Finite Group Databasebeta

← C3×#33012 #10651619

Subgroups — C3×#33012 #10651619

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic, complemented
2C2×C3×#480C211yesmaximal, characteristic, complemented
3C2×C3×#449C211yesmaximal, characteristic
4C3×#2624C211yesmaximal, characteristic, complemented
5C3×#1880C212yesmaximal, complemented
6C3×#1879C212yesmaximal, complemented
7C3×#1863C212yesmaximal, complemented
8C3×#1862C212yesmaximal, complemented
9C3×#1824C211yesmaximal, characteristic, complemented
10C3×#1818C211yesmaximal, characteristic, complemented
11C3×#1788C211yesmaximal, characteristic, complemented
12#33012C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic
15C22×C3×D8C2211yescharacteristic
16C2×C3×#121C2211yescharacteristic
17C3×#520—24no—
18C3×#520—24no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×D4—22no—
21C2×C3×C4×D4C2212yes—
22C2×C3×#117—21no—
23C2×C3×#117C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C22×C3×C4⋊C4C2211yescharacteristic
26C3×#480—24no—
27C3×#449—24no—
28C2×C3×#107C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2211yescharacteristic
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#399—21no—
37C3×#397—21no—
38C3×#394—22no—
39C3×#390C2212yes—
40C3×#389C2212yes—
41C3×#380C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#520C2×C311yescharacteristic, complemented
47C2×#480C2×C311yescharacteristic, complemented
48C2×#449C2×C311yescharacteristic
49#2624C2×C311yescharacteristic, complemented
50#1880C2×C312yescomplemented
51#1879C2×C312yescomplemented
52#1863C2×C312yescomplemented
53#1862C2×C312yescomplemented
54#1824C2×C311yescharacteristic, complemented
55#1818C2×C311yescharacteristic, complemented
56#1788C2×C311yescharacteristic, complemented
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C2×C3×#41D414yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4—22no—
69C23×C3×C4C2311yescharacteristic
70C2×C3×SD16—22no—
71C2×C3×SD16—24no—
72C2×C3×D8—22no—
73C2×C3×D8—24no—
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2312yes—
77C3×#121—22no—
78C3×#121—22no—
79C3×C4×Q8—24no—
80C3×C4×Q8—24no—
81C3×C4×D4—216no—
82C3×C4×D4—24no—
83C3×C4×D4—24no—
84C3×#117—28no—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4—28no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D414yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4D412yes—
104C2×C3×C42—22no—
105C2×C3×C42C2312yes—
106C3×#107—24no—
107C3×#103—22no—
108C3×#103—24no—
109C3×#102—22no—
110C3×#102—24no—
111C3×#102—24no—
112C3×C22⋊C8—22no—
113C3×C22⋊C8—22no—
114C3×#95—21no—
115C3×#95—22no—
116C22×#41C22×C311yescharacteristic
117C22×SD16C22×C311yescharacteristic
118C22×D8C22×C311yescharacteristic
119C2×#121C22×C311yescharacteristic
120#520—24no—
121#520—24no—
122C2×C4×Q8C22×C312yes—
123C2×C4×D4—22no—
124C2×C4×D4C22×C312yes—
125C2×#117—21no—
126C2×#117C22×C311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4C22×C311yescharacteristic
129#480—24no—
130#449—24no—
131C2×#107C22×C312yes—
132C2×#103C22×C311yescharacteristic
133C2×#103C22×C312yes—
134C2×#102C22×C312yes—
135C2×#102C22×C311yescharacteristic
136C2×#102C22×C312yes—
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C311yescharacteristic
139#399—21no—
140#397—21no—
141#394—22no—
142#390C22×C312yes—
143#389C22×C312yes—
144#380C22×C311yescharacteristic
145#340C22×C312yes—
146#340C22×C312yes—
147#336C22×C312yes—
148#336C22×C312yes—
149C24×C3—21no—
150C24×C3—21no—
151C24×C3C2×D411yescharacteristic
152C3×#41—216no—
153C3×#41—28no—
154C3×#41—28no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8C2×D412yes—
158C2×C3×Q8#4114yes—
159C2×C3×D4—28no—
160C2×C3×D4—24no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—42no—
168C2×C3×D4—82no—
169C2×C3×D4C2×D412yes—
170C2×C3×D4#4114yes—
171C22×C3×C4—28no—
172C22×C3×C4—21no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—28no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
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—44no—
191C3×D8—44no—
192C2×C3×C8—22no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C3×C4⋊C4—28no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—82no—
209C3×C4⋊C4C2×D414yes—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—82no—
214C3×C22⋊C4—82no—
215C3×C22⋊C4—81no—
216C3×C42—28no—
217C3×C42—22no—
218C3×C42—22no—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—22no—
222C2×#41C3×D414yes—
223C22×Q8C23×C311yescharacteristic
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C23×C311yescharacteristic
227C22×D4C23×C311yescharacteristic
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4C23×C311yescharacteristic
232C2×SD16—22no—
233C2×SD16—24no—
234C2×D8—22no—
235C2×D8—24no—
236C22×C8C23×C311yescharacteristic
237C22×C8C23×C311yescharacteristic
238C22×C8C23×C312yes—
239#121—22no—
240#121—22no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×D4—216no—
244C4×D4—24no—
245C4×D4—24no—
