Finite Group Databasebeta

← C3×#29815 #10648422

Subgroups — C3×#29815 #10648422

Order 768.

920 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×#497C211yesmaximal, characteristic, complemented
2C2×C3×#432C211yesmaximal, characteristic, complemented
3C3×#2857C211yesmaximal, characteristic, complemented
4C3×#2850C211yesmaximal, characteristic
5C3×#2056C212yesmaximal, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#2000C212yesmaximal
9C3×#1960C211yesmaximal, characteristic, complemented
10C3×#1959C211yesmaximal, characteristic, complemented
11C3×#1836C211yesmaximal, characteristic, complemented
12#29815C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C3×#497—24no—
22C3×#432—24no—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C3×#391C2212yes—
32C3×#391C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#389—21no—
35C3×#389—21no—
36C3×#388—21no—
37C3×#388—21no—
38C3×#386—21no—
39C3×#386—21no—
40C3×#384C2211yescharacteristic
41C3×#382C2211yescharacteristic
42C3×#380—21no—
43C3×#380—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#497C2×C311yescharacteristic, complemented
47C2×#432C2×C311yescharacteristic, complemented
48#2857C2×C311yescharacteristic, complemented
49#2850C2×C311yescharacteristic
50#2056C2×C312yescomplemented
51#2055C2×C312yescomplemented
52#2001C2×C312yescomplemented
53#2000C2×C312yes—
54#1960C2×C311yescharacteristic, complemented
55#1959C2×C311yescharacteristic, complemented
56#1836C2×C311yescharacteristic, complemented
57C22×C3×Q8C2312yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C2×C3×Q16—24no—
65C2×C3×Q16—42no—
66C2×C3×Q16—42no—
67C2×C3×SD16—24no—
68C2×C3×SD16—42no—
69C2×C3×SD16—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8—22no—
74C3×#124—22no—
75C3×#124—22no—
76C3×C22⋊Q8—42no—
77C3×C22⋊Q8—42no—
78C3×C22⋊Q8—42no—
79C3×#121—42no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—22no—
97C2×C3×C42—22no—
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C3×C4⋊C8—24no—
103C3×#103—42no—
104C3×#103—42no—
105C3×#103—42no—
106C3×#102—42no—
107C3×C22⋊C8—24no—
108C3×#97—22no—
109C3×#97D414yescomplemented
110C3×#95—22no—
111C3×#95—22no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C22×Q16C22×C312yes—
117C22×SD16C22×C312yes—
118C2×C4⋊Q8C22×C311yescharacteristic
119C2×#124C22×C311yescharacteristic
120C2×C22⋊Q8C22×C311yescharacteristic
121C2×C22⋊Q8C22×C312yes—
122C2×#121C22×C311yescharacteristic
123C22×C42C22×C311yescharacteristic
124#497—24no—
125#432—24no—
126C2×C4⋊C8C22×C312yes—
127C2×#103C22×C312yes—
128C2×#103C22×C312yes—
129C2×#103C22×C312yes—
130C2×#102C22×C312yes—
131C2×C22⋊C8C22×C312yes—
132C2×#97C22×C311yescharacteristic
133C2×#97C22×C311yescharacteristic
134#391C22×C312yes—
135#391C22×C311yescharacteristic
136#390C22×C311yescharacteristic
137#389—21no—
138#389—21no—
139#388—21no—
140#388—21no—
141#386—21no—
142#386—21no—
143#384C22×C311yescharacteristic
144#382C22×C311yescharacteristic
145#380—21no—
146#380—21no—
147#340C22×C312yes—
148#340C22×C312yes—
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C2×C3×Q8—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—41no—
182C22×C3×C4—41no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C3×Q16—48no—
190C3×SD16—48no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8C2×D414yes—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—42no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—42no—
207C3×C22⋊C4—42no—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—44no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—42no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—42no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42C2×D412yes—
219C3×C42#4112yes—
220C22×Q8C23×C311yescharacteristic
221C22×Q8C23×C312yes—
222C22×D4—21no—
223C22×D4C23×C311yescharacteristic
224C23×C4C3×D411yescharacteristic
225C23×C4C3×D411yescharacteristic
226C23×C4C23×C311yescharacteristic
227C2×Q16—24no—
228C2×Q16—42no—
229C2×Q16—42no—
230C2×SD16—24no—
231C2×SD16—42no—
232C2×SD16—42no—
233C22×C8C23×C312yes—
234C22×C8C23×C312yes—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237#124—22no—
