Finite Group Databasebeta

← C3×#25732 #10644339

Subgroups — C3×#25732 #10644339

Order 768.

1658 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#492C211yesmaximal, characteristic, complemented
4C2×C3×#452C212yesmaximal, complemented
5C2×C3×#450C212yesmaximal, complemented
6C3×#2621C211yesmaximal, characteristic, complemented
7C3×#2015C212yesmaximal, complemented
8C3×#2014C212yesmaximal, complemented
9C3×#1961C211yesmaximal, characteristic, complemented
10C3×#1958C211yesmaximal, characteristic, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25732C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×Q16C2212yescomplemented
16C22×C3×SD16C2212yes—
17C22×C3×SD16C2212yes—
18C22×C3×SD16C2212yescomplemented
19C3×#535—24no—
20C3×#535—24no—
21C2×C3×C4⋊Q8C2211yescharacteristic, complemented
22C2×C3×#127C2211yescharacteristic
23C2×C3×#127C2211yescharacteristic
24C2×C3×#124C2211yescharacteristic
25C2×C3×#124C2211yescharacteristic
26C2×C3×C22⋊Q8C2212yes—
27C2×C3×#121—22no—
28C2×C3×#120—22no—
29C2×C3×#117C2211yescharacteristic
30C3×#497—24no—
31C3×#492—24no—
32C3×#452—28no—
33C3×#450—28no—
34C2×C3×#103C2212yescomplemented
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yescomplemented
38C2×C3×C22⋊C8C2212yes—
39C2×C3×#97C2211yescharacteristic, complemented
40C2×C3×C4×C8C2211yescharacteristic, complemented
41C3×#398—22no—
42C3×#397—22no—
43C3×#391C2212yescomplemented
44C3×#382C2211yescharacteristic, complemented
45C3×#336C2212yescomplemented
46C2×#535C2×C311yescharacteristic
47C2×#497C2×C311yescharacteristic, complemented
48C2×#492C2×C311yescharacteristic, complemented
49C2×#452C2×C312yescomplemented
50C2×#450C2×C312yescomplemented
51#2621C2×C311yescharacteristic, complemented
52#2015C2×C312yescomplemented
53#2014C2×C312yescomplemented
54#1961C2×C311yescharacteristic, complemented
55#1958C2×C311yescharacteristic, complemented
56#1750C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—24no—
61C2×C3×#41—42no—
62C22×C3×Q8C2312yes—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4—21no—
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4—24no—
71C22×C3×D4—42no—
72C22×C3×D4C2311yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C22×C3×D4C2311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C2×C3×Q16—24no—
78C2×C3×Q16—24no—
79C2×C3×Q16—42no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×SD16—24no—
84C2×C3×SD16—24no—
85C2×C3×SD16—24no—
86C2×C3×SD16—42no—
87C2×C3×SD16—42no—
88C2×C3×SD16—42no—
89C22×C3×C8C2312yes—
90C22×C3×C8C2312yes—
91C3×C4⋊Q8—22no—
92C3×C4⋊Q8—22no—
93C3×#127—22no—
94C3×#127—22no—
95C3×#127—22no—
96C3×#127D414yes—
97C3×#124—22no—
98C3×#124—22no—
99C3×#124—22no—
100C3×#124D414yes—
101C3×C22⋊Q8—24no—
102C3×C22⋊Q8—24no—
103C3×#121—216no—
104C3×#120—216no—
105C3×#117—22no—
106C3×#117—22no—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4D412yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4D412yes—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#103—24no—
118C3×#102—24no—
119C3×#102—24no—
120C3×#102—24no—
121C3×C22⋊C8D418yescomplemented
122C3×#97—22no—
123C3×C4×C8—22no—
124C3×#95—22no—
125C3×#95—24no—
126C22×#41C22×C311yescharacteristic
127C23×D4C22×C311yescharacteristic
128C22×Q16C22×C312yescomplemented
129C22×SD16C22×C312yes—
130C22×SD16C22×C312yes—
131C22×SD16C22×C312yescomplemented
132#535—24no—
133#535—24no—
134C2×C4⋊Q8C22×C311yescharacteristic, complemented
135C2×#127C22×C311yescharacteristic
136C2×#127C22×C311yescharacteristic
137C2×#124C22×C311yescharacteristic
138C2×#124C22×C311yescharacteristic
139C2×C22⋊Q8C22×C312yes—
140C2×#121—22no—
141C2×#120—22no—
142C2×#117C22×C311yescharacteristic
143#497—24no—
144#492—24no—
145#452—28no—
146#450—28no—
147C2×#103C22×C312yescomplemented
148C2×#102C22×C312yes—
149C2×#102C22×C312yes—
