Finite Group Databasebeta

← C3×#23591 #10642198

Subgroups — C3×#23591 #10642198

Order 768.

1182 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×#3460C211yesmaximal, characteristic, complemented
2C3×C4×AGL(1,ℤ/8)C212yesmaximal, complemented
3C3×#2328C211yesmaximal, characteristic, complemented
4C3×#2061C211yesmaximal, characteristic, complemented
5C3×#2040C211yesmaximal, characteristic, complemented
6C3×#2037C212yesmaximal, complemented
7C3×#2027C211yesmaximal, characteristic, complemented
8C3×#2006C211yesmaximal, characteristic, complemented
9C3×#2002C212yesmaximal, complemented
10C3×#1759C212yesmaximal, complemented
11C3×C4×#100C211yesmaximal, characteristic, complemented
12#23591C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C3×#544C2211yescharacteristic
15C3×#542—22no—
16C3×#540—22no—
17C3×#538—22no—
18C3×#537C2211yescharacteristic
19C3×#535C2212yes—
20C3×#532—22no—
21C2×C3×#127C2211yescharacteristic
22C3×#529—22no—
23C3×#525—22no—
24C3×C4×#41C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C3×#442—22no—
28C3×#440—22no—
29C3×C4×SD16—22no—
30C3×C4×D8—22no—
31C3×#428C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#418C2212yes—
35C3×#417C2212yes—
36C2×C3×#100C2212yes—
37C3×C4×M16C2212yes—
38C3×#390C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#386—22no—
41C3×#386—22no—
42C3×#382C2211yescharacteristic
43C3×#376C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3460C2×C311yescharacteristic, complemented
49C4×AGL(1,ℤ/8)C2×C312yescomplemented
50#2328C2×C311yescharacteristic, complemented
51#2061C2×C311yescharacteristic, complemented
52#2040C2×C311yescharacteristic, complemented
53#2037C2×C312yescomplemented
54#2027C2×C311yescharacteristic, complemented
55#2006C2×C311yescharacteristic, complemented
56#2002C2×C312yescomplemented
57#1759C2×C312yescomplemented
58C4×#100C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×D4—22no—
62C22×C3×D4C2311yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4D412yes—
66C3×AGL(1,ℤ/8)—24no—
67C3×AGL(1,ℤ/8)—24no—
68C2×C3×SD16—22no—
69C2×C3×D8—22no—
70C2×C3×M16C2312yes—
71C2×C3×M16C2312yes—
72C3×C4⋊Q8—22no—
73C3×#127—22no—
74C3×#127—22no—
75C3×#127—22no—
76C3×#126—24no—
77C3×#125—22no—
78C3×#124—24no—
79C3×#124—22no—
80C3×#123—24no—
81C3×#123—24no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×#121—22no—
86C3×#121—24no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#120—24no—
92C3×C4×Q8D412yescomplemented
93C3×C4×Q8D412yescomplemented
94C3×C4×D4—24no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4D412yescomplemented
102C3×C4×D4D412yescomplemented
103C3×#117—22no—
104C3×#117—22no—
105C3×#117C2312yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—22no—
110C2×C3×C22⋊C4—22no—
111C2×C3×C22⋊C4D412yes—
112C2×C3×C42—22no—
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C3×#107—22no—
117C3×#106—22no—
118C3×#104—24no—
119C3×#104—24no—
120C3×#103—22no—
121C3×#102—22no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#100—24no—
125C3×#97—22no—
126C3×C4×C8—22no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—22no—
130C2×AGL(1,ℤ/8)C22×C312yes—
131#544C22×C311yescharacteristic
132#542—22no—
133#540—22no—
134#538—22no—
135#537C22×C311yescharacteristic
136#535C22×C312yes—
137#532—22no—
138C2×#127C22×C311yescharacteristic
139#529—22no—
140#525—22no—
141C4×#41C22×C312yes—
142C2×C4×D4C22×C311yescharacteristic
143C2×C4×D4C22×C311yescharacteristic
