Finite Group Databasebeta

← C3×#25616 #10644223

Subgroups — C3×#25616 #10644223

Order 768.

1240 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×#3511C211yesmaximal, characteristic, complemented
2C3×#3151C211yesmaximal, characteristic, complemented
3C3×#3038C211yesmaximal, characteristic, complemented
4C3×#2858C211yesmaximal, characteristic, complemented
5C3×#2006C214yesmaximal, complemented
6C3×#1997C211yesmaximal, characteristic, complemented
7C3×#1962C211yesmaximal, characteristic, complemented
8C3×#1883C214yesmaximal, complemented
9C3×C4×#104C211yesmaximal, characteristic, complemented
10#25616C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×AGL(1,ℤ/8)C2214yes—
12C2×C3×AGL(1,ℤ/8)C2212yes—
13C3×#556—21no—
14C3×#555—21no—
15C3×#554—21no—
16C3×#553—22no—
17C3×#548—22no—
18C3×#547—21no—
19C3×#542—22no—
20C3×#537C2212yes—
21C3×#533—21no—
22C3×#532—21no—
23C3×#532C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C2×C3×#127C2211yescharacteristic
26C2×C3×#124—22no—
27C3×#520C2212yes—
28C3×C4×#41C2211yescharacteristic
29C3×#496—22no—
30C3×#493—21no—
31C3×#492—21no—
32C3×#460—21no—
33C3×#460—21no—
34C3×#459—21no—
35C3×#459—21no—
36C3×#423C2211yescharacteristic
37C2×C3×#104C2212yes—
38C2×C3×#104C2214yes—
39C3×#418C2212yes—
40C3×C4×M16C2211yescharacteristic
41C3×#390—22no—
42C3×#390C2214yes—
43C3×C43C2211yescharacteristic
44C3×#343C2214yes—
45C3×#339C2212yes—
46#3511C2×C311yescharacteristic, complemented
47#3151C2×C311yescharacteristic, complemented
48#3038C2×C311yescharacteristic, complemented
49#2858C2×C311yescharacteristic, complemented
50#2006C2×C314yescomplemented
51#1997C2×C311yescharacteristic, complemented
52#1962C2×C311yescharacteristic, complemented
53#1883C2×C314yescomplemented
54C4×#104C2×C311yescharacteristic, complemented
55C2×C3×#41—22no—
56C2×C3×#41C2311yescharacteristic
57C2×C3×#41C2312yes—
58C22×C3×Q8D412yes—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C3×AGL(1,ℤ/8)—28no—
63C3×AGL(1,ℤ/8)—44no—
64C3×AGL(1,ℤ/8)—44no—
65C2×C3×SD16—22no—
66C2×C3×SD16—24no—
67C2×C3×D8—22no—
68C2×C3×D8—24no—
69C2×C3×M16C2312yes—
70C2×C3×M16C2312yes—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8D411yescharacteristic
76C3×#127—21no—
77C3×#127—21no—
78C3×#127—21no—
79C3×#127—21no—
80C3×#127D411yescharacteristic
81C3×#126—24no—
82C3×#125—22no—
83C3×#124—22no—
84C3×#124—24no—
85C3×#124—48no—
86C3×#124D412yes—
87C3×#123—24no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—24no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×#121—22no—
93C3×#121—22no—
94C3×#121—22no—
95C3×#121—21no—
96C3×#121—21no—
97C3×#121—24no—
98C3×C4×Q8—22no—
99C3×C4×Q8—22no—
100C3×C4×Q8D411yescharacteristic
101C3×C4×Q8D411yescharacteristic
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×C4×D4—24no—
107C3×C4×D4—21no—
108C3×C4×D4—21no—
109C3×C4×D4D411yescharacteristic
110C3×C4×D4D411yescharacteristic
111C3×#117—22no—
112C3×#117—22no—
113C3×#117—21no—
114C3×#117C2311yescharacteristic
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4D412yes—
117C2×C3×C4⋊C4C2312yes—
118C2×C3×C22⋊C4—28no—
119C2×C3×C42—21no—
120C2×C3×C42—21no—
121C2×C3×C42—21no—
122C2×C3×C42C2312yes—
123C2×C3×C42C2311yescharacteristic
124C3×C4⋊C8—21no—
125C3×C4⋊C8—21no—
126C3×#104—24no—
127C3×#104—44no—
128C3×#102—22no—
129C3×#102—22no—
130C3×#97—21no—
131C3×C4×C8—21no—
132C2×AGL(1,ℤ/8)C22×C314yes—
133C2×AGL(1,ℤ/8)C22×C312yes—
134#556—21no—
135#555—21no—
136#554—21no—
137#553—22no—
138#548—22no—
139#547—21no—
140#542—22no—
141#537C22×C312yes—
142#533—21no—
143#532—21no—
144#532C22×C311yescharacteristic
145C2×C4⋊Q8C22×C311yescharacteristic
146C2×#127C22×C311yescharacteristic
147C2×#124—22no—
148#520C22×C312yes—
149C4×#41C22×C311yescharacteristic
150#496—22no—
151#493—21no—
152#492—21no—
153#460—21no—
154#460—21no—
