28no—
275C4×D4—28no—
276C4×D4—216no—
277#117—24no—
278#117—24no—
279#117—22no—
280#117—24no—
281#117—22no—
282#117—21no—
283#117—21no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117C2×C3×C412yescomplemented
289#117C2×C3×C412yescomplemented
290#117C3×D414yescomplemented
291#117C3×D414yescomplemented
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4C23×C314yes—
298C2×C22⋊C4C23×C312yes—
299C2×C22⋊C4C23×C314yes—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—22no—
304#99—28no—
305#99—216no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—24no—
310C23×C3—81no—
311C23×C3—82no—
312C23×C3—82no—
313C23×C3C2×C22⋊C414yes—
314C23×C3C23×C411yescharacteristic
315C23×C3C22×D411yescharacteristic
316C23×C3C22×D411yescharacteristic
317C23×C3C22×D412yes—
318C23×C3C2×#4112yes—
319C3×Q8—24no—
320C3×Q8—24no—
321C3×Q8—41no—
322C3×Q8—41no—
323C3×Q8—42no—
324C3×D4—28no—
325C3×D4—28no—
326C3×D4—24no—
327C3×D4—24no—
328C3×D4—42no—
329C3×D4—44no—
330C3×D4—42no—
331C3×D4—41no—
332C3×D4—41no—
333C3×D4—42no—
334C3×D4—88no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—28no—
345C2×C3×C4—42no—
346C2×C3×C4—44no—
347C2×C3×C4—42no—
348C2×C3×C4—44no—
349C2×C3×C4—84no—
350C2×C3×C4—88no—
351C2×C3×C4—82no—
352C2×C3×C4—84no—
353C2×C3×C4C2×C22⋊C414yes—
354C2×C3×C4C2×C22⋊C411yescharacteristic
355C2×C3×C4C2×C22⋊C411yescharacteristic
356C2×C3×C4C2×C22⋊C411yescharacteristic
357C2×C3×C4C2×C22⋊C411yescharacteristic
358C2×C3×C4C4×D412yes—
359C2×C3×C4C4×D412yes—
360C2×C3×C4C4×D412yes—
361C2×C3×C4C4×D412yes—
362C2×C3×C4#12012yes—
363C2×C3×C4#12011yescharacteristic
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4#12112yes—
366C2×C3×C4#12112yes—
367C2×C3×C4#12112yes—
368C2×C3×C4#12112yes—
369C2×C3×C4#12311yescharacteristic
370C2×C3×C4#12312yes—
371C2×C3×C4#12311yescharacteristic
372C24C22×C3×C412yes—
373C24C2×C3×D412yes—
374C24C24×C311yescharacteristic
375#41—24no—
376#41—24no—
377#41—24no—
378#41—24no—
379#41—24no—
380#41—24no—
381#41—24no—
382#41—24no—
383#41—41no—
384#41—42no—
385#41—41no—
386#41—42no—
387#41—42no—
388#41—42no—
389#41—42no—
390#41—41no—
391#41—41no—
392#41—42no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8C3×C22⋊C412yes—
400C2×Q8C3×C22⋊C412yes—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4C3×C22⋊C412yes—
415C2×D4C3×C22⋊C412yes—
416C2×D4C22×C3×C414yes—
417C2×D4C2×C3×D414yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4C3×C22⋊C414yes—
431C22×C4C3×C22⋊C414yes—
432C22×C4C22×C3×C412yes—
433C22×C4C22×C3×C411yescharacteristic
434C22×C4C22×C3×C411yescharacteristic
435C22×C4C2×C3×D412yes—
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C2×C3×D411yescharacteristic
438C22×C4C2×C3×D411yescharacteristic
439C22×C4C2×C3×D411yescharacteristic
440C22×C4C2×C3×D412yes—
441C22×C4C2×C3×D412yes—
442C22×C4C3×#4112yes—
443C22×C4C3×#4112yes—
444C4⋊C4—28no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4C22×C3×C414yes—
460C22⋊C4C2×C3×D418yes—
461C42—28no—
462C42—22no—
463C42—22no—
464C42—24no—
465C42—24no—
466C42—42no—
467C42—42no—
468C22×C3—24no—
469C22×C3—42no—
470C22×C3—41no—
471C22×C3—42no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—42no—
478C22×C3—82no—
479C22×C3—84no—
480C22×C3C22×C22⋊C411yescharacteristic
481C22×C3C2×C4×D412yes—
482C22×C3C2×#12011yescharacteristic
483C22×C3C2×#12112yes—
484C22×C3C2×#12311yescharacteristic
485C3×C4—24no—
486C3×C4—48no—
487C3×C4—44no—