144#442—22no—
145#440—22no—
146C4×SD16—22no—
147C4×D8—22no—
148#428C22×C312yes—
149C2×#104C22×C312yes—
150C2×#104C22×C312yes—
151#418C22×C312yes—
152#417C22×C312yes—
153C2×#100C22×C312yes—
154C4×M16C22×C312yes—
155#390C22×C311yescharacteristic
156#390C22×C311yescharacteristic
157#386—22no—
158#386—22no—
159#382C22×C311yescharacteristic
160#376C22×C311yescharacteristic
161#343C22×C311yescharacteristic
162#343C22×C311yescharacteristic
163#339C22×C312yes—
164#339C22×C312yes—
165C24×C3—21no—
166C24×C3C2×D412yes—
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—24no—
176C2×C3×D4—24no—
177C2×C3×D4—22no—
178C2×C3×D4—28no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—82no—
187C2×C3×D4—82no—
188C2×C3×D4C2×D412yes—
189C2×C3×D4#4112yescomplemented
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—42no—
199C22×C3×C4—82no—
200C22×C3×C4—82no—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C3×SD16—28no—
206C3×D8—28no—
207C3×M16—22no—
208C3×M16—22no—
209C3×M16—22no—
210C3×M16#4114yescomplemented
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4C2×D412yes—
221C3×C4⋊C4C2×D412yes—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—41no—
230C3×C22⋊C4—41no—
231C3×C22⋊C4—82no—
232C3×C22⋊C4—82no—
233C3×C42—22no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42—21no—
237C3×C42—22no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240C3×C42#4112yescomplemented
241C2×#41—22no—
242C2×#41C23×C312yes—
243C22×D4—22no—
244C22×D4C23×C311yescharacteristic
245C22×D4C23×C311yescharacteristic
246C23×C4—21no—
247C23×C4C3×D412yes—
248AGL(1,ℤ/8)—24no—
249AGL(1,ℤ/8)—24no—
250C2×SD16—22no—
251C2×D8—22no—
252C2×M16C23×C312yes—
253C2×M16C23×C312yes—
254C4⋊Q8—22no—
255#127—22no—
256#127—22no—
257#127—22no—
258#126—24no—
259#125—22no—
260#124—24no—
261#124—22no—
262#123—24no—
263#123—24no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267#121—22no—
268#121—24no—
269#121—22no—
270#121—22no—
271#121—22no—
272#121—22no—
273#120—24no—
274C4×Q8C3×D412yescomplemented
275C4×Q8C3×D412yescomplemented
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4C3×D412yescomplemented
284C4×D4C3×D412yescomplemented
285#117—22no—
286#117—22no—
287#117C23×C312yes—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4C3×D412yes—
294C2×C42—22no—
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297C2×C42C23×C311yescharacteristic
298#107—22no—
299#106—22no—
300#104—24no—
301#104—24no—
302#103—22no—
303#102—22no—
304#102—22no—
305#102—22no—
306#100—24no—
307#97—22no—
308C4×C8—22no—
309#95—21no—
310#95—21no—
311#95—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3C22×D411yescharacteristic
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—82no—
327C3×D4—22no—
328C3×D4—24no—
329C3×D4—22no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—84no—
336C3×D4—88no—
337C3×D4—82no—
338C3×D4—84no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—44no—
347C2×C3×C4—44no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—82no—
359C2×C3×C4—82no—
360C2×C3×C4—84no—
361C2×C3×C4#12011yescharacteristic
362C2×C3×C4#12012yes—
363C2×C3×C4#12011yescharacteristic
364C2×C3×C4C22×D412yes—
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4112yes—
367C2×C3×C4C2×#4111yescharacteristic
368C3×C8—24no—
369C3×C8—42no—
370C24—21no—
371C24C2×C3×D412yes—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—84no—
377C2×Q8—22no—
378C2×Q8C2×C3×D412yes—