155#459—21no—
156#459—21no—
157#423C22×C311yescharacteristic
158C2×#104C22×C312yes—
159C2×#104C22×C314yes—
160#418C22×C312yes—
161C4×M16C22×C311yescharacteristic
162#390—22no—
163#390C22×C314yes—
164C43C22×C311yescharacteristic
165#343C22×C314yes—
166#339C22×C312yes—
167C24×C3—22no—
168C3×#41—22no—
169C3×#41—24no—
170C3×#41—24no—
171C3×#41—22no—
172C3×#41—84no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—44no—
176C2×C3×Q8C2×D412yes—
177C2×C3×Q8C2×D411yescharacteristic
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—24no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—24no—
186C2×C3×D4—24no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—41no—
190C2×C3×D4—41no—
191C2×C3×D4—41no—
192C2×C3×D4—41no—
193C2×C3×D4C2×D412yes—
194C2×C3×D4C2×D411yescharacteristic
195C22×C3×C4—22no—
196C22×C3×C4—24no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—22no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D414yes—
204C22×C3×C4C2411yescharacteristic
205C3×SD16—48no—
206C3×SD16—44no—
207C3×D8—48no—
208C3×D8—44no—
209C3×M16—22no—
210C3×M16—24no—
211C3×M16#4114yescomplemented
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4C2×D411yescharacteristic
225C3×C4⋊C4C2×D411yescharacteristic
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—21no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—416no—
235C3×C42—24no—
236C3×C42—24no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42—42no—
243C3×C42C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42C2×D411yescharacteristic
247C2×#41—22no—
248C2×#41C23×C311yescharacteristic
249C2×#41C23×C312yes—
250C22×Q8C3×D412yes—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C23×C312yes—
254AGL(1,ℤ/8)—28no—
255AGL(1,ℤ/8)—44no—
256AGL(1,ℤ/8)—44no—
257C2×SD16—22no—
258C2×SD16—24no—
259C2×D8—22no—
260C2×D8—24no—
261C2×M16C23×C312yes—
262C2×M16C23×C312yes—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8C3×D411yescharacteristic
268#127—21no—
269#127—21no—
270#127—21no—
271#127—21no—
272#127C3×D411yescharacteristic
273#126—24no—
274#125—22no—
275#124—22no—
276#124—24no—
277#124—48no—
278#124C3×D412yes—
279#123—24no—
280C22⋊Q8—22no—
281C22⋊Q8—24no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284#121—22no—
285#121—22no—
286#121—22no—
287#121—21no—
288#121—21no—
289#121—24no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8C3×D411yescharacteristic
293C4×Q8C3×D411yescharacteristic
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—24no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4C3×D411yescharacteristic
302C4×D4C3×D411yescharacteristic
303#117—22no—
304#117—22no—
305#117—21no—
306#117C23×C311yescharacteristic
307C2×C4⋊C4—22no—
308C2×C4⋊C4C3×D412yes—
309C2×C4⋊C4C23×C312yes—
310C2×C22⋊C4—28no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42C23×C312yes—
315C2×C42C23×C311yescharacteristic
316C4⋊C8—21no—
317C4⋊C8—21no—
318#104—24no—
319#104—44no—
320#102—22no—
321#102—22no—
322#97—21no—
323C4×C8—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—44no—
327C23×C3—44no—
328C23×C3—42no—
329C23×C3—44no—
330C23×C3C22×D411yescharacteristic
331C3×Q8—21no—
332C3×Q8—21no—
333C3×Q8—24no—
334C3×Q8—44no—
335C3×Q8—82no—
336C3×D4—21no—
337C3×D4—21no—
338C3×D4—24no—
339C3×D4—42no—
340C3×D4—44no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—44no—
344C3×D4—42no—
345C3×D4—42no—
346C3×D4—82no—
347C3×D4—82no—
348C3×D4—84no—
349C3×D4—82no—
350C3×D4—82no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—24no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—44no—
367C2×C3×C4—84no—
368C2×C3×C4#12011yescharacteristic
369C2×C3×C4#12011yescharacteristic
370C2×C3×C4#12011yescharacteristic
371C2×C3×C4#12011yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4112yes—
375C2×C3×C4H(5,𝔽2)12yes—
