Finite Group Databasebeta

← C3×#18675 #10637282

Subgroups — C3×#18675 #10637282

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3497C211yesmaximal, characteristic
2C3×#3246C211yesmaximal, characteristic, complemented
3C3×#3167C211yesmaximal, characteristic, complemented
4C3×#3078C211yesmaximal, characteristic, complemented
5C3×#1651C212yesmaximal, complemented
6C3×#1642C212yesmaximal, complemented
7C3×#1637C212yesmaximal, complemented
8C3×#1631C212yesmaximal, complemented
9C3×#1547C211yesmaximal, characteristic, complemented
10C3×#1492C211yesmaximal, characteristic, complemented
11C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
12#18675C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4—21no—
14C3×#550—21no—
15C3×#550C2212yes—
16C3×D4×Q8—21no—
17C3×D4×Q8C2212yes—
18C3×#548—22no—
19C3×#548—22no—
20C3×#537—22no—
21C3×#537—21no—
22C3×#533C2211yescharacteristic
23C3×#532—22no—
24C3×#532C2211yescharacteristic
25C3×#532C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic
27C3×#498C2211yescharacteristic
28C3×#492C2211yescharacteristic
29C3×#487C2211yescharacteristic
30C3×#486C2211yescharacteristic
31C3×#479—22no—
32C3×#477C2212yes—
33C3×#476C2212yes—
34C3×#475—21no—
35C3×#475C2212yes—
36C3×#465—22no—
37C3×#465—22no—
38C3×#461C2212yes—
39C3×#460—21no—
40C3×C4×SD16—22no—
41C3×#428—21no—
42C3×#423C2211yescharacteristic
43C2×C3×C4⋊C8C2211yescharacteristic
44C3×#419—21no—
45C2×C3×C4×C8C2211yescharacteristic
46C3×#332C2212yes—
47C3×#331C2212yes—
48C3×Q8⋊C8C2212yes—
49C3×Q8⋊C8C2212yes—
50C3×D4⋊C8C2212yes—
51C3×D4⋊C8C2212yes—
52#3497C2×C311yescharacteristic
53#3246C2×C311yescharacteristic, complemented
54#3167C2×C311yescharacteristic, complemented
55#3078C2×C311yescharacteristic, complemented
56#1651C2×C312yescomplemented
57#1642C2×C312yescomplemented
58#1637C2×C312yescomplemented
59#1631C2×C312yescomplemented
60#1547C2×C311yescharacteristic, complemented
61#1492C2×C311yescharacteristic, complemented
62C2×Q8⋊C8C2×C311yescharacteristic, complemented
63C3×2−1+4—44no—
64C3×2−1+4—44no—
65C2×C3×#41—24no—
66C2×C3×#41—22no—
67C2×C3×#41—22no—
68C2×C3×#41—21no—
69C2×C3×#41—21no—
70C22×C3×Q8—22no—
71C22×C3×Q8—22no—
72C22×C3×Q8D411yescharacteristic
73C2×C3×SD16—22no—
74C2×C3×SD16—22no—
75C2×C3×M16—21no—
76C22×C3×C8—21no—
77C22×C3×C8—21no—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8D411yescharacteristic
80C3×C4⋊Q8C2311yescharacteristic
81C3×C4⋊Q8C2311yescharacteristic
82C3×C4⋊Q8C2311yescharacteristic
83C3×#127—22no—
84C3×#127D411yescharacteristic
85C3×#127C2311yescharacteristic
86C3×#125—21no—
87C3×#124—24no—
88C3×#124—21no—
89C3×#124—21no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—24no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—24no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—21no—
96C3×#121—24no—
97C3×#121—22no—
98C3×#121—24no—
99C3×#121—24no—
100C3×#121—21no—
101C3×#121—21no—
102C3×#121—22no—
103C3×#121D412yes—
104C3×#121D412yes—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8—21no—
109C3×C4×Q8D412yes—
110C3×C4×Q8C2312yes—
111C3×C4×D4—24no—
112C3×C4×D4—22no—
113C3×C4×D4—22no—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×C4×D4D412yes—
117C3×C4×D4C2312yes—
118C3×#117—21no—
119C2×C3×C4⋊C4—21no—
120C2×C3×C4⋊C4D411yescharacteristic
121C2×C3×C42—21no—
122C2×C3×C42C2311yescharacteristic
123C3×#107—22no—
124C3×#106—22no—
125C3×#106—22no—
126C3×#106—22no—
127C3×C4⋊C8—21no—
128C3×C4⋊C8—21no—
129C3×C4⋊C8C2312yes—
130C3×C4⋊C8C2312yes—
131C3×#103—22no—
132C3×#103—22no—
133C3×#103—22no—
134C3×#103—22no—
135C3×#102—22no—
136C3×#102—22no—
137C3×C4×C8—21no—
138C3×C4×C8C2311yescharacteristic
139C3×C4×C8C2311yescharacteristic
140C2×2−1+4—21no—
141#550—21no—
142#550C22×C312yes—
143D4×Q8—21no—
144D4×Q8C22×C312yes—
145#548—22no—
146#548—22no—
147#537—22no—
148#537—21no—
149#533C22×C311yescharacteristic
