Finite Group Databasebeta

← C3×#18676 #10637283

Subgroups — C3×#18676 #10637283

Order 768.

912 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3502C211yesmaximal, characteristic, complemented
2C3×#3246C211yesmaximal, characteristic
3C3×#3167C211yesmaximal, characteristic
4C3×#3037C211yesmaximal, characteristic
5C3×#1653C212yesmaximal, complemented
6C3×#1645C212yesmaximal, complemented
7C3×#1626C212yesmaximal, complemented
8C3×#1623C212yesmaximal, complemented
9C3×#1550C211yesmaximal, characteristic, complemented
10C3×#1493C211yesmaximal, characteristic, complemented
11C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
12#18676C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137—21no—
14C3×Q82—21no—
15C3×Q82C2212yes—
16C3×#557—21no—
17C3×#557—22no—
18C3×#557—22no—
19C3×#557C2212yes—
20C3×#544—22no—
21C3×#544—21no—
22C3×#533—22no—
23C3×#533C2211yescharacteristic
24C3×#533C2211yescharacteristic
25C3×#532C2211yescharacteristic
26C3×#519—21no—
27C2×C3×C4×Q8C2211yescharacteristic
28C3×#498C2211yescharacteristic
29C3×#492C2211yescharacteristic
30C3×#487C2211yescharacteristic
31C3×#486C2211yescharacteristic
32C3×#476—21no—
33C3×#475C2212yes—
34C3×#465—22no—
35C3×#465—22no—
36C3×#464—22no—
37C3×#462C2212yes—
38C3×#461—21no—
39C3×#460C2212yes—
40C3×#460C2212yes—
41C3×C4×SD16—22no—
42C3×#423C2211yescharacteristic
43C2×C3×C4⋊C8C2211yescharacteristic
44C3×#417—21no—
45C2×C3×C4×C8C2211yescharacteristic
46C3×#333C2212yes—
47C3×#332C2212yes—
48C3×Q8⋊C8C2212yes—
49C3×Q8⋊C8C2212yes—
50C3×Q8⋊C8C2212yes—
51C3×Q8⋊C8C2212yes—
52#3502C2×C311yescharacteristic, complemented
53#3246C2×C311yescharacteristic
54#3167C2×C311yescharacteristic
55#3037C2×C311yescharacteristic
56#1653C2×C312yescomplemented
57#1645C2×C312yescomplemented
58#1626C2×C312yescomplemented
59#1623C2×C312yescomplemented
60#1550C2×C311yescharacteristic, complemented
61#1493C2×C311yescharacteristic, complemented
62C2×Q8⋊C8C2×C311yescharacteristic, complemented
63C2×C3×#41—21no—
64C22×C3×Q8D411yescharacteristic
65C3×#137—44no—
66C2×C3×Q16—22no—
67C2×C3×SD16—22no—
68C2×C3×SD16—22no—
69C2×C3×SD16—22no—
70C2×C3×M16—21no—
71C22×C3×C8—21no—
72C22×C3×C8—21no—
73C3×C4⋊Q8—24no—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8—24no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8D411yescharacteristic
80C3×C4⋊Q8D411yescharacteristic
81C3×C4⋊Q8C2311yescharacteristic
82C3×C4⋊Q8C2311yescharacteristic
83C3×C4⋊Q8C2311yescharacteristic
84C3×#127C2311yescharacteristic
85C3×#125—24no—
86C3×#125—24no—
87C3×#125—22no—
88C3×#125—24no—
89C3×#125—22no—
90C3×#125—21no—
91C3×#125—21no—
92C3×#124—21no—
93C3×C22⋊Q8—24no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8D412yes—
97C3×C22⋊Q8D412yes—
98C3×#121—21no—
99C3×#121—21no—
100C3×C4×Q8—22no—
101C3×C4×Q8—22no—
102C3×C4×Q8—24no—
103C3×C4×Q8—22no—
104C3×C4×Q8—22no—
105C3×C4×Q8—22no—
106C3×C4×Q8—22no—
107C3×C4×Q8—21no—
108C3×C4×Q8—21no—
109C3×C4×Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8D412yes—
112C3×C4×Q8D412yes—
113C3×C4×Q8C2312yes—
114C3×C4×Q8C2312yes—
115C3×C4×D4—21no—
116C3×C4×D4—21no—
117C3×#117—24no—
118C3×#117—21no—
119C3×#117—21no—
120C2×C3×C4⋊C4—21no—
121C2×C3×C4⋊C4D411yescharacteristic
122C2×C3×C42—21no—
123C2×C3×C42C2311yescharacteristic
124C3×#106—22no—
125C3×#106—22no—
126C3×C4⋊C8—21no—
127C3×C4⋊C8—21no—
128C3×C4⋊C8C2312yes—
129C3×C4⋊C8C2312yes—
130C3×#103—22no—
131C3×#103—22no—
132C3×#103—22no—
133C3×#102—22no—
134C3×#102—22no—
135C3×#102—22no—
136C3×C4×C8—21no—
137C3×C4×C8C2311yescharacteristic
138C3×C4×C8C2311yescharacteristic
