Finite Group Databasebeta

← C3×#14777 #10633384

Subgroups — C3×#14777 #10633384

Order 768.

2506 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×#399C212yesmaximal, complemented
3C2×C3×#386C211yesmaximal, characteristic
4C2×C3×#386C212yesmaximal
5C2×C3×#385C212yesmaximal, complemented
6C3×#1445C212yesmaximal, complemented
7C3×#1444C212yesmaximal, complemented
8C3×#1439C212yesmaximal, complemented
9C3×C22⋊#95C211yesmaximal, characteristic, complemented
10#14777C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C24×C3×C4C2212yes—
14C22×C3×C4⋊C4—24no—
15C22×C3×C4⋊C4—28no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2214yes—
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2214yes—
23C22×C3×C22⋊C4C2212yes—
24C3×#399—28no—
25C3×#386—24no—
26C3×#386—28no—
27C3×#385—28no—
28C2×C3×#95C2212yes—
29C2×C3×#95C2214yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2214yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C23×C4⋊C4C2×C311yescharacteristic, complemented
35C2×#399C2×C312yescomplemented
36C2×#386C2×C311yescharacteristic
37C2×#386C2×C312yes—
38C2×#385C2×C312yescomplemented
39#1445C2×C312yescomplemented
40#1444C2×C312yescomplemented
41#1439C2×C312yescomplemented
42C22⋊#95C2×C311yescharacteristic, complemented
43C25×C3C2311yescharacteristic
44C25×C3C2311yescharacteristic
45C22×C3×D4—24no—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4—24no—
49C22×C3×D4Q814yes—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—24no—
53C23×C3×C4—24no—
54C23×C3×C4—28no—
55C23×C3×C4—24no—
56C23×C3×C4—24no—
57C23×C3×C4—44no—
58C23×C3×C4D414yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2314yes—
62C23×C3×C4C2314yes—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C2×C3×C4⋊C4—232no—
67C2×C3×C4⋊C4—28no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—28no—
72C2×C3×C4⋊C4—28no—
73C2×C3×C4⋊C4—44no—
74C2×C3×C4⋊C4D418yescomplemented
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—28no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—28no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—44no—
86C2×C3×C22⋊C4D418yes—
87C3×#95—28no—
88C3×#95—28no—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—24no—
92C3×#95—24no—
93C23×D4C22×C311yescharacteristic
94C24×C4C22×C311yescharacteristic
95C24×C4C22×C312yes—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4C22×C312yes—
99C22×C4⋊C4C22×C314yes—
100C22×C4⋊C4C22×C312yes—
101C22×C22⋊C4C22×C311yescharacteristic
102C22×C22⋊C4C22×C311yescharacteristic
103C22×C22⋊C4C22×C311yescharacteristic
104C22×C22⋊C4C22×C314yes—
105C22×C22⋊C4C22×C312yes—
106#399—28no—
107#386—24no—
108#386—28no—
109#385—28no—
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C314yes—
113C2×#95C22×C311yescharacteristic
114C2×#95C22×C314yes—
115C2×#95C22×C311yescharacteristic
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—21no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—21no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3C2×D414yes—
129C24×C3C2×D412yes—
130C24×C3C2×Q812yes—
131C24×C3C2×Q812yes—
132C24×C3#4112yes—
133C24×C3#4114yes—
134C24×C3C2411yescharacteristic
135C2×C3×D4—28no—
136C2×C3×D4—216no—
137C2×C3×D4—48no—
138C2×C3×D4—44no—
139C2×C3×D4—44no—
140C22×C3×C4—232no—
141C22×C3×C4—28no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—28no—
147C22×C3×C4—28no—
148C22×C3×C4—216no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—24no—