150#532—22no—
151#532C22×C311yescharacteristic
152#532C22×C311yescharacteristic
153C2×C4×Q8C22×C311yescharacteristic
154#498C22×C311yescharacteristic
155#492C22×C311yescharacteristic
156#487C22×C311yescharacteristic
157#486C22×C311yescharacteristic
158#479—22no—
159#477C22×C312yes—
160#476C22×C312yes—
161#475—21no—
162#475C22×C312yes—
163#465—22no—
164#465—22no—
165#461C22×C312yes—
166#460—21no—
167C4×SD16—22no—
168#428—21no—
169#423C22×C311yescharacteristic
170C2×C4⋊C8C22×C311yescharacteristic
171#419—21no—
172C2×C4×C8C22×C311yescharacteristic
173#332C22×C312yes—
174#331C22×C312yes—
175Q8⋊C8C22×C312yes—
176Q8⋊C8C22×C312yes—
177D4⋊C8C22×C312yes—
178D4⋊C8C22×C312yes—
179C3×#41—416no—
180C3×#41—44no—
181C3×#41—48no—
182C3×#41—44no—
183C3×#41—42no—
184C3×#41—42no—
185C3×#41—42no—
186C3×#41—42no—
187C2×C3×Q8—24no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—22no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—44no—
195C2×C3×Q8—48no—
196C2×C3×Q8—44no—
197C2×C3×Q8SD1614yescomplemented
198C2×C3×Q8C2×D412yes—
199C2×C3×D4—24no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4C2×D412yes—
208C2×C3×D4C2×D412yes—
209C22×C3×C4—21no—
210C22×C3×C4—24no—
211C22×C3×C4—24no—
212C22×C3×C4—21no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C3×SD16—44no—
219C3×SD16—82no—
220C3×M16—42no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—42no—
226C2×C3×C8—81no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—41no—
239C3×C4⋊C4C2×D412yes—
240C3×C4⋊C4C2×D412yes—
241C3×C4⋊C4C2×D412yes—
242C3×C22⋊C4—24no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—41no—
249C3×C42—22no—
250C3×C42—22no—
251C3×C42—41no—
252C3×C42C2×D411yescharacteristic
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42C2411yescharacteristic
2562−1+4—44no—
2572−1+4—44no—
258C2×#41—24no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—21no—
262C2×#41—21no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8C3×D411yescharacteristic
266C2×SD16—22no—
267C2×SD16—22no—
268C2×M16—21no—
269C22×C8—21no—
270C22×C8—21no—
271C4⋊Q8—22no—
272C4⋊Q8C3×D411yescharacteristic
273C4⋊Q8C23×C311yescharacteristic
274C4⋊Q8C23×C311yescharacteristic
275C4⋊Q8C23×C311yescharacteristic
276#127—22no—
277#127C3×D411yescharacteristic
278#127C23×C311yescharacteristic
279#125—21no—
280#124—24no—
281#124—21no—
282#124—21no—
283C22⋊Q8—22no—
284C22⋊Q8—24no—
285C22⋊Q8—22no—
286C22⋊Q8—24no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289#121—24no—
290#121—22no—
291#121—24no—
292#121—24no—
293#121—21no—
294#121—21no—
295#121—22no—
296#121C3×D412yes—
297#121C3×D412yes—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8C3×D412yes—
303C4×Q8C23×C312yes—
304C4×D4—24no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4C3×D412yes—
310C4×D4C23×C312yes—
311#117—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4C3×D411yescharacteristic
314C2×C42—21no—
315C2×C42C23×C311yescharacteristic
316#107—22no—
317#106—22no—
318#106—22no—
319#106—22no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8C23×C312yes—
323C4⋊C8C23×C312yes—
324#103—22no—
325#103—22no—
326#103—22no—
327#103—22no—
328#102—22no—
329#102—22no—
330C4×C8—21no—
331C4×C8C23×C311yescharacteristic
332C4×C8C23×C311yescharacteristic
333C23×C3—22no—
334C23×C3—22no—
335C23×C3#12011yescharacteristic
336C3×Q8—22no—
337C3×Q8—24no—
338C3×Q8—48no—
339C3×Q8—42no—
340C3×Q8—41no—
341C3×Q8—41no—
342C3×Q8—42no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×Q8C2×SD1614yes—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—42no—
349C3×D4—48no—
350C3×D4—44no—
351C3×D4—44no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—41no—
356C3×D4—42no—
357C3×D4—81no—
358C3×D4—84no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—41no—
367C2×C3×C4—48no—
368C2×C3×C4—48no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—44no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—81no—