139C2×#137—21no—
140Q82—21no—
141Q82C22×C312yes—
142#557—21no—
143#557—22no—
144#557—22no—
145#557C22×C312yes—
146#544—22no—
147#544—21no—
148#533—22no—
149#533C22×C311yescharacteristic
150#533C22×C311yescharacteristic
151#532C22×C311yescharacteristic
152#519—21no—
153C2×C4×Q8C22×C311yescharacteristic
154#498C22×C311yescharacteristic
155#492C22×C311yescharacteristic
156#487C22×C311yescharacteristic
157#486C22×C311yescharacteristic
158#476—21no—
159#475C22×C312yes—
160#465—22no—
161#465—22no—
162#464—22no—
163#462C22×C312yes—
164#461—21no—
165#460C22×C312yes—
166#460C22×C312yes—
167C4×SD16—22no—
168#423C22×C311yescharacteristic
169C2×C4⋊C8C22×C311yescharacteristic
170#417—21no—
171C2×C4×C8C22×C311yescharacteristic
172#333C22×C312yes—
173#332C22×C312yes—
174Q8⋊C8C22×C312yes—
175Q8⋊C8C22×C312yes—
176Q8⋊C8C22×C312yes—
177Q8⋊C8C22×C312yes—
178C3×#41—42no—
179C3×#41—42no—
180C2×C3×Q8—24no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—22no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8SD1614yescomplemented
187C2×C3×Q8C2×D412yes—
188C2×C3×Q8C2×D412yes—
189C2×C3×Q8C2×D412yes—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—41no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—41no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C3×Q16—44no—
201C3×SD16—44no—
202C3×SD16—44no—
203C3×SD16—82no—
204C3×M16—42no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—42no—
210C2×C3×C8—81no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C4⋊C4C2×D412yes—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—41no—
237C3×C22⋊C4—41no—
238C3×C42—24no—
239C3×C42—24no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42—22no—
243C3×C42—22no—
244C3×C42C2×D411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42C2×D411yescharacteristic
247C3×C42C2411yescharacteristic
248C2×#41—21no—
249C22×Q8C3×D411yescharacteristic
250#137—44no—
251C2×Q16—22no—
252C2×SD16—22no—
253C2×SD16—22no—
254C2×SD16—22no—
255C2×M16—21no—
256C22×C8—21no—
257C22×C8—21no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8C3×D411yescharacteristic
265C4⋊Q8C3×D411yescharacteristic
266C4⋊Q8C23×C311yescharacteristic
267C4⋊Q8C23×C311yescharacteristic
268C4⋊Q8C23×C311yescharacteristic
269#127C23×C311yescharacteristic
270#125—24no—
271#125—24no—
272#125—22no—
273#125—24no—
274#125—22no—
275#125—21no—
276#125—21no—
277#124—21no—
278C22⋊Q8—24no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8C3×D412yes—
282C22⋊Q8C3×D412yes—
283#121—21no—
284#121—21no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—24no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8C3×D412yes—
297C4×Q8C3×D412yes—
298C4×Q8C23×C312yes—
299C4×Q8C23×C312yes—
300C4×D4—21no—
301C4×D4—21no—
302#117—24no—
303#117—21no—
304#117—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C3×D411yescharacteristic
307C2×C42—21no—
308C2×C42C23×C311yescharacteristic
309#106—22no—
310#106—22no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8C23×C312yes—
314C4⋊C8C23×C312yes—
315#103—22no—
316#103—22no—
317#103—22no—
318#102—22no—
319#102—22no—
320#102—22no—
321C4×C8—21no—
322C4×C8C23×C311yescharacteristic
323C4×C8C23×C311yescharacteristic
324C23×C3—41no—
325C23×C3#12011yescharacteristic
326C3×Q8—22no—
327C3×Q8—24no—
328C3×Q8—22no—
329C3×Q8—22no—
330C3×Q8—42no—
331C3×Q8—41no—
332C3×Q8—41no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—81no—
336C3×Q8—84no—
337C3×Q8C2×SD1614yes—
338C3×D4—42no—
339C3×D4—41no—
340C3×D4—41no—
341C3×D4—41no—
342C3×D4—41no—
343C3×D4—82no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—81no—
