Finite Group Databasebeta

← C3×#33051 #10651658

Subgroups — C3×#33051 #10651658

Order 768.

952 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×#489C211yesmaximal, characteristic, complemented
2C22×C3×#97C211yesmaximal, characteristic, complemented
3C3×#2857C211yesmaximal, characteristic, complemented
4C3×#2850C211yesmaximal, characteristic
5C3×#1993C211yesmaximal, characteristic, complemented
6C3×#1959C211yesmaximal, characteristic, complemented
7C3×#1936C212yesmaximal, complemented
8C3×#1935C212yesmaximal, complemented
9C3×#1893C212yesmaximal
10C3×#1892C212yesmaximal, complemented
11C3×#1836C211yesmaximal, characteristic, complemented
12#33051C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×C8C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C3×#489—24no—
21C2×C3×#103C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#97—24no—
26C2×C3×#97C2211yescharacteristic
27C2×C3×#97C2212yes—
28C2×C3×#97C2211yescharacteristic
29C3×#391—21no—
30C3×#391C2211yescharacteristic
31C3×#390C2211yescharacteristic
32C3×#389—21no—
33C3×#389C2212yes—
34C3×#388—21no—
35C3×#388—21no—
36C3×#386—21no—
37C3×#386—21no—
38C3×#384C2211yescharacteristic
39C3×#382C2211yescharacteristic
40C3×#380—21no—
41C3×#380C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#489C2×C311yescharacteristic, complemented
47C22×#97C2×C311yescharacteristic, complemented
48#2857C2×C311yescharacteristic, complemented
49#2850C2×C311yescharacteristic
50#1993C2×C311yescharacteristic, complemented
51#1959C2×C311yescharacteristic, complemented
52#1936C2×C312yescomplemented
53#1935C2×C312yescomplemented
54#1893C2×C312yes—
55#1892C2×C312yescomplemented
56#1836C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×C3×C8—24no—
65C22×C3×C8—24no—
66C22×C3×C8—24no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#124—22no—
72C3×#124—22no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—44no—
75C3×#121—24no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—22no—
93C2×C3×C42—22no—
94C2×C3×C42D411yescharacteristic
95C2×C3×C42D411yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C3×#103—24no—
99C3×#103—24no—
100C3×#102—24no—
101C3×#102—24no—
102C3×#97—24no—
103C3×#97—22no—
104C3×#97D414yescomplemented
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111C23×C8C22×C312yes—
112C2×C4⋊Q8C22×C311yescharacteristic
113C2×#124C22×C311yescharacteristic
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C22×C311yescharacteristic
116C2×#121C22×C311yescharacteristic
117C22×C42C22×C311yescharacteristic
118#489—24no—
119C2×#103C22×C312yes—
120C2×#103C22×C312yes—
121C2×#102C22×C312yes—
122C2×#102C22×C312yes—
123C2×#97—24no—
124C2×#97C22×C312yes—
125C2×#97C22×C311yescharacteristic
126C2×#97C22×C311yescharacteristic
127#391—21no—
128#391C22×C311yescharacteristic
129#390C22×C311yescharacteristic
130#389—21no—
131#389C22×C312yes—
132#388—21no—
133#388—21no—
134#386—21no—
135#386—21no—
136#384C22×C311yescharacteristic
137#382C22×C311yescharacteristic
138#380—21no—
139#380C22×C312yes—
140#340C22×C312yes—
141#340C22×C312yes—
142#340C22×C312yes—
143#340C22×C312yes—
144C24×C3—21no—
145C24×C3C2×D411yescharacteristic
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—21no—
150C2×C3×Q8—42no—
151C2×C3×Q8—42no—
152C2×C3×Q8—42no—
153C2×C3×D4—21no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—41no—
173C22×C3×C4—41no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4#4114yes—
180C22×C3×C4C2411yescharacteristic
181C2×C3×C8—216no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8C2×D414yes—
187C2×C3×C8#4114yes—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—44no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—42no—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—42no—
202C3×C22⋊C4—44no—
203C3×C22⋊C4—42no—
204C3×C22⋊C4—42no—
205C3×C22⋊C4—42no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42C2×D412yes—
211C3×C42#4112yes—
212C22×Q8—21no—
213C22×Q8C23×C311yescharacteristic
214C22×D4—21no—
215C22×D4C23×C311yescharacteristic
216C23×C4C3×D411yescharacteristic
217C23×C4C3×D411yescharacteristic
218C23×C4C23×C311yescharacteristic
219C22×C8—24no—
220C22×C8—24no—
221C22×C8—24no—
222C22×C8C23×C312yes—
223C22×C8C23×C312yes—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226#124—22no—
227#124—22no—
228C22⋊Q8—24no—
229C22⋊Q8—44no—
