Finite Group Databasebeta

← C3×#25727 #10644334

Subgroups — C3×#25727 #10644334

Order 768.

1612 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#496C211yesmaximal, characteristic, complemented
4C2×C3×#452C212yesmaximal, complemented
5C2×C3×#449C212yesmaximal, complemented
6C3×#2850C211yesmaximal, characteristic, complemented
7C3×#2015C212yesmaximal, complemented
8C3×#2012C212yesmaximal, complemented
9C3×#1961C211yesmaximal, characteristic, complemented
10C3×#1960C211yesmaximal, characteristic, complemented
11C3×#1749C211yesmaximal, characteristic, complemented
12#25727C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C22×C3×SD16C2212yes—
17C22×C3×D8C2212yes—
18C3×#532—24no—
19C3×#532—24no—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#127C2211yescharacteristic
22C2×C3×#124C2211yescharacteristic
23C2×C3×#124C2212yes—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×#121—22no—
26C2×C3×#121C2211yescharacteristic
27C2×C3×C4×D4—22no—
28C22×C3×C42C2211yescharacteristic
29C3×#497—24no—
30C3×#496—24no—
31C3×#452—28no—
32C3×#449—28no—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C2×C3×#97C2211yescharacteristic
39C2×C3×#97C2211yescharacteristic
40C3×#391C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#388—22no—
43C3×#386—22no—
44C3×#382C2211yescharacteristic
45C3×#336C2212yes—
46C2×#532C2×C311yescharacteristic, complemented
47C2×#497C2×C311yescharacteristic, complemented
48C2×#496C2×C311yescharacteristic, complemented
49C2×#452C2×C312yescomplemented
50C2×#449C2×C312yescomplemented
51#2850C2×C311yescharacteristic, complemented
52#2015C2×C312yescomplemented
53#2012C2×C312yescomplemented
54#1961C2×C311yescharacteristic, complemented
55#1960C2×C311yescharacteristic, complemented
56#1749C2×C311yescharacteristic, complemented
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—42no—
62C2×C3×#41—42no—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×Q8C2312yes—
65C22×C3×D4—22no—
66C22×C3×D4—21no—
67C22×C3×D4—22no—
68C22×C3×D4C2311yescharacteristic
69C22×C3×D4C2312yes—
70C23×C3×C4—22no—
71C23×C3×C4D412yes—
72C23×C3×C4C2311yescharacteristic
73C2×C3×Q16—24no—
74C2×C3×Q16—24no—
75C2×C3×Q16—42no—
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—42no—
81C2×C3×SD16—42no—
82C2×C3×D8—24no—
83C2×C3×D8—24no—
84C2×C3×D8—42no—
85C22×C3×C8C2312yes—
86C22×C3×C8C2312yes—
87C3×C4⋊Q8—22no—
88C3×C4⋊Q8D414yescomplemented
89C3×#127—22no—
90C3×#127D414yescomplemented
91C3×#124—24no—
92C3×#124—24no—
93C3×#124—22no—
94C3×#124—22no—
95C3×C22⋊Q8—24no—
96C3×#121—216no—
97C3×#121—24no—
98C3×C4×D4—216no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—24no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—22no—
109C2×C3×C42—22no—
110C2×C3×C42D412yes—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C3×#103—24no—
114C3×#103—24no—
115C3×#102—24no—
116C3×#102—24no—
117C3×C22⋊C8D418yescomplemented
118C3×#97—22no—
119C3×#97—22no—
120C3×#95—22no—
121C3×#95—22no—
122C22×#41C22×C312yes—
123C22×Q16C22×C312yes—
124C22×SD16C22×C312yes—
125C22×SD16C22×C312yes—
126C22×D8C22×C312yes—
127#532—24no—
128#532—24no—
129C2×C4⋊Q8C22×C311yescharacteristic
130C2×#127C22×C311yescharacteristic
131C2×#124C22×C311yescharacteristic
132C2×#124C22×C312yes—
133C2×C22⋊Q8C22×C311yescharacteristic
134C2×#121—22no—
135C2×#121C22×C311yescharacteristic
136C2×C4×D4—22no—
137C22×C42C22×C311yescharacteristic
138#497—24no—
139#496—24no—
140#452—28no—
141#449—28no—
142C2×#103C22×C312yes—