356C2×C3×C4—82no—
357C2×C3×C4#12011yescharacteristic
358C2×C3×C4#12011yescharacteristic
359C2×C3×C4#12011yescharacteristic
360C2×C3×C4C2×SD1612yes—
361C2×C3×C4#13712yes—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C3×C8—22no—
366C3×C8—41no—
367C3×C8—42no—
368C3×C8—42no—
369#41—42no—
370#41—42no—
371C2×Q8—24no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8C3×SD1614yescomplemented
378C2×Q8C2×C3×D412yes—
379C2×Q8C2×C3×D412yes—
380C2×Q8C2×C3×D412yes—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—41no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D411yescharacteristic
391Q16—44no—
392SD16—44no—
393SD16—44no—
394SD16—82no—
395M16—42no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—42no—
401C2×C8—81no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4C2×C3×D412yes—
423C4⋊C4C2×C3×D412yes—
424C4⋊C4C2×C3×D412yes—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C42—24no—
430C42—24no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42C2×C3×D411yescharacteristic
436C42C2×C3×D411yescharacteristic
437C42C2×C3×D411yescharacteristic
438C42C24×C311yescharacteristic
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3—81no—
444C22×C3#44812yes—
445C22×C3C2×#12011yescharacteristic
446C3×C4—24no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—81no—
456C3×C4#44812yes—
457C3×C4C22×SD1611yescharacteristic
458C3×C4C2×#13711yescharacteristic
459C3×C4#57711yescharacteristic
460C3×C4#57811yescharacteristic
461C23—41no—
462C23C3×#12011yescharacteristic
463Q8—22no—
464Q8—24no—
465Q8—22no—
466Q8—22no—
467Q8—42no—
468Q8—41no—
469Q8—41no—
470Q8—41no—
471Q8—41no—
472Q8—81no—
473Q8—84no—
474Q8C2×C3×SD1614yes—
475D4—42no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—82no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4C3×#12011yescharacteristic
495C2×C4C3×#12011yescharacteristic
496C2×C4C3×#12011yescharacteristic
497C2×C4C2×C3×SD1612yes—
498C2×C4C3×#13712yes—
499C2×C4C22×C3×D411yescharacteristic
500C2×C4C22×C3×D411yescharacteristic
501C2×C4C22×C3×D411yescharacteristic
502C8—22no—
503C8—41no—
504C8—42no—
505C8—42no—
506C2×C3—22no—
507C2×C3—161no—
508C2×C3C2×#44811yescharacteristic
509C2×C3#301311yescharacteristic
510C2×C3#301811yescharacteristic
511C22—21no—
512C22—21no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22C3×#44812yes—
517C22C2×C3×#12011yescharacteristic
518C4—24no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—81no—
528C4C3×#44812yes—
529C4C22×C3×SD1611yescharacteristic
530C4C2×C3×#13711yescharacteristic
531C4C3×#57711yescharacteristic
532C4C3×#57811yescharacteristic
533C3#1867611yescharacteristic, direct product factor, Sylow
534C2—22no—
535C2—161no—
536C2C2×C3×#44811yescharacteristic
537C2C3×#301311yescharacteristic
538C2C3×#301811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#532C22
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×#3502C2semidirect
5. C3×#1653C2semidirect
6. C3×#1645C2semidirect
7. C3×#1626C2semidirect
8. C3×#1623C2semidirect
9. C3×#1550C2semidirect
10. C3×#1493C2semidirect
11. C2×C3×Q8⋊C8C2semidirect
12. #18676C3direct (N × H)
52. #3502C2×C3semidirect
56. #1653C2×C3semidirect
57. #1645C2×C3semidirect
58. #1626C2×C3semidirect
59. #1623C2×C3semidirect
60. #1550C2×C3semidirect
61. #1493C2×C3semidirect
62. C2×Q8⋊C8C2×C3semidirect
186. C2×C3×Q8SD16semidirect
377. C2×Q8C3×SD16semidirect
533. C3#18676direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35022
C3×#32462
C3×#31672
C3×#30372
C3×#16532
C3×#16452
C3×#16262
C3×#16232
C3×#15502
C3×#14932
C2×C3×Q8⋊C82
#186763