Finite Group Databasebeta

← C3×#14506 #10633113

Subgroups — C3×#14506 #10633113

Order 768.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic
2C2×C3×#388C212yesmaximal, complemented
3C3×#1445C212yesmaximal, complemented
4C3×#1439C212yesmaximal, complemented
5C3×#1438C211yesmaximal, characteristic
6C3×#1438C211yesmaximal, characteristic
7C3×#1437C212yesmaximal, complemented
8C3×C4⋊#95C212yesmaximal, complemented
9C3×C22⋊#95C212yesmaximal, complemented
10#14506C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C2×C3×#123—28no—
14C2×C3×#123C418yescomplemented
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4C2211yescharacteristic
24C22×C3×C42C2212yes—
25C3×#388—28no—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3×#95C2211yescharacteristic
35C22×#123C2×C311yescharacteristic
36C2×#388C2×C312yescomplemented
37#1445C2×C312yescomplemented
38#1439C2×C312yescomplemented
39#1438C2×C311yescharacteristic
40#1438C2×C311yescharacteristic
41#1437C2×C312yescomplemented
42C4⋊#95C2×C312yescomplemented
43C22⋊#95C2×C312yescomplemented
44C25×C3C2311yescharacteristic
45C25×C3C2311yescharacteristic
46C22×C3×D4—24no—
47C22×C3×D4—42no—
48C22×C3×D4—41no—
49C22×C3×D4—41no—
50C22×C3×D4C2×C414yes—
51C23×C3×C4—24no—
52C23×C3×C4—42no—
53C23×C3×C4—41no—
54C23×C3×C4—41no—
55C23×C3×C4C2×C414yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2312yes—
63C3×#123—232no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4C2×C418yes—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—41no—
79C2×C3×C22⋊C4—42no—
80C2×C3×C22⋊C4—42no—
81C2×C3×C22⋊C4—42no—
82C2×C3×C22⋊C4—42no—
83C2×C3×C22⋊C4—41no—
84C2×C3×C22⋊C4—42no—
85C2×C3×C22⋊C4C2×C418yes—
86C2×C3×C22⋊C4C2×C414yes—
87C2×C3×C22⋊C4D418yes—
88C2×C3×C42—24no—
89C2×C3×C42—24no—
90C2×C3×C42—42no—
91C3×#95—24no—
92C3×#95—24no—
93C3×#95—41no—
94C3×#95—42no—
95C3×#95—42no—
96C3×#95—42no—
97C3×#95—42no—
98C3×#95—42no—
99C3×#95—41no—
100C23×D4C22×C311yescharacteristic
101C24×C4C22×C311yescharacteristic
102C2×#123—28no—
103C2×#123C3×C418yescomplemented
104C22×C4⋊C4C22×C312yes—
105C22×C4⋊C4C22×C312yes—
106C22×C22⋊C4C22×C312yes—
107C22×C22⋊C4C22×C312yes—
108C22×C22⋊C4C22×C311yescharacteristic
109C22×C22⋊C4C22×C312yes—
110C22×C22⋊C4C22×C312yes—
111C22×C22⋊C4C22×C311yescharacteristic
112C22×C22⋊C4C22×C311yescharacteristic
113C22×C42C22×C312yes—
114#388—28no—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C312yes—
118C2×#95C22×C312yes—
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C311yescharacteristic
122C2×#95C22×C312yes—
123C2×#95C22×C311yescharacteristic
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—21no—
129C24×C3—21no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—21no—
137C24×C3—21no—
138C24×C3—22no—
139C24×C3C22×C412yes—
140C24×C3C22×C412yes—
141C24×C3C2×D414yes—
142C24×C3C2411yescharacteristic
143C2×C3×D4—28no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—48no—
147C2×C3×D4—48no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—28no—
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—21no—
170C22×C3×C4—21no—
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—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—44no—
190C22×C3×C4—44no—
191C22×C3×C4—48no—
192C22×C3×C4—48no—
193C22×C3×C4C22×C414yes—
194C22×C3×C4C22×C412yes—
195C22×C3×C4C22×C414yes—
196C22×C3×C4C2×D414yes—
197C22×C3×C4C2×D414yes—
198C22×C3×C4#4114yes—
199C22×C3×C4#4114yes—
200C3×C4⋊C4—216no—
201C3×C4⋊C4—48no—
202C3×C22⋊C4—216no—
203C3×C22⋊C4—44no—
204C3×C22⋊C4—48no—
205C3×C22⋊C4—48no—
206C3×C22⋊C4—48no—
207C3×C22⋊C4—44no—
208C3×C22⋊C4C22⋊C4116yescomplemented
209C3×C42—48no—
210C25C23×C311yescharacteristic
211C25C23×C311yescharacteristic
212C22×D4—24no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4—42no—
216C22×D4C2×C3×C414yes—
217C23×C4—24no—
218C23×C4—41no—
219C23×C4—41no—
220C23×C4—42no—
221C23×C4C2×C3×C414yes—
222C23×C4C23×C312yes—
223C23×C4C23×C312yes—
224C23×C4C23×C312yes—
225C23×C4C23×C311yescharacteristic
226C23×C4C23×C312yes—
227C23×C4C23×C312yes—
228C23×C4C23×C312yes—
229#123—232no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C2×C3×C418yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4C2×C3×C418yes—
252C2×C22⋊C4C2×C3×C414yes—
253C2×C22⋊C4C3×D418yes—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—42no—
257#95—24no—
258#95—24no—
259#95—41no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—41no—
266C23×C3—24no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—24no—
270C23×C3—24no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—22no—
277C23×C3—24no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—24no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—41no—
286C23×C3—42no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—42no—
