Finite Group Databasebeta

← C3×#25660 #10644267

Subgroups — C3×#25660 #10644267

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic
2C2×C3×#489C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic, complemented
4C3×#2567C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2041C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1983C211yesmaximal, characteristic, complemented
9C3×#1863C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25660C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#124C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×#123—21no—
21C2×C3×#123—21no—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C3×#489—24no—
28C3×#474—24no—
29C2×C3×#107C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#97C2211yescharacteristic
38C3×#395—21no—
39C3×#395—21no—
40C3×#391—21no—
41C3×#389C2212yes—
42C3×#389C2212yes—
43C3×#384—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#559C2×C311yescharacteristic
49C2×#489C2×C311yescharacteristic, complemented
50C2×#474C2×C311yescharacteristic, complemented
51#2567C2×C311yescharacteristic, complemented
52#2042C2×C312yescomplemented
53#2041C2×C312yes—
54#1985C2×C311yescharacteristic, complemented
55#1983C2×C311yescharacteristic, complemented
56#1863C2×C312yescomplemented
57#1862C2×C312yescomplemented
58#1762C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8D411yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×#127—22no—
73C3×#127—22no—
74C3×#124—28no—
75C3×#124—22no—
76C3×#124—22no—
77C3×#124—22no—
78C3×#124D414yes—
79C3×#123—28no—
80C3×#123—28no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×D4—28no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4D414yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C3×#107—24no—
112C3×C4⋊C8—22no—
113C3×#103—24no—
114C3×#103—24no—
115C3×#102—24no—
116C3×#102—24no—
117C3×#102—24no—
118C3×#102—24no—
119C3×#97—22no—
120C3×#95—21no—
121C3×#95—22no—
122C3×#95—22no—
123#559—24no—
124#559—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×#127C22×C311yescharacteristic
127C2×#124—21no—
128C2×#124C22×C311yescharacteristic
129C2×#124C22×C311yescharacteristic
130C2×#123—21no—
131C2×#123—21no—
132C2×C4×Q8C22×C311yescharacteristic
133C2×C4×Q8C22×C311yescharacteristic
134C2×C4×D4—21no—
135C2×C4×D4C22×C311yescharacteristic
136C2×C4×D4C22×C311yescharacteristic
137#489—24no—
138#474—24no—
139C2×#107C22×C312yes—
140C2×C4⋊C8C22×C311yescharacteristic
141C2×#103C22×C312yes—
142C2×#103C22×C312yes—
143C2×#102C22×C312yes—
144C2×#102C22×C312yes—
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×#97C22×C311yescharacteristic
148#395—21no—
149#395—21no—
150#391—21no—
151#389C22×C312yes—
152#389C22×C312yes—
153#384—21no—
154C4×C4⋊C4C22×C311yescharacteristic
155#340C22×C312yes—
156#336C22×C312yes—
157#336C22×C312yes—
158C24×C3—21no—
159C24×C3—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—41no—
166C2×C3×Q8—42no—
167C2×C3×Q8—82no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4—82no—
180C2×C3×D4C2×D412yes—
181C2×C3×D4C2×D412yes—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—82no—
189C22×C3×C4—82no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—82no—
213C3×C4⋊C4—82no—
214C3×C4⋊C4C2×D412yes—
215C3×C4⋊C4#4114yes—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C42—24no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—42no—
228C3×C42C2×D412yes—
229C3×C42C2×D412yes—
230C22×Q8—21no—
231C22×Q8C3×D411yescharacteristic
232C22×Q8C23×C311yescharacteristic
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C23×C311yescharacteristic
236C22×D4C23×C311yescharacteristic
237C23×C4—21no—
238C23×C4—21no—
239C22×C8C23×C312yes—
240C22×C8C23×C312yes—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243#127—22no—
244#127—22no—
245#124—28no—
246#124—22no—
247#124—22no—
248#124—22no—
249#124C3×D414yes—
250#123—28no—
251#123—28no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×D4—28no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4C3×D414yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C3×D411yescharacteristic
267C2×C4⋊C4C3×D411yescharacteristic
268C2×C4⋊C4C23×C312yes—
