Finite Group Databasebeta

← C3×#18687 #10637294

Subgroups — C3×#18687 #10637294

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3548C211yesmaximal, characteristic, complemented
2C3×#3254C211yesmaximal, characteristic, complemented
3C3×#3253C211yesmaximal, characteristic, complemented
4C3×#3147C211yesmaximal, characteristic, complemented
5C3×#1632C212yesmaximal, complemented
6C3×#1631C212yesmaximal, complemented
7C3×#1622C212yesmaximal, complemented
8C3×#1621C212yesmaximal, complemented
9C3×#1560C211yesmaximal, characteristic, complemented
10C3×#1493C211yesmaximal, characteristic, complemented
11C3×#1492C211yesmaximal, characteristic, complemented
12#18687C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#135—22no—
14C3×#562—22no—
15C3×#554—21no—
16C3×#554C2212yes—
17C3×#550—21no—
18C3×#550C2212yes—
19C3×#547—21no—
20C3×#547—21no—
21C3×#540—22no—
22C3×#537—22no—
23C3×#537—22no—
24C3×#533C2211yescharacteristic
25C3×#532C2211yescharacteristic
26C3×#532C2211yescharacteristic
27C3×#532C2211yescharacteristic
28C3×#495—22no—
29C3×#494C2211yescharacteristic
30C3×#493C2211yescharacteristic
31C3×#492C2211yescharacteristic
32C3×#492C2211yescharacteristic
33C3×#477C2212yes—
34C3×#476—21no—
35C3×#476C2212yes—
36C3×#474—21no—
37C3×#469—22no—
38C3×#468—22no—
39C3×#461—21no—
40C3×#460C2212yes—
41C3×#459—21no—
42C3×#459C2212yes—
43C3×#423C2211yescharacteristic
44C3×#423C2211yescharacteristic
45C2×C3×C4×C8C2211yescharacteristic
46C3×#333C2212yes—
47C3×#331C2212yes—
48C3×Q8⋊C8C2212yes—
49C3×Q8⋊C8C2212yes—
50C3×D4⋊C8C2212yes—
51C3×D4⋊C8C2212yes—
52#3548C2×C311yescharacteristic, complemented
53#3254C2×C311yescharacteristic, complemented
54#3253C2×C311yescharacteristic, complemented
55#3147C2×C311yescharacteristic, complemented
56#1632C2×C312yescomplemented
57#1631C2×C312yescomplemented
58#1622C2×C312yescomplemented
59#1621C2×C312yescomplemented
60#1560C2×C311yescharacteristic, complemented
61#1493C2×C311yescharacteristic, complemented
62#1492C2×C311yescharacteristic, complemented
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—21no—
66C2×C3×#41—21no—
67C2×C3×#41D412yes—
68C3×#135—48no—
69C2×C3×Q16—22no—
70C2×C3×SD16—22no—
71C2×C3×SD16—22no—
72C2×C3×SD16—22no—
73C2×C3×D8—22no—
74C2×C3×D8—22no—
75C2×C3×M16—21no—
76C2×C3×M16—21no—
77C22×C3×C8—21no—
78C3×C4⋊Q8D411yescharacteristic
79C3×C4⋊Q8D411yescharacteristic
80C3×C4⋊Q8D411yescharacteristic
81C3×C4⋊Q8C2311yescharacteristic
82C3×C4⋊Q8C2311yescharacteristic
83C3×#127—22no—
84C3×#127D411yescharacteristic
85C3×#127C2311yescharacteristic
86C3×#127C2311yescharacteristic
87C3×#125—22no—
88C3×#125—21no—
89C3×#124—21no—
90C3×#124—21no—
91C3×#124D412yes—
92C3×#123—24no—
93C3×#123—24no—
94C3×C22⋊Q8—24no—
95C3×C22⋊Q8—22no—
96C3×C22⋊Q8—24no—
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—21no—
99C3×C22⋊Q8D412yes—
100C3×#121—24no—
101C3×#121—24no—
102C3×#121—24no—
103C3×#121—22no—
104C3×#121—24no—
105C3×#121—21no—
106C3×#121—21no—
107C3×#121—21no—
108C3×#121—22no—
109C3×#121—21no—
110C3×#121—21no—
111C3×#121D412yes—
112C3×C4×Q8—21no—
113C3×C4×Q8C2312yes—
114C3×C4×D4—22no—
115C3×C4×D4—22no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4—21no—
119C3×C4×D4—21no—
120C3×C4×D4—21no—
121C3×C4×D4C2312yes—
122C3×#117—21no—
123C2×C3×C4⋊C4—22no—
124C2×C3×C4⋊C4—22no—
125C2×C3×C42C2311yescharacteristic
126C3×#107—22no—
127C3×#106—22no—
128C3×C4⋊C8—21no—
129C3×C4⋊C8—21no—
130C3×C4⋊C8C2312yes—
131C3×C4⋊C8C2312yes—
132C3×#103—22no—
133C3×#102—22no—
134C3×#102—22no—
135C3×#102—22no—
136C3×C4×C8—21no—
137C3×C4×C8C2311yescharacteristic