157C22×C3×C4—24no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—416no—
174C22×C3×C4—416no—
175C22×C3×C4—48no—
176C22×C3×C4—48no—
177C22×C3×C4C2×D418yes—
178C22×C3×C4C2×D418yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×Q812yes—
183C22×C3×C4#4114yes—
184C3×C4⋊C4—216no—
185C3×C4⋊C4—216no—
186C3×C4⋊C4—416no—
187C3×C22⋊C4—216no—
188C3×C22⋊C4—416no—
189C3×C22⋊C4—44no—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—44no—
192C25C23×C311yescharacteristic
193C25C23×C311yescharacteristic
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4C3×Q814yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—44no—
207C23×C4C3×D414yes—
208C23×C4C23×C311yescharacteristic
209C23×C4C23×C311yescharacteristic
210C23×C4C23×C314yes—
211C23×C4C23×C314yes—
212C23×C4C23×C311yescharacteristic
213C23×C4C23×C311yescharacteristic
214C23×C4C23×C311yescharacteristic
215C2×C4⋊C4—232no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C3×D418yescomplemented
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4C3×D418yes—
236#95—28no—
237#95—28no—
238#95—22no—
239#95—22no—
240#95—24no—
241#95—24no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—28no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—28no—
248C23×C3—28no—
249C23×C3—28no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—42no—
262C23×C3—44no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—44no—
267C23×C3—44no—
268C23×C3—44no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3C22⋊Q814yes—
272C23×C3C22⋊Q818yes—
273C23×C3#12318yes—
274C23×C3#12314yes—
275C23×C3#12714yes—
276C23×C3C22×D411yescharacteristic
277C23×C3C22×D412yes—
278C23×C3C22×D412yes—
279C23×C3C22×Q811yescharacteristic
280C23×C3C2×#4112yes—
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4112yes—
283C23×C3H(5,𝔽2)12yes—
284C23×C32−1+412yes—
285C3×D4—416no—
286C2×C3×C4—232no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—28no—
292C2×C3×C4—28no—
293C2×C3×C4—28no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—28no—
297C2×C3×C4—28no—
298C2×C3×C4—28no—
299C2×C3×C4—28no—
300C2×C3×C4—28no—
301C2×C3×C4—432no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—44no—
305C2×C3×C4—44no—
306C2×C3×C4—41no—
307C2×C3×C4—44no—
308C2×C3×C4—44no—
309C2×C3×C4—44no—
310C2×C3×C4—44no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—44no—
320C2×C3×C4—44no—
321C2×C3×C4#12014yes—
322C2×C3×C4#12314yes—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24C2×C3×D414yes—
336C24C2×C3×D412yes—
337C24C2×C3×Q812yes—
338C24C2×C3×Q812yes—
339C24C3×#4112yes—
340C24C3×#4114yes—
341C24C24×C311yescharacteristic
342C2×D4—28no—
343C2×D4—216no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—44no—
347C22×C4—232no—
348C22×C4—28no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—416no—
381C22×C4—416no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4C2×C3×D418yes—
385C22×C4C2×C3×D418yes—
386C22×C4C2×C3×D412yes—
387C22×C4C2×C3×D414yes—
388C22×C4C2×C3×D412yes—
389C22×C4C2×C3×Q812yes—
390C22×C4C3×#4114yes—
391C4⋊C4—216no—
392C4⋊C4—216no—
393C4⋊C4—416no—
394C22⋊C4—216no—
395C22⋊C4—416no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—216no—
404C22×C3—216no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—48no—
410C22×C3—42no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3—84no—
415C22×C3—81no—
416C22×C3#39918yescomplemented
417C22×C3C2×#12011yescharacteristic
418C22×C3C2×C22⋊Q811yescharacteristic