230#121—24no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C23×C311yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C312yes—
239C2×C4⋊C4C23×C312yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42C3×D411yescharacteristic
250C2×C42C3×D411yescharacteristic
251C2×C42C23×C311yescharacteristic
252C2×C42C23×C311yescharacteristic
253#103—24no—
254#103—24no—
255#102—24no—
256#102—24no—
257#97—24no—
258#97—22no—
259#97C3×D414yescomplemented
260#95—22no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3#12412yes—
278C23×C3AGL(1,ℤ/8)12yes—
279C23×C3#13712yes—
280C23×C3C22×D411yescharacteristic
281C3×Q8—42no—
282C3×Q8—42no—
283C3×Q8—84no—
284C3×D4—42no—
285C3×D4—42no—
286C3×D4—84no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—44no—
302C2×C3×C4—44no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4#12112yes—
319C2×C3×C4#12112yes—
320C2×C3×C4#12412yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4112yes—
323C2×C3×C4C2×#4112yes—
324C2×C3×C4C2×#4111yescharacteristic
325C3×C8—24no—
326C3×C8—24no—
327C24—21no—
328C24C2×C3×D411yescharacteristic
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—21no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C3×#4114yes—
363C22×C4C24×C311yescharacteristic
364C2×C8—216no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8C2×C3×D414yes—
370C2×C8C3×#4114yes—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C42—22no—
390C42—22no—
391C42—21no—
392C42—21no—
393C42C2×C3×D412yes—
394C42C3×#4112yes—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3#48914yescomplemented
409C22×C3C2×#12111yescharacteristic
410C22×C3C2×#12411yescharacteristic
411C22×C3#53211yescharacteristic
412C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
413C22×C3C2×#13711yescharacteristic
414C22×C3#57511yescharacteristic
415C22×C3#57511yescharacteristic
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—81no—
422C3×C4—81no—
423C3×C4—81no—
424C3×C4—82no—
425C3×C4—82no—
426C3×C4#52512yes—
427C3×C4#52912yes—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23C3×#12412yes—
440C23C3×AGL(1,ℤ/8)12yes—
441C23C3×#13712yes—
442C23C22×C3×D411yescharacteristic
443Q8—42no—
444Q8—42no—
445Q8—84no—
446D4—42no—
447D4—42no—
448D4—84no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4C3×#12112yes—
481C2×C4C3×#12112yes—
482C2×C4C3×#12412yes—
483C2×C4C22×C3×D411yescharacteristic
484C2×C4C2×C3×#4112yes—
485C2×C4C2×C3×#4112yes—
486C2×C4C2×C3×#4111yescharacteristic
487C8—24no—
488C8—24no—
489C2×C3—24no—
490C2×C3—161no—
491C2×C3#285011yescharacteristic
492C2×C3#305511yescharacteristic
493C2×C3#305611yescharacteristic
494C2×C3C2×#48911yescharacteristic
495C2×C3#313511yescharacteristic
496C2×C3#318011yescharacteristic
497C2×C3#318111yescharacteristic
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22C3×#48914yescomplemented
512C22C2×C3×#12111yescharacteristic
513C22C2×C3×#12411yescharacteristic
514C22C3×#53211yescharacteristic
515C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
516C22C2×C3×#13711yescharacteristic
517C22C3×#57511yescharacteristic
518C22C3×#57511yescharacteristic
519C4—22no—
520C4—22no—
521C4—22no—
522C4—41no—
523C4—41no—
524C4—81no—
525C4—81no—
526C4—82no—
527C4—82no—
528C4—81no—
529C4C3×#52512yes—
530C4C3×#52912yes—
531C3#3305111yescharacteristic, direct product factor, Sylow
532C2—24no—
533C2—161no—
534C2C3×#285011yescharacteristic
535C2C3×#305511yescharacteristic
536C2C3×#305611yescharacteristic
537C2C2×C3×#48911yescharacteristic
538C2C3×#313511yescharacteristic
539C2C3×#318011yescharacteristic
540C2C3×#318111yescharacteristic
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×#489C2semidirect
2. C22×C3×#97C2semidirect
3. C3×#2857C2semidirect
5. C3×#1993C2semidirect
6. C3×#1959C2semidirect
7. C3×#1936C2semidirect
8. C3×#1935C2semidirect
10. C3×#1892C2semidirect
11. C3×#1836C2semidirect
12. #33051C3direct (N × H)
46. C2×#489C2×C3semidirect
47. C22×#97C2×C3semidirect
48. #2857C2×C3semidirect
50. #1993C2×C3semidirect
51. #1959C2×C3semidirect
52. #1936C2×C3semidirect
53. #1935C2×C3semidirect
55. #1892C2×C3semidirect
56. #1836C2×C3semidirect
104. C3×#97D4semidirect
259. #97C3×D4semidirect
408. C22×C3#489semidirect
511. C22C3×#489semidirect
531. C3#33051direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4892
C22×C3×#972
C3×#28572
C3×#28502
C3×#19932
C3×#19592
C3×#19362
C3×#19352
C3×#18932
C3×#18922
C3×#18362
#330513