Finite Group Databasebeta

← C3×#18315 #10636922

Subgroups — C3×#18315 #10636922

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic
2C3×#3232C212yesmaximal, complemented
3C3×#2634C211yesmaximal, characteristic, complemented
4C3×#1617C211yesmaximal, characteristic, complemented
5C3×#1616C211yesmaximal, characteristic
6C3×#1615C212yesmaximal, complemented
7C3×#1611C211yesmaximal, characteristic, complemented
8C3×#1610C211yesmaximal, characteristic
9C3×#1609C212yesmaximal, complemented
10C3×#1521C212yesmaximal, complemented
11C3×#1461C211yesmaximal, characteristic, complemented
12#18315C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#544C2211yescharacteristic
14C3×#542—22no—
15C3×#541—22no—
16C3×#539—21no—
17C3×#539C2212yes—
18C3×#537C2211yescharacteristic
19C3×#536—21no—
20C3×#533—22no—
21C3×#529—22no—
22C3×#529—21no—
23C3×#529C2212yes—
24C2×C3×#125C2211yescharacteristic
25C3×C4×#41—21no—
26C2×C3×C4×Q8C2211yescharacteristic
27C3×#499—22no—
28C3×#488—22no—
29C3×#484C2212yes—
30C3×#483C2212yes—
31C3×#481C2212yes—
32C3×#480C2212yes—
33C3×#479—22no—
34C3×#478—22no—
35C3×#431C2212yes—
36C3×#400—21no—
37C3×#400C2211yescharacteristic
38C3×#400C2211yescharacteristic
39C3×#398C2211yescharacteristic
40C3×#398C2211yescharacteristic
41C3×#395—21no—
42C3×#387—21no—
43C3×#387—21no—
44C3×#383C2211yescharacteristic
45C3×#328C2212yes—
46C3×#328C2212yes—
47C3×#327C2212yes—
48C3×#327C2212yes—
49C3×#324C2212yes—
50C3×#324C2211yescharacteristic
51C3×#324C2211yescharacteristic
52#3461C2×C311yescharacteristic
53#3232C2×C312yescomplemented
54#2634C2×C311yescharacteristic, complemented
55#1617C2×C311yescharacteristic, complemented
56#1616C2×C311yescharacteristic
57#1615C2×C312yescomplemented
58#1611C2×C311yescharacteristic, complemented
59#1610C2×C311yescharacteristic
60#1609C2×C312yescomplemented
61#1521C2×C312yescomplemented
62#1461C2×C311yescharacteristic, complemented
63C2×C3×#41—21no—
64C22×C3×Q8D411yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×#126—22no—
68C3×#126—22no—
69C3×#126—22no—
70C3×#126—22no—
71C3×#125—21no—
72C3×#125—22no—
73C3×#125—22no—
74C3×#125—22no—
75C3×#125—22no—
76C3×#125—22no—
77C3×#125D412yes—
78C3×#124—22no—
79C3×#124—21no—
80C3×#124D412yes—
81C3×#123—22no—
82C3×#123—22no—
83C3×#123—22no—
84C3×#123—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8C2312yes—
92C3×#121—22no—
93C3×#121—21no—
94C3×#121C2312yes—
95C3×C4×Q8—21no—
96C3×C4×Q8—21no—
97C3×C4×Q8—21no—
98C3×C4×Q8—22no—
99C3×C4×Q8—22no—
100C3×C4×Q8D412yes—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×C4×D4—22no—
105C3×C4×D4—22no—
106C3×C4×D4D412yes—
107C3×#117—22no—
108C3×#117—22no—
109C3×#117—21no—
110C3×#117—22no—
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4—21no—
115C2×C3×C4⋊C4D411yescharacteristic
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C4⋊C4C2311yescharacteristic
118C2×C3×C4⋊C4C2311yescharacteristic
119C2×C3×C4⋊C4C2311yescharacteristic
120C2×C3×C42—22no—
121C2×C3×C42D411yescharacteristic
122C2×C3×C42D411yescharacteristic
123C2×C3×C42C2311yescharacteristic
124C3×#107—22no—
125C3×#107—22no—
126C3×#106—22no—
127C3×#106—22no—
128C3×C4⋊C8—22no—
129C3×#103—22no—
130C3×#103—22no—
131C3×#102—22no—
132C3×#102—22no—
133C3×C22⋊C8C2312yes—
134C3×C22⋊C8C2312yes—
135C3×C4×C8—22no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95—21no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95C2311yescharacteristic
145C3×#95C2311yescharacteristic
146#544C22×C311yescharacteristic
147#542—22no—
148#541—22no—
149#539—21no—
150#539C22×C312yes—
151#537C22×C311yescharacteristic
152#536—21no—
153#533—22no—
154#529—22no—
155#529—21no—
156#529C22×C312yes—
157C2×#125C22×C311yescharacteristic
158C4×#41—21no—
159C2×C4×Q8C22×C311yescharacteristic
160#499—22no—
161#488—22no—
162#484C22×C312yes—
163#483C22×C312yes—
164#481C22×C312yes—
165#480C22×C312yes—
166#479—22no—
167#478—22no—
168#431C22×C312yes—
169#400—21no—
170#400C22×C311yescharacteristic
171#400C22×C311yescharacteristic
172#398C22×C311yescharacteristic
173#398C22×C311yescharacteristic
174#395—21no—
175#387—21no—
176#387—21no—
177#383C22×C311yescharacteristic
178#328C22×C312yes—
179#328C22×C312yes—
180#327C22×C312yes—
181#327C22×C312yes—
182#324C22×C312yes—
183#324C22×C311yescharacteristic
184#324C22×C311yescharacteristic
185C3×#41—42no—
186C3×#41—42no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—41no—
192C2×C3×Q8C2×D412yes—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4C2×D412yes—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4C2×D412yes—
236C3×C4⋊C4C2×D412yes—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C42—22no—
247C3×C42—22no—
248C3×C42—21no—
249C3×C42—22no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42C2×D412yes—