238#124—22no—
239C22⋊Q8—42no—
240C22⋊Q8—42no—
241C22⋊Q8—42no—
242#121—42no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C311yescharacteristic
251C2×C4⋊C4C23×C312yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C3×D411yescharacteristic
262C2×C42C3×D411yescharacteristic
263C2×C42C23×C311yescharacteristic
264C2×C42C23×C311yescharacteristic
265C4⋊C8—24no—
266#103—42no—
267#103—42no—
268#103—42no—
269#102—42no—
270C22⋊C8—24no—
271#97—22no—
272#97C3×D414yescomplemented
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3#13712yes—
293C23×C3C22×D411yescharacteristic
294C3×Q8—42no—
295C3×Q8—42no—
296C3×Q8—42no—
297C3×Q8—42no—
298C3×Q8—42no—
299C3×Q8—42no—
300C3×D4—42no—
301C3×D4—42no—
302C3×D4—84no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—44no—
316C2×C3×C4—44no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4#12112yes—
335C2×C3×C4#12112yes—
336C2×C3×C4#13712yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4H(5,𝔽2)12yes—
340C2×C3×C4H(5,𝔽2)12yes—
341C3×C8—24no—
342C3×C8—42no—
343C24—21no—
344C24C2×C3×D411yescharacteristic
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C24×C311yescharacteristic
383Q16—48no—
384SD16—48no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8C2×C3×D414yes—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C42—22no—
409C42—22no—
410C42—21no—
411C42—21no—
412C42C2×C3×D412yes—
413C42C3×#4112yes—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3C2×#12111yescharacteristic
429C22×C3#53511yescharacteristic
430C22×C3#53511yescharacteristic
431C22×C3C2×#13711yescharacteristic
432C22×C3C2×#13711yescharacteristic
433C22×C3#57511yescharacteristic
434C22×C3#57511yescharacteristic
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—81no—
441C3×C4—82no—
442C3×C4—81no—
443C3×C4—81no—
444C3×C4—82no—
445C3×C4#53512yes—
446C3×C4#54012yes—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23C3×#13712yes—
461C23C22×C3×D411yescharacteristic
462Q8—42no—
463Q8—42no—
464Q8—42no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468D4—42no—
469D4—42no—
470D4—84no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4C3×#12112yes—
503C2×C4C3×#12112yes—
504C2×C4C3×#13712yes—
505C2×C4C22×C3×D411yescharacteristic
506C2×C4C2×C3×#4111yescharacteristic
507C2×C4C3×H(5,𝔽2)12yes—
508C2×C4C3×H(5,𝔽2)12yes—
509C8—24no—
510C8—42no—
511C2×C3—42no—
512C2×C3—161no—
513C2×C3#261911yescharacteristic
514C2×C3#305611yescharacteristic
515C2×C3#305611yescharacteristic
516C2×C3#321511yescharacteristic
517C2×C3#321611yescharacteristic
518C2×C3#321811yescharacteristic
519C2×C3#321911yescharacteristic
520C22—22no—
521C22—22no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22C2×C3×#12111yescharacteristic
535C22C3×#53511yescharacteristic
536C22C3×#53511yescharacteristic
537C22C2×C3×#13711yescharacteristic
538C22C2×C3×#13711yescharacteristic
539C22C3×#57511yescharacteristic
540C22C3×#57511yescharacteristic
541C4—22no—
542C4—22no—
543C4—22no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—82no—
548C4—81no—
549C4—81no—
550C4—82no—
551C4C3×#53512yes—
552C4C3×#54012yes—
553C3#2981511yescharacteristic, direct product factor, Sylow
554C2—42no—
555C2—161no—
556C2C3×#261911yescharacteristic
557C2C3×#305611yescharacteristic
558C2C3×#305611yescharacteristic
559C2C3×#321511yescharacteristic
560C2C3×#321611yescharacteristic
561C2C3×#321811yescharacteristic
562C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#497C2semidirect
2. C2×C3×#432C2semidirect
3. C3×#2857C2semidirect
5. C3×#2056C2semidirect
6. C3×#2055C2semidirect
7. C3×#2001C2semidirect
9. C3×#1960C2semidirect
10. C3×#1959C2semidirect
11. C3×#1836C2semidirect
12. #29815C3direct (N × H)
46. C2×#497C2×C3semidirect
47. C2×#432C2×C3semidirect
48. #2857C2×C3semidirect
50. #2056C2×C3semidirect
51. #2055C2×C3semidirect
52. #2001C2×C3semidirect
54. #1960C2×C3semidirect
55. #1959C2×C3semidirect
56. #1836C2×C3semidirect
109. C3×#97D4semidirect
272. #97C3×D4semidirect
553. C3#29815direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4972
C2×C3×#4322
C3×#28572
C3×#28502
C3×#20562
C3×#20552
C3×#20012
C3×#20002
C3×#19602
C3×#19592
C3×#18362
#298153