Finite Group Databasebeta

← C3×#25517 #10644124

Subgroups — C3×#25517 #10644124

Order 768.

1174 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3492C211yesmaximal, characteristic, complemented
2C3×#3055C212yesmaximal, complemented
3C3×#2501C211yesmaximal, characteristic, complemented
4C3×#2060C211yesmaximal, characteristic, complemented
5C3×#2039C211yesmaximal, characteristic, complemented
6C3×#2027C211yesmaximal, characteristic, complemented
7C3×#2006C211yesmaximal, characteristic, complemented
8C3×#1883C212yesmaximal, complemented
9C3×#1871C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C2×C3×#343C211yesmaximal, characteristic, complemented
12#25517C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C22×C3×M16C2212yes—
15C3×#554—22no—
16C3×#553—22no—
17C3×#552—22no—
18C3×#547—22no—
19C3×#544C2211yescharacteristic
20C3×#539—22no—
21C3×#537C2211yescharacteristic
22C3×#525—22no—
23C3×#525C2212yes—
24C2×C3×#123—22no—
25C2×C3×#121C2211yescharacteristic
26C3×#520C2212yes—
27C22×C3×C4⋊C4C2211yescharacteristic
28C3×#469—22no—
29C3×#468—22no—
30C3×#466—22no—
31C3×#465—22no—
32C3×#428C2212yes—
33C2×C3×#104C2212yes—
34C2×C3×#104C2212yes—
35C3×#418C2212yes—
36C3×#417C2212yes—
37C3×#399—22no—
38C3×#390C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#384C2211yescharacteristic
41C3×#384C2211yescharacteristic
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#343C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2212yes—
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3492C2×C311yescharacteristic, complemented
49#3055C2×C312yescomplemented
50#2501C2×C311yescharacteristic, complemented
51#2060C2×C311yescharacteristic, complemented
52#2039C2×C311yescharacteristic, complemented
53#2027C2×C311yescharacteristic, complemented
54#2006C2×C311yescharacteristic, complemented
55#1883C2×C312yescomplemented
56#1871C2×C312yescomplemented
57#1859C2×C312yescomplemented
58C2×#343C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4D412yes—
64C23×C3×C4C2311yescharacteristic
65C3×AGL(1,ℤ/8)—24no—
66C3×AGL(1,ℤ/8)—42no—
67C2×C3×SD16—22no—
68C2×C3×D8—22no—
69C2×C3×M16—22no—
70C2×C3×M16—24no—
71C2×C3×M16—42no—
72C2×C3×M16C2312yes—
73C2×C3×M16C2312yes—
74C22×C3×C8—22no—
75C3×C4⋊Q8—22no—
76C3×#126—24no—
77C3×#125—24no—
78C3×#125—22no—
79C3×#123—24no—
80C3×#123—24no—
81C3×#123—22no—
82C3×#123—48no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8D412yes—
88C3×C22⋊Q8D412yes—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—22no—
93C3×#121—22no—
94C3×#121D412yes—
95C3×#121D412yes—
96C3×#120—22no—
97C3×C4×Q8—22no—
98C3×C4×D4—24no—
99C3×C4×D4—22no—
100C3×C4×D4—24no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×#117—22no—
104C3×#117—22no—
105C3×#117C2312yes—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4—22no—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4D412yes—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4—22no—
118C2×C3×C22⋊C4—21no—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C3×#107—22no—
122C3×#106—22no—
123C3×#104—24no—
124C3×#104—42no—
125C3×#103—22no—
126C3×#102—22no—
127C3×#102—22no—
128C3×#102—22no—
129C3×#95—21no—
130C2×AGL(1,ℤ/8)C22×C312yes—
131C22×M16C22×C312yes—
132#554—22no—
133#553—22no—
134#552—22no—
135#547—22no—
136#544C22×C311yescharacteristic
137#539—22no—
138#537C22×C311yescharacteristic
139#525—22no—
140#525C22×C312yes—
141C2×#123—22no—
142C2×#121C22×C311yescharacteristic
143#520C22×C312yes—
144C22×C4⋊C4C22×C311yescharacteristic
145#469—22no—
146#468—22no—
147#466—22no—
148#465—22no—
149#428C22×C312yes—
150C2×#104C22×C312yes—
151C2×#104C22×C312yes—
152#418C22×C312yes—
153#417C22×C312yes—
154#399—22no—
155#390C22×C311yescharacteristic
156#390C22×C311yescharacteristic
157#384C22×C311yescharacteristic
158#384C22×C311yescharacteristic
159C4×C22⋊C4C22×C311yescharacteristic
160#343C22×C311yescharacteristic
161#343C22×C312yes—
162#343C22×C311yescharacteristic
163#339C22×C312yes—
164#339C22×C312yes—
165C24×C3—21no—
166C24×C3C2×D411yescharacteristic
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—84no—
172C2×C3×Q8—22no—
173C2×C3×Q8C2×D412yes—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—44no—
183C2×C3×D4—41no—
184C2×C3×D4—41no—
185C2×C3×D4C2×D412yes—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—24no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—44no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2411yescharacteristic
203C3×SD16—44no—
204C3×D8—44no—
205C3×M16—22no—
206C3×M16—22no—
207C3×M16—22no—
208C3×M16—44no—
209C3×M16#4114yescomplemented
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—44no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—44no—
224C3×C4⋊C4—44no—
225C3×C4⋊C4C2×D412yes—
226C3×C4⋊C4C2×D412yes—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—44no—
236C3×C22⋊C4—44no—
237C3×C22⋊C4—42no—
238C3×C42—24no—
239C3×C42—22no—
240C3×C42—22no—
241C3×C42—21no—
242C3×C42—21no—
243C2×#41—22no—
244C2×#41C23×C312yes—
245C22×D4—21no—
246C22×D4C23×C311yescharacteristic
247C23×C4C3×D412yes—
248C23×C4C23×C311yescharacteristic
