Finite Group Databasebeta

← C3×#16682 #10635289

Subgroups — C3×#16682 #10635289

Order 768.

1534 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic, complemented
2C2×C3×#399C211yesmaximal, characteristic
3C2×C3×#397C211yesmaximal, characteristic
4C2×C3×#388C211yesmaximal, characteristic
5C3×#1445C211yesmaximal, characteristic, complemented
6C3×#1445C211yesmaximal, characteristic, complemented
7C3×#1444C212yesmaximal, complemented
8C3×#1441C212yesmaximal, complemented
9C3×#1439C212yesmaximal
10C3×#1439C212yesmaximal
11C3×#1434C211yesmaximal, characteristic, complemented
12#16682C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#125—24no—
14C2×C3×#125—24no—
15C2×C3×#125—24no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4C2211yescharacteristic
24C22×C3×C22⋊C4C2212yes—
25C22×C3×C22⋊C4C2212yes—
26C22×C3×C22⋊C4C2211yescharacteristic
27C22×C3×C42C2211yescharacteristic
28C3×#399—24no—
29C3×#397—24no—
30C3×#388—24no—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2211yescharacteristic
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×C3×#95C2212yes—
41C2×C3×#95C2212yes—
42C22×#125C2×C311yescharacteristic, complemented
43C2×#399C2×C311yescharacteristic
44C2×#397C2×C311yescharacteristic
45C2×#388C2×C311yescharacteristic
46#1445C2×C311yescharacteristic, complemented
47#1445C2×C311yescharacteristic, complemented
48#1444C2×C312yescomplemented
49#1441C2×C312yescomplemented
50#1439C2×C312yes—
51#1439C2×C312yes—
52#1434C2×C311yescharacteristic, complemented
53C25×C3C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C3×#125—232no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—42no—
79C2×C3×C4⋊C4—42no—
80C2×C3×C4⋊C4D414yes—
81C2×C3×C4⋊C4D414yes—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—24no—
93C2×C3×C22⋊C4—42no—
94C2×C3×C22⋊C4—42no—
95C2×C3×C22⋊C4—42no—
96C2×C3×C42—22no—
97C2×C3×C42—22no—
98C2×C3×C42D414yes—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—24no—
102C3×#95—22no—
103C3×#95—22no—
104C3×#95—24no—
105C3×#95—24no—
106C3×#95—42no—
107C3×#95—42no—
108C3×#95—42no—
109C3×#95—42no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—24no—
113C22×C4⋊C4C22×C312yes—
114C22×C4⋊C4C22×C312yes—
115C22×C4⋊C4C22×C312yes—
116C22×C4⋊C4C22×C311yescharacteristic
117C22×C4⋊C4C22×C311yescharacteristic
118C22×C4⋊C4C22×C311yescharacteristic
119C22×C22⋊C4C22×C311yescharacteristic
120C22×C22⋊C4C22×C311yescharacteristic
121C22×C22⋊C4C22×C312yes—
122C22×C22⋊C4C22×C312yes—
123C22×C22⋊C4C22×C311yescharacteristic
124C22×C42C22×C311yescharacteristic
125#399—24no—
126#397—24no—
127#388—24no—
128C2×#95C22×C311yescharacteristic
129C2×#95C22×C312yes—
130C2×#95C22×C311yescharacteristic
131C2×#95C22×C312yes—
132C2×#95C22×C312yes—
133C2×#95C22×C311yescharacteristic
134C2×#95C22×C311yescharacteristic
135C2×#95C22×C312yes—
136C2×#95C22×C312yes—
137C2×#95C22×C312yes—
138C2×#95C22×C312yes—
139C24×C3—21no—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—21no—
146C24×C3—22no—
147C24×C3—21no—
148C24×C3—22no—
149C24×C3#4112yes—
150C24×C3C2411yescharacteristic
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D414yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D414yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4#4112yes—
210C22×C3×C4#4114yes—
211C3×C4⋊C4—216no—
212C3×C4⋊C4—216no—
213C3×C4⋊C4—28no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—48no—
216C3×C4⋊C4—44no—
217C3×C22⋊C4—28no—
218C3×C22⋊C4—48no—
219C3×C22⋊C4—48no—
220C3×C22⋊C4—44no—
221C3×C22⋊C4—44no—
222C3×C42—28no—
223C25C23×C311yescharacteristic
224C23×C4C23×C311yescharacteristic
225C23×C4C23×C312yes—
226C23×C4C23×C311yescharacteristic
227C23×C4C23×C312yes—
228C23×C4C23×C312yes—
229C23×C4C23×C311yescharacteristic
230C23×C4C23×C311yescharacteristic
231C23×C4C23×C312yes—
232C23×C4C23×C311yescharacteristic
233C23×C4C23×C311yescharacteristic
234#125—232no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C3×D414yes—
251C2×C4⋊C4C3×D414yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42C3×D414yes—
269#95—22no—
270#95—22no—
271#95—24no—
272#95—22no—
273#95—22no—
274#95—24no—
275#95—24no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—42no—
280C23×C3—22no—
281C23×C3—24no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—24no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D411yescharacteristic
305C23×C3C22×D411yescharacteristic
