Finite Group Databasebeta

← C3×#18640 #10637247

Subgroups — C3×#18640 #10637247

Order 768.

1210 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3499C211yesmaximal, characteristic, complemented
2C3×#3205C211yesmaximal, characteristic, complemented
3C2×C3×#486C211yesmaximal, characteristic, complemented
4C3×#3038C211yesmaximal, characteristic, complemented
5C3×#1627C214yesmaximal, complemented
6C3×#1625C214yesmaximal, complemented
7C3×#1546C211yesmaximal, characteristic, complemented
8C3×#1492C211yesmaximal, characteristic, complemented
9C3×#1476C211yesmaximal, characteristic, complemented
10#18640C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×2−1+4—21no—
12C2×C3×AGL(1,ℤ/8)—21no—
13C3×D4×Q8—21no—
14C3×D4×Q8C2212yescomplemented
15C3×#547—22no—
16C3×#547—21no—
17C3×#547—22no—
18C3×#547C2212yescomplemented
19C3×#536—22no—
20C3×#536—21no—
21C3×#532—21no—
22C3×#532C2211yescharacteristic
23C2×C3×C4⋊Q8—21no—
24C2×C3×C4⋊Q8C2211yescharacteristic
25C2×C3×#127C2211yescharacteristic
26C3×C4×#41C2211yescharacteristic
27C3×#493—21no—
28C3×#486—22no—
29C3×#486—21no—
30C3×#486C2214yes—
31C3×#460—21no—
32C3×#460C2212yes—
33C3×#459—21no—
34C3×#459C2214yes—
35C3×#459C2212yes—
36C3×#447—22no—
37C3×#431C2211yescharacteristic
38C3×#423C2211yescharacteristic
39C3×#418—21no—
40C2×C3×#102—22no—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#332C2214yescomplemented
43C3×Q8⋊C8C2212yescomplemented
44C3×D4⋊C8C2214yescomplemented
45C3×D4⋊C8C2212yescomplemented
46#3499C2×C311yescharacteristic, complemented
47#3205C2×C311yescharacteristic, complemented
48C2×#486C2×C311yescharacteristic, complemented
49#3038C2×C311yescharacteristic, complemented
50#1627C2×C314yescomplemented
51#1625C2×C314yescomplemented
52#1546C2×C311yescharacteristic, complemented
53#1492C2×C311yescharacteristic, complemented
54#1476C2×C311yescharacteristic, complemented
55C3×2−1+4—44no—
56C3×2−1+4—44no—
57C2×C3×#41—22no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C2×C3×#41—21no—
61C2×C3×#41D411yescharacteristic
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8—21no—
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4—21no—
68C3×AGL(1,ℤ/8)—44no—
69C2×C3×SD16—22no—
70C2×C3×D8—22no—
71C2×C3×D8—24no—
72C2×C3×M16—21no—
73C22×C3×C8—21no—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—21no—
78C3×C4⋊Q8—21no—
79C3×C4⋊Q8D411yescharacteristic
80C3×C4⋊Q8D412yes—
81C3×C4⋊Q8C2312yes—
82C3×#127—21no—
83C3×#127—21no—
84C3×#127—21no—
85C3×#127D411yescharacteristic
86C3×#127C2312yes—
87C3×#124—22no—
88C3×#124—21no—
89C3×#123—24no—
90C3×#123—24no—
91C3×#123—24no—
92C3×C22⋊Q8—24no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—24no—
95C3×C22⋊Q8—22no—
96C3×#121—22no—
97C3×#121—22no—
98C3×#121—21no—
99C3×#121D412yes—
100C3×C4×Q8D411yescharacteristic
101C3×C4×Q8C2311yescharacteristic
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×C4×D4—21no—
107C3×C4×D4D411yescharacteristic
108C3×C4×D4C2311yescharacteristic
109C3×C4×D4C2312yes—
110C3×#117—21no—
111C3×#117D411yescharacteristic
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4—22no—
115C2×C3×C4⋊C4D412yes—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C3×C4⋊C8—21no—
119C3×C4⋊C8C2312yes—
120C3×C4⋊C8C2312yes—
121C3×#102—28no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—24no—
125C3×C22⋊C8—21no—
126C3×C4×C8—21no—
127C3×C4×C8—21no—
128C3×C4×C8C2312yes—
129C2×2−1+4—21no—
130C2×AGL(1,ℤ/8)—21no—
131D4×Q8—21no—
132D4×Q8C22×C312yescomplemented
133#547—22no—
134#547—21no—
135#547—22no—
136#547C22×C312yescomplemented
137#536—22no—
138#536—21no—
139#532—21no—
140#532C22×C311yescharacteristic
141C2×C4⋊Q8—21no—
142C2×C4⋊Q8C22×C311yescharacteristic
143C2×#127C22×C311yescharacteristic
144C4×#41C22×C311yescharacteristic
145#493—21no—
146#486—22no—
147#486—21no—
148#486C22×C314yes—
149#460—21no—
150#460C22×C312yes—
151#459—21no—
