Finite Group Databasebeta

← C3×#14926 #10633533

Subgroups — C3×#14926 #10633533

Order 768.

3030 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C212yesmaximal, complemented
2C22×C3×#121C211yesmaximal, characteristic
3C2×C3×#386C212yesmaximal, complemented
4C22×C3×#95C211yesmaximal, characteristic, complemented
5C3×#1444C212yesmaximal, complemented
6C3×#1439C214yesmaximal, complemented
7C3×#1438C212yesmaximal, complemented
8C3×C22⋊#95C211yesmaximal, characteristic, complemented
9#14926C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2212yes—
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C24×C3×C4C2211yescharacteristic
14C24×C3×C4C2211yescharacteristic
15C2×C3×#123—28no—
16C2×C3×#123—28no—
17C2×C3×#123—28no—
18C2×C3×#121—28no—
19C2×C3×#121—24no—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2214yes—
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C22⋊C4C2214yes—
24C22×C3×C22⋊C4C2214yes—
25C22×C3×C22⋊C4C2212yes—
26C3×#386—28no—
27C2×C3×#95—24no—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2214yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C22×#123C2×C312yescomplemented
34C22×#121C2×C311yescharacteristic
35C2×#386C2×C312yescomplemented
36C22×#95C2×C311yescharacteristic, complemented
37#1444C2×C312yescomplemented
38#1439C2×C314yescomplemented
39#1438C2×C312yescomplemented
40C22⋊#95C2×C311yescharacteristic, complemented
41C25×C3C2312yes—
42C25×C3C2311yescharacteristic
43C22×C3×D4—24no—
44C22×C3×D4—24no—
45C22×C3×D4—24no—
46C22×C3×D4—24no—
47C22×C3×D4—24no—
48C22×C3×D4—24no—
49C22×C3×D4—24no—
50C22×C3×D4—22no—
51C22×C3×D4—24no—
52C22×C3×D4—42no—
53C22×C3×D4D418yes—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—24no—
57C23×C3×C4—24no—
58C23×C3×C4—24no—
59C23×C3×C4—22no—
60C23×C3×C4—24no—
61C23×C3×C4—24no—
62C23×C3×C4—24no—
63C23×C3×C4—42no—
64C23×C3×C4—42no—
65C23×C3×C4D414yes—
66C23×C3×C4D414yescomplemented
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2314yes—
71C23×C3×C4C2312yes—
72C23×C3×C4C2311yescharacteristic
73C3×#123—264no—
74C3×#121—232no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—28no—
78C2×C3×C4⋊C4—28no—
79C2×C3×C4⋊C4—28no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—28no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—28no—
87C2×C3×C22⋊C4—28no—
88C2×C3×C22⋊C4—28no—
89C2×C3×C22⋊C4—28no—
90C2×C3×C22⋊C4—44no—
91C2×C3×C22⋊C4D418yescomplemented
92C3×#95—24no—
93C3×#95—24no—
94C3×#95—28no—
95C3×#95—24no—
96C3×#95—42no—
97C23×D4C22×C312yes—
98C23×D4C22×C311yescharacteristic
99C24×C4C22×C311yescharacteristic
100C24×C4C22×C311yescharacteristic
101C24×C4C22×C311yescharacteristic
102C2×#123—28no—
103C2×#123—28no—
104C2×#123—28no—
105C2×#121—28no—
106C2×#121—24no—
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C4⋊C4C22×C314yes—
109C22×C22⋊C4C22×C312yes—
110C22×C22⋊C4C22×C314yes—
111C22×C22⋊C4C22×C314yes—
112C22×C22⋊C4C22×C312yes—
113#386—28no—
114C2×#95—24no—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C312yes—
118C2×#95C22×C314yes—
119C2×#95C22×C312yes—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—24no—
130C24×C3—22no—
131C24×C3—24no—
132C24×C3—24no—
133C24×C3—24no—
134C24×C3C2×D414yes—
135C24×C3C2×D412yes—
136C24×C3C2×D414yes—
137C24×C3C2×D414yes—
138C24×C3C2×D412yes—
139C24×C3#4112yes—
140C24×C3C2411yescharacteristic
141C2×C3×D4—28no—
142C2×C3×D4—216no—
143C2×C3×D4—216no—
144C2×C3×D4—216no—
145C2×C3×D4—416no—
146C2×C3×D4—44no—
147C2×C3×D4—48no—
