Finite Group Databasebeta

← C3×#61664 #10680271

Subgroups — C3×#61664 #10680271

Order 768.

2136 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3549C211yesmaximal, characteristic, complemented
2C3×#3499C211yesmaximal, characteristic, complemented
3C3×#3329C214yesmaximal, complemented
4C3×#3303C214yesmaximal, complemented
5C3×D4×SD16C214yesmaximal, complemented
6C3×#3272C214yesmaximal, complemented
7C3×#3206C212yesmaximal, complemented
8C3×#3200C214yesmaximal, complemented
9C3×#3106C212yesmaximal, complemented
10C3×#3010C214yesmaximal, complemented
11C3×#2979C211yesmaximal, characteristic, complemented
12#61664C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4C2212yescomplemented
14C2×C3×AGL(1,ℤ/8)C2214yescomplemented
15C22×C3×SD16C2214yescomplemented
16C3×H(3,𝔽4)—22no—
17C3×#555—21no—
18C3×#555C2214yes—
19C3×D4×Q8C2214yescomplemented
20C3×#547—24no—
21C3×#547C2214yescomplemented
22C3×D42—21no—
23C3×D42C2214yes—
24C3×#536—22no—
25C3×#536C2212yescomplemented
26C3×#535—24no—
27C3×#535C2214yes—
28C3×#532—21no—
29C3×#532C2212yes—
30C3×#532C2211yescharacteristic
31C2×C3×C4⋊Q8—21no—
32C2×C3×C4⋊Q8C2211yescharacteristic, complemented
33C3×C4×#41C2211yescharacteristic, complemented
34C3×#492C2214yes—
35C3×#487C2214yescomplemented
36C3×#485C2218yescomplemented
37C3×#482C2214yescomplemented
38C3×#477C2212yescomplemented
39C3×#476C2212yescomplemented
40C3×#468C2218yescomplemented
41C3×#465C2218yescomplemented
42C3×#461C2214yescomplemented
43C3×#461C2214yescomplemented
44C3×#460C2212yes—
45C3×#459C2212yes—
46C3×#450C2218yescomplemented
47C3×#449C2214yescomplemented
48C3×#449C2218yescomplemented
49C3×#448C2218yescomplemented
50C3×#448C2218yescomplemented
51C3×C4×SD16C2214yescomplemented
52C3×C4×D8C2214yescomplemented
53C3×#436C2214yescomplemented
54C3×C8×D4C2214yescomplemented
55C3×#431C2212yescomplemented
56C3×#419C2214yescomplemented
57C3×#416C2214yescomplemented
58C3×#408C2214yescomplemented
59#3549C2×C311yescharacteristic, complemented
60#3499C2×C311yescharacteristic, complemented
61#3329C2×C314yescomplemented
62#3303C2×C314yescomplemented
63D4×SD16C2×C314yescomplemented
64#3272C2×C314yescomplemented
65#3206C2×C312yescomplemented
66#3200C2×C314yescomplemented
67#3106C2×C312yescomplemented
68#3010C2×C314yescomplemented
69#2979C2×C311yescharacteristic, complemented
70C3×2−1+4—44no—
71C3×2−1+4—44no—
72C2×C3×#41—22no—
73C2×C3×#41—24no—
74C2×C3×#41—22no—
75C2×C3×#41C2311yescharacteristic
76C2×C3×#41C2314yes—
77C2×C3×#41C2312yes—
78C22×C3×Q8—22no—
79C22×C3×Q8C2314yes—
80C22×C3×Q8C2312yes—
81C22×C3×D4—24no—
82C22×C3×D4C2314yes—
83C3×AGL(1,ℤ/8)—48no—
84C2×C3×SD16—48no—
85C2×C3×SD16C2314yes—
86C2×C3×SD16C2318yes—
87C2×C3×SD16C2318yes—
88C2×C3×D8C2314yes—
89C2×C3×M16C2314yes—
90C22×C3×C8C2314yes—
91C3×C4⋊Q8—22no—
92C3×C4⋊Q8—24no—
93C3×C4⋊Q8—21no—
94C3×C4⋊Q8—21no—
95C3×C4⋊Q8D411yescharacteristic
96C3×C4⋊Q8C2314yes—
97C3×C4⋊Q8C2312yes—