375C2×C3×C4#12011yescharacteristic
376C2×C3×C4#12011yescharacteristic
377C2×C3×C4#12011yescharacteristic
378C2×C3×C4C2×SD1612yes—
379C2×C3×C4AGL(1,ℤ/8)12yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C3×C8—22no—
384C3×C8—41no—
385C3×C8—42no—
386C3×C8—42no—
387#41—416no—
388#41—44no—
389#41—48no—
390#41—44no—
391#41—42no—
392#41—42no—
393#41—42no—
394#41—42no—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—44no—
403C2×Q8—48no—
404C2×Q8—44no—
405C2×Q8C3×SD1614yescomplemented
406C2×Q8C2×C3×D412yes—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4C2×C3×D412yes—
416C2×D4C2×C3×D412yes—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426SD16—44no—
427SD16—82no—
428M16—42no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—42no—
434C2×C8—81no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4C2×C3×D412yes—
448C4⋊C4C2×C3×D412yes—
449C4⋊C4C2×C3×D412yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—41no—
457C42—22no—
458C42—22no—
459C42—41no—
460C42C2×C3×D411yescharacteristic
461C42C2×C3×D411yescharacteristic
462C42C2×C3×D411yescharacteristic
463C42C24×C311yescharacteristic
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3#45012yes—
473C22×C3C2×#12011yescharacteristic
474C3×C4—24no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4#45012yes—
484C3×C4C22×SD1611yescharacteristic
485C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
486C3×C4#57711yescharacteristic
487C3×C4#57811yescharacteristic
488C23—22no—
489C23—22no—
490C23C3×#12011yescharacteristic
491Q8—22no—
492Q8—24no—
493Q8—48no—
494Q8—42no—
495Q8—41no—
496Q8—41no—
497Q8—42no—
498Q8—41no—
499Q8—41no—
500Q8C2×C3×SD1614yes—
501D4—22no—
502D4—22no—
503D4—42no—
504D4—48no—
505D4—44no—
506D4—44no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—42no—
512D4—81no—
513D4—84no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—81no—
530C2×C4C3×#12011yescharacteristic
531C2×C4C3×#12011yescharacteristic
532C2×C4C3×#12011yescharacteristic
533C2×C4C2×C3×SD1612yes—
534C2×C4C3×AGL(1,ℤ/8)12yes—
535C2×C4C22×C3×D411yescharacteristic
536C2×C4C22×C3×D411yescharacteristic
537C2×C4C22×C3×D411yescharacteristic
538C8—22no—
539C8—41no—
540C8—42no—
541C8—42no—
542C2×C3—22no—
543C2×C3—82no—
544C2×C3—82no—
545C2×C3C2×#45011yescharacteristic
546C2×C3#301311yescharacteristic
547C2×C3#301811yescharacteristic
548C22—21no—
549C22—21no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22C3×#45012yes—
557C22C2×C3×#12011yescharacteristic
558C4—24no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—81no—
567C4C3×#45012yes—
568C4C22×C3×SD1611yescharacteristic
569C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
570C4C3×#57711yescharacteristic
571C4C3×#57811yescharacteristic
572C3#1867511yescharacteristic, direct product factor, Sylow
573C2—22no—
574C2—82no—
575C2—82no—
576C2C2×C3×#45011yescharacteristic
577C2C3×#301311yescharacteristic
578C2C3×#301811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3497C2
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
2. C3×#3246C2semidirect
3. C3×#3167C2semidirect
4. C3×#3078C2semidirect
5. C3×#1651C2semidirect
6. C3×#1642C2semidirect
7. C3×#1637C2semidirect
8. C3×#1631C2semidirect
9. C3×#1547C2semidirect
10. C3×#1492C2semidirect
11. C2×C3×Q8⋊C8C2semidirect
12. #18675C3direct (N × H)
53. #3246C2×C3semidirect
54. #3167C2×C3semidirect
55. #3078C2×C3semidirect
56. #1651C2×C3semidirect
57. #1642C2×C3semidirect
58. #1637C2×C3semidirect
59. #1631C2×C3semidirect
60. #1547C2×C3semidirect
61. #1492C2×C3semidirect
62. C2×Q8⋊C8C2×C3semidirect
197. C2×C3×Q8SD16semidirect
405. C2×Q8C3×SD16semidirect
572. C3#18675direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34972
C3×#32462
C3×#31672
C3×#30782
C3×#16512
C3×#16422
C3×#16372
C3×#16312
C3×#15472
C3×#14922
C2×C3×Q8⋊C82
#186753