143C2×#103C22×C312yes—
144C2×#102C22×C312yes—
145C2×#102C22×C312yes—
146C2×C22⋊C8C22×C312yes—
147C2×#97C22×C311yescharacteristic
148C2×#97C22×C311yescharacteristic
149#391C22×C311yescharacteristic
150#390C22×C311yescharacteristic
151#388—22no—
152#386—22no—
153#382C22×C311yescharacteristic
154#336C22×C312yes—
155C24×C3—22no—
156C24×C3—21no—
157C24×C3C2×D411yescharacteristic
158C3×#41—48no—
159C3×#41—48no—
160C3×#41—48no—
161C3×#41—48no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8C2×D414yes—
172C2×C3×D4—28no—
173C2×C3×D4—28no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—42no—
189C2×C3×D4C2×D414yes—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—22no—
193C22×C3×C4—28no—
194C22×C3×C4—22no—
195C22×C3×C4—24no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4—44no—
202C22×C3×C4—44no—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D414yes—
207C22×C3×C4C2411yescharacteristic
208C3×Q16—48no—
209C3×SD16—48no—
210C3×SD16—48no—
211C3×D8—48no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8C2×D414yes—
217C2×C3×C8C2×D414yes—
218C3×C4⋊C4—28no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—42no—
222C3×C22⋊C4—216no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—44no—
226C3×C22⋊C4—42no—
227C3×C42—24no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42C2×D412yes—
231C3×C42C2×D412yes—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C2×#41—42no—
238C22×Q8C23×C311yescharacteristic
239C22×Q8C23×C312yes—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4C23×C311yescharacteristic
244C22×D4C23×C312yes—
245C23×C4—22no—
246C23×C4C3×D412yes—
247C23×C4C23×C311yescharacteristic
248C2×Q16—24no—
249C2×Q16—24no—
250C2×Q16—42no—
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—42no—
256C2×SD16—42no—
257C2×D8—24no—
258C2×D8—24no—
259C2×D8—42no—
260C22×C8C23×C312yes—
261C22×C8C23×C312yes—
262C4⋊Q8—22no—
263C4⋊Q8C3×D414yescomplemented
264#127—22no—
265#127C3×D414yescomplemented
266#124—24no—
267#124—24no—
268#124—22no—
269#124—22no—
270C22⋊Q8—24no—
271#121—216no—
272#121—24no—
273C4×D4—216no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C22⋊C4—24no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42C3×D412yes—
286C2×C42C23×C311yescharacteristic
287C2×C42C23×C311yescharacteristic
288#103—24no—
289#103—24no—
290#102—24no—
291#102—24no—
292C22⋊C8C3×D418yescomplemented
293#97—22no—
294#97—22no—
295#95—22no—
296#95—22no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—44no—
305C23×C3—44no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3#12712yes—
312C23×C3C22×D411yescharacteristic
313C3×Q8—24no—
314C3×Q8—24no—
315C3×Q8—42no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×Q8—42no—
319C3×D4—24no—
320C3×D4—24no—
321C3×D4—416no—
322C3×D4—416no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—84no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—48no—
339C2×C3×C4—416no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4#12014yes—
350C2×C3×C4#12712yes—
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D412yes—
354C2×C3×C4H(5,𝔽2)12yes—
355C3×C8—24no—
356C3×C8—24no—
357C24—22no—
358C24—21no—
359C24C2×C3×D411yescharacteristic
360#41—48no—
361#41—48no—
362#41—48no—
363#41—48no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8C2×C3×D414yes—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4C2×C3×D414yes—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4C2×C3×D412yes—
406C22×C4C2×C3×D412yes—