254C3×C42C2×D412yes—
255C2×#41—21no—
256C22×Q8C3×D411yescharacteristic
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259#126—22no—
260#126—22no—
261#126—22no—
262#126—22no—
263#125—21no—
264#125—22no—
265#125—22no—
266#125—22no—
267#125—22no—
268#125—22no—
269#125C3×D412yes—
270#124—22no—
271#124—21no—
272#124C3×D412yes—
273#123—22no—
274#123—22no—
275#123—22no—
276#123—22no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281C22⋊Q8—21no—
282C22⋊Q8—22no—
283C22⋊Q8C23×C312yes—
284#121—22no—
285#121—21no—
286#121C23×C312yes—
287C4×Q8—21no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8C3×D412yes—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4C3×D412yes—
299#117—22no—
300#117—22no—
301#117—21no—
302#117—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C3×D411yescharacteristic
308C2×C4⋊C4C23×C311yescharacteristic
309C2×C4⋊C4C23×C311yescharacteristic
310C2×C4⋊C4C23×C311yescharacteristic
311C2×C4⋊C4C23×C311yescharacteristic
312C2×C42—22no—
313C2×C42C3×D411yescharacteristic
314C2×C42C3×D411yescharacteristic
315C2×C42C23×C311yescharacteristic
316#107—22no—
317#107—22no—
318#106—22no—
319#106—22no—
320C4⋊C8—22no—
321#103—22no—
322#103—22no—
323#102—22no—
324#102—22no—
325C22⋊C8C23×C312yes—
326C22⋊C8C23×C312yes—
327C4×C8—22no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95C23×C311yescharacteristic
337#95C23×C311yescharacteristic
338C23×C3—22no—
339C23×C3C22×D411yescharacteristic
340C3×Q8—44no—
341C3×Q8—42no—
342C3×Q8—42no—
343C3×Q8—81no—
344C3×D4—42no—
345C3×D4—42no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—81no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—44no—
364C2×C3×C4—44no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—42no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—82no—
379C2×C3×C4#12011yescharacteristic
380C2×C3×C4#12011yescharacteristic
381C2×C3×C4#12012yes—
382C2×C3×C4#13512yes—
383C2×C3×C4C22×D412yes—
384C3×C8—42no—
385C3×C8—42no—
386#41—42no—
387#41—42no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—41no—
393C2×Q8C2×C3×D412yes—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4C2×C3×D412yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C24×C311yescharacteristic
412C2×C8—22no—
413C2×C8—22no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4C2×C3×D412yes—
437C4⋊C4C2×C3×D412yes—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C42—22no—
448C42—22no—
449C42—21no—
450C42—22no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42C2×C3×D412yes—
455C42C2×C3×D412yes—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3C2×#12011yescharacteristic
462C22×C3C2×#13511yescharacteristic
463C22×C3#57511yescharacteristic
464C3×C4—22no—
465C3×C4—22no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—22no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—42no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4—81no—
479C23—22no—
480C23C22×C3×D411yescharacteristic
481Q8—44no—
482Q8—42no—
483Q8—42no—
484Q8—81no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—81no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4C3×#12011yescharacteristic
521C2×C4C3×#12011yescharacteristic
522C2×C4C3×#12012yes—
523C2×C4C3×#13512yes—
524C2×C4C22×C3×D412yes—
525C8—42no—
526C8—42no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—82no—
530C2×C3#300211yescharacteristic
531C2×C3#302011yescharacteristic
532C2×C3#302811yescharacteristic
533C22—21no—
534C22—21no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22C2×C3×#12011yescharacteristic
539C22C2×C3×#13511yescharacteristic
540C22C3×#57511yescharacteristic
541C4—22no—
542C4—22no—
543C4—21no—
544C4—21no—
545C4—22no—
546C4—41no—
547C4—42no—
548C4—42no—
549C4—41no—
550C4—41no—
551C4—42no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C3#1831511yescharacteristic, direct product factor, Sylow
557C2—21no—
558C2—21no—
559C2—82no—
560C2C3×#300211yescharacteristic
561C2C3×#302011yescharacteristic
562C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#537C22
LCSG ▷ C22×C4 ▷ 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×#3232C2semidirect
3. C3×#2634C2semidirect
4. C3×#1617C2semidirect
6. C3×#1615C2semidirect
7. C3×#1611C2semidirect
9. C3×#1609C2semidirect
10. C3×#1521C2semidirect
11. C3×#1461C2semidirect
12. #18315C3direct (N × H)
53. #3232C2×C3semidirect
54. #2634C2×C3semidirect
55. #1617C2×C3semidirect
57. #1615C2×C3semidirect
58. #1611C2×C3semidirect
60. #1609C2×C3semidirect
61. #1521C2×C3semidirect
62. #1461C2×C3semidirect
556. C3#18315direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34612
C3×#32322
C3×#26342
C3×#16172
C3×#16162
C3×#16152
C3×#16112
C3×#16102
C3×#16092
C3×#15212
C3×#14612
#183153