98C3×#127—24no—
99C3×#127D411yescharacteristic
100C3×#127C2312yes—
101C3×#124—24no—
102C3×#124—22no—
103C3×#124—28no—
104C3×#124—22no—
105C3×#124D412yes—
106C3×#123—24no—
107C3×#123—28no—
108C3×C22⋊Q8—28no—
109C3×C22⋊Q8C2318yes—
110C3×#121—24no—
111C3×#121—24no—
112C3×#121—22no—
113C3×#121—24no—
114C3×#121D414yes—
115C3×#121C2314yes—
116C3×#121C2318yes—
117C3×#120—24no—
118C3×#120—24no—
119C3×#120—28no—
120C3×C4×Q8C2312yes—
121C3×C4×D4—24no—
122C3×C4×D4—22no—
123C3×C4×D4—22no—
124C3×C4×D4C2312yes—
125C3×C4×D4C2314yes—
126C3×C4×D4C2314yes—
127C3×#117—24no—
128C3×#117—21no—
129C3×#117C2314yes—
130C3×#117C2311yescharacteristic
131C2×C3×C4⋊C4—22no—
132C2×C3×C4⋊C4—24no—
133C2×C3×C42—21no—
134C2×C3×C42C2311yescharacteristic
135C3×#107C2314yes—
136C3×#106C2314yes—
137C3×C4⋊C8C2312yes—
138C3×C4⋊C8C2312yes—
139C3×#103C2318yes—
140C3×#103C2314yes—
141C3×#103C2318yes—
142C3×#102C2314yes—
143C3×#102C2314yes—
144C3×#102C2314yes—
145C3×C22⋊C8C2314yes—
146C3×C22⋊C8C2318yes—
147C3×C4×C8C2314yes—
148C2×2−1+4C22×C312yescomplemented
149C2×AGL(1,ℤ/8)C22×C314yescomplemented
150C22×SD16C22×C314yescomplemented
151H(3,𝔽4)—22no—
152#555—21no—
153#555C22×C314yes—
154D4×Q8C22×C314yescomplemented
155#547—24no—
156#547C22×C314yescomplemented
157D42—21no—
158D42C22×C314yes—
159#536—22no—
160#536C22×C312yescomplemented
161#535—24no—
162#535C22×C314yes—
163#532—21no—
164#532C22×C312yes—
165#532C22×C311yescharacteristic
166C2×C4⋊Q8—21no—
167C2×C4⋊Q8C22×C311yescharacteristic, complemented
168C4×#41C22×C311yescharacteristic, complemented
169#492C22×C314yes—
170#487C22×C314yescomplemented
171#485C22×C318yescomplemented
172#482C22×C314yescomplemented
173#477C22×C312yescomplemented
174#476C22×C312yescomplemented
175#468C22×C318yescomplemented
176#465C22×C318yescomplemented
177#461C22×C314yescomplemented
178#461C22×C314yescomplemented
179#460C22×C312yes—
180#459C22×C312yes—
181#450C22×C318yescomplemented
182#449C22×C314yescomplemented
183#449C22×C318yescomplemented
184#448C22×C318yescomplemented
185#448C22×C318yescomplemented
186C4×SD16C22×C314yescomplemented
187C4×D8C22×C314yescomplemented
188#436C22×C314yescomplemented
189C8×D4C22×C314yescomplemented
190#431C22×C312yescomplemented
191#419C22×C314yescomplemented
192#416C22×C314yescomplemented
193#408C22×C314yescomplemented
194C24×C3—24no—
195C3×#41—22no—
196C3×#41—22no—
197C3×#41—28no—
198C3×#41—42no—
199C3×#41—42no—
200C3×#41—44no—
201C3×#41—48no—
202C3×#41—416no—
203C3×#41—84no—
204C2×C3×Q8—24no—
205C2×C3×Q8—22no—
206C2×C3×Q8—22no—
207C2×C3×Q8—28no—
208C2×C3×Q8—44no—
209C2×C3×Q8—42no—
210C2×C3×Q8—42no—
211C2×C3×Q8—48no—
212C2×C3×Q8C2×D412yes—
213C2×C3×Q8C2414yes—
214C2×C3×Q8C2411yescharacteristic
215C2×C3×D4—21no—
216C2×C3×D4—24no—
217C2×C3×D4—28no—
218C2×C3×D4—24no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—24no—
222C2×C3×D4—44no—
223C2×C3×D4—44no—
224C2×C3×D4—88no—
225C2×C3×D4C2×D414yes—
226C2×C3×D4C2×D412yes—