306C23×C3C2×#4111yescharacteristic
307C23×C3C2×#4111yescharacteristic
308C23×C3C2×#4112yes—
309C23×C3H(5,𝔽2)12yes—
310C23×C3H(5,𝔽2)11yescharacteristic
311C23×C32−1+412yes—
312C23×C32−1+412yes—
313C23×C32−1+411yescharacteristic
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4#12014yes—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—21no—
378C24—22no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—21no—
383C24—22no—
384C24C3×#4112yes—
385C24C24×C311yescharacteristic
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4C2×C3×D412yes—
438C22×C4C2×C3×D414yes—
439C22×C4C2×C3×D412yes—
440C22×C4C2×C3×D414yes—
441C22×C4C2×C3×D412yes—
442C22×C4C2×C3×D412yes—
443C22×C4C2×C3×D412yes—
444C22×C4C3×#4112yes—
445C22×C4C3×#4114yes—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C22⋊C4—28no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C42—28no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—44no—
462C22×C3—44no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—81no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3C2×#12011yescharacteristic
470C22×C3#52511yescharacteristic
471C22×C3#53211yescharacteristic
472C22×C3#53511yescharacteristic
473C22×C3#53611yescharacteristic
474C22×C3#53611yescharacteristic
475C22×C3#53612yes—
476C22×C3#53612yes—
477C22×C3#53711yescharacteristic
478C22×C3#53912yes—
479C22×C3#53912yes—
480C22×C3#53912yes—
481C22×C3#54112yes—
482C22×C3#54611yescharacteristic
483C22×C3#54612yes—
484C22×C3#54711yescharacteristic
485C22×C3#54811yescharacteristic
486C22×C3#55011yescharacteristic
487C22×C3#55012yes—
488C22×C3#55212yes—
489C22×C3#55212yes—
490C22×C3#56212yes—
491C22×C3#56312yes—
492C3×C4—24no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4—42no—
497C3×C4—44no—
498C3×C4—44no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—82no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23C22×C3×D411yescharacteristic
526C23C22×C3×D411yescharacteristic
527C23C22×C3×D411yescharacteristic
528C23C2×C3×#4111yescharacteristic
529C23C2×C3×#4111yescharacteristic
530C23C2×C3×#4112yes—
531C23C3×H(5,𝔽2)11yescharacteristic
532C23C3×H(5,𝔽2)12yes—
533C23C3×2−1+412yes—
534C23C3×2−1+411yescharacteristic
535C23C3×2−1+412yes—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4C3×#12014yes—
596C2×C3—82no—
597C2×C3#240711yescharacteristic
598C2×C3#243111yescharacteristic
599C2×C3#261111yescharacteristic
600C2×C3#261411yescharacteristic
601C2×C3#263311yescharacteristic
602C2×C3#266411yescharacteristic
603C2×C3#269012yes—
604C2×C3#269412yes—
605C2×C3#269412yes—
606C2×C3#272112yes—
607C2×C3#282011yescharacteristic
608C22—42no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—82no—
616C22—82no—
617C22—81no—
618C22—81no—
619C22C2×C3×#12011yescharacteristic
620C22C3×#52511yescharacteristic
621C22C3×#53211yescharacteristic
622C22C3×#53511yescharacteristic
623C22C3×#53611yescharacteristic
624C22C3×#53612yes—
625C22C3×#53612yes—
626C22C3×#53611yescharacteristic
627C22C3×#53711yescharacteristic
628C22C3×#53912yes—
629C22C3×#53912yes—
630C22C3×#53912yes—
631C22C3×#54112yes—
632C22C3×#54611yescharacteristic
633C22C3×#54612yes—
634C22C3×#54711yescharacteristic
635C22C3×#54811yescharacteristic
636C22C3×#55011yescharacteristic
637C22C3×#55012yes—
638C22C3×#55212yes—
639C22C3×#55212yes—
640C22C3×#56212yes—
641C22C3×#56312yes—
642C4—24no—
643C4—42no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—44no—
648C4—44no—
649C4—81no—
650C4—82no—
651C4—81no—
652C3#1668211yescharacteristic, direct product factor, Sylow
653C2—82no—
654C2C3×#240711yescharacteristic
655C2C3×#243111yescharacteristic
656C2C3×#261111yescharacteristic
657C2C3×#261411yescharacteristic
658C2C3×#263311yescharacteristic
659C2C3×#266411yescharacteristic
660C2C3×#269012yes—
661C2C3×#269412yes—
662C2C3×#269412yes—
663C2C3×#272112yes—
664C2C3×#282011yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#125C2semidirect
5. C3×#1445C2semidirect
6. C3×#1445C2semidirect
7. C3×#1444C2semidirect
8. C3×#1441C2semidirect
11. C3×#1434C2semidirect
12. #16682C3direct (N × H)
42. C22×#125C2×C3semidirect
46. #1445C2×C3semidirect
47. #1445C2×C3semidirect
48. #1444C2×C3semidirect
49. #1441C2×C3semidirect
52. #1434C2×C3semidirect
652. C3#16682direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×C3×#3992
C2×C3×#3972
C2×C3×#3882
C3×#14452
C3×#14442
C3×#14412
C3×#14392
C3×#14342
#166823