Finite Group Databasebeta

← C3×#29612 #10648219

Subgroups — C3×#29612 #10648219

Order 768.

1464 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic
2C3×#3095C211yesmaximal, characteristic, complemented
3C3×#3007C211yesmaximal, characteristic, complemented
4C3×#2435C211yesmaximal, characteristic, complemented
5C3×#2037C214yesmaximal, complemented
6C3×#2004C214yesmaximal, complemented
7C3×#1826C211yesmaximal, characteristic, complemented
8C3×#1789C211yesmaximal, characteristic, complemented
9C3×#1783C211yesmaximal, characteristic, complemented
10#29612C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C2×C3×#137C2212yescomplemented
14C3×#535—22no—
15C3×#535—22no—
16C3×#535—22no—
17C3×#535C2214yes—
18C2×C3×#127—21no—
19C2×C3×#124—21no—
20C2×C3×#124C2212yescomplemented
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121—21no—
23C2×C3×#121—21no—
24C2×C3×#120—22no—
25C2×C3×#117C2211yescharacteristic
26C3×#485—22no—
27C3×#484—21no—
28C3×#483—21no—
29C3×#452—22no—
30C3×#451—21no—
31C3×#450—21no—
32C3×#428C2212yescomplemented
33C2×C3×#104C2214yescomplemented
34C3×#419C2212yescomplemented
35C3×#417C2212yescomplemented
36C2×C3×#100C2214yescomplemented
37C3×#407C2211yescharacteristic
38C3×#407C2211yescharacteristic
39C3×#394—22no—
40C3×#390C2212yescomplemented
41C3×#379—22no—
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#339C2214yescomplemented
44C2×#535C2×C311yescharacteristic
45#3095C2×C311yescharacteristic, complemented
46#3007C2×C311yescharacteristic, complemented
47#2435C2×C311yescharacteristic, complemented
48#2037C2×C314yescomplemented
49#2004C2×C314yescomplemented
50#1826C2×C311yescharacteristic, complemented
51#1789C2×C311yescharacteristic, complemented
52#1783C2×C311yescharacteristic, complemented
53C25×C3D412yes—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41—44no—
57C2×C3×#41D412yes—
58C2×C3×#41C2312yes—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4—21no—
68C22×C3×D4—44no—
69C22×C3×D4D412yes—
70C22×C3×D4D411yescharacteristic
71C22×C3×D4C2312yes—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C3×#137—44no—
75C2×C3×Q16—22no—
76C2×C3×SD16—22no—
77C2×C3×M16C2312yes—
78C2×C3×M16C2312yes—
79C3×#127—24no—
80C3×#127—24no—
81C3×#124—24no—
82C3×#124—24no—
83C3×#124—44no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8—21no—
86C3×C22⋊Q8—21no—
87C3×C22⋊Q8—21no—
88C3×#121—24no—
89C3×#121—24no—
90C3×#121—24no—
91C3×#121—24no—
92C3×#120—28no—
93C3×#120—28no—
94C3×#117—21no—
95C3×#117—21no—
96C3×#117D412yes—
97C3×#117C2312yes—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—24no—
108C2×C3×C22⋊C4—24no—
109C2×C3×C42—21no—
110C2×C3×C42C2312yes—
111C3×#107—22no—
112C3×#106—22no—
113C3×#104—44no—
114C3×#103—22no—
115C3×#103—22no—
116C3×#103—22no—
117C3×#102—22no—
118C3×#100—44no—
119C3×C22⋊C8—21no—
120C3×C22⋊C8—21no—
121C3×C22⋊C8—21no—
122C3×C22⋊C8—21no—
123C3×#95—21no—
124C22×#41C22×C311yescharacteristic
125C23×D4C22×C311yescharacteristic
126C2×#137C22×C312yescomplemented
127#535—22no—
128#535—22no—
129#535—22no—
130#535C22×C314yes—
131C2×#127—21no—
132C2×#124—21no—
133C2×#124C22×C312yescomplemented
134C2×C22⋊Q8C22×C311yescharacteristic
135C2×#121—21no—
136C2×#121—21no—
137C2×#120—22no—
138C2×#117C22×C311yescharacteristic
139#485—22no—
140#484—21no—
141#483—21no—
142#452—22no—
143#451—21no—
144#450—21no—
145#428C22×C312yescomplemented
146C2×#104C22×C314yescomplemented
147#419C22×C312yescomplemented
148#417C22×C312yescomplemented
149C2×#100C22×C314yescomplemented
150#407C22×C311yescharacteristic
151#407C22×C311yescharacteristic
152#394—22no—
153#390C22×C312yescomplemented
