Finite Group Databasebeta

← C3×#16665 #10635272

Subgroups — C3×#16665 #10635272

Order 768.

1410 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#393C211yesmaximal, characteristic
2C2×C3×#391C211yesmaximal, characteristic, complemented
3C2×C3×#388C211yesmaximal, characteristic
4C2×C3×#383C211yesmaximal, characteristic, complemented
5C3×#1446C211yesmaximal, characteristic, complemented
6C3×#1444C211yesmaximal, characteristic, complemented
7C3×#1443C212yesmaximal, complemented
8C3×#1441C212yesmaximal, complemented
9C3×#1440C212yesmaximal
10C3×#1439C212yesmaximal
11C3×#1439C211yesmaximal, characteristic
12#16665C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C3×#393—24no—
24C3×#391—24no—
25C3×#388—24no—
26C3×#383—24no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2211yescharacteristic
40C2×#393C2×C311yescharacteristic
41C2×#391C2×C311yescharacteristic, complemented
42C2×#388C2×C311yescharacteristic
43C2×#383C2×C311yescharacteristic, complemented
44#1446C2×C311yescharacteristic, complemented
45#1444C2×C311yescharacteristic, complemented
46#1443C2×C312yescomplemented
47#1441C2×C312yescomplemented
48#1440C2×C312yes—
49#1439C2×C311yescharacteristic
50#1439C2×C312yes—
51C25×C3C2311yescharacteristic
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—24no—
55C22×C3×Q8—22no—
56C22×C3×Q8—42no—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—42no—
71C2×C3×C4⋊C4—42no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C4⋊C4—42no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—42no—
85C2×C3×C22⋊C4—42no—
86C2×C3×C22⋊C4—42no—
87C2×C3×C22⋊C4—42no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42D414yes—
91C3×#95—24no—
92C3×#95—24no—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—24no—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—24no—
100C3×#95—42no—
101C3×#95—42no—
102C3×#95—42no—
103C3×#95D414yes—
104C23×Q8C22×C311yescharacteristic
105C22×C4⋊C4C22×C312yes—
106C22×C4⋊C4C22×C311yescharacteristic
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C22⋊C4C22×C311yescharacteristic
109C22×C22⋊C4C22×C311yescharacteristic
110C22×C22⋊C4C22×C312yes—
111C22×C22⋊C4C22×C312yes—
112C22×C22⋊C4C22×C311yescharacteristic
113C22×C42C22×C311yescharacteristic
114#393—24no—
115#391—24no—
116#388—24no—
117#383—24no—
118C2×#95C22×C312yes—
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C311yescharacteristic
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C2×#95C22×C312yes—
125C2×#95C22×C311yescharacteristic
126C2×#95C22×C311yescharacteristic
127C2×#95C22×C312yes—
128C2×#95C22×C312yes—
129C2×#95C22×C312yes—
130C2×#95C22×C311yescharacteristic
131C24×C3—21no—
132C24×C3—21no—
133C24×C3—21no—
134C24×C3—21no—
135C24×C3—22no—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3—21no—
139C24×C3—22no—
140C24×C3—21no—
141C24×C3#4112yes—
142C24×C3C2411yescharacteristic
143C2×C3×Q8—28no—
144C2×C3×Q8—48no—
145C2×C3×Q8—48no—
146C2×C3×Q8—44no—
147C2×C3×Q8—44no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
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—21no—
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—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4C2×D414yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4#4114yes—
206C22×C3×C4#4112yes—
207C22×C3×C4#4112yes—
208C3×C4⋊C4—44no—
209C3×C4⋊C4—44no—
210C3×C4⋊C4—48no—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—48no—
213C3×C22⋊C4—44no—
214C3×C22⋊C4—48no—
215C3×C22⋊C4—44no—
216C3×C42—28no—
217C25C23×C311yescharacteristic
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—22no—
222C22×Q8—42no—
223C23×C4C23×C312yes—
224C23×C4C23×C311yescharacteristic
225C23×C4C23×C312yes—
226C23×C4C23×C311yescharacteristic
227C23×C4C23×C312yes—
228C23×C4C23×C311yescharacteristic
229C23×C4C23×C311yescharacteristic
230C23×C4C23×C311yescharacteristic
231C23×C4C23×C312yes—
232C23×C4C23×C311yescharacteristic
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42C3×D414yes—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95C3×D414yes—
270C23×C3—22no—
271C23×C3—24no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—22no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—41no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3C22×D411yescharacteristic
294C23×C3C22×D411yescharacteristic
295C23×C3C2×#4111yescharacteristic
296C23×C3C2×#4111yescharacteristic
297C23×C3C2×#4111yescharacteristic
298C23×C3C2×#4112yes—
299C23×C3H(5,𝔽2)12yes—