376C3×C8—24no—
377C3×C8—42no—
378C24—22no—
379#41—22no—
380#41—24no—
381#41—24no—
382#41—22no—
383#41—84no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—44no—
387C2×Q8C2×C3×D412yes—
388C2×Q8C2×C3×D411yescharacteristic
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4C2×C3×D412yes—
405C2×D4C2×C3×D411yescharacteristic
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D414yes—
415C22×C4C24×C311yescharacteristic
416SD16—48no—
417SD16—44no—
418D8—48no—
419D8—44no—
420M16—22no—
421M16—24no—
422M16C3×#4114yescomplemented
423C2×C8—22no—
424C2×C8—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4C2×C3×D411yescharacteristic
436C4⋊C4C2×C3×D411yescharacteristic
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—416no—
446C42—24no—
447C42—24no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—22no—
452C42—22no—
453C42—42no—
454C42C2×C3×D411yescharacteristic
455C42C2×C3×D411yescharacteristic
456C42C2×C3×D411yescharacteristic
457C42C2×C3×D411yescharacteristic
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—42no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—41no—
465C22×C3—42no—
466C22×C3—88no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3—82no—
470C22×C3C2×#12011yescharacteristic
471C22×C3#55012yes—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—44no—
475C3×C4—42no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—44no—
479C3×C4—41no—
480C3×C4#45312yes—
481C3×C4#45412yes—
482C3×C4#53211yescharacteristic
483C3×C4#53511yescharacteristic
484C3×C4#54612yes—
485C23—21no—
486C23—22no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—44no—
491C23C22×C3×D411yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—24no—
495Q8—44no—
496Q8—82no—
497D4—21no—
498D4—21no—
499D4—24no—
500D4—42no—
501D4—44no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—44no—
507D4—82no—
508D4—82no—
509D4—84no—
510D4—82no—
511D4—82no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—84no—
529C2×C4C3×#12011yescharacteristic
530C2×C4C3×#12011yescharacteristic
531C2×C4C3×#12011yescharacteristic
532C2×C4C3×#12011yescharacteristic
533C2×C4C22×C3×D411yescharacteristic
534C2×C4C22×C3×D411yescharacteristic
535C2×C4C2×C3×#4112yes—
536C2×C4C3×H(5,𝔽2)12yes—
537C8—24no—
538C8—42no—
539C2×C3—22no—
540C2×C3—81no—
541C2×C3—82no—
542C2×C3—162no—
543C2×C3#243111yescharacteristic
544C2×C3C2×#45311yescharacteristic
545C2×C3C2×#45411yescharacteristic
546C22—21no—
547C22—21no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—41no—
553C22—42no—
554C22—88no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22C2×C3×#12011yescharacteristic
559C22C3×#55012yes—
560C4—21no—
561C4—21no—
562C4—44no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—44no—
567C4—41no—
568C4C3×#45312yes—
569C4C3×#45412yes—
570C4C3×#53211yescharacteristic
571C4C3×#53511yescharacteristic
572C4C3×#54612yes—
573C3#2561611yescharacteristic, direct product factor, Sylow
574C2—22no—
575C2—81no—
576C2—82no—
577C2—162no—
578C2C3×#243111yescharacteristic
579C2C2×C3×#45311yescharacteristic
580C2C2×C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3511C2semidirect
2. C3×#3151C2semidirect
3. C3×#3038C2semidirect
4. C3×#2858C2semidirect
5. C3×#2006C2semidirect
6. C3×#1997C2semidirect
7. C3×#1962C2semidirect
8. C3×#1883C2semidirect
9. C3×C4×#104C2semidirect
10. #25616C3direct (N × H)
46. #3511C2×C3semidirect
47. #3151C2×C3semidirect
48. #3038C2×C3semidirect
49. #2858C2×C3semidirect
50. #2006C2×C3semidirect
51. #1997C2×C3semidirect
52. #1962C2×C3semidirect
53. #1883C2×C3semidirect
54. C4×#104C2×C3semidirect
211. C3×M16#41semidirect
422. M16C3×#41semidirect
573. C3#25616direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35112
C3×#31512
C3×#30382
C3×#28582
C3×#20062
C3×#19972
C3×#19622
C3×#18832
C3×C4×#1042
#256163