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C312yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C23×C311yescharacteristic
280C2×C42C23×C311yescharacteristic
281C2×C42C23×C311yescharacteristic
282#107—24no—
283C4⋊C8—22no—
284#103—24no—
285#103—24no—
286#102—24no—
287#102—24no—
288#102—24no—
289#102—24no—
290#97—22no—
291#95—21no—
292#95—22no—
293#95—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3C22×D411yescharacteristic
305C3×Q8—24no—
306C3×Q8—44no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×Q8—84no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—24no—
313C3×D4—48no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—42no—
317C3×D4—84no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—24no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—84no—
341C2×C3×C4—84no—
342C2×C3×C4#12012yes—
343C2×C3×C4#12012yes—
344C2×C3×C4#13712yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4112yes—
348C2×C3×C42−1+412yes—
349C3×C8—42no—
350C3×C8—42no—
351C24—21no—
352C24—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—42no—
360C2×Q8—82no—
361C2×Q8C2×C3×D412yes—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—82no—
373C2×D4C2×C3×D412yes—
374C2×D4C2×C3×D412yes—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D411yescharacteristic
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C24×C311yescharacteristic
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—42no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4C2×C3×D412yes—
408C4⋊C4C3×#4114yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C42—24no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—42no—
421C42C2×C3×D412yes—
422C42C2×C3×D412yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3C2×#12011yescharacteristic
434C22×C3#54812yes—
435C22×C3C2×#13711yescharacteristic
436C22×C3#57511yescharacteristic
437C22×C3#57612yes—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—44no—
442C3×C4—42no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—42no—
447C3×C4—82no—
448C3×C4#53211yescharacteristic
449C3×C4#53611yescharacteristic
450C3×C4#54712yes—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23C22×C3×D411yescharacteristic
462Q8—24no—
463Q8—44no—
464Q8—41no—
465Q8—41no—
466Q8—84no—
467D4—22no—
468D4—22no—
469D4—24no—
470D4—48no—
471D4—41no—
472D4—41no—
473D4—42no—
474D4—84no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4C3×#12012yes—
500C2×C4C3×#12012yes—
501C2×C4C3×#13712yes—
502C2×C4C22×C3×D411yescharacteristic
503C2×C4C22×C3×D411yescharacteristic
504C2×C4C2×C3×#4112yes—
505C2×C4C3×2−1+412yes—
506C8—42no—
507C8—42no—
508C2×C3—82no—
509C2×C3—82no—
510C2×C3#243211yescharacteristic
511C2×C3#300711yescharacteristic
512C2×C3#300811yescharacteristic
513C2×C3#337012yes—
514C2×C3#337512yes—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22C2×C3×#12011yescharacteristic
526C22C3×#54812yes—
527C22C2×C3×#13711yescharacteristic
528C22C3×#57511yescharacteristic
529C22C3×#57612yes—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—42no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—42no—
538C4—44no—
539C4—82no—
540C4C3×#53211yescharacteristic
541C4C3×#53611yescharacteristic
542C4C3×#54712yes—
543C3#2566011yescharacteristic, direct product factor, Sylow
544C2—82no—
545C2—82no—
546C2C3×#243211yescharacteristic
547C2C3×#300711yescharacteristic
548C2C3×#300811yescharacteristic
549C2C3×#337012yes—
550C2C3×#337512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#489C2semidirect
3. C2×C3×#474C2semidirect
4. C3×#2567C2semidirect
5. C3×#2042C2semidirect
7. C3×#1985C2semidirect
8. C3×#1983C2semidirect
9. C3×#1863C2semidirect
10. C3×#1862C2semidirect
11. C3×#1762C2semidirect
12. #25660C3direct (N × H)
49. C2×#489C2×C3semidirect
50. C2×#474C2×C3semidirect
51. #2567C2×C3semidirect
52. #2042C2×C3semidirect
54. #1985C2×C3semidirect
55. #1983C2×C3semidirect
56. #1863C2×C3semidirect
57. #1862C2×C3semidirect
58. #1762C2×C3semidirect
543. C3#25660direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4892
C2×C3×#4742
C3×#25672
C3×#20422
C3×#20412
C3×#19852
C3×#19832
C3×#18632
C3×#18622
C3×#17622
#256603