379C2×D4—22no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4C2×C3×D412yes—
394C2×D4C3×#4112yescomplemented
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—42no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4C2×C3×D412yes—
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C24×C311yescharacteristic
410SD16—28no—
411D8—28no—
412M16—22no—
413M16—22no—
414M16—22no—
415M16C3×#4114yescomplemented
416C2×C8—22no—
417C2×C8—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4C2×C3×D412yes—
426C4⋊C4C2×C3×D412yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—21no—
442C42—22no—
443C42C2×C3×D412yes—
444C42C2×C3×D412yes—
445C42C3×#4112yescomplemented
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3C2×#12011yescharacteristic
462C22×C3#52511yescharacteristic
463C22×C3#53211yescharacteristic
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4#52512yes—
475C3×C4#53212yes—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23C22×C3×D411yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—82no—
491D4—22no—
492D4—24no—
493D4—22no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—88no—
500D4—82no—
501D4—84no—
502D4—84no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—84no—
525C2×C4C3×#12011yescharacteristic
526C2×C4C3×#12012yes—
527C2×C4C3×#12011yescharacteristic
528C2×C4C22×C3×D412yes—
529C2×C4C2×C3×#4111yescharacteristic
530C2×C4C2×C3×#4112yes—
531C2×C4C2×C3×#4111yescharacteristic
532C8—24no—
533C8—42no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—82no—
537C2×C3—82no—
538C2×C3—82no—
539C2×C3#238811yescharacteristic
540C2×C3#301911yescharacteristic
541C2×C3#302111yescharacteristic
542C22—21no—
543C22—21no—
544C22—44no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22—81no—
556C22—81no—
557C22C2×C3×#12011yescharacteristic
558C22C3×#52511yescharacteristic
559C22C3×#53211yescharacteristic
560C4—21no—
561C4—21no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—81no—
569C4—81no—
570C4C3×#52512yes—
571C4C3×#53212yes—
572C3#2359111yescharacteristic, direct product factor, Sylow
573C2—21no—
574C2—21no—
575C2—82no—
576C2—82no—
577C2—82no—
578C2C3×#238811yescharacteristic
579C2C3×#301911yescharacteristic
580C2C3×#302111yescharacteristic
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×#3460C2semidirect
2. C3×C4×AGL(1,ℤ/8)C2semidirect
3. C3×#2328C2semidirect
4. C3×#2061C2semidirect
5. C3×#2040C2semidirect
6. C3×#2037C2semidirect
7. C3×#2027C2semidirect
8. C3×#2006C2semidirect
9. C3×#2002C2semidirect
10. C3×#1759C2semidirect
11. C3×C4×#100C2semidirect
12. #23591C3direct (N × H)
48. #3460C2×C3semidirect
49. C4×AGL(1,ℤ/8)C2×C3semidirect
50. #2328C2×C3semidirect
51. #2061C2×C3semidirect
52. #2040C2×C3semidirect
53. #2037C2×C3semidirect
54. #2027C2×C3semidirect
55. #2006C2×C3semidirect
56. #2002C2×C3semidirect
57. #1759C2×C3semidirect
58. C4×#100C2×C3semidirect
92. C3×C4×Q8D4semidirect
93. C3×C4×Q8D4semidirect
101. C3×C4×D4D4semidirect
102. C3×C4×D4D4semidirect
189. C2×C3×D4#41semidirect
210. C3×M16#41semidirect
240. C3×C42#41semidirect
274. C4×Q8C3×D4semidirect
275. C4×Q8C3×D4semidirect
283. C4×D4C3×D4semidirect
284. C4×D4C3×D4semidirect
394. C2×D4C3×#41semidirect
415. M16C3×#41semidirect
445. C42C3×#41semidirect
572. C3#23591direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34602
C3×C4×AGL(1,ℤ/8)2
C3×#23282
C3×#20612
C3×#20402
C3×#20372
C3×#20272
C3×#20062
C3×#20022
C3×#17592
C3×C4×#1002
#235913