Finite Group Databasebeta

← C3×#14666 #10633273

Subgroups — C3×#14666 #10633273

Order 768.

1602 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#386C211yesmaximal, characteristic
2C3×#1443C212yesmaximal, complemented
3C3×#1442C212yesmaximal, complemented
4C3×#1440C212yesmaximal, complemented
5C3×#1439C212yesmaximal, complemented
6C3×#1439C212yesmaximal, complemented
7C3×#1438C211yesmaximal, characteristic
8C3×#1438C211yesmaximal, characteristic
9C3×C22⋊#95C212yesmaximal, complemented
10#14666C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C3×#386C418yescomplemented
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×#386C2×C311yescharacteristic
34#1443C2×C312yescomplemented
35#1442C2×C312yescomplemented
36#1440C2×C312yescomplemented
37#1439C2×C312yescomplemented
38#1439C2×C312yescomplemented
39#1438C2×C311yescharacteristic
40#1438C2×C311yescharacteristic
41C22⋊#95C2×C312yescomplemented
42C25×C3C2311yescharacteristic
43C25×C3C2311yescharacteristic
44C22×C3×D4—24no—
45C22×C3×D4—42no—
46C22×C3×D4—41no—
47C22×C3×D4—41no—
48C22×C3×D4C2×C414yes—
49C23×C3×C4—41no—
50C23×C3×C4—41no—
51C23×C3×C4—42no—
52C23×C3×C4—42no—
53C23×C3×C4C2×C414yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C2×C3×C4⋊C4—42no—
62C2×C3×C4⋊C4—42no—
63C2×C3×C4⋊C4—42no—
64C2×C3×C22⋊C4—24no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—42no—
67C2×C3×C22⋊C4—42no—
68C2×C3×C22⋊C4—42no—
69C2×C3×C22⋊C4—42no—
70C2×C3×C22⋊C4—42no—
71C2×C3×C22⋊C4—42no—
72C2×C3×C22⋊C4—42no—
73C2×C3×C22⋊C4—41no—
74C2×C3×C22⋊C4—41no—
75C2×C3×C22⋊C4—42no—
76C2×C3×C22⋊C4—42no—
77C2×C3×C22⋊C4—42no—
78C2×C3×C22⋊C4—42no—
79C2×C3×C22⋊C4—42no—
80C2×C3×C22⋊C4C2×C414yes—
81C2×C3×C22⋊C4C2×C418yes—
82C3×#95—42no—
83C3×#95—42no—
84C3×#95—42no—
85C3×#95—42no—
86C3×#95—42no—
87C3×#95—42no—
88C3×#95—42no—
89C3×#95—41no—
90C3×#95—41no—
91C3×#95—42no—
92C3×#95C2×C418yes—
93C23×D4C22×C311yescharacteristic
94C24×C4C22×C311yescharacteristic
95C22×C4⋊C4C22×C312yes—
96C22×C22⋊C4C22×C312yes—
97C22×C22⋊C4C22×C312yes—
98C22×C22⋊C4C22×C311yescharacteristic
99C22×C22⋊C4C22×C312yes—
100C22×C22⋊C4C22×C311yescharacteristic
101C22×C22⋊C4C22×C311yescharacteristic
102C22×C22⋊C4C22×C312yes—
103#386C3×C418yescomplemented
104C2×#95C22×C312yes—
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C311yescharacteristic
109C2×#95C22×C312yes—
110C2×#95C22×C311yescharacteristic
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—21no—
127C24×C3—22no—
128C24×C3—21no—
129C24×C3—22no—
130C24×C3C22×C412yes—
131C24×C3C22×C412yes—
132C24×C3#4114yes—
133C24×C3C2411yescharacteristic
134C2×C3×D4—28no—
135C2×C3×D4—44no—
136C2×C3×D4—44no—
137C2×C3×D4—48no—
138C2×C3×D4—48no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
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—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
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—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—48no—
184C22×C3×C4—44no—
185C22×C3×C4—48no—
186C22×C3×C4—44no—
187C22×C3×C4—44no—
188C22×C3×C4C22×C412yes—
189C22×C3×C4C22×C414yes—
190C22×C3×C4C22×C414yes—
191C3×C4⋊C4—48no—
192C3×C22⋊C4—48no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—48no—
195C3×C22⋊C4—48no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—48no—
198C3×C22⋊C4—44no—
199C25C23×C311yescharacteristic
200C25C23×C311yescharacteristic
201C22×D4—24no—
202C22×D4—42no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4C2×C3×C414yes—
206C23×C4—41no—
207C23×C4—41no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4C2×C3×C414yes—
211C23×C4C23×C312yes—
212C23×C4C23×C312yes—
213C23×C4C23×C312yes—
214C23×C4C23×C312yes—
215C23×C4C23×C312yes—
216C23×C4C23×C312yes—
217C23×C4C23×C311yescharacteristic
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4C2×C3×C414yes—
238C2×C22⋊C4C2×C3×C418yes—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—42no—
245#95—42no—
246#95—41no—
247#95—41no—
248#95—42no—
249#95C2×C3×C418yes—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—22no—
261C23×C3—24no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—24no—
265C23×C3—42no—
266C23×C3—41no—
267C23×C3—42no—
