Finite Group Databasebeta

← C3×#24628 #10643235

Subgroups — C3×#24628 #10643235

Order 768.

1034 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#460C211yesmaximal, characteristic, complemented
2C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
3C3×#2541C211yesmaximal, characteristic
4C3×#2355C211yesmaximal, characteristic, complemented
5C3×#2055C212yesmaximal, complemented
6C3×#2014C212yesmaximal, complemented
7C3×#1982C212yesmaximal, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1926C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24628C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#127C2212yes—
17C2×C3×#121—21no—
18C2×C3×#121—21no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C3×#460—24no—
22C3×C4×SD16—24no—
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2212yes—
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C4×C8C2211yescharacteristic
33C3×#392—22no—
34C3×#391C2212yes—
35C3×#390C2212yes—
36C3×#387—22no—
37C3×#384C2211yescharacteristic
38C3×#382—21no—
39C3×C4×C4⋊C4—21no—
40C3×C4×C4⋊C4C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#460C2×C311yescharacteristic, complemented
45C2×C4×SD16C2×C311yescharacteristic, complemented
46#2541C2×C311yescharacteristic
47#2355C2×C311yescharacteristic, complemented
48#2055C2×C312yescomplemented
49#2014C2×C312yescomplemented
50#1982C2×C312yescomplemented
51#1957C2×C311yescharacteristic, complemented
52#1939C2×C311yescharacteristic, complemented
53#1926C2×C312yescomplemented
54#1802C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4C2312yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C2×C3×SD16—24no—
66C2×C3×SD16—22no—
67C2×C3×SD16—42no—
68C2×C3×SD16—42no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×#127—24no—
73C3×#127—22no—
74C3×#127—22no—
75C3×#127—42no—
76C3×#121—44no—
77C3×#121—44no—
78C3×C4×Q8—22no—
79C3×C4×Q8D414yescomplemented
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×C3×C42—22no—
95C2×C3×C42C2312yes—
96C2×C3×C42C2311yescharacteristic
97C3×#106—22no—
98C3×C4⋊C8—22no—
99C3×C4⋊C8—42no—
100C3×#103—22no—
101C3×#102—24no—
102C3×#102—22no—
103C3×#102—24no—
104C3×#102—42no—
105C3×#102—42no—
106C3×C4×C8—22no—
107C3×#95—22no—
108C3×#95—22no—
109C3×#95—22no—
110C3×#95D412yes—
111C22×SD16C22×C311yescharacteristic
112C22×SD16C22×C312yes—
113C2×#127C22×C311yescharacteristic
114C2×#127C22×C312yes—
115C2×#121—21no—
116C2×#121—21no—
117C2×C4×Q8C22×C311yescharacteristic
118C2×C4×D4C22×C311yescharacteristic
119#460—24no—
120C4×SD16—24no—
121C2×#106C22×C311yescharacteristic
122C2×C4⋊C8C22×C311yescharacteristic
123C2×C4⋊C8C22×C312yes—
124C2×#103C22×C311yescharacteristic
125C2×#102C22×C311yescharacteristic
126C2×#102C22×C312yes—
127C2×#102C22×C312yes—
128C2×#102C22×C312yes—
129C2×#102C22×C312yes—
130C2×C4×C8C22×C311yescharacteristic
131#392—22no—
132#391C22×C312yes—
133#390C22×C312yes—
134#387—22no—
135#384C22×C311yescharacteristic
136#382—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C22×C312yes—
139C4×C4⋊C4C22×C311yescharacteristic
140#340C22×C312yes—
141#336C22×C312yes—
142C24×C3—21no—
143C24×C3—22no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—21no—
149C2×C3×Q8C2×D412yes—
150C2×C3×D4—21no—
151C2×C3×D4—22no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—44no—
162C2×C3×D4—44no—
163C2×C3×D4—44no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—44no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C2×C3×D4—82no—
170C2×C3×D4—81no—
171C2×C3×D4—81no—
172C2×C3×D4—82no—
173C2×C3×D4#4112yes—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C3×SD16—28no—
187C3×SD16—48no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8#4112yes—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—81no—
206C3×C4⋊C4SD1614yescomplemented
207C3×C4⋊C4C2×D412yes—
208C3×C22⋊C4—44no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—44no—
211C3×C42—24no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—82no—
215C3×C42C2×D412yes—
216C3×C42#4112yes—
217C22×Q8C23×C311yescharacteristic
218C22×D4—21no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4C23×C312yes—
223C22×D4C23×C311yescharacteristic
224C23×C4—21no—
225C2×SD16—22no—
226C2×SD16—22no—
227C2×SD16—24no—
228C2×SD16—22no—
229C2×SD16—42no—
230C2×SD16—42no—
231C22×C8C23×C312yes—
232C22×C8C23×C311yescharacteristic
233C22×C8C23×C311yescharacteristic
234#127—24no—
235#127—22no—
236#127—22no—
237#127—42no—
238#121—44no—
239#121—44no—
240C4×Q8—22no—
241C4×Q8C3×D414yescomplemented
242C4×D4—22no—
243C4×D4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C3×D411yescharacteristic
248C2×C4⋊C4C3×D411yescharacteristic
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C312yes—
251C2×C4⋊C4C23×C311yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42C23×C312yes—
258C2×C42C23×C311yescharacteristic
259#106—22no—
260C4⋊C8—22no—
261C4⋊C8—42no—
262#103—22no—
263#102—24no—
264#102—22no—
265#102—24no—
266#102—42no—
267#102—42no—
268C4×C8—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95C3×D412yes—