227C2×C3×D4C2×D414yes—
228C2×C3×D4C2414yes—
229C2×C3×D4C2411yescharacteristic
230C22×C3×C4—21no—
231C22×C3×C4—22no—
232C22×C3×C4—24no—
233C22×C3×C4—22no—
234C22×C3×C4—24no—
235C22×C3×C4—24no—
236C22×C3×C4C2×D412yes—
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2414yes—
239C3×SD16—28no—
240C3×SD16—48no—
241C3×SD16—48no—
242C3×D8—28no—
243C3×M16—44no—
244C2×C3×C8—44no—
245C2×C3×C8C2414yes—
246C2×C3×C8C2414yes—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—28no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4C2×D414yes—
253C3×C4⋊C4C2412yes—
254C3×C4⋊C4C2414yes—
255C3×C22⋊C4—21no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—28no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—28no—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—24no—
262C3×C22⋊C4C2×D418yes—
263C3×C42—21no—
264C3×C42—22no—
265C3×C42—24no—
266C3×C42C2×D411yescharacteristic
267C3×C42C2×D411yescharacteristic
268C3×C42C2412yes—
2692−1+4—44no—
2702−1+4—44no—
271C2×#41—22no—
272C2×#41—24no—
273C2×#41—22no—
274C2×#41C23×C311yescharacteristic
275C2×#41C23×C314yes—
276C2×#41C23×C312yes—
277C22×Q8—22no—
278C22×Q8C23×C314yes—
279C22×Q8C23×C312yes—
280C22×D4—24no—
281C22×D4C23×C314yes—
282AGL(1,ℤ/8)—48no—
283C2×SD16—48no—
284C2×SD16C23×C314yes—
285C2×SD16C23×C318yes—
286C2×SD16C23×C318yes—
287C2×D8C23×C314yes—
288C2×M16C23×C314yes—
289C22×C8C23×C314yes—
290C4⋊Q8—22no—
291C4⋊Q8—24no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C3×D411yescharacteristic
295C4⋊Q8C23×C314yes—
296C4⋊Q8C23×C312yes—
297#127—24no—
298#127C3×D411yescharacteristic
299#127C23×C312yes—
300#124—24no—
301#124—22no—
302#124—28no—
303#124—22no—
304#124C3×D412yes—
305#123—24no—
306#123—28no—
307C22⋊Q8—28no—
308C22⋊Q8C23×C318yes—
309#121—24no—
310#121—24no—
311#121—22no—
312#121—24no—
313#121C3×D414yes—
314#121C23×C314yes—
315#121C23×C318yes—
316#120—24no—
317#120—24no—
318#120—28no—
319C4×Q8C23×C312yes—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4C23×C312yes—
324C4×D4C23×C314yes—
325C4×D4C23×C314yes—
326#117—24no—
327#117—21no—
328#117C23×C311yescharacteristic
329#117C23×C314yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C42—21no—
333C2×C42C23×C311yescharacteristic
334#107C23×C314yes—
335#106C23×C314yes—
336C4⋊C8C23×C312yes—
337C4⋊C8C23×C312yes—
338#103C23×C318yes—
339#103C23×C314yes—
340#103C23×C318yes—
341#102C23×C314yes—
342#102C23×C314yes—
343#102C23×C314yes—
344C22⋊C8C23×C314yes—
345C22⋊C8C23×C318yes—
346C4×C8C23×C314yes—
347C23×C3—21no—
348C23×C3—24no—
349C23×C3—84no—
350C23×C3—88no—
351C23×C3C22×D414yes—
352C23×C3C22×D411yescharacteristic
353C3×Q8—21no—
354C3×Q8—24no—
355C3×Q8—24no—
356C3×Q8—42no—
357C3×Q8—48no—
358C3×Q8—44no—
359C3×Q8—82no—
360C3×Q8H(5,𝔽2)12yes—
361C3×D4—24no—
362C3×D4—28no—
363C3×D4—21no—
364C3×D4—24no—
365C3×D4—24no—
366C3×D4—42no—
367C3×D4—44no—
368C3×D4—48no—
369C3×D4—44no—
370C3×D4—84no—
371C3×D4—82no—
372C3×D4—88no—
373C3×D4—84no—