152#459C22×C312yes—
153#459C22×C314yes—
154#447—22no—
155#431C22×C311yescharacteristic
156#423C22×C311yescharacteristic
157#418—21no—
158C2×#102—22no—
159C2×C4×C8C22×C311yescharacteristic
160#332C22×C314yescomplemented
161Q8⋊C8C22×C312yescomplemented
162D4⋊C8C22×C312yescomplemented
163D4⋊C8C22×C314yescomplemented
164C24×C3—41no—
165C3×#41—22no—
166C3×#41—48no—
167C3×#41—416no—
168C3×#41—44no—
169C3×#41—44no—
170C3×#41—42no—
171C3×#41—42no—
172C3×#41D814yescomplemented
173C2×C3×Q8—24no—
174C2×C3×Q8—22no—
175C2×C3×Q8—21no—
176C2×C3×Q8—22no—
177C2×C3×Q8—48no—
178C2×C3×Q8—44no—
179C2×C3×Q8—44no—
180C2×C3×Q8—42no—
181C2×C3×Q8—42no—
182C2×C3×Q8C2×D411yescharacteristic
183C2×C3×D4—21no—
184C2×C3×D4—24no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—44no—
192C2×C3×D4—41no—
193C2×C3×D4—41no—
194C2×C3×D4—41no—
195C2×C3×D4—42no—
196C2×C3×D4—42no—
197C2×C3×D4—82no—
198C2×C3×D4C2×D412yes—
199C2×C3×D4C2×D411yescharacteristic
200C22×C3×C4—21no—
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C3×SD16—44no—
211C3×D8—44no—
212C3×D8—84no—
213C3×M16—42no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—42no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4C2×D412yes—
227C3×C4⋊C4C2×D414yes—
228C3×C4⋊C4C2×D411yescharacteristic
229C3×C4⋊C4C2×D411yescharacteristic
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—22no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42C2411yescharacteristic
2442−1+4—44no—
2452−1+4—44no—
246C2×#41—22no—
247C2×#41—24no—
248C2×#41—22no—
249C2×#41—21no—
250C2×#41C3×D411yescharacteristic
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—21no—
257AGL(1,ℤ/8)—44no—
258C2×SD16—22no—
259C2×D8—22no—
260C2×D8—24no—
261C2×M16—21no—
262C22×C8—21no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8C3×D411yescharacteristic
269C4⋊Q8C3×D412yes—
270C4⋊Q8C23×C312yes—
271#127—21no—
272#127—21no—
273#127—21no—
274#127C3×D411yescharacteristic
275#127C23×C312yes—
276#124—22no—
277#124—21no—
278#123—24no—
279#123—24no—
280#123—24no—
281C22⋊Q8—24no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284C22⋊Q8—22no—
285#121—22no—
286#121—22no—
287#121—21no—
288#121C3×D412yes—
289C4×Q8C3×D411yescharacteristic
290C4×Q8C23×C311yescharacteristic
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4C3×D411yescharacteristic
297C4×D4C23×C311yescharacteristic
298C4×D4C23×C312yes—
299#117—21no—
300#117C3×D411yescharacteristic
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C3×D412yes—
305C2×C42—21no—
306C2×C42C23×C311yescharacteristic
307C4⋊C8—21no—
308C4⋊C8C23×C312yes—
309C4⋊C8C23×C312yes—
310#102—28no—
311#102—22no—
312#102—22no—
313#102—24no—
314C22⋊C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8C23×C312yes—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—42no—
321C23×C3—82no—
322C23×C3—82no—
323C23×C3—82no—
324C23×C3#12011yescharacteristic
325C3×Q8—21no—
326C3×Q8—48no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—44no—
331C3×Q8C2×D812yes—
332C3×D4—24no—
333C3×D4—21no—
334C3×D4—24no—
335C3×D4—44no—
336C3×D4—48no—
337C3×D4—44no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—44no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—82no—
344C3×D4—82no—
345C3×D4—82no—
346C3×D4C2×D812yes—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—48no—
356C2×C3×C4—48no—
357C2×C3×C4—44no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4#12011yescharacteristic
363C2×C3×C4#12011yescharacteristic
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4C2×D812yes—
366C2×C3×C4AGL(1,ℤ/8)12yes—
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C3×C8—42no—
371C3×C8—42no—
372C3×C8—42no—
373C24—41no—
374#41—22no—
375#41—48no—
376#41—416no—
377#41—44no—
378#41—44no—
379#41—42no—
380#41—42no—
381#41C3×D814yescomplemented
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—48no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8C2×C3×D411yescharacteristic
