Finite Group Databasebeta

← C3×#20989 #10639596

Subgroups — C3×#20989 #10639596

Order 768.

1874 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C211yesmaximal, characteristic, complemented
2C2×C3×#525C211yesmaximal, characteristic
3C2×C3×#413C212yesmaximal, complemented
4C2×C3×#407C211yesmaximal, characteristic, complemented
5C3×#1901C214yesmaximal, complemented
6C3×#1900C214yesmaximal, complemented
7C3×#1742C212yesmaximal, complemented
8#20989C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2211yescharacteristic
10C24×C3×C4C2211yescharacteristic
11C22×C3×M16—22no—
12C22×C3×M16—21no—
13C22×C3×M16—28no—
14C22×C3×M16C2212yes—
15C22×C3×M16C2214yes—
16C22×C3×M16C2212yes—
17C23×C3×C8—21no—
18C3×#525—21no—
19C3×#525—22no—
20C3×#525—21no—
21C3×#525—22no—
22C3×#525—21no—
23C3×#525—21no—
24C2×C3×#123C412yes—
25C2×C3×C22⋊Q8C2212yes—
26C2×C3×#121C2212yes—
27C2×C3×#120C412yes—
28C2×C3×C4×D4C414yes—
29C2×C3×#117C2211yescharacteristic
30C3×#413—28no—
31C2×C3×#101C2214yescomplemented
32C2×C3×#100C2214yescomplemented
33C3×#407—21no—
34C3×#407—21no—
35C3×#407C2214yes—
36C2×C3×C22⋊C8—22no—
37C3×#343C2218yescomplemented
38C23×M16C2×C311yescharacteristic, complemented
39C2×#525C2×C311yescharacteristic
40C2×#413C2×C312yescomplemented
41C2×#407C2×C311yescharacteristic, complemented
42#1901C2×C314yescomplemented
43#1900C2×C314yescomplemented
44#1742C2×C312yescomplemented
45C25×C3C2×C411yescharacteristic
46C2×C3×#41—24no—
47C2×C3×#41—21no—
48C2×C3×#41—21no—
49C2×C3×#41—21no—
50C2×C3×#41—21no—
51C2×C3×#41—24no—
52C22×C3×Q8C2311yescharacteristic
53C22×C3×D4C2×C412yes—
54C22×C3×D4C2×C414yes—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—24no—
57C23×C3×C4—24no—
58C23×C3×C4C2×C412yes—
59C23×C3×C4C2×C412yes—
60C23×C3×C4C2×C411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D412yes—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D412yes—
65C23×C3×C4D411yescharacteristic
66C23×C3×C4D411yescharacteristic
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C2×C3×M16—216no—
70C2×C3×M16—28no—
71C2×C3×M16—24no—
72C2×C3×M16—24no—
73C2×C3×M16—22no—
74C2×C3×M16—216no—
75C2×C3×M16—22no—
76C2×C3×M16—24no—
77C2×C3×M16—24no—
78C2×C3×M16D418yescomplemented
79C2×C3×M16C2314yes—
80C2×C3×M16C2314yes—
81C22×C3×C8—28no—
82C22×C3×C8—24no—
83C22×C3×C8—22no—
84C22×C3×C8—22no—
85C3×#123—22no—
86C3×#123—24no—
87C3×#123—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×#121—22no—
93C3×#121—22no—
94C3×#121—22no—
95C3×#121—22no—
96C3×#120—22no—
97C3×#120—24no—
98C3×#120—22no—
99C3×C4×D4—24no—
100C3×C4×D4—24no—
101C3×C4×D4—24no—
102C3×C4×D4—24no—
103C3×#117—21no—
104C3×#117—21no—
105C3×#117—21no—
106C3×#117—21no—
107C2×C3×C4⋊C4C2×C414yes—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C22⋊C4C2×C412yes—
110C2×C3×C22⋊C4C2×C414yes—
111C2×C3×C22⋊C4C2×C414yes—
112C2×C3×C42C2×C412yes—
113C3×#101—28no—
114C3×#100—28no—
115C3×C22⋊C8—24no—
116C3×C22⋊C8—24no—
