Finite Group Databasebeta

← C3×#25776 #10644383

Subgroups — C3×#25776 #10644383

Order 768.

1656 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#495C212yesmaximal, complemented
3C2×C3×#465C211yesmaximal, characteristic, complemented
4C2×C3×#451C211yesmaximal, characteristic, complemented
5C2×C3×#447C211yesmaximal, characteristic
6C3×#2436C211yesmaximal, characteristic, complemented
7C3×#2435C211yesmaximal, characteristic, complemented
8C3×#2015C212yesmaximal, complemented
9C3×#2012C212yesmaximal, complemented
10C3×#1968C212yesmaximal, complemented
11C3×C8×C22⋊C4C211yesmaximal, characteristic, complemented
12#25776C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×Q16C2212yescomplemented
16C22×C3×SD16—28no—
17C22×C3×SD16—24no—
18C22×C3×SD16C2212yes—
19C22×C3×SD16C2212yes—
20C22×C3×SD16C2211yescharacteristic, complemented
21C22×C3×D8C2212yes—
22C23×C3×C8C2211yescharacteristic
23C2×C3×#124C2212yes—
24C2×C3×#124C2212yescomplemented
25C2×C3×C22⋊Q8C2211yescharacteristic
26C2×C3×#121C2211yescharacteristic
27C3×#495—28no—
28C3×#465—24no—
29C3×#451—24no—
30C3×#447—24no—
31C2×C3×#106C2211yescharacteristic, complemented
32C2×C3×#103C2212yescomplemented
33C2×C3×#103C2211yescharacteristic, complemented
34C2×C3×#102C2212yes—
35C2×C3×#102C2211yescharacteristic, complemented
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C4×C8C2212yescomplemented
38C3×#394—22no—
39C3×#393—22no—
40C3×#391C2212yescomplemented
41C3×#390C2212yes—
42C3×#386—22no—
43C3×#379—22no—
44C3×C4×C22⋊C4C2211yescharacteristic
45C3×#336C2212yescomplemented
46C23×SD16C2×C311yescharacteristic, complemented
47C2×#495C2×C312yescomplemented
48C2×#465C2×C311yescharacteristic, complemented
49C2×#451C2×C311yescharacteristic, complemented
50C2×#447C2×C311yescharacteristic
51#2436C2×C311yescharacteristic, complemented
52#2435C2×C311yescharacteristic, complemented
53#2015C2×C312yescomplemented
54#2012C2×C312yescomplemented
55#1968C2×C312yescomplemented
56C8×C22⋊C4C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8—24no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—42no—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×Q8C2312yes—
65C22×C3×D4—24no—
66C22×C3×D4—21no—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—21no—
70C22×C3×D4—42no—
71C22×C3×D4C2311yescharacteristic
72C22×C3×D4C2312yes—
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C2×C3×Q16—24no—
76C2×C3×Q16—24no—
77C2×C3×Q16—42no—
78C2×C3×SD16—232no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×SD16—24no—
84C2×C3×SD16—22no—
85C2×C3×SD16—24no—
86C2×C3×SD16D418yescomplemented
87C2×C3×D8—24no—
88C2×C3×D8—24no—
89C2×C3×D8—42no—
90C22×C3×C8—24no—
91C22×C3×C8D414yes—
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2312yes—
94C22×C3×C8C2311yescharacteristic
95C3×#124—24no—
96C3×#124—24no—
97C3×#124—42no—
98C3×#124—42no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8—22no—
101C3×#121—22no—
102C3×#121—22no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—24no—
108C2×C3×C22⋊C4—24no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—24no—
111C2×C3×C22⋊C4D412yes—
112C2×C3×C42C2312yes—
113C3×#106—22no—
114C3×#103—24no—
115C3×#103—22no—
116C3×#102—24no—
117C3×#102—22no—
118C3×C22⋊C8D414yes—
119C3×C4×C8—24no—
120C3×#95—24no—
121C3×#95D412yes—
122C23×Q8C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C22×Q16C22×C312yescomplemented
125C22×SD16—28no—
126C22×SD16—24no—
127C22×SD16C22×C312yes—
128C22×SD16C22×C312yes—
129C22×SD16C22×C311yescharacteristic, complemented
130C22×D8C22×C312yes—
131C23×C8C22×C311yescharacteristic
132C2×#124C22×C312yes—
133C2×#124C22×C312yescomplemented
134C2×C22⋊Q8C22×C311yescharacteristic
135C2×#121C22×C311yescharacteristic
136#495—28no—
137#465—24no—
138#451—24no—
139#447—24no—
140C2×#106C22×C311yescharacteristic, complemented
141C2×#103C22×C312yescomplemented
142C2×#103C22×C311yescharacteristic, complemented
143C2×#102C22×C312yes—
