Finite Group Databasebeta

← C3×#18321 #10636928

Subgroups — C3×#18321 #10636928

Order 768.

982 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#539C211yesmaximal, characteristic
2C3×#3195C211yesmaximal, characteristic, complemented
3C3×#3193C211yesmaximal, characteristic
4C3×#2583C211yesmaximal, characteristic, complemented
5C3×#1609C214yesmaximal, complemented
6C3×#1603C214yesmaximal, complemented
7C3×#1508C211yesmaximal, characteristic, complemented
8C3×#1507C211yesmaximal, characteristic
9C3×#1463C211yesmaximal, characteristic, complemented
10#18321C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#539—22no—
12C3×#539—22no—
13C3×#539—22no—
14C3×#539C2214yes—
15C2×C3×#126—21no—
16C2×C3×#125—21no—
17C2×C3×#123—21no—
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C22×C3×C4⋊C4C2211yescharacteristic
25C3×#484—21no—
26C3×#483—21no—
27C3×#483C2212yescomplemented
28C3×#480C2212yes—
29C3×#470—22no—
30C3×#468—22no—
31C3×#451—21no—
32C3×#451C2212yescomplemented
33C3×#450—21no—
34C3×#448C2212yes—
35C3×#407C2211yescharacteristic
36C3×#407C2211yescharacteristic
37C3×#398C2212yescomplemented
38C3×#395C2212yescomplemented
39C3×#389—22no—
40C3×#385—21no—
41C3×#385—21no—
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#328C2214yescomplemented
44C3×#327C2214yes—
45C3×#324C2214yescomplemented
46C2×#539C2×C311yescharacteristic
47#3195C2×C311yescharacteristic, complemented
48#3193C2×C311yescharacteristic
49#2583C2×C311yescharacteristic, complemented
50#1609C2×C314yescomplemented
51#1603C2×C314yescomplemented
52#1508C2×C311yescharacteristic, complemented
53#1507C2×C311yescharacteristic
54#1463C2×C311yescharacteristic, complemented
55C22×C3×Q8D411yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—21no—
58C22×C3×D4D411yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C2×C3×Q16—22no—
64C2×C3×SD16—22no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C3×#126—24no—
68C3×#126—24no—
69C3×#125—24no—
70C3×#125—24no—
71C3×#123—24no—
72C3×#123—24no—
73C3×#123—24no—
74C3×#123—24no—
75C3×C22⋊Q8—24no—
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8—21no—
78C3×C22⋊Q8—21no—
79C3×C22⋊Q8—21no—
80C3×C22⋊Q8—21no—
81C3×C22⋊Q8—21no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8D412yes—
84C3×C22⋊Q8C2312yes—
85C3×#121—21no—
86C3×#121—21no—
87C3×#121D412yes—
88C3×#121C2312yes—
89C3×C4×D4—24no—
90C3×C4×D4—24no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4D412yes—
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C42—21no—
115C2×C3×C42—21no—
116C3×#107—22no—
117C3×#106—22no—
118C3×#103—22no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#102—22no—
122C3×C22⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×C22⋊C8C2312yes—
125C3×C22⋊C8C2312yes—
126C3×#95—22no—
127C3×#95—22no—
128C3×#95C2312yes—
129#539—22no—
130#539—22no—
131#539—22no—
132#539C22×C314yes—
133C2×#126—21no—
134C2×#125—21no—
135C2×#123—21no—
136C2×#123—21no—
137C2×C22⋊Q8—21no—
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×C22⋊Q8C22×C311yescharacteristic
140C2×#121C22×C311yescharacteristic
141C2×C4×D4—21no—
142C22×C4⋊C4C22×C311yescharacteristic
143#484—21no—
144#483—21no—
145#483C22×C312yescomplemented
146#480C22×C312yes—
147#470—22no—
148#468—22no—
149#451—21no—
150#451C22×C312yescomplemented
151#450—21no—
152#448C22×C312yes—
153#407C22×C311yescharacteristic
154#407C22×C311yescharacteristic
155#398C22×C312yescomplemented
156#395C22×C312yescomplemented
157#389—22no—
158#385—21no—
159#385—21no—
160C4×C22⋊C4C22×C311yescharacteristic
161#328C22×C314yescomplemented
162#327C22×C314yes—
163#324C22×C314yescomplemented
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—81no—
170C2×C3×Q8—81no—
171C2×C3×Q8C2×D412yes—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—21no—
175C2×C3×D4—44no—
176C2×C3×D4—81no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—24no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—42no—
190C22×C3×C4—81no—
191C22×C3×C4—81no—
192C22×C3×C4—82no—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C3×Q16—82no—
202C3×SD16—82no—
203C3×M16—81no—
204C3×M16—81no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—44no—
221C3×C4⋊C4—41no—
222C3×C4⋊C4—41no—
223C3×C4⋊C4—82no—
224C3×C4⋊C4C2×D412yes—
225C3×C4⋊C4C2×D412yes—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—41no—
235C3×C22⋊C4—41no—
236C3×C42—24no—
237C3×C42—81no—
238C22×Q8C3×D411yescharacteristic
239C22×Q8C23×C311yescharacteristic
240C22×D4—21no—
241C22×D4C3×D411yescharacteristic
242C23×C4—21no—
243C23×C4C3×D411yescharacteristic