117C22×#41C22×C311yescharacteristic
118C24×C4C22×C311yescharacteristic
119C22×M16—22no—
120C22×M16—21no—
121C22×M16—28no—
122C22×M16C22×C312yes—
123C22×M16C22×C314yes—
124C22×M16C22×C312yes—
125C23×C8—21no—
126#525—21no—
127#525—21no—
128#525—22no—
129#525—22no—
130#525—21no—
131#525—21no—
132C2×#123C3×C412yes—
133C2×C22⋊Q8C22×C312yes—
134C2×#121C22×C312yes—
135C2×#120C3×C412yes—
136C2×C4×D4C3×C414yes—
137C2×#117C22×C311yescharacteristic
138#413—28no—
139C2×#101C22×C314yescomplemented
140C2×#100C22×C314yescomplemented
141#407—21no—
142#407—21no—
143#407C22×C314yes—
144C2×C22⋊C8—22no—
145#343C22×C318yescomplemented
146C24×C3—24no—
147C24×C3—44no—
148C24×C3C22⋊C411yescharacteristic
149C24×C3C22⋊C412yes—
150C24×C3C22⋊C411yescharacteristic
151C24×C3C22×C412yes—
152C24×C3C22×C412yes—
153C24×C3C22×C411yescharacteristic
154C3×#41—42no—
155C3×#41—42no—
156C3×#41—44no—
157C3×#41—44no—
158C3×#41—42no—
159C3×#41—42no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—28no—
169C2×C3×D4—22no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—48no—
174C22×C3×C4—24no—
175C22×C3×C4—28no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—28no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—28no—
190C22×C3×C4C22⋊C412yes—
191C22×C3×C4C22⋊C412yes—
192C22×C3×C4C22⋊C411yescharacteristic
193C22×C3×C4C22⋊C412yes—
194C22×C3×C4C22⋊C411yescharacteristic
195C22×C3×C4C22⋊C412yes—
196C22×C3×C4C22⋊C412yes—
197C22×C3×C4C22×C412yes—
198C22×C3×C4C22×C412yes—
199C22×C3×C4C22×C414yes—
200C22×C3×C4C22×C411yescharacteristic
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D414yes—
210C22×C3×C4#4114yes—
211C22×C3×C4C2411yescharacteristic
212C3×M16—216no—
213C3×M16—28no—
214C3×M16—24no—
215C3×M16—24no—
216C3×M16—24no—
217C3×M16—44no—
218C3×M16C2×D418yes—
219C2×C3×C8—216no—
220C2×C3×C8—28no—
221C2×C3×C8—24no—
222C2×C3×C8—24no—
223C2×C3×C8—44no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C42—22no—
235C3×C42—22no—
236C25C2×C3×C411yescharacteristic
237C2×#41—21no—
238C2×#41—21no—
239C2×#41—21no—
240C2×#41—21no—
241C2×#41—24no—
242C2×#41—24no—
243C22×Q8C23×C311yescharacteristic
244C22×D4C2×C3×C412yes—
245C22×D4C2×C3×C414yes—
246C22×D4C23×C311yescharacteristic
247C23×C4—24no—
248C23×C4—24no—
249C23×C4C2×C3×C412yes—
250C23×C4C2×C3×C412yes—
251C23×C4C2×C3×C411yescharacteristic
252C23×C4C3×D411yescharacteristic
253C23×C4C3×D412yes—
254C23×C4C3×D411yescharacteristic
255C23×C4C3×D412yes—
256C23×C4C3×D411yescharacteristic
257C23×C4C3×D411yescharacteristic
258C23×C4C23×C312yes—
259C23×C4C23×C311yescharacteristic
260C2×M16—216no—
261C2×M16—28no—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—22no—
265C2×M16—216no—
266C2×M16—22no—
267C2×M16—24no—
268C2×M16—24no—
269C2×M16C3×D418yescomplemented
270C2×M16C23×C314yes—
271C2×M16C23×C314yes—
272C22×C8—28no—
273C22×C8—24no—
274C22×C8—22no—
275C22×C8—22no—
276#123—22no—
277#123—24no—
278#123—22no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283#121—22no—