300C23×C3H(5,𝔽2)12yes—
301C23×C32−1+412yes—
302C23×C32−1+411yescharacteristic
303C23×C32−1+411yescharacteristic
304C3×Q8—416no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
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—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4#12114yes—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—22no—
374C24—21no—
375C24C3×#4112yes—
376C24C24×C311yescharacteristic
377C2×Q8—28no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—44no—
381C2×Q8—44no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
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—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4C2×C3×D414yes—
435C22×C4C2×C3×D412yes—
436C22×C4C2×C3×D412yes—
437C22×C4C2×C3×D412yes—
438C22×C4C2×C3×D412yes—
439C22×C4C3×#4114yes—
440C22×C4C3×#4112yes—
441C22×C4C3×#4112yes—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C22⋊C4—28no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—44no—
450C42—28no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—81no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3C2×#12111yescharacteristic
463C22×C3#52511yescharacteristic
464C22×C3#52511yescharacteristic
465C22×C3#53511yescharacteristic
466C22×C3#53511yescharacteristic
467C22×C3#53611yescharacteristic
468C22×C3#53611yescharacteristic
469C22×C3#53712yes—
470C22×C3#53812yes—
471C22×C3#53812yes—
472C22×C3#54212yes—
473C22×C3#54212yes—
474C22×C3#54212yes—
475C22×C3#54612yes—
476C22×C3#54711yescharacteristic
477C22×C3#54811yescharacteristic
478C22×C3#55112yes—
479C22×C3#55212yes—
480C22×C3#55211yescharacteristic
481C22×C3#55612yes—
482C22×C3#55611yescharacteristic
483C22×C3#56312yes—
484C22×C3#56312yes—
485C3×C4—24no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—44no—
489C3×C4—42no—
490C3×C4—44no—
491C3×C4—81no—
492C3×C4—82no—
493C3×C4—81no—
494C3×C4—82no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23C22×C3×D411yescharacteristic
519C23C22×C3×D411yescharacteristic
520C23C2×C3×#4111yescharacteristic
521C23C2×C3×#4111yescharacteristic
522C23C2×C3×#4111yescharacteristic
523C23C2×C3×#4112yes—
524C23C3×H(5,𝔽2)12yes—
525C23C3×H(5,𝔽2)12yes—
526C23C3×2−1+412yes—
527C23C3×2−1+411yescharacteristic
528C23C3×2−1+411yescharacteristic
529Q8—416no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4C3×#12114yes—
590C2×C3—82no—
591C2×C3#241111yescharacteristic
592C2×C3#241511yescharacteristic
593C2×C3#259911yescharacteristic
594C2×C3#262011yescharacteristic
595C2×C3#263111yescharacteristic
596C2×C3#263211yescharacteristic
597C2×C3#270212yes—
598C2×C3#271512yes—
599C2×C3#275912yes—
600C2×C3#276412yes—
601C2×C3#282111yescharacteristic
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—82no—
611C22—82no—
612C22—81no—
613C22C2×C3×#12111yescharacteristic
614C22C3×#52511yescharacteristic
615C22C3×#52511yescharacteristic
616C22C3×#53511yescharacteristic
617C22C3×#53511yescharacteristic
618C22C3×#53611yescharacteristic
619C22C3×#53611yescharacteristic
620C22C3×#53712yes—
621C22C3×#53812yes—
622C22C3×#53812yes—
623C22C3×#54212yes—
624C22C3×#54212yes—
625C22C3×#54212yes—
626C22C3×#54612yes—
627C22C3×#54711yescharacteristic
628C22C3×#54811yescharacteristic
629C22C3×#55112yes—
630C22C3×#55211yescharacteristic
631C22C3×#55212yes—
632C22C3×#55611yescharacteristic
633C22C3×#55612yes—
634C22C3×#56312yes—
635C22C3×#56312yes—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—42no—
641C4—44no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4—82no—
646C3#1666511yescharacteristic, direct product factor, Sylow
647C2—82no—
648C2C3×#241111yescharacteristic
649C2C3×#241511yescharacteristic
650C2C3×#259911yescharacteristic
651C2C3×#262011yescharacteristic
652C2C3×#263111yescharacteristic
653C2C3×#263211yescharacteristic
654C2C3×#270212yes—
655C2C3×#271512yes—
656C2C3×#275912yes—
657C2C3×#276412yes—
658C2C3×#282111yescharacteristic
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
2. C2×C3×#391C2semidirect
4. C2×C3×#383C2semidirect
5. C3×#1446C2semidirect
6. C3×#1444C2semidirect
7. C3×#1443C2semidirect
8. C3×#1441C2semidirect
12. #16665C3direct (N × H)
41. C2×#391C2×C3semidirect
43. C2×#383C2×C3semidirect
44. #1446C2×C3semidirect
45. #1444C2×C3semidirect
46. #1443C2×C3semidirect
47. #1441C2×C3semidirect
646. C3#16665direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3932
C2×C3×#3912
C2×C3×#3882
C2×C3×#3832
C3×#14462
C3×#14442
C3×#14432
C3×#14412
C3×#14402
C3×#14392
#166653