392C2×D4—21no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—44no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—82no—
407C2×D4C2×C3×D412yes—
408C2×D4C2×C3×D411yescharacteristic
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419SD16—44no—
420D8—44no—
421D8—84no—
422M16—42no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—42no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4C2×C3×D412yes—
436C4⋊C4C2×C3×D414yes—
437C4⋊C4C2×C3×D411yescharacteristic
438C4⋊C4C2×C3×D411yescharacteristic
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—21no—
445C22⋊C4—22no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42C2×C3×D411yescharacteristic
450C42C2×C3×D411yescharacteristic
451C42C2×C3×D411yescharacteristic
452C42C24×C311yescharacteristic
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—22no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3—164no—
465C22×C3#44712yes—
466C22×C3C2×#12011yescharacteristic
467C3×C4—22no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—44no—
471C3×C4—42no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4#44712yes—
475C3×C4#45412yes—
476C3×C4C22×D811yescharacteristic
477C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
478C3×C4#57712yes—
479C23—21no—
480C23—22no—
481C23—42no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23C3×#12011yescharacteristic
486Q8—21no—
487Q8—48no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—44no—
492Q8C2×C3×D812yes—
493D4—24no—
494D4—21no—
495D4—24no—
496D4—44no—
497D4—48no—
498D4—44no—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—42no—
503D4—42no—
504D4—82no—
505D4—82no—
506D4—82no—
507D4C2×C3×D812yes—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4C3×#12011yescharacteristic
524C2×C4C3×#12011yescharacteristic
525C2×C4C3×#12011yescharacteristic
526C2×C4C2×C3×D812yes—
527C2×C4C3×AGL(1,ℤ/8)12yes—
528C2×C4C22×C3×D411yescharacteristic
529C2×C4C22×C3×D411yescharacteristic
530C2×C4C22×C3×D411yescharacteristic
531C8—42no—
532C8—42no—
533C8—42no—
534C2×C3—22no—
535C2×C3—42no—
536C2×C3—82no—
537C2×C3—162no—
538C2×C3C2×#44711yescharacteristic
539C2×C3#301011yescharacteristic
540C2×C3C2×#45411yescharacteristic
541C22—21no—
542C22—21no—
543C22—22no—
544C22—41no—
545C22—41no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—164no—
553C22C3×#44712yes—
554C22C2×C3×#12011yescharacteristic
555C4—22no—
556C4—21no—
557C4—21no—
558C4—44no—
559C4—42no—
560C4—41no—
561C4—41no—
562C4C3×#44712yes—
563C4C3×#45412yes—
564C4C22×C3×D811yescharacteristic
565C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
566C4C3×#57712yes—
567C3#1864011yescharacteristic, direct product factor, Sylow
568C2—22no—
569C2—42no—
570C2—82no—
571C2—162no—
572C2C2×C3×#44711yescharacteristic
573C2C3×#301011yescharacteristic
574C2C2×C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3499C2semidirect
2. C3×#3205C2semidirect
3. C2×C3×#486C2semidirect
4. C3×#3038C2semidirect
5. C3×#1627C2semidirect
6. C3×#1625C2semidirect
7. C3×#1546C2semidirect
8. C3×#1492C2semidirect
9. C3×#1476C2semidirect
10. #18640C3direct (N × H)
14. C3×D4×Q8C22semidirect
18. C3×#547C22semidirect
42. C3×#332C22semidirect
43. C3×Q8⋊C8C22semidirect
44. C3×D4⋊C8C22semidirect
45. C3×D4⋊C8C22semidirect
46. #3499C2×C3semidirect
47. #3205C2×C3semidirect
48. C2×#486C2×C3semidirect
49. #3038C2×C3semidirect
50. #1627C2×C3semidirect
51. #1625C2×C3semidirect
52. #1546C2×C3semidirect
53. #1492C2×C3semidirect
54. #1476C2×C3semidirect
132. D4×Q8C22×C3semidirect
136. #547C22×C3semidirect
160. #332C22×C3semidirect
161. Q8⋊C8C22×C3semidirect
162. D4⋊C8C22×C3semidirect
163. D4⋊C8C22×C3semidirect
172. C3×#41D8semidirect
381. #41C3×D8semidirect
567. C3#18640direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34992
C3×#32052
C2×C3×#4862
C3×#30382
C3×#16272
C3×#16252
C3×#15462
C3×#14922
C3×#14762
#186403