154#379—22no—
155C4×C22⋊C4C22×C311yescharacteristic
156#339C22×C314yescomplemented
157C24×C3—22no—
158C24×C3—21no—
159C24×C3—22no—
160C24×C3—44no—
161C24×C3—44no—
162C24×C3—82no—
163C24×C3C2×D411yescharacteristic
164C24×C3C2×D414yes—
165C3×#41—24no—
166C3×#41—24no—
167C3×#41—42no—
168C3×#41—42no—
169C3×#41—44no—
170C3×#41—44no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—24no—
175C2×C3×Q8—42no—
176C2×C3×Q8C2×D412yes—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—28no—
180C2×C3×D4—28no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—24no—
187C2×C3×D4—24no—
188C2×C3×D4—42no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—44no—
192C2×C3×D4—44no—
193C2×C3×D4—44no—
194C2×C3×D4—44no—
195C2×C3×D4—84no—
196C2×C3×D4—84no—
197C2×C3×D4C2×D412yes—
198C22×C3×C4—24no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—44no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C3×Q16—44no—
215C3×SD16—44no—
216C3×M16—24no—
217C3×M16—24no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—41no—
226C3×C4⋊C4C2×D412yes—
227C3×C4⋊C4C2×D412yes—
228C3×C22⋊C4—28no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—48no—
234C3×C22⋊C4—41no—
235C3×C22⋊C4—41no—
236C3×C22⋊C4—84no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42—24no—
240C25C3×D412yes—
241C2×#41—21no—
242C2×#41—21no—
243C2×#41—44no—
244C2×#41C3×D412yes—
245C2×#41C23×C312yes—
246C22×Q8C3×D411yescharacteristic
247C22×Q8C23×C311yescharacteristic
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—44no—
256C22×D4C3×D412yes—
257C22×D4C3×D411yescharacteristic
258C22×D4C23×C312yes—
259C23×C4—21no—
260C23×C4C23×C311yescharacteristic
261#137—44no—
262C2×Q16—22no—
263C2×SD16—22no—
264C2×M16C23×C312yes—
265C2×M16C23×C312yes—
266#127—24no—
267#127—24no—
268#124—24no—
269#124—24no—
270#124—44no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275#121—24no—
276#121—24no—
277#121—24no—
278#121—24no—
279#120—28no—
280#120—28no—
281#117—21no—
282#117—21no—
283#117C3×D412yes—
284#117C23×C312yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4C3×D411yescharacteristic
287C2×C4⋊C4C3×D411yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—24no—
296C2×C42—21no—
297C2×C42C23×C312yes—
298#107—22no—
299#106—22no—
300#104—44no—
301#103—22no—
302#103—22no—
303#103—22no—
304#102—22no—
305#100—44no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310#95—21no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—22no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—41no—
318C23×C3—44no—
319C23×C3—44no—
320C23×C3—42no—
321C23×C3—44no—
322C23×C3—44no—
323C23×C3—42no—
324C23×C3—44no—
325C23×C3—42no—
326C23×C3—44no—
327C23×C3—82no—
328C23×C3—82no—
329C23×C3—816no—
330C23×C3#12011yescharacteristic
331C23×C3#12011yescharacteristic
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—42no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×D4—22no—
339C3×D4—22no—
340C3×D4—44no—
341C3×D4—44no—
342C3×D4—42no—
343C3×D4—44no—
344C3×D4—44no—
345C3×D4—44no—
346C3×D4—44no—
347C3×D4—88no—
348C3×D4—88no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—44no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—44no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4#12011yescharacteristic
370C2×C3×C4#12011yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4H(5,𝔽2)14yes—
374C3×C8—42no—
375C3×C8—42no—
376C24—22no—
377C24—21no—
378C24—22no—
379C24—44no—
380C24—44no—
381C24—82no—
382C24C2×C3×D411yescharacteristic
383C24C2×C3×D414yes—
384#41—24no—
385#41—24no—
386#41—42no—
387#41—42no—
388#41—44no—
389#41—44no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—24no—
