Finite Group Databasebeta

← C3×#14776 #10633383

Subgroups — C3×#14776 #10633383

Order 768.

2362 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×#397C212yesmaximal, complemented
3C2×C3×#386C211yesmaximal, characteristic
4C2×C3×#386C212yesmaximal
5C3×#1445C212yesmaximal, complemented
6C3×#1442C212yesmaximal, complemented
7C3×#1439C212yesmaximal, complemented
8C3×C22⋊#95C211yesmaximal, characteristic, complemented
9C3×C22⋊#95C212yesmaximal, complemented
10#14776C311yesmaximal, 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⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2214yes—
23C3×#397—28no—
24C3×#386—28no—
25C3×#386—24no—
26C2×C3×#95C2214yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2214yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2214yes—
33C23×C4⋊C4C2×C311yescharacteristic, complemented
34C2×#397C2×C312yescomplemented
35C2×#386C2×C311yescharacteristic
36C2×#386C2×C312yes—
37#1445C2×C312yescomplemented
38#1442C2×C312yescomplemented
39#1439C2×C312yescomplemented
40C22⋊#95C2×C311yescharacteristic, complemented
41C22⋊#95C2×C312yescomplemented
42C25×C3C2311yescharacteristic
43C25×C3C2311yescharacteristic
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—24no—
47C22×C3×D4—24no—
48C22×C3×D4D414yes—
49C23×C3×C4—24no—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—24no—
53C23×C3×C4—28no—
54C23×C3×C4—24no—
55C23×C3×C4—24no—
56C23×C3×C4—44no—
57C23×C3×C4D414yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2314yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2314yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×C4⋊C4—232no—
66C2×C3×C4⋊C4—28no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4D418yescomplemented
71C2×C3×C22⋊C4—28no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—28no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—44no—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—28no—
86C3×#95—28no—
87C3×#95—24no—
88C3×#95—24no—
89C3×#95—44no—
90C23×D4C22×C311yescharacteristic
91C24×C4C22×C311yescharacteristic
92C24×C4C22×C312yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4C22×C312yes—
96C22×C4⋊C4C22×C312yes—
97C22×C22⋊C4C22×C311yescharacteristic
98C22×C22⋊C4C22×C312yes—
99C22×C22⋊C4C22×C311yescharacteristic
100C22×C22⋊C4C22×C311yescharacteristic
101C22×C22⋊C4C22×C314yes—
102#397—28no—
103#386—28no—
104#386—24no—
105C2×#95C22×C314yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C311yescharacteristic
109C2×#95C22×C314yes—
110C2×#95C22×C311yescharacteristic
111C2×#95C22×C314yes—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—21no—
118C24×C3—22no—
119C24×C3—21no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3C2×D412yes—
125C24×C3C2×D412yes—
126C24×C3C2×D412yes—
127C24×C3#4112yes—
128C24×C3#4114yes—
129C24×C3#4114yes—
130C24×C3C2411yescharacteristic
131C2×C3×D4—28no—
132C2×C3×D4—216no—
133C2×C3×D4—44no—
134C2×C3×D4—44no—
135C2×C3×D4—48no—
136C22×C3×C4—232no—
137C22×C3×C4—28no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—28no—
143C22×C3×C4—28no—
144C22×C3×C4—216no—
145C22×C3×C4—21no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—22no—
156C22×C3×C4—24no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—24no—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—416no—
171C22×C3×C4—416no—
172C22×C3×C4—48no—
173C22×C3×C4—48no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D418yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4#4114yes—
180C3×C4⋊C4—216no—
181C3×C4⋊C4—216no—
182C3×C22⋊C4—416no—
183C3×C22⋊C4—48no—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—44no—
187C25C23×C311yescharacteristic
188C25C23×C311yescharacteristic
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4C3×D414yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4C3×D414yes—
203C23×C4C23×C311yescharacteristic
204C23×C4C23×C314yes—
205C23×C4C23×C311yescharacteristic
206C23×C4C23×C314yes—
207C23×C4C23×C311yescharacteristic
208C23×C4C23×C311yescharacteristic
209C23×C4C23×C311yescharacteristic
210C2×C4⋊C4—232no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4C3×D418yescomplemented
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—44no—
228#95—22no—
229#95—22no—
230#95—28no—
231#95—28no—
232#95—24no—
233#95—24no—
234#95—44no—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—28no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—28no—
241C23×C3—24no—
242C23×C3—24no—
243C23×C3—24no—
244C23×C3—28no—
245C23×C3—24no—
246C23×C3—28no—
247C23×C3—22no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—44no—
255C23×C3—41no—
256C23×C3—42no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—44no—
260C23×C3—42no—
261C23×C3—44no—
262C23×C3—42no—
263C23×C3—44no—
264C23×C3#12114yes—
265C23×C3#12318yes—
266C23×C3#12314yes—
267C23×C3#12414yes—
268C23×C3#12618yes—
269C23×C3C22×D411yescharacteristic
270C23×C3C22×D412yes—
271C23×C3C22×D411yescharacteristic
272C23×C3C2×#4111yescharacteristic
273C23×C3C2×#4112yes—
274C23×C3C2×#4112yes—
275C23×C3C2×#4112yes—
276C23×C3H(5,𝔽2)12yes—
277C23×C32−1+412yes—
278C3×D4—416no—