246#117—28no—
247#117—22no—
248#117—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C3×D411yescharacteristic
255C2×C4⋊C4C3×D411yescharacteristic
256C2×C4⋊C4C3×D414yes—
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C4⋊C4C23×C312yes—
259C2×C4⋊C4C23×C312yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4C3×D412yes—
266C2×C42—22no—
267C2×C42C23×C312yes—
268#107—24no—
269#103—22no—
270#103—24no—
271#102—22no—
272#102—24no—
273#102—24no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276#95—21no—
277#95—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—24no—
281C23×C3—22no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—42no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3#12012yes—
293C23×C3C22×D411yescharacteristic
294C3×Q8—22no—
295C3×Q8—24no—
296C3×Q8#12314yes—
297C3×D4—216no—
298C3×D4—22no—
299C3×D4—24no—
300C3×D4—41no—
301C3×D4—41no—
302C3×D4—42no—
303C3×D4—84no—
304C3×D4#12314yes—
305C2×C3×C4—216no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—84no—
327C2×C3×C4—88no—
328C2×C3×C4#12012yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4112yes—
332C2×C3×C4C2×#4112yes—
333C3×C8—41no—
334C3×C8—41no—
335C3×C8—42no—
336C24—21no—
337C24—21no—
338C24C2×C3×D411yescharacteristic
339#41—216no—
340#41—28no—
341#41—28no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8C2×C3×D412yes—
345C2×Q8C3×#4114yes—
346C2×D4—28no—
347C2×D4—24no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—42no—
355C2×D4—82no—
356C2×D4C2×C3×D412yes—
357C2×D4C3×#4114yes—
358C22×C4—28no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4C2×C3×D412yes—
372C22×C4C2×C3×D412yes—
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D412yes—
375C22×C4C2×C3×D412yes—
376C22×C4C24×C311yescharacteristic
377SD16—44no—
378D8—44no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C4⋊C4—28no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—82no—
396C4⋊C4C2×C3×D414yes—
397C22⋊C4—28no—
398C22⋊C4—24no—
399C22⋊C4—42no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—81no—
403C42—28no—
404C42—22no—
405C42—22no—
406C22×C3—22no—
407C22×C3—24no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—42no—
411C22×C3—44no—
412C22×C3—42no—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—81no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3—81no—
420C22×C3—81no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#52512yes—
423C22×C3#57612yes—
424C22×C3#57711yescharacteristic
425C22×C3#57811yescharacteristic
426C3×C4—22no—
427C3×C4—24no—
428C3×C4—42no—
429C3×C4—44no—
430C3×C4—42no—
431C3×C4—82no—
432C3×C4—81no—
433C3×C4C2×#12311yescharacteristic
434C3×C4C2×#12311yescharacteristic
435C3×C4#52512yes—
436C23—22no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23C3×#12012yes—
451C23C22×C3×D411yescharacteristic
452Q8—22no—
453Q8—24no—
454Q8C3×#12314yes—
455D4—216no—
456D4—22no—
457D4—24no—
458D4—41no—
459D4—41no—
460D4—42no—
461D4—84no—
462D4C3×#12314yes—
463C2×C4—216no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4—88no—
486C2×C4C3×#12012yes—
487C2×C4C22×C3×D411yescharacteristic
488C2×C4C22×C3×D411yescharacteristic
489C2×C4C2×C3×#4112yes—
490C2×C4C2×C3×#4112yes—
491C8—41no—
492C8—41no—
493C8—42no—
494C2×C3—42no—
495C2×C3—44no—
496C2×C3—161no—
497C2×C3#284911yescharacteristic
498C2×C3#300811yescharacteristic
499C2×C3#301311yescharacteristic
500C2×C3#311912yes—
501C2×C3#312412yes—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—44no—
510C22—42no—
511C22—44no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22C2×C3×#12011yescharacteristic
518C22C3×#52512yes—
519C22C3×#57612yes—
520C22C3×#57711yescharacteristic
521C22C3×#57811yescharacteristic
522C4—22no—
523C4—24no—
524C4—42no—
525C4—44no—
526C4—42no—
527C4—81no—
528C4—82no—
529C4C2×C3×#12311yescharacteristic
530C4C2×C3×#12311yescharacteristic
531C4C3×#52512yes—
532C3#3301211yescharacteristic, direct product factor, Sylow
533C2—42no—
534C2—44no—
535C2—161no—
536C2C3×#284911yescharacteristic
537C2C3×#300811yescharacteristic
538C2C3×#301311yescharacteristic
539C2C3×#311912yes—
540C2C3×#312412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#449C2
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
1. C2×C3×#520C2semidirect
2. C2×C3×#480C2semidirect
4. C3×#2624C2semidirect
5. C3×#1880C2semidirect
6. C3×#1879C2semidirect
7. C3×#1863C2semidirect
8. C3×#1862C2semidirect
9. C3×#1824C2semidirect
10. C3×#1818C2semidirect
11. C3×#1788C2semidirect
12. #33012C3direct (N × H)
46. C2×#520C2×C3semidirect
47. C2×#480C2×C3semidirect
49. #2624C2×C3semidirect
50. #1880C2×C3semidirect
51. #1879C2×C3semidirect
52. #1863C2×C3semidirect
53. #1862C2×C3semidirect
54. #1824C2×C3semidirect
55. #1818C2×C3semidirect
56. #1788C2×C3semidirect
532. C3#33012direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×C3×#4802
C2×C3×#4492
C3×#26242
C3×#18802
C3×#18792
C3×#18632
C3×#18622
C3×#18242
C3×#18182
C3×#17882
#330123