148C2×C3×D4—48no—
149C2×C3×D4—48no—
150C2×C3×D4—48no—
151C2×C3×D4—44no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—28no—
155C22×C3×C4—28no—
156C22×C3×C4—216no—
157C22×C3×C4—28no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—28no—
182C22×C3×C4—44no—
183C22×C3×C4—416no—
184C22×C3×C4—416no—
185C22×C3×C4—44no—
186C22×C3×C4—44no—
187C22×C3×C4—48no—
188C22×C3×C4—44no—
189C22×C3×C4—44no—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D418yes—
192C22×C3×C4C2×D414yes—
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4#4114yes—
196C22×C3×C4#4114yes—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—232no—
199C3×C22⋊C4—216no—
200C3×C22⋊C4—232no—
201C3×C22⋊C4—216no—
202C3×C22⋊C4—416no—
203C25C23×C312yes—
204C25C23×C311yescharacteristic
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—42no—
215C22×D4C3×D418yes—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4C3×D414yes—
228C23×C4C3×D414yescomplemented
229C23×C4C23×C314yes—
230C23×C4C23×C312yes—
231C23×C4C23×C311yescharacteristic
232C23×C4C23×C312yes—
233C23×C4C23×C312yes—
234C23×C4C23×C311yescharacteristic
235#123—264no—
236#121—232no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4C3×D418yescomplemented
254#95—24no—
255#95—24no—
256#95—28no—
257#95—24no—
258#95—42no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—28no—
268C23×C3—28no—
269C23×C3—28no—
270C23×C3—28no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—28no—
275C23×C3—28no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—44no—
282C23×C3—42no—
283C23×C3—44no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—44no—
287C23×C3—44no—
288C23×C3—44no—
289C23×C3—44no—
290C23×C3—44no—
291C23×C3—44no—
292C23×C3—44no—
293C23×C3—44no—
294C23×C3#12018yes—
295C23×C3#12014yes—
296C23×C3#12114yes—
297C23×C3#12118yes—
298C23×C3#12714yes—
299C23×C3C22×D411yescharacteristic
300C23×C3C22×D412yes—
301C23×C3C22×D411yescharacteristic
302C23×C3C22×D412yes—
303C23×C3C2×#4111yescharacteristic
304C23×C3C2×#4112yes—
305C23×C3C2×#4112yes—
306C23×C3H(5,𝔽2)12yes—
307C23×C32−1+412yes—
308C3×D4—416no—
309C3×D4—432no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—28no—
313C2×C3×C4—28no—
314C2×C3×C4—22no—
315C2×C3×C4—28no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—28no—
324C2×C3×C4—28no—
325C2×C3×C4—28no—
326C2×C3×C4—416no—
327C2×C3×C4—416no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—42no—
338C2×C3×C4—44no—
339C2×C3×C4—44no—
340C2×C3×C4—44no—
341C2×C3×C4—44no—
342C2×C3×C4—44no—
343C2×C3×C4—416no—
344C2×C3×C4#12314yescomplemented
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24C2×C3×D414yes—
360C24C2×C3×D412yes—
361C24C2×C3×D414yes—
362C24C2×C3×D414yes—
363C24C2×C3×D412yes—
364C24C3×#4112yes—
365C24C24×C311yescharacteristic
366C2×D4—28no—
367C2×D4—216no—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—416no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—44no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—44no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4C2×C3×D412yes—
416C22×C4C2×C3×D418yes—
417C22×C4C2×C3×D414yes—
418C22×C4C2×C3×D414yes—
419C22×C4C2×C3×D412yes—
420C22×C4C3×#4114yes—
421C22×C4C3×#4114yes—
422C4⋊C4—28no—
423C4⋊C4—232no—
424C22⋊C4—216no—
425C22⋊C4—232no—
426C22⋊C4—216no—
427C22⋊C4—416no—
428C22×C3—24no—
429C22×C3—24no—
430C22×C3—24no—
431C22×C3—24no—
432C22×C3—216no—
433C22×C3—216no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—48no—
