Finite Group Databasebeta

← C3×#33018 #10651625

Subgroups — C3×#33018 #10651625

Order 768.

1432 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic
2C2×C3×#485C211yesmaximal, characteristic, complemented
3C2×C3×#450C211yesmaximal, characteristic
4C3×#2625C211yesmaximal, characteristic, complemented
5C3×#1877C212yesmaximal, complemented
6C3×#1877C212yesmaximal, complemented
7C3×#1864C212yesmaximal, complemented
8C3×#1862C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1818C211yesmaximal, characteristic, complemented
11C3×#1786C211yesmaximal, characteristic
12#33018C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C3×#518—24no—
17C3×#518—24no—
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2212yes—
22C2×C3×#117C2211yescharacteristic
23C2×C3×#117C2211yescharacteristic
24C22×C3×C22⋊C4—22no—
25C3×#485—24no—
26C3×#450—24no—
27C2×C3×#107C2211yescharacteristic, complemented
28C2×C3×#106C2211yescharacteristic, complemented
29C2×C3×#103C2211yescharacteristic, complemented
30C2×C3×#103C2211yescharacteristic, complemented
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#400—21no—
37C3×#398—21no—
38C3×#397—22no—
39C3×#391C2212yescomplemented
40C3×#389C2212yescomplemented
41C3×#380C2211yescharacteristic
42C3×#340C2212yescomplemented
43C3×#340C2212yescomplemented
44C3×#336C2212yescomplemented
45C3×#336C2212yescomplemented
46C2×#518C2×C311yescharacteristic
47C2×#485C2×C311yescharacteristic, complemented
48C2×#450C2×C311yescharacteristic
49#2625C2×C311yescharacteristic, complemented
50#1877C2×C312yescomplemented
51#1877C2×C312yescomplemented
52#1864C2×C312yescomplemented
53#1862C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1818C2×C311yescharacteristic, complemented
56#1786C2×C311yescharacteristic
57C25×C3—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—24no—
60C22×C3×D4—24no—
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4D414yes—
64C22×C3×D4C2312yes—
65C23×C3×C4—22no—
66C23×C3×C4—22no—
67C23×C3×C4C2311yescharacteristic
68C2×C3×SD16—24no—
69C2×C3×SD16—24no—
70C2×C3×SD16—24no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C4×D4—28no—
77C3×C4×D4—28no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×C4×D4—24no—
82C3×#117—22no—
83C3×#117—22no—
84C3×#117—22no—
85C3×#117D414yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—216no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4D412yes—
101C2×C3×C42—21no—
102C2×C3×C42C2312yes—
103C3×#107—22no—
104C3×#106—22no—
105C3×#103—22no—
106C3×#103—22no—
107C3×#102—24no—
108C3×#102—24no—
109C3×#102—24no—
110C3×C22⋊C8—22no—
111C3×C22⋊C8—22no—
112C3×#95—21no—
113C3×#95—22no—
114C3×#95—22no—
115C3×#95—22no—
116C23×D4C22×C311yescharacteristic
117C22×SD16C22×C312yes—
118C2×C22⋊Q8C22×C311yescharacteristic
119#518—24no—
120#518—24no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C22×C312yes—
124C2×C4×D4C22×C312yes—
125C2×#117C22×C311yescharacteristic
126C2×#117C22×C311yescharacteristic
127C22×C22⋊C4—22no—
128#485—24no—
129#450—24no—
130C2×#107C22×C311yescharacteristic, complemented
131C2×#106C22×C311yescharacteristic, complemented
132C2×#103C22×C311yescharacteristic, complemented
133C2×#103C22×C311yescharacteristic, complemented
134C2×#102C22×C312yes—
135C2×#102C22×C312yes—
136C2×#102C22×C312yes—
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C311yescharacteristic
139#400—21no—
140#398—21no—
141#397—22no—
142#391C22×C312yescomplemented
143#389C22×C312yescomplemented
144#380C22×C311yescharacteristic
145#340C22×C312yescomplemented
146#340C22×C312yescomplemented
147#336C22×C312yescomplemented
148#336C22×C312yescomplemented
149C24×C3—28no—
150C24×C3—22no—
151C24×C3—24no—
152C24×C3—81no—
153C24×C3—82no—
154C24×C3—81no—
155C24×C3C2×D411yescharacteristic
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×D4—28no—
161C2×C3×D4—28no—
162C2×C3×D4—28no—
163C2×C3×D4—216no—
164C2×C3×D4—24no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4C2×D414yes—
169C2×C3×D4#4114yes—
170C2×C3×D4#4114yes—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—28no—
176C22×C3×C4—28no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—82no—
181C22×C3×C4—82no—
182C22×C3×C4—82no—
183C22×C3×C4—82no—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2411yescharacteristic
190C3×SD16—48no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—82no—
208C3×C4⋊C4C2×D412yes—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—81no—
216C3×C42—24no—
217C3×C42—22no—
218C3×C42—22no—
219C25—21no—
220C22×Q8C23×C311yescharacteristic
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—21no—
224C22×D4—22no—
225C22×D4C3×D414yes—
226C22×D4C23×C312yes—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4C23×C311yescharacteristic
230C2×SD16—24no—
231C2×SD16—24no—
232C2×SD16—24no—
233C22×C8C23×C312yes—
234C22×C8C23×C311yescharacteristic
235C22×C8C23×C311yescharacteristic
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117C3×D414yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C3×D411yescharacteristic
251C2×C4⋊C4C3×D411yescharacteristic