374C3×D4H(5,𝔽2)12yes—
375C2×C3×C4—24no—
376C2×C3×C4—21no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—24no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—24no—
385C2×C3×C4—48no—
386C2×C3×C4—48no—
387C2×C3×C4—44no—
388C2×C3×C4—44no—
389C2×C3×C4—48no—
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C22×D414yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4C22×D412yes—
394C2×C3×C4C2511yescharacteristic
395C3×C8—24no—
396C3×C8—24no—
397C24—24no—
398#41—22no—
399#41—22no—
400#41—28no—
401#41—42no—
402#41—42no—
403#41—44no—
404#41—48no—
405#41—416no—
406#41—84no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—28no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—48no—
415C2×Q8C2×C3×D412yes—
416C2×Q8C24×C314yes—
417C2×Q8C24×C311yescharacteristic
418C2×D4—21no—
419C2×D4—24no—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—88no—
428C2×D4C2×C3×D414yes—
429C2×D4C2×C3×D412yes—
430C2×D4C2×C3×D414yes—
431C2×D4C24×C314yes—
432C2×D4C24×C311yescharacteristic
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4C2×C3×D412yes—
440C22×C4C24×C311yescharacteristic
441C22×C4C24×C314yes—
442SD16—28no—
443SD16—48no—
444SD16—48no—
445D8—28no—
446M16—44no—
447C2×C8—44no—
448C2×C8C24×C314yes—
449C2×C8C24×C314yes—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—28no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4C2×C3×D414yes—
456C4⋊C4C24×C312yes—
457C4⋊C4C24×C314yes—
458C22⋊C4—21no—
459C22⋊C4—24no—
460C22⋊C4—28no—
461C22⋊C4—22no—
462C22⋊C4—28no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4C2×C3×D418yes—
466C42—21no—
467C42—22no—
468C42—24no—
469C42C2×C3×D411yescharacteristic
470C42C2×C3×D411yescharacteristic
471C42C24×C312yes—
472C22×C3—21no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—22no—
476C22×C3—28no—
477C22×C3—42no—
478C22×C3—44no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—168no—
482C22×C3C23×D411yescharacteristic
483C3×C4—21no—
484C3×C4—22no—
485C3×C4—24no—
486C3×C4—22no—
487C3×C4—44no—
488C3×C4—42no—
489C3×C4—44no—
490C3×C4#57714yes—
491C3×C4C2×H(5,𝔽2)11yescharacteristic
492C3×C4C2×H(5,𝔽2)11yescharacteristic
493C23—21no—
494C23—24no—
495C23—84no—
496C23—88no—
497C23C22×C3×D414yes—
498C23C22×C3×D411yescharacteristic
499Q8—21no—
500Q8—24no—
501Q8—24no—
502Q8—42no—
503Q8—48no—
504Q8—44no—
505Q8—82no—
506Q8C3×H(5,𝔽2)12yes—
507D4—24no—
508D4—28no—
509D4—21no—
510D4—24no—
511D4—24no—
512D4—42no—
513D4—44no—
514D4—48no—
515D4—44no—
516D4—84no—
517D4—82no—
518D4—88no—
519D4—84no—
520D4C3×H(5,𝔽2)12yes—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4C22×C3×D411yescharacteristic
537C2×C4C22×C3×D414yes—
538C2×C4C22×C3×D412yes—
539C2×C4C22×C3×D412yes—
540C2×C4C25×C311yescharacteristic
541C8—24no—
542C8—24no—
543C2×C3—41no—
544C2×C3—44no—
545C2×C3—42no—
546C2×C3—84no—
547C2×C3#348411yescharacteristic
548C2×C3C2×#57712yes—
549C22—21no—
550C22—22no—
551C22—24no—
552C22—22no—