150C2×#102C22×C312yescomplemented
151C2×C22⋊C8C22×C312yes—
152C2×#97C22×C311yescharacteristic, complemented
153C2×C4×C8C22×C311yescharacteristic, complemented
154#398—22no—
155#397—22no—
156#391C22×C312yescomplemented
157#382C22×C311yescharacteristic, complemented
158#336C22×C312yescomplemented
159C24×C3—24no—
160C24×C3—21no—
161C24×C3—21no—
162C24×C3—21no—
163C24×C3—42no—
164C24×C3—42no—
165C24×C3—84no—
166C24×C3C2×D411yescharacteristic
167C3×#41—48no—
168C3×#41—48no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8C2×D412yes—
179C2×C3×D4—24no—
180C2×C3×D4—24no—
181C2×C3×D4—28no—
182C2×C3×D4—28no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—48no—
193C2×C3×D4—48no—
194C2×C3×D4—42no—
195C2×C3×D4—42no—
196C2×C3×D4—42no—
197C2×C3×D4—42no—
198C2×C3×D4—84no—
199C2×C3×D4—84no—
200C2×C3×D4C2×D412yes—
201C2×C3×D4C2×D412yes—
202C2×C3×D4C2×D412yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—24no—
207C22×C3×C4—21no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D414yes—
216C22×C3×C4C2411yescharacteristic
217C3×Q16—48no—
218C3×SD16—48no—
219C3×SD16—48no—
220C3×SD16—48no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8C2×D414yes—
226C2×C3×C8C2×D414yes—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—44no—
231C3×C4⋊C4—44no—
232C3×C22⋊C4—28no—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—44no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240C25—21no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—42no—
245C22×Q8C23×C312yes—
246C22×Q8C23×C311yescharacteristic
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—24no—
254C22×D4—42no—
255C22×D4C23×C311yescharacteristic
256C22×D4C23×C311yescharacteristic
257C22×D4C23×C311yescharacteristic
258C23×C4—21no—
259C23×C4C23×C311yescharacteristic
260C2×Q16—24no—
261C2×Q16—24no—
262C2×Q16—42no—
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—24no—
267C2×SD16—24no—
268C2×SD16—24no—
269C2×SD16—42no—
270C2×SD16—42no—
271C2×SD16—42no—
272C22×C8C23×C312yes—
273C22×C8C23×C312yes—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276#127—22no—
277#127—22no—
278#127—22no—
279#127C3×D414yes—
280#124—22no—
281#124—22no—
282#124—22no—
283#124C3×D414yes—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286#121—216no—
287#120—216no—
288#117—22no—
289#117—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C3×D412yes—
293C2×C4⋊C4C23×C312yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4C3×D412yes—
298C2×C42C23×C311yescharacteristic
299C2×C42C23×C311yescharacteristic
300#103—24no—
301#102—24no—
302#102—24no—
303#102—24no—
304C22⋊C8C3×D418yescomplemented
305#97—22no—
306C4×C8—22no—
307#95—22no—
308#95—24no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—48no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—44no—
323C23×C3—44no—
324C23×C3—44no—
325C23×C3—88no—
326C23×C3—88no—
327C23×C3—88no—
328C23×C3#12712yes—
329C23×C3C22×D411yescharacteristic
330C3×Q8—24no—
331C3×Q8—42no—
332C3×Q8—42no—
333C3×Q8—42no—
334C3×Q8—42no—
335C3×Q8—42no—
336C3×D4—24no—
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—48no—
340C3×D4—48no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—88no—
345C3×D4—88no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—48no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—44no—
362C2×C3×C4#12014yes—
363C2×C3×C4#12712yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D412yes—
367C2×C3×C4H(5,𝔽2)12yes—
368C3×C8—24no—
369C3×C8—24no—
370C24—24no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24—42no—
375C24—42no—
376C24—84no—
377C24C2×C3×D411yescharacteristic
378#41—48no—
379#41—48no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8C2×C3×D412yes—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4C2×C3×D412yes—
412C2×D4C2×C3×D412yes—