394C2×Q8—42no—
395C2×Q8C2×C3×D412yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4C2×C3×D412yes—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—44no—
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C24×C311yescharacteristic
433Q16—44no—
434SD16—44no—
435M16—24no—
436M16—24no—
437C2×C8—22no—
438C2×C8—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4C2×C3×D412yes—
446C4⋊C4C2×C3×D412yes—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—48no—
452C22⋊C4—42no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—84no—
456C42—22no—
457C42—22no—
458C42—24no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—44no—
467C22×C3—48no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—42no—
471C22×C3—44no—
472C22×C3—42no—
473C22×C3—44no—
474C22×C3—84no—
475C22×C3—84no—
476C22×C3—88no—
477C22×C3—88no—
478C22×C3#45512yes—
479C22×C3C2×#12011yescharacteristic
480C22×C3#53412yes—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—82no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4#53412yes—
490C3×C4#53512yes—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—41no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—82no—
508C23—82no—
509C23—816no—
510C23C3×#12011yescharacteristic
511C23C3×#12011yescharacteristic
512C23C22×C3×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518D4—22no—
519D4—22no—
520D4—44no—
521D4—44no—
522D4—42no—
523D4—44no—
524D4—44no—
525D4—44no—
526D4—44no—
527D4—88no—
528D4—88no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4C3×#12011yescharacteristic
550C2×C4C3×#12011yescharacteristic
551C2×C4C22×C3×D411yescharacteristic
552C2×C4C22×C3×D411yescharacteristic
553C2×C4C3×H(5,𝔽2)14yes—
554C8—42no—
555C8—42no—
556C2×C3—22no—
557C2×C3—22no—
558C2×C3—41no—
559C2×C3—84no—
560C2×C3—82no—
561C2×C3#261311yescharacteristic
562C2×C3C2×#45511yescharacteristic
563C2×C3#301911yescharacteristic
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—44no—
572C22—48no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—84no—
580C22—88no—
581C22—84no—
582C22—88no—
583C22C3×#45512yes—
584C22C2×C3×#12011yescharacteristic
585C22C3×#53412yes—
586C4—21no—
587C4—21no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—82no—
592C4—81no—
593C4—81no—
594C4C3×#53412yes—
595C4C3×#53512yes—
596C3#2961211yescharacteristic, direct product factor, Sylow
597C2—22no—
598C2—22no—
599C2—41no—
600C2—84no—
601C2—82no—
602C2C3×#261311yescharacteristic
603C2C2×C3×#45511yescharacteristic
604C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3095C2semidirect
3. C3×#3007C2semidirect
4. C3×#2435C2semidirect
5. C3×#2037C2semidirect
6. C3×#2004C2semidirect
7. C3×#1826C2semidirect
8. C3×#1789C2semidirect
9. C3×#1783C2semidirect
10. #29612C3direct (N × H)
13. C2×C3×#137C22semidirect
20. C2×C3×#124C22semidirect
32. C3×#428C22semidirect
33. C2×C3×#104C22semidirect
34. C3×#419C22semidirect
35. C3×#417C22semidirect
36. C2×C3×#100C22semidirect
40. C3×#390C22semidirect
43. C3×#339C22semidirect
45. #3095C2×C3semidirect
46. #3007C2×C3semidirect
47. #2435C2×C3semidirect
48. #2037C2×C3semidirect
49. #2004C2×C3semidirect
50. #1826C2×C3semidirect
51. #1789C2×C3semidirect
52. #1783C2×C3semidirect
126. C2×#137C22×C3semidirect
133. C2×#124C22×C3semidirect
145. #428C22×C3semidirect
146. C2×#104C22×C3semidirect
147. #419C22×C3semidirect
148. #417C22×C3semidirect
149. C2×#100C22×C3semidirect
153. #390C22×C3semidirect
156. #339C22×C3semidirect
596. C3#29612direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C3×#30952
C3×#30072
C3×#24352
C3×#20372
C3×#20042
C3×#18262
C3×#17892
C3×#17832
#296123