273C23×C3—22no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3C22×D411yescharacteristic
288C3×Q8—22no—
289C3×Q8—22no—
290C3×Q8—41no—
291C3×Q8—41no—
292C3×D4—22no—
293C3×D4—22no—
294C3×D4—42no—
295C3×D4—41no—
296C3×D4—42no—
297C3×D4—42no—
298C3×D4—41no—
299C3×D4—42no—
300C3×D4—88no—
301C3×D4—84no—
302C3×D4—88no—
303C3×D4—84no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—84no—
324C2×C3×C4#12112yes—
325C2×C3×C4#12112yes—
326C2×C3×C4C2×SD1614yes—
327C2×C3×C4C2×SD1612yes—
328C2×C3×C4AGL(1,ℤ/8)12yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C2×#4111yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334C2×C3×C4H(5,𝔽2)11yescharacteristic
335C3×C8—22no—
336C3×C8—41no—
337C3×C8—42no—
338C24—21no—
339C24—22no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8C2×C3×D412yes—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—82no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4—82no—
369C2×D4C3×#4112yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4C2×C3×D412yes—
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D412yes—
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C24×C311yescharacteristic
382SD16—28no—
383SD16—48no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8C3×#4112yes—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—81no—
402C4⋊C4C3×SD1614yescomplemented
403C4⋊C4C2×C3×D412yes—
404C22⋊C4—44no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C42—24no—
408C42—22no—
409C42—22no—
410C42—82no—
411C42C2×C3×D412yes—
412C42C3×#4112yes—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—81no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3C2×#12111yescharacteristic
428C22×C3#54611yescharacteristic
429C22×C3#55011yescharacteristic
430C22×C3C22×SD1611yescharacteristic
431C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
432C22×C3#57611yescharacteristic
433C22×C3#57711yescharacteristic
434C3×C4—24no—
435C3×C4—22no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—42no—
439C3×C4—82no—
440C3×C4—81no—
441C3×C4#46014yes—
442C3×C4#52511yescharacteristic
443C3×C4#53511yescharacteristic
444C3×C4#55311yescharacteristic
445C3×C4#55511yescharacteristic
446C23—22no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23C22×C3×D411yescharacteristic
461Q8—22no—
462Q8—22no—
463Q8—41no—
464Q8—41no—
465D4—22no—
466D4—22no—
467D4—42no—
468D4—41no—
469D4—42no—
470D4—42no—
471D4—41no—
472D4—42no—
473D4—88no—
474D4—84no—
475D4—88no—
476D4—84no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4C3×#12112yes—
498C2×C4C3×#12112yes—
499C2×C4C2×C3×SD1614yes—
500C2×C4C2×C3×SD1612yes—
501C2×C4C3×AGL(1,ℤ/8)12yes—
502C2×C4C22×C3×D411yescharacteristic
503C2×C4C2×C3×#4111yescharacteristic
504C2×C4C2×C3×#4111yescharacteristic
505C2×C4C2×C3×#4111yescharacteristic
506C2×C4C3×H(5,𝔽2)11yescharacteristic
507C2×C4C3×H(5,𝔽2)11yescharacteristic
508C8—22no—
509C8—42no—
510C8—41no—
511C2×C3—82no—
512C2×C3—162no—
513C2×C3#241411yescharacteristic
514C2×C3C2×#46011yescharacteristic
515C2×C3#304511yescharacteristic
516C2×C3#329511yescharacteristic
517C2×C3#332411yescharacteristic
518C2×C3#339211yescharacteristic
519C2×C3#339911yescharacteristic
520C22—42no—
521C22—42no—
522C22—42no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22C2×C3×#12111yescharacteristic
535C22C3×#54611yescharacteristic
536C22C3×#55011yescharacteristic
537C22C22×C3×SD1611yescharacteristic
538C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
539C22C3×#57611yescharacteristic
540C22C3×#57711yescharacteristic
541C4—24no—
542C4—22no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—82no—
547C4—81no—
548C4C3×#46014yes—
549C4C3×#52511yescharacteristic
550C4C3×#53511yescharacteristic
551C4C3×#55311yescharacteristic
552C4C3×#55511yescharacteristic
553C3#2462811yescharacteristic, direct product factor, Sylow
554C2—82no—
555C2—162no—
556C2C3×#241411yescharacteristic
557C2C2×C3×#46011yescharacteristic
558C2C3×#304511yescharacteristic
559C2C3×#329511yescharacteristic
560C2C3×#332411yescharacteristic
561C2C3×#339211yescharacteristic
562C2C3×#339911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#460C2semidirect
2. C2×C3×C4×SD16C2semidirect
4. C3×#2355C2semidirect
5. C3×#2055C2semidirect
6. C3×#2014C2semidirect
7. C3×#1982C2semidirect
8. C3×#1957C2semidirect
9. C3×#1939C2semidirect
10. C3×#1926C2semidirect
11. C3×#1802C2semidirect
12. #24628C3direct (N × H)
44. C2×#460C2×C3semidirect
45. C2×C4×SD16C2×C3semidirect
47. #2355C2×C3semidirect
48. #2055C2×C3semidirect
49. #2014C2×C3semidirect
50. #1982C2×C3semidirect
51. #1957C2×C3semidirect
52. #1939C2×C3semidirect
53. #1926C2×C3semidirect
54. #1802C2×C3semidirect
79. C3×C4×Q8D4semidirect
206. C3×C4⋊C4SD16semidirect
241. C4×Q8C3×D4semidirect
402. C4⋊C4C3×SD16semidirect
553. C3#24628direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4602
C2×C3×C4×SD162
C3×#25412
C3×#23552
C3×#20552
C3×#20142
C3×#19822
C3×#19572
C3×#19392
C3×#19262
C3×#18022
#246283