413C2×D4C2×C3×D412yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4C2×C3×D412yes—
424C22×C4C2×C3×D412yes—
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C2×C3×D414yes—
427C22×C4C24×C311yescharacteristic
428Q16—48no—
429SD16—48no—
430SD16—48no—
431SD16—48no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8C2×C3×D414yes—
437C2×C8C2×C3×D414yes—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—24no—
446C22⋊C4—44no—
447C42—21no—
448C42—21no—
449C42C2×C3×D412yes—
450C42C2×C3×D412yes—
451C22×C3—24no—
452C22×C3—22no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—816no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3C2×#12011yescharacteristic
469C22×C3C2×#12711yescharacteristic
470C22×C3#53511yescharacteristic
471C22×C3#57714yes—
472C3×C4—22no—
473C3×C4—24no—
474C3×C4—42no—
475C3×C4—42no—
476C3×C4—82no—
477C3×C4—82no—
478C3×C4D4212yes—
479C3×C4D4212yes—
480C23—24no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—48no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—88no—
497C23—88no—
498C23—88no—
499C23C3×#12712yes—
500C23C22×C3×D411yescharacteristic
501Q8—24no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507D4—24no—
508D4—24no—
509D4—24no—
510D4—48no—
511D4—48no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—88no—
516D4—88no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—48no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4C3×#12014yes—
534C2×C4C3×#12712yes—
535C2×C4C22×C3×D411yescharacteristic
536C2×C4C22×C3×D411yescharacteristic
537C2×C4C22×C3×D412yes—
538C2×C4C3×H(5,𝔽2)12yes—
539C8—24no—
540C8—24no—
541C2×C3—42no—
542C2×C3—82no—
543C2×C3—82no—
544C2×C3—82no—
545C2×C3#243311yescharacteristic
546C2×C3#300912yes—
547C2×C3#315412yes—
548C2×C3#322212yes—
549C22—24no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—816no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22C2×C3×#12011yescharacteristic
567C22C2×C3×#12711yescharacteristic
568C22C3×#53511yescharacteristic
569C22C3×#57714yes—
570C4—22no—
571C4—24no—
572C4—42no—
573C4—42no—
574C4—82no—
575C4—82no—
576C4C3×D4212yes—
577C4C3×D4212yes—
578C3#2573211yescharacteristic, direct product factor, Sylow
579C2—42no—
580C2—82no—
581C2—82no—
582C2—82no—
583C2C3×#243311yescharacteristic
584C2C3×#300912yes—
585C2C3×#315412yes—
586C2C3×#322212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#535C2
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
2. C2×C3×#497C2semidirect
3. C2×C3×#492C2semidirect
4. C2×C3×#452C2semidirect
5. C2×C3×#450C2semidirect
6. C3×#2621C2semidirect
7. C3×#2015C2semidirect
8. C3×#2014C2semidirect
9. C3×#1961C2semidirect
10. C3×#1958C2semidirect
11. C3×#1750C2semidirect
12. #25732C3direct (N × H)
15. C22×C3×Q16C22semidirect
18. C22×C3×SD16C22semidirect
21. C2×C3×C4⋊Q8C22semidirect
34. C2×C3×#103C22semidirect
37. C2×C3×#102C22semidirect
39. C2×C3×#97C22semidirect
40. C2×C3×C4×C8C22semidirect
43. C3×#391C22semidirect
44. C3×#382C22semidirect
45. C3×#336C22semidirect
47. C2×#497C2×C3semidirect
48. C2×#492C2×C3semidirect
49. C2×#452C2×C3semidirect
50. C2×#450C2×C3semidirect
51. #2621C2×C3semidirect
52. #2015C2×C3semidirect
53. #2014C2×C3semidirect
54. #1961C2×C3semidirect
55. #1958C2×C3semidirect
56. #1750C2×C3semidirect
121. C3×C22⋊C8D4semidirect
128. C22×Q16C22×C3semidirect
131. C22×SD16C22×C3semidirect
134. C2×C4⋊Q8C22×C3semidirect
147. C2×#103C22×C3semidirect
150. C2×#102C22×C3semidirect
152. C2×#97C22×C3semidirect
153. C2×C4×C8C22×C3semidirect
156. #391C22×C3semidirect
157. #382C22×C3semidirect
158. #336C22×C3semidirect
304. C22⋊C8C3×D4semidirect
578. C3#25732direct (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
C2×C3×#4972
C2×C3×#4922
C2×C3×#4522
C2×C3×#4502
C3×#26212
C3×#20152
C3×#20142
C3×#19612
C3×#19582
C3×#17502
#257323