138C3×C4×C8C2311yescharacteristic
139C2×#135—22no—
140#562—22no—
141#554—21no—
142#554C22×C312yes—
143#550—21no—
144#550C22×C312yes—
145#547—21no—
146#547—21no—
147#540—22no—
148#537—22no—
149#537—22no—
150#533C22×C311yescharacteristic
151#532C22×C311yescharacteristic
152#532C22×C311yescharacteristic
153#532C22×C311yescharacteristic
154#495—22no—
155#494C22×C311yescharacteristic
156#493C22×C311yescharacteristic
157#492C22×C311yescharacteristic
158#492C22×C311yescharacteristic
159#477C22×C312yes—
160#476—21no—
161#476C22×C312yes—
162#474—21no—
163#469—22no—
164#468—22no—
165#461—21no—
166#460C22×C312yes—
167#459—21no—
168#459C22×C312yes—
169#423C22×C311yescharacteristic
170#423C22×C311yescharacteristic
171C2×C4×C8C22×C311yescharacteristic
172#333C22×C312yes—
173#331C22×C312yes—
174Q8⋊C8C22×C312yes—
175Q8⋊C8C22×C312yes—
176D4⋊C8C22×C312yes—
177D4⋊C8C22×C312yes—
178C3×#41—44no—
179C3×#41—44no—
180C3×#41—81no—
181C3×#41—81no—
182C3×#41—82no—
183C3×#41—82no—
184C3×#41—84no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8C2×D412yes—
189C2×C3×Q8C2×D412yes—
190C2×C3×D4—24no—
191C2×C3×D4—24no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—41no—
202C2×C3×D4C2×D412yes—
203C2×C3×D4C2×D412yes—
204C22×C3×C4—24no—
205C22×C3×C4—24no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—41no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C3×Q16—44no—
213C3×SD16—44no—
214C3×SD16—44no—
215C3×SD16—82no—
216C3×D8—44no—
217C3×D8—82no—
218C3×M16—81no—
219C3×M16—81no—
220C2×C3×C8—24no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4C2×D412yes—
238C3×C4⋊C4C2×D412yes—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C42—22no—
249C3×C42C2×D411yescharacteristic
250C3×C42C2×D411yescharacteristic
251C3×C42C2×D411yescharacteristic
252C3×C42C2411yescharacteristic
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—21no—
256C2×#41—21no—
257C2×#41C3×D412yes—
258#135—48no—
259C2×Q16—22no—
260C2×SD16—22no—
261C2×SD16—22no—
262C2×SD16—22no—
263C2×D8—22no—
264C2×D8—22no—
265C2×M16—21no—
266C2×M16—21no—
267C22×C8—21no—
268C4⋊Q8C3×D411yescharacteristic
269C4⋊Q8C3×D411yescharacteristic
270C4⋊Q8C3×D411yescharacteristic
271C4⋊Q8C23×C311yescharacteristic
272C4⋊Q8C23×C311yescharacteristic
273#127—22no—
274#127C3×D411yescharacteristic
275#127C23×C311yescharacteristic
276#127C23×C311yescharacteristic
277#125—22no—
278#125—21no—
279#124—21no—
280#124—21no—
281#124C3×D412yes—
282#123—24no—
283#123—24no—
284C22⋊Q8—22no—
285C22⋊Q8—24no—
286C22⋊Q8—22no—
287C22⋊Q8—24no—
288C22⋊Q8—21no—
289C22⋊Q8C3×D412yes—
290#121—24no—
291#121—24no—
292#121—22no—
293#121—24no—
294#121—24no—
295#121—21no—
296#121—21no—
297#121—21no—
298#121—22no—
299#121—21no—
300#121—21no—
301#121C3×D412yes—
302C4×Q8—21no—
303C4×Q8C23×C312yes—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4C23×C312yes—
312#117—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C42C23×C311yescharacteristic
316#107—22no—
317#106—22no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8C23×C312yes—
321C4⋊C8C23×C312yes—
322#103—22no—
323#102—22no—
324#102—22no—
325#102—22no—
326C4×C8—21no—
327C4×C8C23×C311yescharacteristic
328C4×C8C23×C311yescharacteristic
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3#12011yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—42no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×Q8—81no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—41no—
344C3×D4—41no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4—41no—
348C3×D4—41no—
349C3×D4—41no—
350C3×D4—41no—
351C3×D4—42no—
352C3×D4—81no—
353C3×D4—82no—