284#121—22no—
285#121—22no—
286#121—22no—
287#120—22no—
288#120—24no—
289#120—22no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298C2×C4⋊C4C2×C3×C414yes—
299C2×C4⋊C4C23×C312yes—
300C2×C22⋊C4C2×C3×C412yes—
301C2×C22⋊C4C2×C3×C414yes—
302C2×C22⋊C4C2×C3×C414yes—
303C2×C42C2×C3×C412yes—
304#101—28no—
305#100—28no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C23×C3—22no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—24no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—416no—
322C23×C3—44no—
323C23×C3C2×C22⋊C412yes—
324C23×C3C2×C22⋊C411yescharacteristic
325C23×C3C2×C22⋊C411yescharacteristic
326C23×C3C2×C22⋊C412yes—
327C23×C3C4×D414yes—
328C23×C3C23×C411yescharacteristic
329C3×Q8—41no—
330C3×Q8—42no—
331C3×Q8—41no—
332C3×D4—48no—
333C3×D4—44no—
334C3×D4—44no—
335C3×D4—48no—
336C3×D4—41no—
337C3×D4—42no—
338C3×D4—41no—
339C2×C3×C4—24no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—44no—
357C2×C3×C4—44no—
358C2×C3×C4—44no—
359C2×C3×C4C2×C22⋊C412yes—
360C2×C3×C4C2×C22⋊C412yes—
361C2×C3×C4C2×C22⋊C411yescharacteristic
362C2×C3×C4C2×C22⋊C411yescharacteristic
363C2×C3×C4C4×D414yes—
364C2×C3×C4#12012yes—
365C2×C3×C4#12012yes—
366C2×C3×C4#12114yes—
367C2×C3×C4#12114yes—
368C2×C3×C4#12312yes—
369C2×C3×C4#12312yes—
370C2×C3×C4C22×D412yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4112yes—
374C3×C8—28no—
375C3×C8—44no—
376C24—24no—
377C24—44no—
378C24C3×C22⋊C411yescharacteristic
379C24C3×C22⋊C412yes—
380C24C3×C22⋊C411yescharacteristic
381C24C22×C3×C412yes—
382C24C22×C3×C412yes—
383C24C22×C3×C411yescharacteristic
384#41—42no—
385#41—42no—
386#41—44no—
387#41—44no—
388#41—42no—
389#41—42no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—28no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—48no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4C3×C22⋊C412yes—
421C22×C4C3×C22⋊C412yes—
422C22×C4C3×C22⋊C412yes—
423C22×C4C3×C22⋊C412yes—
424C22×C4C3×C22⋊C411yescharacteristic
425C22×C4C3×C22⋊C412yes—
426C22×C4C3×C22⋊C411yescharacteristic
427C22×C4C22×C3×C412yes—
428C22×C4C22×C3×C412yes—
429C22×C4C22×C3×C414yes—
430C22×C4C22×C3×C411yescharacteristic
431C22×C4C2×C3×D412yes—
432C22×C4C2×C3×D412yes—
433C22×C4C2×C3×D412yes—
434C22×C4C2×C3×D412yes—
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C2×C3×D411yescharacteristic
438C22×C4C2×C3×D411yescharacteristic
439C22×C4C2×C3×D414yes—
440C22×C4C3×#4114yes—
441C22×C4C24×C311yescharacteristic
442M16—216no—
443M16—28no—
444M16—24no—
445M16—24no—
446M16—24no—
447M16—44no—
448M16C2×C3×D418yes—
449C2×C8—216no—
450C2×C8—28no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—44no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C42—22no—
465C42—22no—
466C22×C3—28no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—44no—
476C22×C3—48no—
477C22×C3—48no—
478C22×C3—44no—
479C22×C3—44no—
480C22×C3—42no—
481C22×C3—84no—
482C22×C3#38612yes—
483C22×C3#38612yes—
