Finite Group Databasebeta

← C3×#24627 #10643234

Subgroups — C3×#24627 #10643234

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic, complemented
2C2×C3×C4×D8C211yesmaximal, characteristic
3C3×#2547C211yesmaximal, characteristic, complemented
4C3×#2354C211yesmaximal, characteristic, complemented
5C3×#2055C212yesmaximal, complemented
6C3×#2012C212yesmaximal
7C3×#1983C212yesmaximal, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1928C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24627C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2212yes—
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117—21no—
20C3×#461—24no—
21C3×C4×D8—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2212yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#397—22no—
33C3×#391C2212yes—
34C3×#390C2212yes—
35C3×#388—22no—
36C3×#384C2211yescharacteristic
37C3×#380—21no—
38C3×#380—21no—
39C3×#380—21no—
40C3×C4×C4⋊C4C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#461C2×C311yescharacteristic, complemented
45C2×C4×D8C2×C311yescharacteristic
46#2547C2×C311yescharacteristic, complemented
47#2354C2×C311yescharacteristic, complemented
48#2055C2×C312yescomplemented
49#2012C2×C312yes—
50#1983C2×C312yescomplemented
51#1956C2×C311yescharacteristic, complemented
52#1939C2×C311yescharacteristic, complemented
53#1928C2×C312yescomplemented
54#1802C2×C311yescharacteristic, complemented
55C22×C3×Q8C2312yes—
56C22×C3×D4C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C2×C3×SD16—24no—
61C2×C3×SD16—42no—
62C2×C3×SD16—42no—
63C2×C3×D8—22no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C22×C3×C8C2312yes—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#124—24no—
72C3×#124—42no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4D414yes—
77C3×#117—44no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C42—22no—
94C2×C3×C42C2312yes—
95C2×C3×C42C2311yescharacteristic
96C3×#107—22no—
97C3×C4⋊C8—22no—
98C3×C4⋊C8—42no—
99C3×#103—24no—
100C3×#103—24no—
101C3×#103—42no—
102C3×#102—22no—
103C3×#102—22no—
104C3×#102—42no—
105C3×C4×C8—22no—
106C3×#95—22no—
107C3×#95—22no—
108C3×#95—22no—
109C3×#95D412yes—
110C22×SD16C22×C312yes—
111C22×D8C22×C311yescharacteristic
112C2×C4⋊Q8C22×C311yescharacteristic
113C2×#124C22×C312yes—
114C2×C4×D4C22×C311yescharacteristic
115C2×C4×D4C22×C311yescharacteristic
116C2×#117—21no—
117#461—24no—
118C4×D8—24no—
119C2×#107C22×C311yescharacteristic
120C2×C4⋊C8C22×C311yescharacteristic
121C2×C4⋊C8C22×C312yes—
122C2×#103C22×C312yes—
123C2×#103C22×C312yes—
124C2×#103C22×C312yes—
125C2×#102C22×C311yescharacteristic
126C2×#102C22×C311yescharacteristic
127C2×#102C22×C312yes—
128C2×C4×C8C22×C311yescharacteristic
129#397—22no—
130#391C22×C312yes—
131#390C22×C312yes—
132#388—22no—
133#384C22×C311yescharacteristic
134#380—21no—
135#380—21no—
136#380—21no—
137C4×C4⋊C4C22×C312yes—
138C4×C4⋊C4C22×C311yescharacteristic
139#340C22×C312yes—
140#336C22×C312yes—
141C24×C3—21no—
142C24×C3—21no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4C2×D412yes—
160C2×C3×D4#4112yes—
161C22×C3×C4—21no—
162C22×C3×C4—24no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C3×SD16—48no—
177C3×D8—28no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—22no—
186C2×C3×C8#4112yes—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—81no—
198C3×C4⋊C4C2×D412yes—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—44no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—44no—
203C3×C22⋊C4—42no—
204C3×C22⋊C4—82no—
205C3×C42—22no—
206C3×C42—22no—
207C3×C42—44no—
208C3×C42C2×D412yes—
209C3×C42#4112yes—
210C22×Q8C23×C312yes—
211C22×D4C23×C311yescharacteristic
212C22×D4C23×C311yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C2×SD16—24no—
216C2×SD16—42no—
217C2×SD16—42no—
218C2×D8—22no—
219C2×D8—22no—
220C2×D8—22no—
221C22×C8C23×C312yes—
222C22×C8C23×C311yescharacteristic
223C22×C8C23×C311yescharacteristic
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226#124—24no—
227#124—42no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4C3×D414yes—
232#117—44no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C3×D411yescharacteristic
238C2×C4⋊C4C3×D411yescharacteristic
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C312yes—
241C2×C4⋊C4C23×C311yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C42—22no—
249C2×C42C23×C312yes—
250C2×C42C23×C311yescharacteristic
251#107—22no—
252C4⋊C8—22no—
253C4⋊C8—42no—
254#103—24no—
255#103—24no—
256#103—42no—
257#102—22no—