488C3×C4—42no—
489C3×C4—84no—
490C3×C4#38611yescharacteristic
491C3×C4#38611yescharacteristic
492C3×C4#38611yescharacteristic
493C3×C4#38612yes—
494C3×C4#38612yes—
495C3×C4#38611yescharacteristic
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—81no—
501C23—82no—
502C23—82no—
503C23C2×C3×C22⋊C414yes—
504C23C23×C3×C411yescharacteristic
505C23C22×C3×D411yescharacteristic
506C23C22×C3×D411yescharacteristic
507C23C22×C3×D412yes—
508C23C2×C3×#4112yes—
509Q8—24no—
510Q8—24no—
511Q8—41no—
512Q8—41no—
513Q8—42no—
514D4—28no—
515D4—28no—
516D4—24no—
517D4—24no—
518D4—42no—
519D4—44no—
520D4—42no—
521D4—41no—
522D4—41no—
523D4—42no—
524D4—88no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—84no—
540C2×C4—88no—
541C2×C4—82no—
542C2×C4—84no—
543C2×C4C2×C3×C22⋊C414yes—
544C2×C4C2×C3×C22⋊C411yescharacteristic
545C2×C4C2×C3×C22⋊C411yescharacteristic
546C2×C4C2×C3×C22⋊C411yescharacteristic
547C2×C4C2×C3×C22⋊C411yescharacteristic
548C2×C4C3×C4×D412yes—
549C2×C4C3×C4×D412yes—
550C2×C4C3×C4×D412yes—
551C2×C4C3×C4×D412yes—
552C2×C4C3×#12011yescharacteristic
553C2×C4C3×#12012yes—
554C2×C4C3×#12011yescharacteristic
555C2×C4C3×#12112yes—
556C2×C4C3×#12112yes—
557C2×C4C3×#12112yes—
558C2×C4C3×#12112yes—
559C2×C4C3×#12311yescharacteristic
560C2×C4C3×#12311yescharacteristic
561C2×C4C3×#12312yes—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—22no—
565C2×C3—22no—
566C2×C3—21no—
567C2×C3—44no—
568C2×C3—82no—
569C2×C3C2×#38611yescharacteristic
570C22—24no—
571C22—42no—
572C22—41no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—82no—
581C22—84no—
582C22C22×C3×C22⋊C411yescharacteristic
583C22C2×C3×C4×D412yes—
584C22C2×C3×#12011yescharacteristic
585C22C2×C3×#12112yes—
586C22C2×C3×#12311yescharacteristic
587C4—24no—
588C4—48no—
589C4—44no—
590C4—42no—
591C4—84no—
592C4C3×#38611yescharacteristic
593C4C3×#38611yescharacteristic
594C4C3×#38611yescharacteristic
595C4C3×#38611yescharacteristic
596C4C3×#38612yes—
597C4C3×#38612yes—
598C3#2101411yescharacteristic, direct product factor, Sylow
599C2—21no—
600C2—21no—
601C2—22no—
602C2—22no—
603C2—21no—
604C2—44no—
605C2—82no—
606C2C2×C3×#38611yescharacteristic
subgroupG/H
centerC3×C4#386
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
socleC2×C3C2×#386
prime radicalC3×#3449C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3428C2semidirect
3. C2×C3×#410C2semidirect
4. C2×C3×#410C2semidirect
5. C3×#1913C2semidirect
6. C3×#1740C2semidirect
7. #21014C3direct (N × H)
12. C3×#536C4semidirect
14. C3×#535C4semidirect
18. C3×#525C4semidirect
36. #3428C2×C3semidirect
37. C2×#410C2×C3semidirect
38. C2×#410C2×C3semidirect
39. #1913C2×C3semidirect
40. #1740C2×C3semidirect
57. C22×C3×Q8C2×C4semidirect
59. C22×C3×D4C2×C4semidirect
63. C23×C3×C4C2×C4semidirect
71. C3×#123C2×C4semidirect
78. C3×#120C2×C4semidirect
94. C3×#117C2×C4semidirect
95. C3×#117C2×C4semidirect
96. C3×#117D4semidirect
97. C3×#117D4semidirect
116. #536C3×C4semidirect
118. #535C3×C4semidirect
122. #525C3×C4semidirect
251. C22×Q8C2×C3×C4semidirect
253. C22×D4C2×C3×C4semidirect
257. C23×C4C2×C3×C4semidirect
265. #123C2×C3×C4semidirect
272. #120C2×C3×C4semidirect
288. #117C2×C3×C4semidirect
289. #117C2×C3×C4semidirect
290. #117C3×D4semidirect
291. #117C3×D4semidirect
598. C3#21014direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34492
C3×#34282
C2×C3×#4102
C3×#19132
C3×#17402
#210143