440C22×C3—48no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—84no—
444C22×C3—84no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3#39218yes—
448C22×C3C2×#12011yescharacteristic
449C22×C3C2×#12012yes—
450C22×C3C2×#12111yescharacteristic
451C22×C3C2×#12112yes—
452C22×C3C2×#12311yescharacteristic
453C22×C3#52511yescharacteristic
454C22×C3#52912yes—
455C22×C3C2×#12711yescharacteristic
456C22×C3#53212yes—
457C22×C3#53411yescharacteristic
458C22×C3#53611yescharacteristic
459C22×C3#53712yes—
460C22×C3#53912yes—
461C22×C3#54614yes—
462C22×C3#54614yes—
463C22×C3#54714yes—
464C22×C3#54714yes—
465C3×C4—24no—
466C3×C4—44no—
467C3×C4—44no—
468C3×C4—42no—
469C3×C4—48no—
470C3×C4—44no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C3×#12018yes—
507C23C3×#12014yes—
508C23C3×#12114yes—
509C23C3×#12118yes—
510C23C3×#12714yes—
511C23C22×C3×D411yescharacteristic
512C23C22×C3×D412yes—
513C23C22×C3×D411yescharacteristic
514C23C22×C3×D412yes—
515C23C2×C3×#4111yescharacteristic
516C23C2×C3×#4112yes—
517C23C2×C3×#4112yes—
518C23C3×H(5,𝔽2)12yes—
519C23C3×2−1+412yes—
520D4—416no—
521D4—432no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—416no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4C3×#12314yescomplemented
557C2×C3—28no—
558C2×C3—84no—
559C2×C3C2×#39211yescharacteristic
560C2×C3#240511yescharacteristic
561C2×C3#240611yescharacteristic
562C2×C3#242514yes—
563C2×C3#243014yes—
564C2×C3#250111yescharacteristic
565C2×C3#253911yescharacteristic
566C2×C3#261811yescharacteristic
567C2×C3#285411yescharacteristic
568C22—24no—
569C22—24no—
570C22—24no—
571C22—216no—
572C22—216no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—48no—
580C22—48no—
581C22—82no—
582C22—84no—
583C22—84no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22C3×#39218yes—
588C22C2×C3×#12011yescharacteristic
589C22C2×C3×#12012yes—
590C22C2×C3×#12111yescharacteristic
591C22C2×C3×#12112yes—
592C22C2×C3×#12311yescharacteristic
593C22C3×#52511yescharacteristic
594C22C3×#52912yes—
595C22C2×C3×#12711yescharacteristic
596C22C3×#53212yes—
597C22C3×#53411yescharacteristic
598C22C3×#53611yescharacteristic
599C22C3×#53712yes—
600C22C3×#53912yes—
601C22C3×#54614yes—
602C22C3×#54614yes—
603C22C3×#54714yes—
604C22C3×#54714yes—
605C4—24no—
606C4—44no—
607C4—44no—
608C4—42no—
609C4—48no—
610C4—44no—
611C3#1492611yescharacteristic, direct product factor, Sylow
612C2—28no—
613C2—84no—
614C2C2×C3×#39211yescharacteristic
615C2C3×#240511yescharacteristic
616C2C3×#240611yescharacteristic
617C2C3×#242514yes—
618C2C3×#243014yes—
619C2C3×#250111yescharacteristic
620C2C3×#253911yescharacteristic
621C2C3×#261811yescharacteristic
622C2C3×#285411yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#123C2semidirect
3. C2×C3×#386C2semidirect
4. C22×C3×#95C2semidirect
5. C3×#1444C2semidirect
6. C3×#1439C2semidirect
7. C3×#1438C2semidirect
8. C3×C22⋊#95C2semidirect
9. #14926C3direct (N × H)
33. C22×#123C2×C3semidirect
35. C2×#386C2×C3semidirect
36. C22×#95C2×C3semidirect
37. #1444C2×C3semidirect
38. #1439C2×C3semidirect
39. #1438C2×C3semidirect
40. C22⋊#95C2×C3semidirect
66. C23×C3×C4D4semidirect
91. C2×C3×C22⋊C4D4semidirect
228. C23×C4C3×D4semidirect
253. C2×C22⋊C4C3×D4semidirect
344. C2×C3×C4#123semidirect
556. C2×C4C3×#123semidirect
611. C3#14926direct (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
C22×C3×#1212
C2×C3×#3862
C22×C3×#952
C3×#14442
C3×#14392
C3×#14382
C3×C22⋊#952
#149263