258#102—22no—
259#102—42no—
260C4×C8—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95C3×D412yes—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3C22×D411yescharacteristic
280C3×Q8—42no—
281C3×Q8—42no—
282C3×Q8—42no—
283C3×Q8—42no—
284C3×D4—22no—
285C3×D4—22no—
286C3×D4—22no—
287C3×D4—22no—
288C3×D4—41no—
289C3×D4—41no—
290C3×D4—41no—
291C3×D4—41no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—48no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—84no—
314C2×C3×C4#12112yes—
315C2×C3×C4#12112yes—
316C2×C3×C4#13512yes—
317C2×C3×C4AGL(1,ℤ/8)12yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C4H(5,𝔽2)11yescharacteristic
324C3×C8—22no—
325C3×C8—41no—
326C3×C8—42no—
327C24—21no—
328C24—21no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4C2×C3×D412yes—
346C2×D4C3×#4112yes—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4C2×C3×D412yes—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C24×C311yescharacteristic
362SD16—48no—
363D8—28no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8C3×#4112yes—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—81no—
384C4⋊C4C2×C3×D412yes—
385C22⋊C4—24no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—42no—
390C22⋊C4—82no—
391C42—22no—
392C42—22no—
393C42—44no—
394C42C2×C3×D412yes—
395C42C3×#4112yes—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3C2×#12111yescharacteristic
411C22×C3#54611yescharacteristic
412C22×C3#55011yescharacteristic
413C22×C3C2×#13511yescharacteristic
414C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
415C22×C3#57711yescharacteristic
416C22×C3#57811yescharacteristic
417C3×C4—22no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—82no—
422C3×C4—82no—
423C3×C4—81no—
424C3×C4#46314yes—
425C3×C4#52511yescharacteristic
426C3×C4#53511yescharacteristic
427C3×C4#55311yescharacteristic
428C3×C4#55511yescharacteristic
429C23—22no—
430C23—22no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23C22×C3×D411yescharacteristic
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448D4—22no—
449D4—22no—
450D4—22no—
451D4—22no—
452D4—41no—
453D4—41no—
454D4—41no—
455D4—41no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—48no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—84no—
478C2×C4C3×#12112yes—
479C2×C4C3×#12112yes—
480C2×C4C3×#13512yes—
481C2×C4C3×AGL(1,ℤ/8)12yes—
482C2×C4C22×C3×D411yescharacteristic
483C2×C4C2×C3×#4111yescharacteristic
484C2×C4C2×C3×#4111yescharacteristic
485C2×C4C2×C3×#4111yescharacteristic
486C2×C4C3×H(5,𝔽2)11yescharacteristic
487C2×C4C3×H(5,𝔽2)11yescharacteristic
488C8—22no—
489C8—42no—
490C8—41no—
491C2×C3—82no—
492C2×C3—82no—
493C2×C3#241411yescharacteristic
494C2×C3C2×#46311yescharacteristic
495C2×C3#304611yescharacteristic
496C2×C3#330311yescharacteristic
497C2×C3#333011yescharacteristic
498C2×C3#339311yescharacteristic
499C2×C3#339711yescharacteristic
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22C2×C3×#12111yescharacteristic
515C22C3×#54611yescharacteristic
516C22C3×#55011yescharacteristic
517C22C2×C3×#13511yescharacteristic
518C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
519C22C3×#57711yescharacteristic
520C22C3×#57811yescharacteristic
521C4—22no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—82no—
526C4—82no—
527C4—81no—
528C4C3×#46314yes—
529C4C3×#52511yescharacteristic
530C4C3×#53511yescharacteristic
531C4C3×#55311yescharacteristic
532C4C3×#55511yescharacteristic
533C3#2462711yescharacteristic, direct product factor, Sylow
534C2—82no—
535C2—82no—
536C2C3×#241411yescharacteristic
537C2C2×C3×#46311yescharacteristic
538C2C3×#304611yescharacteristic
539C2C3×#330311yescharacteristic
540C2C3×#333011yescharacteristic
541C2C3×#339311yescharacteristic
542C2C3×#339711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
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×#461C2semidirect
3. C3×#2547C2semidirect
4. C3×#2354C2semidirect
5. C3×#2055C2semidirect
7. C3×#1983C2semidirect
8. C3×#1956C2semidirect
9. C3×#1939C2semidirect
10. C3×#1928C2semidirect
11. C3×#1802C2semidirect
12. #24627C3direct (N × H)
44. C2×#461C2×C3semidirect
46. #2547C2×C3semidirect
47. #2354C2×C3semidirect
48. #2055C2×C3semidirect
50. #1983C2×C3semidirect
51. #1956C2×C3semidirect
52. #1939C2×C3semidirect
53. #1928C2×C3semidirect
54. #1802C2×C3semidirect
533. C3#24627direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4612
C2×C3×C4×D82
C3×#25472
C3×#23542
C3×#20552
C3×#20122
C3×#19832
C3×#19562
C3×#19392
C3×#19282
C3×#18022
#246273