290C23×C3—41no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—42no—
306C23×C3—41no—
307C23×C3C2×C22⋊C414yes—
308C23×C3C23×C411yescharacteristic
309C23×C3C22×D412yes—
310C23×C3C2×#4112yes—
311C23×C3C2×#4112yes—
312C23×C3H(5,𝔽2)12yes—
313C23×C3H(5,𝔽2)11yescharacteristic
314C23×C3H(5,𝔽2)11yescharacteristic
315C23×C3H(5,𝔽2)12yes—
316C23×C32−1+412yes—
317C3×D4—416no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—416no—
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—
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—42no—
365C2×C3×C4C2×C22⋊C418yes—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—22no—
381C24C22×C3×C412yes—
382C24C22×C3×C412yes—
383C24C2×C3×D414yes—
384C24C24×C311yescharacteristic
385C2×D4—28no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—48no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
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—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
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—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4C22×C3×C414yes—
436C22×C4C22×C3×C412yes—
437C22×C4C22×C3×C414yes—
438C22×C4C2×C3×D414yes—
439C22×C4C2×C3×D414yes—
440C22×C4C3×#4114yes—
441C22×C4C3×#4114yes—
442C4⋊C4—216no—
443C4⋊C4—48no—
444C22⋊C4—216no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—44no—
450C22⋊C4C3×C22⋊C4116yescomplemented
451C42—48no—
452C22×C3—24no—
453C22×C3—28no—
454C22×C3—44no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—42no—
462C22×C3—44no—
463C22×C3—44no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3—82no—
470C22×C3C22×C22⋊C411yescharacteristic
471C22×C3C4×#4112yes—
472C22×C3#51811yescharacteristic
473C22×C3#51811yescharacteristic
474C22×C3#52012yes—
475C22×C3#53512yes—
476C22×C3#53512yes—
477C22×C3#53812yes—
478C22×C3#53912yes—
479C22×C3#54012yes—
480C22×C3#54212yes—
481C22×C3#54612yes—
482C22×C3#54612yes—
483C22×C3#54712yes—
484C22×C3#54712yes—
485C22×C3#55111yescharacteristic
486C22×C3#55312yes—
487C22×C3#55911yescharacteristic
488C22×C3#56011yescharacteristic
489C22×C3#56212yes—
490C22×C3#56311yescharacteristic
491C3×C4—28no—
492C3×C4—44no—
493C3×C4—42no—
494C3×C4—44no—
495C3×C4—44no—
496C3×C4—82no—
497C3×C4—82no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23C2×C3×C22⋊C414yes—
540C23C23×C3×C411yescharacteristic
541C23C22×C3×D412yes—
542C23C2×C3×#4112yes—
543C23C2×C3×#4112yes—
544C23C3×H(5,𝔽2)11yescharacteristic
545C23C3×H(5,𝔽2)12yes—
546C23C3×H(5,𝔽2)11yescharacteristic
547C23C3×H(5,𝔽2)12yes—
548C23C3×2−1+412yes—
549D4—416no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—416no—
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×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×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4C2×C3×C22⋊C418yes—
598C2×C3—44no—
599C2×C3—82no—
600C2×C3#230911yescharacteristic
601C2×C3#233512yes—
602C2×C3#233912yes—
603C2×C3#237411yescharacteristic
604C2×C3#237611yescharacteristic
605C2×C3#270312yes—
606C2×C3#270612yes—
607C2×C3#271012yes—
608C2×C3#271812yes—
609C22—24no—
610C22—28no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22C22×C3×C22⋊C411yescharacteristic
628C22C3×C4×#4112yes—
629C22C3×#51811yescharacteristic
630C22C3×#51811yescharacteristic
631C22C3×#52012yes—
632C22C3×#53512yes—
633C22C3×#53512yes—
634C22C3×#53812yes—
635C22C3×#53912yes—
636C22C3×#54012yes—
637C22C3×#54212yes—
638C22C3×#54612yes—
639C22C3×#54612yes—
640C22C3×#54712yes—
641C22C3×#54712yes—
642C22C3×#55111yescharacteristic
643C22C3×#55312yes—
644C22C3×#55911yescharacteristic
645C22C3×#56011yescharacteristic
646C22C3×#56212yes—
647C22C3×#56311yescharacteristic
648C4—28no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—82no—
654C4—82no—
655C3#1450611yescharacteristic, direct product factor, Sylow
656C2—44no—
657C2—82no—
658C2C3×#230911yescharacteristic
659C2C3×#233512yes—
660C2C3×#233912yes—
661C2C3×#237411yescharacteristic
662C2C3×#237611yescharacteristic
663C2C3×#270312yes—
664C2C3×#270612yes—
665C2C3×#271012yes—
666C2C3×#271812yes—
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. C2×C3×#388C2semidirect
3. C3×#1445C2semidirect
4. C3×#1439C2semidirect
7. C3×#1437C2semidirect
8. C3×C4⋊#95C2semidirect
9. C3×C22⋊#95C2semidirect
10. #14506C3direct (N × H)
14. C2×C3×#123C4semidirect
36. C2×#388C2×C3semidirect
37. #1445C2×C3semidirect
38. #1439C2×C3semidirect
41. #1437C2×C3semidirect
42. C4⋊#95C2×C3semidirect
43. C22⋊#95C2×C3semidirect
103. C2×#123C3×C4semidirect
208. C3×C22⋊C4C22⋊C4semidirect
450. C22⋊C4C3×C22⋊C4semidirect
655. C3#14506direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C2×C3×#3882
C3×#14452
C3×#14392
C3×#14382
C3×#14372
C3×C4⋊#952
C3×C22⋊#952
#145063