553C22—28no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—168no—
559C22C23×C3×D411yescharacteristic
560C4—21no—
561C4—22no—
562C4—24no—
563C4—22no—
564C4—44no—
565C4—44no—
566C4—42no—
567C4C3×#57714yes—
568C4C2×C3×H(5,𝔽2)11yescharacteristic
569C4C2×C3×H(5,𝔽2)11yescharacteristic
570C3#6166411yescharacteristic, direct product factor, Sylow
571C2—44no—
572C2—41no—
573C2—42no—
574C2—84no—
575C2C3×#348411yescharacteristic
576C2C2×C3×#57712yes—
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×#532 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3549C2semidirect
2. C3×#3499C2semidirect
3. C3×#3329C2semidirect
4. C3×#3303C2semidirect
5. C3×D4×SD16C2semidirect
6. C3×#3272C2semidirect
7. C3×#3206C2semidirect
8. C3×#3200C2semidirect
9. C3×#3106C2semidirect
10. C3×#3010C2semidirect
11. C3×#2979C2semidirect
12. #61664C3direct (N × H)
13. C2×C3×2−1+4C22semidirect
14. C2×C3×AGL(1,ℤ/8)C22semidirect
15. C22×C3×SD16C22semidirect
19. C3×D4×Q8C22semidirect
21. C3×#547C22semidirect
25. C3×#536C22semidirect
32. C2×C3×C4⋊Q8C22semidirect
33. C3×C4×#41C22semidirect
35. C3×#487C22semidirect
36. C3×#485C22semidirect
37. C3×#482C22semidirect
38. C3×#477C22semidirect
39. C3×#476C22semidirect
40. C3×#468C22semidirect
41. C3×#465C22semidirect
42. C3×#461C22semidirect
43. C3×#461C22semidirect
46. C3×#450C22semidirect
47. C3×#449C22semidirect
48. C3×#449C22semidirect
49. C3×#448C22semidirect
50. C3×#448C22semidirect
51. C3×C4×SD16C22semidirect
52. C3×C4×D8C22semidirect
53. C3×#436C22semidirect
54. C3×C8×D4C22semidirect
55. C3×#431C22semidirect
56. C3×#419C22semidirect
57. C3×#416C22semidirect
58. C3×#408C22semidirect
59. #3549C2×C3semidirect
60. #3499C2×C3semidirect
61. #3329C2×C3semidirect
62. #3303C2×C3semidirect
63. D4×SD16C2×C3semidirect
64. #3272C2×C3semidirect
65. #3206C2×C3semidirect
66. #3200C2×C3semidirect
67. #3106C2×C3semidirect
68. #3010C2×C3semidirect
69. #2979C2×C3semidirect
148. C2×2−1+4C22×C3semidirect
149. C2×AGL(1,ℤ/8)C22×C3semidirect
150. C22×SD16C22×C3semidirect
154. D4×Q8C22×C3semidirect
156. #547C22×C3semidirect
160. #536C22×C3semidirect
167. C2×C4⋊Q8C22×C3semidirect
168. C4×#41C22×C3semidirect
170. #487C22×C3semidirect
171. #485C22×C3semidirect
172. #482C22×C3semidirect
173. #477C22×C3semidirect
174. #476C22×C3semidirect
175. #468C22×C3semidirect
176. #465C22×C3semidirect
177. #461C22×C3semidirect
178. #461C22×C3semidirect
181. #450C22×C3semidirect
182. #449C22×C3semidirect
183. #449C22×C3semidirect
184. #448C22×C3semidirect
185. #448C22×C3semidirect
186. C4×SD16C22×C3semidirect
187. C4×D8C22×C3semidirect
188. #436C22×C3semidirect
189. C8×D4C22×C3semidirect
190. #431C22×C3semidirect
191. #419C22×C3semidirect
192. #416C22×C3semidirect
193. #408C22×C3semidirect
570. C3#61664direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35492
C3×#34992
C3×#33292
C3×#33032
C3×D4×SD162
C3×#32722
C3×#32062
C3×#32002
C3×#31062
C3×#30102
C3×#29792
#616643