484C22×C3#41314yescomplemented
485C22×C3C22×C22⋊C411yescharacteristic
486C22×C3C2×C4×D412yes—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—44no—
490C3×C4—42no—
491C3×C4#38612yes—
492C3×C4#38612yes—
493C3×C4C2×#12011yescharacteristic
494C3×C4C2×#12112yes—
495C3×C4C2×#12311yescharacteristic
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—416no—
510C23—44no—
511C23C2×C3×C22⋊C412yes—
512C23C2×C3×C22⋊C411yescharacteristic
513C23C2×C3×C22⋊C411yescharacteristic
514C23C2×C3×C22⋊C412yes—
515C23C3×C4×D414yes—
516C23C23×C3×C411yescharacteristic
517Q8—41no—
518Q8—42no—
519Q8—41no—
520D4—48no—
521D4—44no—
522D4—44no—
523D4—48no—
524D4—41no—
525D4—42no—
526D4—41no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4C2×C3×C22⋊C412yes—
548C2×C4C2×C3×C22⋊C412yes—
549C2×C4C2×C3×C22⋊C411yescharacteristic
550C2×C4C2×C3×C22⋊C411yescharacteristic
551C2×C4C3×C4×D414yes—
552C2×C4C3×#12012yes—
553C2×C4C3×#12012yes—
554C2×C4C3×#12114yes—
555C2×C4C3×#12114yes—
556C2×C4C3×#12312yes—
557C2×C4C3×#12312yes—
558C2×C4C22×C3×D412yes—
559C2×C4C22×C3×D411yescharacteristic
560C2×C4C22×C3×D411yescharacteristic
561C2×C4C2×C3×#4112yes—
562C8—28no—
563C8—44no—
564C2×C3—22no—
565C2×C3—24no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—42no—
569C2×C3—82no—
570C2×C3C2×#38611yescharacteristic
571C2×C3C2×#41312yes—
572C22—28no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—44no—
582C22—48no—
583C22—48no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—84no—
588C22C3×#38612yes—
589C22C3×#38612yes—
590C22C3×#41314yescomplemented
591C22C22×C3×C22⋊C411yescharacteristic
592C22C2×C3×C4×D412yes—
593C4—22no—
594C4—22no—
595C4—44no—
596C4—42no—
597C4C3×#38612yes—
598C4C3×#38612yes—
599C4C2×C3×#12011yescharacteristic
600C4C2×C3×#12112yes—
601C4C2×C3×#12311yescharacteristic
602C3#2098911yescharacteristic, direct product factor, Sylow
603C2—22no—
604C2—24no—
605C2—22no—
606C2—22no—
607C2—42no—
608C2—82no—
609C2C2×C3×#38611yescharacteristic
610C2C2×C3×#41312yes—
subgroupG/H
centerC2×C3×C4C2×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC22×C4C24×C3
socleC22×C3C22×C22⋊C4
prime radicalC2×C3×#525C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C3×M16C2semidirect
3. C2×C3×#413C2semidirect
4. C2×C3×#407C2semidirect
5. C3×#1901C2semidirect
6. C3×#1900C2semidirect
7. C3×#1742C2semidirect
8. #20989C3direct (N × H)
31. C2×C3×#101C22semidirect
32. C2×C3×#100C22semidirect
37. C3×#343C22semidirect
38. C23×M16C2×C3semidirect
40. C2×#413C2×C3semidirect
41. C2×#407C2×C3semidirect
42. #1901C2×C3semidirect
43. #1900C2×C3semidirect
44. #1742C2×C3semidirect
78. C2×C3×M16D4semidirect
139. C2×#101C22×C3semidirect
140. C2×#100C22×C3semidirect
145. #343C22×C3semidirect
269. C2×M16C3×D4semidirect
484. C22×C3#413semidirect
590. C22C3×#413semidirect
602. C3#20989direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×M162
C2×C3×#5252
C2×C3×#4132
C2×C3×#4072
C3×#19012
C3×#19002
C3×#17422
#209893