354C3×D4—82no—
355C3×D4—84no—
356C3×D4—84no—
357C3×D4—82no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—44no—
370C2×C3×C4—41no—
371C2×C3×C4—84no—
372C2×C3×C4—84no—
373C2×C3×C4—82no—
374C2×C3×C4#12011yescharacteristic
375C2×C3×C4#12011yescharacteristic
376C2×C3×C4#12011yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C3×C8—22no—
381C3×C8—22no—
382C3×C8—42no—
383C3×C8—42no—
384#41—44no—
385#41—44no—
386#41—81no—
387#41—81no—
388#41—82no—
389#41—82no—
390#41—84no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8C2×C3×D412yes—
395C2×Q8C2×C3×D412yes—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—41no—
408C2×D4C2×C3×D412yes—
409C2×D4C2×C3×D412yes—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418Q16—44no—
419SD16—44no—
420SD16—44no—
421SD16—82no—
422D8—44no—
423D8—82no—
424M16—81no—
425M16—81no—
426C2×C8—24no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4C2×C3×D412yes—
444C4⋊C4C2×C3×D412yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C42—22no—
455C42C2×C3×D411yescharacteristic
456C42C2×C3×D411yescharacteristic
457C42C2×C3×D411yescharacteristic
458C42C24×C311yescharacteristic
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3C2×#12011yescharacteristic
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—81no—
480C3×C4#45312yes—
481C3×C4#57611yescharacteristic
482C3×C4#57711yescharacteristic
483C3×C4#57711yescharacteristic
484C3×C4#57811yescharacteristic
485C23—22no—
486C23—22no—
487C23—41no—
488C23C3×#12011yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—81no—
495D4—22no—
496D4—22no—
497D4—42no—
498D4—42no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—42no—
508D4—81no—
509D4—82no—
510D4—82no—
511D4—84no—
512D4—84no—
513D4—82no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—82no—
530C2×C4C3×#12011yescharacteristic
531C2×C4C3×#12011yescharacteristic
532C2×C4C3×#12011yescharacteristic
533C2×C4C22×C3×D411yescharacteristic
534C2×C4C22×C3×D411yescharacteristic
535C2×C4C22×C3×D411yescharacteristic
536C8—22no—
537C8—22no—
538C8—42no—
539C8—42no—
540C2×C3—41no—
541C2×C3—82no—
542C2×C3—82no—
543C2×C3—161no—
544C2×C3#301211yescharacteristic
545C2×C3#301311yescharacteristic
546C2×C3C2×#45311yescharacteristic
547C22—21no—
548C22—21no—
549C22—21no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×C3×#12011yescharacteristic
560C4—21no—
561C4—21no—
562C4—21no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4C3×#45312yes—
569C4C3×#57611yescharacteristic
570C4C3×#57711yescharacteristic
571C4C3×#57711yescharacteristic
572C4C3×#57811yescharacteristic
573C3#1868711yescharacteristic, direct product factor, Sylow
574C2—41no—
575C2—82no—
576C2—82no—
577C2—161no—
578C2C3×#301211yescharacteristic
579C2C3×#301311yescharacteristic
580C2C2×C3×#45311yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3548C2semidirect
2. C3×#3254C2semidirect
3. C3×#3253C2semidirect
4. C3×#3147C2semidirect
5. C3×#1632C2semidirect
6. C3×#1631C2semidirect
7. C3×#1622C2semidirect
8. C3×#1621C2semidirect
9. C3×#1560C2semidirect
10. C3×#1493C2semidirect
11. C3×#1492C2semidirect
12. #18687C3direct (N × H)
52. #3548C2×C3semidirect
53. #3254C2×C3semidirect
54. #3253C2×C3semidirect
55. #3147C2×C3semidirect
56. #1632C2×C3semidirect
57. #1631C2×C3semidirect
58. #1622C2×C3semidirect
59. #1621C2×C3semidirect
60. #1560C2×C3semidirect
61. #1493C2×C3semidirect
62. #1492C2×C3semidirect
573. C3#18687direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35482
C3×#32542
C3×#32532
C3×#31472
C3×#16322
C3×#16312
C3×#16222
C3×#16212
C3×#15602
C3×#14932
C3×#14922
#186873