249AGL(1,ℤ/8)—24no—
250AGL(1,ℤ/8)—42no—
251C2×SD16—22no—
252C2×D8—22no—
253C2×M16—22no—
254C2×M16—24no—
255C2×M16—42no—
256C2×M16C23×C312yes—
257C2×M16C23×C312yes—
258C22×C8—22no—
259C4⋊Q8—22no—
260#126—24no—
261#125—24no—
262#125—22no—
263#123—24no—
264#123—24no—
265#123—22no—
266#123—48no—
267C22⋊Q8—22no—
268C22⋊Q8—24no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8C3×D412yes—
272C22⋊Q8C3×D412yes—
273#121—22no—
274#121—22no—
275#121—22no—
276#121—22no—
277#121—22no—
278#121C3×D412yes—
279#121C3×D412yes—
280#120—22no—
281C4×Q8—22no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287#117—22no—
288#117—22no—
289#117C23×C312yes—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4C3×D412yes—
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C42C23×C311yescharacteristic
304C2×C42C23×C311yescharacteristic
305#107—22no—
306#106—22no—
307#104—24no—
308#104—42no—
309#103—22no—
310#102—22no—
311#102—22no—
312#102—22no—
313#95—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—41no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3#12011yescharacteristic
325C23×C3#12011yescharacteristic
326C23×C3C22×D411yescharacteristic
327C3×Q8—22no—
328C3×Q8—22no—
329C3×Q8—82no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—44no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—44no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—82no—
339C3×D4—82no—
340C3×D4—82no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—44no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—82no—
366C2×C3×C4—84no—
367C2×C3×C4#12012yes—
368C2×C3×C4C22×D412yes—
369C2×C3×C4C2×#4112yes—
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×C3×C42−1+411yescharacteristic
372C3×C8—24no—
373C3×C8—42no—
374C24—21no—
375C24C2×C3×D411yescharacteristic
376#41—22no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—84no—
381C2×Q8—22no—
382C2×Q8C2×C3×D412yes—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—44no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4C2×C3×D412yes—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—44no—
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D412yes—
408C22×C4C2×C3×D412yes—
409C22×C4C2×C3×D412yes—
410C22×C4C2×C3×D412yes—
411C22×C4C24×C311yescharacteristic
412SD16—44no—
413D8—44no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—44no—
418M16C3×#4114yescomplemented
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—44no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4C2×C3×D412yes—
435C4⋊C4C2×C3×D412yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C42—24no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3#45312yes—
470C22×C3C2×#12011yescharacteristic
471C22×C3#53211yescharacteristic
472C22×C3#53711yescharacteristic
473C3×C4—22no—
474C3×C4—42no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—81no—
482C3×C4#54612yes—
483C3×C4#54712yes—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C3×#12011yescharacteristic
495C23C3×#12011yescharacteristic
496C23C22×C3×D411yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—82no—
500D4—22no—
501D4—22no—
502D4—44no—
503D4—44no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—82no—
509D4—82no—
510D4—82no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—84no—
537C2×C4C3×#12012yes—
538C2×C4C22×C3×D412yes—
539C2×C4C2×C3×#4112yes—
540C2×C4C3×H(5,𝔽2)11yescharacteristic
541C2×C4C3×2−1+411yescharacteristic
542C8—24no—
543C8—42no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—22no—
547C2×C3—41no—
548C2×C3—82no—
549C2×C3—161no—
550C2×C3#243011yescharacteristic
551C2×C3C2×#45311yescharacteristic
552C2×C3#302011yescharacteristic
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—82no—
566C22—82no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22C3×#45312yes—
571C22C2×C3×#12011yescharacteristic
572C22C3×#53211yescharacteristic
573C22C3×#53711yescharacteristic
574C4—22no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4C3×#54612yes—
584C4C3×#54712yes—
585C3#2551711yescharacteristic, direct product factor, Sylow
586C2—21no—
587C2—21no—
588C2—22no—
589C2—41no—
590C2—82no—
591C2—161no—
592C2C3×#243011yescharacteristic
593C2C2×C3×#45311yescharacteristic
594C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C3×#3492C2semidirect
2. C3×#3055C2semidirect
3. C3×#2501C2semidirect
4. C3×#2060C2semidirect
5. C3×#2039C2semidirect
6. C3×#2027C2semidirect
7. C3×#2006C2semidirect
8. C3×#1883C2semidirect
9. C3×#1871C2semidirect
10. C3×#1859C2semidirect
11. C2×C3×#343C2semidirect
12. #25517C3direct (N × H)
48. #3492C2×C3semidirect
49. #3055C2×C3semidirect
50. #2501C2×C3semidirect
51. #2060C2×C3semidirect
52. #2039C2×C3semidirect
53. #2027C2×C3semidirect
54. #2006C2×C3semidirect
55. #1883C2×C3semidirect
56. #1871C2×C3semidirect
57. #1859C2×C3semidirect
58. C2×#343C2×C3semidirect
209. C3×M16#41semidirect
418. M16C3×#41semidirect
585. C3#25517direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34922
C3×#30552
C3×#25012
C3×#20602
C3×#20392
C3×#20272
C3×#20062
C3×#18832
C3×#18712
C3×#18592
C2×C3×#3432
#255173