144C2×#102C22×C311yescharacteristic, complemented
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C4×C8C22×C312yescomplemented
147#394—22no—
148#393—22no—
149#391C22×C312yescomplemented
150#390C22×C312yes—
151#386—22no—
152#379—22no—
153C4×C22⋊C4C22×C311yescharacteristic
154#336C22×C312yescomplemented
155C24×C3—22no—
156C24×C3—21no—
157C24×C3—44no—
158C24×C3—42no—
159C24×C3—44no—
160C24×C3—82no—
161C24×C3C2×D411yescharacteristic
162C2×C3×Q8—22no—
163C2×C3×Q8—28no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—44no—
173C2×C3×Q8—48no—
174C2×C3×Q8—44no—
175C2×C3×Q8—42no—
176C2×C3×Q8C2×D414yes—
177C2×C3×D4—28no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—48no—
188C2×C3×D4—44no—
189C2×C3×D4—44no—
190C2×C3×D4—42no—
191C2×C3×D4—44no—
192C2×C3×D4—42no—
193C2×C3×D4C2×D414yes—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—44no—
201C22×C3×C4—42no—
202C22×C3×C4C2×D414yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2411yescharacteristic
207C3×Q16—48no—
208C3×SD16—216no—
209C3×SD16—216no—
210C3×SD16—44no—
211C3×D8—48no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—28no—
216C2×C3×C8C2×D414yes—
217C2×C3×C8C2×D412yes—
218C2×C3×C8C2×D414yes—
219C2×C3×C8C2×D412yes—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—42no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—48no—
227C3×C22⋊C4—48no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—84no—
231C3×C42—22no—
232C3×C42—22no—
233C25—21no—
234C22×Q8—21no—
235C22×Q8—24no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8C23×C311yescharacteristic
240C22×Q8C23×C312yes—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—42no—
247C22×D4C23×C311yescharacteristic
248C22×D4C23×C312yes—
249C23×C4—21no—
250C23×C4C23×C311yescharacteristic
251C2×Q16—24no—
252C2×Q16—24no—
253C2×Q16—42no—
254C2×SD16—232no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—22no—
261C2×SD16—24no—
262C2×SD16C3×D418yescomplemented
263C2×D8—24no—
264C2×D8—24no—
265C2×D8—42no—
266C22×C8—24no—
267C22×C8C3×D414yes—
268C22×C8C23×C311yescharacteristic
269C22×C8C23×C312yes—
270C22×C8C23×C311yescharacteristic
271#124—24no—
272#124—24no—
273#124—42no—
274#124—42no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277#121—22no—
278#121—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4C3×D412yes—
288C2×C42C23×C312yes—
289#106—22no—
290#103—24no—
291#103—22no—
292#102—24no—
293#102—22no—
294C22⋊C8C3×D414yes—
295C4×C8—24no—
296#95—24no—
297#95C3×D412yes—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—48no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—44no—
305C23×C3—44no—
306C23×C3—41no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—816no—
311C23×C3—84no—
312C23×C3—84no—
313C23×C3#12712yes—
314C23×C3#13514yes—
315C23×C3C22×D411yescharacteristic
316C3×Q8—24no—
317C3×Q8—24no—
318C3×Q8—48no—
319C3×Q8—42no—
320C3×Q8—42no—
321C3×Q8—41no—
322C3×Q8—41no—
323C3×Q8—42no—
324C3×D4—24no—
325C3×D4—24no—
326C3×D4—48no—
327C3×D4—41no—
328C3×D4—42no—
329C3×D4—41no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—84no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—48no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4#12712yes—
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D412yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4H(5,𝔽2)12yes—
355C3×C8—22no—
356C3×C8—24no—
357C3×C8#12014yes—
358C24—22no—
359C24—21no—
360C24—44no—
361C24—42no—
362C24—44no—
363C24—82no—
364C24C2×C3×D411yescharacteristic
365C2×Q8—22no—
366C2×Q8—28no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—44no—
376C2×Q8—48no—
377C2×Q8—44no—
378C2×Q8—42no—
379C2×Q8C2×C3×D414yes—
380C2×D4—28no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4C2×C3×D414yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4C2×C3×D414yes—
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D412yes—
408C22×C4C2×C3×D412yes—
409C22×C4C24×C311yescharacteristic
410Q16—48no—