244C23×C4C3×D411yescharacteristic
245C23×C4C23×C311yescharacteristic
246C2×Q16—22no—
247C2×SD16—22no—
248C2×M16—21no—
249C2×M16—21no—
250#126—24no—
251#126—24no—
252#125—24no—
253#125—24no—
254#123—24no—
255#123—24no—
256#123—24no—
257#123—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—21no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8C3×D412yes—
267C22⋊Q8C23×C312yes—
268#121—21no—
269#121—21no—
270#121C3×D412yes—
271#121C23×C312yes—
272C4×D4—24no—
273C4×D4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C3×D412yes—
286C2×C4⋊C4C3×D411yescharacteristic
287C2×C4⋊C4C3×D411yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C312yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C42—21no—
298C2×C42—21no—
299#107—22no—
300#106—22no—
301#103—22no—
302#103—22no—
303#103—22no—
304#102—22no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8C23×C312yes—
308C22⋊C8C23×C312yes—
309#95—22no—
310#95—22no—
311#95C23×C312yes—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3#12011yescharacteristic
320C23×C3#12011yescharacteristic
321C23×C3C22×D411yescharacteristic
322C3×Q8—42no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—81no—
326C3×D4—42no—
327C3×D4—44no—
328C3×D4—44no—
329C3×D4—81no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—81no—
358C2×C3×C4—84no—
359C2×C3×C4#12011yescharacteristic
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C3×C8—42no—
364C3×C8—42no—
365C24—21no—
366C24C2×C3×D411yescharacteristic
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—81no—
371C2×Q8—81no—
372C2×Q8C2×C3×D412yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—44no—
377C2×D4—81no—
378C2×D4C2×C3×D412yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—42no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—82no—
394C22×C4C2×C3×D412yes—
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C24×C311yescharacteristic
402Q16—82no—
403SD16—82no—
404M16—81no—
405M16—81no—
406C2×C8—22no—
407C2×C8—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—44no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—82no—
425C4⋊C4C2×C3×D412yes—
426C4⋊C4C2×C3×D412yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—24no—
438C42—81no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—44no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—84no—
451C22×C3#45612yes—
452C22×C3C2×#12011yescharacteristic
453C22×C3#57512yes—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—44no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—81no—
465C3×C4—81no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C3×#12011yescharacteristic
474C23C3×#12011yescharacteristic
475C23C22×C3×D411yescharacteristic
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—81no—
480D4—42no—
481D4—44no—
482D4—44no—
483D4—81no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—84no—
513C2×C4C3×#12011yescharacteristic
514C2×C4C3×#12011yescharacteristic
515C2×C4C22×C3×D411yescharacteristic
516C2×C4C22×C3×D411yescharacteristic
517C8—42no—
518C8—42no—
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—41no—
522C2×C3—82no—
523C2×C3#300511yescharacteristic
524C2×C3C2×#45611yescharacteristic
525C2×C3#302611yescharacteristic
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—44no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—84no—
538C22C3×#45612yes—
539C22C2×C3×#12011yescharacteristic
540C22C3×#57512yes—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—44no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—81no—
552C4—81no—
553C3#1832111yescharacteristic, direct product factor, Sylow
554C2—22no—
555C2—22no—
556C2—41no—
557C2—82no—
558C2C3×#300511yescharacteristic
559C2C2×C3×#45611yescharacteristic
560C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3195C2semidirect
4. C3×#2583C2semidirect
5. C3×#1609C2semidirect
6. C3×#1603C2semidirect
7. C3×#1508C2semidirect
9. C3×#1463C2semidirect
10. #18321C3direct (N × H)
27. C3×#483C22semidirect
32. C3×#451C22semidirect
37. C3×#398C22semidirect
38. C3×#395C22semidirect
43. C3×#328C22semidirect
45. C3×#324C22semidirect
47. #3195C2×C3semidirect
49. #2583C2×C3semidirect
50. #1609C2×C3semidirect
51. #1603C2×C3semidirect
52. #1508C2×C3semidirect
54. #1463C2×C3semidirect
145. #483C22×C3semidirect
150. #451C22×C3semidirect
155. #398C22×C3semidirect
156. #395C22×C3semidirect
161. #328C22×C3semidirect
163. #324C22×C3semidirect
553. C3#18321direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5392
C3×#31952
C3×#31932
C3×#25832
C3×#16092
C3×#16032
C3×#15082
C3×#15072
C3×#14632
#183213