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D414yes—
409C22×C4C24×C311yescharacteristic
410Q16—48no—
411SD16—48no—
412SD16—48no—
413D8—48no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8C2×C3×D414yes—
419C2×C8C2×C3×D414yes—
420C4⋊C4—28no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C22⋊C4—216no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C42—24no—
430C42—21no—
431C42—21no—
432C42C2×C3×D412yes—
433C42C2×C3×D412yes—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—44no—
440C22×C3—48no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—84no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3C2×#12011yescharacteristic
450C22×C3C2×#12711yescharacteristic
451C22×C3#53511yescharacteristic
452C22×C3#57512yes—
453C22×C3#57512yes—
454C3×C4—22no—
455C3×C4—24no—
456C3×C4—44no—
457C3×C4—42no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4—81no—
461C3×C4D4212yes—
462C3×C4D4212yes—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—24no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23C3×#12712yes—
478C23C22×C3×D411yescharacteristic
479Q8—24no—
480Q8—24no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485D4—24no—
486D4—24no—
487D4—416no—
488D4—416no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—84no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—416no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4C3×#12014yes—
516C2×C4C3×#12712yes—
517C2×C4C22×C3×D411yescharacteristic
518C2×C4C22×C3×D411yescharacteristic
519C2×C4C22×C3×D412yes—
520C2×C4C3×H(5,𝔽2)12yes—
521C8—24no—
522C8—24no—
523C2×C3—42no—
524C2×C3—84no—
525C2×C3—161no—
526C2×C3#243311yescharacteristic
527C2×C3#300512yes—
528C2×C3#315011yescharacteristic
529C2×C3#315311yescharacteristic
530C2×C3#321912yes—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—44no—
537C22—48no—
538C22—82no—
539C22—82no—
540C22—84no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—82no—
545C22—82no—
546C22C2×C3×#12011yescharacteristic
547C22C2×C3×#12711yescharacteristic
548C22C3×#53511yescharacteristic
549C22C3×#57512yes—
550C22C3×#57512yes—
551C4—22no—
552C4—24no—
553C4—44no—
554C4—42no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4C3×D4212yes—
559C4C3×D4212yes—
560C3#2572711yescharacteristic, direct product factor, Sylow
561C2—42no—
562C2—84no—
563C2—161no—
564C2C3×#243311yescharacteristic
565C2C3×#300512yes—
566C2C3×#315011yescharacteristic
567C2C3×#315311yescharacteristic
568C2C3×#321912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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×#532C2semidirect
2. C2×C3×#497C2semidirect
3. C2×C3×#496C2semidirect
4. C2×C3×#452C2semidirect
5. C2×C3×#449C2semidirect
6. C3×#2850C2semidirect
7. C3×#2015C2semidirect
8. C3×#2012C2semidirect
9. C3×#1961C2semidirect
10. C3×#1960C2semidirect
11. C3×#1749C2semidirect
12. #25727C3direct (N × H)
46. C2×#532C2×C3semidirect
47. C2×#497C2×C3semidirect
48. C2×#496C2×C3semidirect
49. C2×#452C2×C3semidirect
50. C2×#449C2×C3semidirect
51. #2850C2×C3semidirect
52. #2015C2×C3semidirect
53. #2012C2×C3semidirect
54. #1961C2×C3semidirect
55. #1960C2×C3semidirect
56. #1749C2×C3semidirect
88. C3×C4⋊Q8D4semidirect
90. C3×#127D4semidirect
117. C3×C22⋊C8D4semidirect
263. C4⋊Q8C3×D4semidirect
265. #127C3×D4semidirect
292. C22⋊C8C3×D4semidirect
560. C3#25727direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4972
C2×C3×#4962
C2×C3×#4522
C2×C3×#4492
C3×#28502
C3×#20152
C3×#20122
C3×#19612
C3×#19602
C3×#17492
#257273