268C23×C3—41no—
269C23×C3—42no—
270C23×C3—41no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—41no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—41no—
291C23×C3#11714yes—
292C23×C3C23×C411yescharacteristic
293C23×C3C2×#4112yes—
294C23×C3H(5,𝔽2)11yescharacteristic
295C23×C3H(5,𝔽2)12yes—
296C23×C3H(5,𝔽2)11yescharacteristic
297C23×C3H(5,𝔽2)12yes—
298C23×C3H(5,𝔽2)12yes—
299C23×C3H(5,𝔽2)12yes—
300C23×C32−1+412yes—
301C3×D4—416no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—416no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
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—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—22no—
363C24—21no—
364C24—22no—
365C24C22×C3×C412yes—
366C24C22×C3×C412yes—
367C24C3×#4114yes—
368C24C24×C311yescharacteristic
369C2×D4—28no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
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—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4C22×C3×C412yes—
424C22×C4C22×C3×C414yes—
425C22×C4C22×C3×C414yes—
426C4⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22×C3—24no—
435C22×C3—28no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—44no—
443C22×C3—42no—
444C22×C3—44no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—44no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3C2×#11711yescharacteristic
453C22×C3#51811yescharacteristic
454C22×C3#51811yescharacteristic
455C22×C3#51812yes—
456C22×C3#52012yes—
457C22×C3#53812yes—
458C22×C3#53812yes—
459C22×C3#53812yes—
460C22×C3#53812yes—
461C22×C3#53912yes—
462C22×C3#53912yes—
463C22×C3#56012yes—
464C22×C3#56011yescharacteristic
465C22×C3#56012yes—
466C22×C3#56012yes—
467C22×C3#56011yescharacteristic
468C22×C3H(3,𝔽4)11yescharacteristic
469C22×C3#56212yes—
470C22×C3#56212yes—
471C22×C3#56212yes—
472C22×C3#56311yescharacteristic
473C3×C4—42no—
474C3×C4—44no—
475C3×C4—44no—
476C3×C4—82no—
477C3×C4—82no—
478C3×C4—82no—
479C3×C4—82no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C3×#11714yes—
522C23C23×C3×C411yescharacteristic
523C23C2×C3×#4112yes—
524C23C3×H(5,𝔽2)12yes—
525C23C3×H(5,𝔽2)12yes—
526C23C3×H(5,𝔽2)11yescharacteristic
527C23C3×H(5,𝔽2)12yes—
528C23C3×H(5,𝔽2)12yes—
529C23C3×H(5,𝔽2)11yescharacteristic
530C23C3×2−1+412yes—
531D4—416no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C3—44no—
581C2×C3—82no—
582C2×C3#231211yescharacteristic
583C2×C3#231812yes—
584C2×C3#237511yescharacteristic
585C2×C3#237611yescharacteristic
586C2×C3#237712yes—
587C2×C3#282412yes—
588C2×C3#282512yes—
589C2×C3#282612yes—
590C2×C3#282712yes—
591C22—24no—
592C22—28no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22C2×C3×#11711yescharacteristic
610C22C3×#51811yescharacteristic
611C22C3×#51812yes—
612C22C3×#51811yescharacteristic
613C22C3×#52012yes—
614C22C3×#53812yes—
615C22C3×#53812yes—
616C22C3×#53812yes—
617C22C3×#53812yes—
618C22C3×#53912yes—
619C22C3×#53912yes—
620C22C3×#56012yes—
621C22C3×#56011yescharacteristic
622C22C3×#56011yescharacteristic
623C22C3×#56012yes—
624C22C3×#56012yes—
625C22C3×H(3,𝔽4)11yescharacteristic
626C22C3×#56212yes—
627C22C3×#56212yes—
628C22C3×#56212yes—
629C22C3×#56311yescharacteristic
630C4—44no—
631C4—44no—
632C4—42no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C3#1466611yescharacteristic, direct product factor, Sylow
638C2—44no—
639C2—82no—
640C2C3×#231211yescharacteristic
641C2C3×#231812yes—
642C2C3×#237511yescharacteristic
643C2C3×#237611yescharacteristic
644C2C3×#237712yes—
645C2C3×#282412yes—
646C2C3×#282512yes—
647C2C3×#282612yes—
648C2C3×#282712yes—
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#1443C2semidirect
3. C3×#1442C2semidirect
4. C3×#1440C2semidirect
5. C3×#1439C2semidirect
6. C3×#1439C2semidirect
9. C3×C22⋊#95C2semidirect
10. #14666C3direct (N × H)
21. C3×#386C4semidirect
34. #1443C2×C3semidirect
35. #1442C2×C3semidirect
36. #1440C2×C3semidirect
37. #1439C2×C3semidirect
38. #1439C2×C3semidirect
41. C22⋊#95C2×C3semidirect
103. #386C3×C4semidirect
637. C3#14666direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3862
C3×#14432
C3×#14422
C3×#14402
C3×#14392
C3×#14382
C3×C22⋊#952
#146663