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C4⋊C4C23×C312yes—
254C2×C4⋊C4C23×C311yescharacteristic
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C22⋊C4—216no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4C3×D412yes—
263C2×C42—21no—
264C2×C42C23×C312yes—
265#107—22no—
266#106—22no—
267#103—22no—
268#103—22no—
269#102—24no—
270#102—24no—
271#102—24no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—22no—
278C23×C3—216no—
279C23×C3—24no—
280C23×C3—22no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—21no—
284C23×C3—22no—
285C23×C3—21no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—84no—
291C23×C3—84no—
292C23×C3—88no—
293C23×C3—88no—
294C23×C3#12012yes—
295C23×C3C22×D411yescharacteristic
296C3×Q8—42no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×D4—216no—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—24no—
303C3×D4#12318yes—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—88no—
328C2×C3×C4—88no—
329C2×C3×C4#12012yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4112yes—
333C2×C3×C4C2×#4112yes—
334C3×C8—42no—
335C3×C8—41no—
336C3×C8—41no—
337C24—28no—
338C24—22no—
339C24—24no—
340C24—81no—
341C24—82no—
342C24—81no—
343C24C2×C3×D411yescharacteristic
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—216no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4C2×C3×D414yes—
357C2×D4C3×#4114yes—
358C2×D4C3×#4114yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4C2×C3×D412yes—
373C22×C4C2×C3×D412yes—
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C2×C3×D412yes—
376C22×C4C2×C3×D412yes—
377C22×C4C24×C311yescharacteristic
378SD16—48no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—42no—
395C4⋊C4—82no—
396C4⋊C4C2×C3×D412yes—
397C4⋊C4C2×C3×D412yes—
398C22⋊C4—28no—
399C22⋊C4—28no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—81no—
404C42—24no—
405C42—22no—
406C42—22no—
407C22×C3—22no—
408C22×C3—28no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—42no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—816no—
420C22×C3C2×#12011yescharacteristic
421C22×C3#52512yes—
422C22×C3#57611yescharacteristic
423C22×C3#57712yes—
424C22×C3#57711yescharacteristic
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—81no—
431C3×C4—82no—
432C3×C4—81no—
433C3×C4C2×#12312yes—
434C3×C4#52511yescharacteristic
435C3×C4#52511yescharacteristic
436C23—216no—
437C23—21no—
438C23—22no—
439C23—21no—
440C23—24no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—84no—
449C23—84no—
450C23—88no—
451C23—88no—
452C23C3×#12012yes—
453C23C22×C3×D411yescharacteristic
454Q8—42no—
455Q8—41no—
456Q8—41no—
457D4—216no—
458D4—24no—
459D4—24no—
460D4—24no—
461D4C3×#12318yes—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—88no—
486C2×C4—88no—
487C2×C4C3×#12012yes—
488C2×C4C22×C3×D411yescharacteristic
489C2×C4C22×C3×D411yescharacteristic
490C2×C4C2×C3×#4112yes—
491C2×C4C2×C3×#4112yes—
492C8—42no—
493C8—41no—
494C8—41no—
495C2×C3—42no—
496C2×C3—48no—
497C2×C3#284911yescharacteristic
498C2×C3#300911yescharacteristic
499C2×C3#301211yescharacteristic
500C2×C3#311812yes—
501C2×C3#312312yes—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—28no—
506C22—44no—
507C22—44no—
508C22—42no—
509C22—44no—
510C22—44no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22—816no—
515C22C2×C3×#12011yescharacteristic
516C22C3×#52512yes—
517C22C3×#57611yescharacteristic
518C22C3×#57712yes—
519C22C3×#57711yescharacteristic
520C4—22no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—81no—
526C4—81no—
527C4—82no—
528C4C2×C3×#12312yes—
529C4C3×#52511yescharacteristic
530C4C3×#52511yescharacteristic
531C3#3301811yescharacteristic, direct product factor, Sylow
532C2—42no—
533C2—48no—
534C2C3×#284911yescharacteristic
535C2C3×#300911yescharacteristic
536C2C3×#301211yescharacteristic
537C2C3×#311812yes—
538C2C3×#312312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#485C2semidirect
4. C3×#2625C2semidirect
5. C3×#1877C2semidirect
6. C3×#1877C2semidirect
7. C3×#1864C2semidirect
8. C3×#1862C2semidirect
9. C3×#1825C2semidirect
10. C3×#1818C2semidirect
12. #33018C3direct (N × H)
27. C2×C3×#107C22semidirect
28. C2×C3×#106C22semidirect
29. C2×C3×#103C22semidirect
30. C2×C3×#103C22semidirect
39. C3×#391C22semidirect
40. C3×#389C22semidirect
42. C3×#340C22semidirect
43. C3×#340C22semidirect
44. C3×#336C22semidirect
45. C3×#336C22semidirect
47. C2×#485C2×C3semidirect
49. #2625C2×C3semidirect
50. #1877C2×C3semidirect
51. #1877C2×C3semidirect
52. #1864C2×C3semidirect
53. #1862C2×C3semidirect
54. #1825C2×C3semidirect
55. #1818C2×C3semidirect
130. C2×#107C22×C3semidirect
131. C2×#106C22×C3semidirect
132. C2×#103C22×C3semidirect
133. C2×#103C22×C3semidirect
142. #391C22×C3semidirect
143. #389C22×C3semidirect
145. #340C22×C3semidirect
146. #340C22×C3semidirect
147. #336C22×C3semidirect
148. #336C22×C3semidirect
531. C3#33018direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C2×C3×#4852
C2×C3×#4502
C3×#26252
C3×#18772
C3×#18642
C3×#18622
C3×#18252
C3×#18182
C3×#17862
#330183