419C22×C3C2×C22⋊Q812yes—
420C22×C3C2×#12311yescharacteristic
421C22×C3C2×#12311yescharacteristic
422C22×C3C2×#12312yes—
423C22×C3#52514yes—
424C22×C3C2×#12711yescharacteristic
425C22×C3#53511yescharacteristic
426C22×C3#53611yescharacteristic
427C22×C3#53914yes—
428C22×C3D4214yes—
429C22×C3#54714yes—
430C22×C3#54812yes—
431C22×C3D4×Q812yes—
432C22×C3#55312yes—
433C22×C3#55412yes—
434C3×C4—24no—
435C3×C4—24no—
436C3×C4—42no—
437C3×C4—48no—
438C3×C4—48no—
439C3×C4—81no—
440C3×C4—81no—
441C23—22no—
442C23—22no—
443C23—28no—
444C23—22no—
445C23—22no—
446C23—28no—
447C23—28no—
448C23—28no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—44no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23C3×C22⋊Q814yes—
471C23C3×C22⋊Q818yes—
472C23C3×#12318yes—
473C23C3×#12314yes—
474C23C3×#12714yes—
475C23C22×C3×D411yescharacteristic
476C23C22×C3×D412yes—
477C23C22×C3×D412yes—
478C23C22×C3×Q811yescharacteristic
479C23C2×C3×#4112yes—
480C23C2×C3×#4111yescharacteristic
481C23C2×C3×#4112yes—
482C23C3×H(5,𝔽2)12yes—
483C23C3×2−1+412yes—
484D4—416no—
485C2×C4—232no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—432no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4C3×#12014yes—
521C2×C4C3×#12314yes—
522C2×C3—28no—
523C2×C3—82no—
524C2×C3C2×#39911yescharacteristic
525C2×C3#240911yescharacteristic
526C2×C3#243311yescharacteristic
527C2×C3#244314yes—
528C2×C3#244914yes—
529C2×C3#248711yescharacteristic
530C2×C3#250111yescharacteristic
531C2×C3#262311yescharacteristic
532C2×C3#284911yescharacteristic
533C22—24no—
534C22—24no—
535C22—24no—
536C22—216no—
537C22—216no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—42no—
542C22—42no—
543C22—48no—
544C22—42no—
545C22—81no—
546C22—81no—
547C22—84no—
548C22—81no—
549C22—81no—
550C22C3×#39918yescomplemented
551C22C2×C3×#12011yescharacteristic
552C22C2×C3×C22⋊Q811yescharacteristic
553C22C2×C3×C22⋊Q812yes—
554C22C2×C3×#12311yescharacteristic
555C22C2×C3×#12312yes—
556C22C2×C3×#12311yescharacteristic
557C22C3×#52514yes—
558C22C2×C3×#12711yescharacteristic
559C22C3×#53511yescharacteristic
560C22C3×#53611yescharacteristic
561C22C3×#53914yes—
562C22C3×D4214yes—
563C22C3×#54714yes—
564C22C3×#54812yes—
565C22C3×D4×Q812yes—
566C22C3×#55312yes—
567C22C3×#55412yes—
568C4—24no—
569C4—24no—
570C4—42no—
571C4—48no—
572C4—48no—
573C4—81no—
574C4—81no—
575C3#1477711yescharacteristic, direct product factor, Sylow
576C2—28no—
577C2—82no—
578C2C2×C3×#39911yescharacteristic
579C2C3×#240911yescharacteristic
580C2C3×#243311yescharacteristic
581C2C3×#244314yes—
582C2C3×#244914yes—
583C2C3×#248711yescharacteristic
584C2C3×#250111yescharacteristic
585C2C3×#262311yescharacteristic
586C2C3×#284911yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C3×C4⋊C4C2semidirect
2. C2×C3×#399C2semidirect
5. C2×C3×#385C2semidirect
6. C3×#1445C2semidirect
7. C3×#1444C2semidirect
8. C3×#1439C2semidirect
9. C3×C22⋊#95C2semidirect
10. #14777C3direct (N × H)
34. C23×C4⋊C4C2×C3semidirect
35. C2×#399C2×C3semidirect
38. C2×#385C2×C3semidirect
39. #1445C2×C3semidirect
40. #1444C2×C3semidirect
41. #1439C2×C3semidirect
42. C22⋊#95C2×C3semidirect
74. C2×C3×C4⋊C4D4semidirect
223. C2×C4⋊C4C3×D4semidirect
416. C22×C3#399semidirect
550. C22C3×#399semidirect
575. C3#14777direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C2×C3×#3992
C2×C3×#3862
C2×C3×#3852
C3×#14452
C3×#14442
C3×#14392
C3×C22⋊#952
#147773