411SD16—216no—
412SD16—216no—
413SD16—44no—
414D8—48no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—28no—
419C2×C8C2×C3×D414yes—
420C2×C8C2×C3×D412yes—
421C2×C8C2×C3×D414yes—
422C2×C8C2×C3×D412yes—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—42no—
428C22⋊C4—24no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—84no—
434C42—22no—
435C42—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—28no—
439C22×C3—44no—
440C22×C3—48no—
441C22×C3—816no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3—84no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—84no—
449C22×C3#49514yescomplemented
450C22×C3C2×#12711yescharacteristic
451C22×C3D4212yes—
452C22×C3C2×#13512yes—
453C22×C3#57712yes—
454C3×C4—22no—
455C3×C4—44no—
456C3×C4—44no—
457C3×C4—81no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4C2×#12011yescharacteristic
461C3×C4#53511yescharacteristic
462C3×C4D4212yes—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—48no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—816no—
476C23—84no—
477C23—84no—
478C23C3×#12712yes—
479C23C3×#13514yes—
480C23C22×C3×D411yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—48no—
484Q8—42no—
485Q8—42no—
486Q8—41no—
487Q8—41no—
488Q8—42no—
489D4—24no—
490D4—24no—
491D4—48no—
492D4—41no—
493D4—42no—
494D4—41no—
495D4—42no—
496D4—42no—
497D4—84no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4C3×#12712yes—
516C2×C4C22×C3×D411yescharacteristic
517C2×C4C22×C3×D412yes—
518C2×C4C22×C3×D411yescharacteristic
519C2×C4C3×H(5,𝔽2)12yes—
520C8—22no—
521C8—24no—
522C8C3×#12014yes—
523C2×C3—24no—
524C2×C3—84no—
525C2×C3—161no—
526C2×C3#243311yescharacteristic
527C2×C3C2×#49511yescharacteristic
528C2×C3#315411yescharacteristic
529C2×C3#328214yes—
530C22—22no—
531C22—22no—
532C22—28no—
533C22—44no—
534C22—48no—
535C22—816no—
536C22—82no—
537C22—82no—
538C22—84no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—84no—
543C22C3×#49514yescomplemented
544C22C2×C3×#12711yescharacteristic
545C22C3×D4212yes—
546C22C2×C3×#13512yes—
547C22C3×#57712yes—
548C4—22no—
549C4—44no—
550C4—44no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4C2×C3×#12011yescharacteristic
555C4C3×#53511yescharacteristic
556C4C3×D4212yes—
557C3#2577611yescharacteristic, direct product factor, Sylow
558C2—24no—
559C2—84no—
560C2—161no—
561C2C3×#243311yescharacteristic
562C2C2×C3×#49511yescharacteristic
563C2C3×#315411yescharacteristic
564C2C3×#328214yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C3×SD16C2semidirect
2. C2×C3×#495C2semidirect
3. C2×C3×#465C2semidirect
4. C2×C3×#451C2semidirect
6. C3×#2436C2semidirect
7. C3×#2435C2semidirect
8. C3×#2015C2semidirect
9. C3×#2012C2semidirect
10. C3×#1968C2semidirect
11. C3×C8×C22⋊C4C2semidirect
12. #25776C3direct (N × H)
15. C22×C3×Q16C22semidirect
20. C22×C3×SD16C22semidirect
24. C2×C3×#124C22semidirect
31. C2×C3×#106C22semidirect
32. C2×C3×#103C22semidirect
33. C2×C3×#103C22semidirect
35. C2×C3×#102C22semidirect
37. C2×C3×C4×C8C22semidirect
40. C3×#391C22semidirect
45. C3×#336C22semidirect
46. C23×SD16C2×C3semidirect
47. C2×#495C2×C3semidirect
48. C2×#465C2×C3semidirect
49. C2×#451C2×C3semidirect
51. #2436C2×C3semidirect
52. #2435C2×C3semidirect
53. #2015C2×C3semidirect
54. #2012C2×C3semidirect
55. #1968C2×C3semidirect
56. C8×C22⋊C4C2×C3semidirect
86. C2×C3×SD16D4semidirect
124. C22×Q16C22×C3semidirect
129. C22×SD16C22×C3semidirect
133. C2×#124C22×C3semidirect
140. C2×#106C22×C3semidirect
141. C2×#103C22×C3semidirect
142. C2×#103C22×C3semidirect
144. C2×#102C22×C3semidirect
146. C2×C4×C8C22×C3semidirect
149. #391C22×C3semidirect
154. #336C22×C3semidirect
262. C2×SD16C3×D4semidirect
449. C22×C3#495semidirect
543. C22C3×#495semidirect
557. C3#25776direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SD162
C2×C3×#4952
C2×C3×#4652
C2×C3×#4512
C2×C3×#4472
C3×#24362
C3×#24352
C3×#20152
C3×#20122
C3×#19682
C3×C8×C22⋊C42
#257763