279C2×C3×C4—232no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—28no—
285C2×C3×C4—28no—
286C2×C3×C4—28no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—28no—
290C2×C3×C4—28no—
291C2×C3×C4—432no—
292C2×C3×C4—44no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4—44no—
299C2×C3×C4—41no—
300C2×C3×C4—44no—
301C2×C3×C4—41no—
302C2×C3×C4—44no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
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—44no—
314C2×C3×C4#12014yes—
315C2×C3×C4#12314yes—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—22no—
323C24—21no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24C2×C3×D412yes—
329C24C2×C3×D412yes—
330C24C2×C3×D412yes—
331C24C3×#4112yes—
332C24C3×#4114yes—
333C24C3×#4114yes—
334C24C24×C311yescharacteristic
335C2×D4—28no—
336C2×D4—216no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—48no—
340C22×C4—232no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—216no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—416no—
375C22×C4—416no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4C2×C3×D412yes—
379C22×C4C2×C3×D418yes—
380C22×C4C2×C3×D412yes—
381C22×C4C2×C3×D414yes—
382C22×C4C2×C3×D412yes—
383C22×C4C3×#4114yes—
384C4⋊C4—216no—
385C4⋊C4—216no—
386C22⋊C4—416no—
387C22⋊C4—48no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—216no—
396C22×C3—216no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—48no—
402C22×C3—42no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—84no—
407C22×C3—81no—
408C22×C3#39718yescomplemented
409C22×C3C2×#12011yescharacteristic
410C22×C3C2×#12111yescharacteristic
411C22×C3C2×#12311yescharacteristic
412C22×C3C2×#12311yescharacteristic
413C22×C3C2×#12312yes—
414C22×C3#52514yes—
415C22×C3C2×#12411yescharacteristic
416C22×C3C2×#12612yes—
417C22×C3#53511yescharacteristic
418C22×C3#53611yescharacteristic
419C22×C3#53914yes—
420C22×C3D4212yes—
421C22×C3#54612yes—
422C22×C3#54712yes—
423C22×C3#54814yes—
424C22×C3#55212yes—
425C22×C3#55314yes—
426C3×C4—24no—
427C3×C4—24no—
428C3×C4—42no—
429C3×C4—48no—
430C3×C4—81no—
431C3×C4—84no—
432C3×C4—81no—
433C23—22no—
434C23—22no—
435C23—28no—
436C23—22no—
437C23—22no—
438C23—28no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—28no—
443C23—24no—
444C23—28no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—44no—
453C23—41no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—44no—
458C23—42no—
459C23—44no—
460C23—42no—
461C23—44no—
462C23C3×#12114yes—
463C23C3×#12318yes—
464C23C3×#12314yes—
465C23C3×#12414yes—
466C23C3×#12618yes—
467C23C22×C3×D411yescharacteristic
468C23C22×C3×D412yes—
469C23C22×C3×D411yescharacteristic
470C23C2×C3×#4111yescharacteristic
471C23C2×C3×#4112yes—
472C23C2×C3×#4112yes—
473C23C2×C3×#4112yes—
474C23C3×H(5,𝔽2)12yes—
475C23C3×2−1+412yes—
476D4—416no—
477C2×C4—232no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—432no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4C3×#12014yes—
513C2×C4C3×#12314yes—
514C2×C3—28no—
515C2×C3—82no—
516C2×C3C2×#39711yescharacteristic
517C2×C3#240811yescharacteristic
518C2×C3#243611yescharacteristic
519C2×C3#244314yes—
520C2×C3#248114yes—
521C2×C3#248611yescharacteristic
522C2×C3#250411yescharacteristic
523C2×C3#262311yescharacteristic
524C2×C3#284911yescharacteristic
525C22—24no—
526C22—24no—
527C22—24no—
528C22—216no—
529C22—216no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—48no—
536C22—42no—
537C22—81no—
538C22—81no—
539C22—84no—
540C22—81no—
541C22—81no—
542C22C3×#39718yescomplemented
543C22C2×C3×#12011yescharacteristic
544C22C2×C3×#12111yescharacteristic
545C22C2×C3×#12311yescharacteristic
546C22C2×C3×#12312yes—
547C22C2×C3×#12311yescharacteristic
548C22C3×#52514yes—
549C22C2×C3×#12411yescharacteristic
550C22C2×C3×#12612yes—
551C22C3×#53511yescharacteristic
552C22C3×#53611yescharacteristic
553C22C3×#53914yes—
554C22C3×D4212yes—
555C22C3×#54612yes—
556C22C3×#54712yes—
557C22C3×#54814yes—
558C22C3×#55212yes—
559C22C3×#55314yes—
560C4—24no—
561C4—24no—
562C4—42no—
563C4—48no—
564C4—81no—
565C4—84no—
566C4—81no—
567C3#1477611yescharacteristic, direct product factor, Sylow
568C2—28no—
569C2—82no—
570C2C2×C3×#39711yescharacteristic
571C2C3×#240811yescharacteristic
572C2C3×#243611yescharacteristic
573C2C3×#244314yes—
574C2C3×#248114yes—
575C2C3×#248611yescharacteristic
576C2C3×#250411yescharacteristic
577C2C3×#262311yescharacteristic
578C2C3×#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×#397C2semidirect
5. C3×#1445C2semidirect
6. C3×#1442C2semidirect
7. C3×#1439C2semidirect
8. C3×C22⋊#95C2semidirect
9. C3×C22⋊#95C2semidirect
10. #14776C3direct (N × H)
33. C23×C4⋊C4C2×C3semidirect
34. C2×#397C2×C3semidirect
37. #1445C2×C3semidirect
38. #1442C2×C3semidirect
39. #1439C2×C3semidirect
40. C22⋊#95C2×C3semidirect
41. C22⋊#95C2×C3semidirect
70. C2×C3×C4⋊C4D4semidirect
215. C2×C4⋊C4C3×D4semidirect
408. C22×C3#397semidirect
542. C22C3×#397semidirect
567. C3#14776direct (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×#3972
C2×C3×#3862
C3×#14452
C3×#14422
C3×#14392
C3×C22⋊#952
#147763