Finite Group Databasebeta

← C3×#32905 #10651512

Subgroups — C3×#32905 #10651512

Order 768.

1868 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×#461C212yesmaximal, complemented
3C2×C3×#450C212yesmaximal
4C3×#2857C211yesmaximal, characteristic, complemented
5C3×#1877C214yesmaximal, complemented
6C3×#1872C212yesmaximal, complemented
7C3×#1802C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#32905C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4—21no—
12C22×C3×SD16C2214yes—
13C2×C3×C4⋊Q8C2211yescharacteristic, complemented
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×C4×D4—28no—
16C2×C3×C4×D4—21no—
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4C2212yes—
19C2×C3×C4×D4C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4—21no—
22C22×C3×C42C2211yescharacteristic
23C3×#461—28no—
24C3×#450—28no—
25C2×C3×C4⋊C8C2212yescomplemented
26C2×C3×#103C2214yescomplemented
27C2×C3×#102C2214yes—
28C2×C3×C22⋊C8C2212yes—
29C3×#391C2212yescomplemented
30C3×#389—21no—
31C3×#389—21no—
32C3×#384C2211yescharacteristic, complemented
33C3×#380—21no—
34C3×#380—21no—
35C3×#340C2214yescomplemented
36C3×#336C2212yescomplemented
37C22×C4×D4C2×C311yescharacteristic
38C2×#461C2×C312yescomplemented
39C2×#450C2×C312yes—
40#2857C2×C311yescharacteristic, complemented
41#1877C2×C314yescomplemented
42#1872C2×C312yescomplemented
43#1802C2×C312yescomplemented
44#1797C2×C311yescharacteristic, complemented
45C25×C3—21no—
46C22×C3×Q8C2312yes—
47C22×C3×D4—24no—
48C22×C3×D4—24no—
49C22×C3×D4—21no—
50C22×C3×D4—22no—
51C22×C3×D4D414yes—
52C22×C3×D4C2312yes—
53C23×C3×C4—28no—
54C23×C3×C4—21no—
55C23×C3×C4—22no—
56C23×C3×C4—22no—
57C23×C3×C4—44no—
58C23×C3×C4—44no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C2×C3×SD16—28no—
63C2×C3×SD16—28no—
64C2×C3×SD16—28no—
65C22×C3×C8C2314yes—
66C3×C4⋊Q8—24no—
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8—24no—
69C3×C4×D4—28no—
70C3×C4×D4—28no—
71C3×C4×D4—24no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×C4×D4D418yes—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—28no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C42—24no—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C3×C4⋊C8—24no—
94C3×#103—28no—
95C3×#102—28no—
96C3×C22⋊C8—24no—
97C3×#95—22no—
98C3×#95—22no—
99C23×D4C22×C311yescharacteristic
100C24×C4—21no—
101C22×SD16C22×C314yes—
102C2×C4⋊Q8C22×C311yescharacteristic, complemented
103C2×C22⋊Q8C22×C312yes—
104C2×C4×D4—28no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C22×C312yes—
108C2×C4×D4C22×C312yes—
109C22×C4⋊C4C22×C311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C42C22×C311yescharacteristic
112#461—28no—
113#450—28no—
114C2×C4⋊C8C22×C312yescomplemented
115C2×#103C22×C314yescomplemented
116C2×#102C22×C314yes—
117C2×C22⋊C8C22×C312yes—
118#391C22×C312yescomplemented
119#389—21no—
120#389—21no—
121#384C22×C311yescharacteristic, complemented
122#380—21no—
123#380—21no—
124#340C22×C314yescomplemented
125#336C22×C312yescomplemented
126C24×C3—24no—
127C24×C3—24no—
128C24×C3—24no—
129C24×C3—22no—
130C24×C3—42no—
131C24×C3—42no—
132C24×C3—42no—
133C24×C3—42no—
134C24×C3C2×D411yescharacteristic
135C2×C3×Q8—24no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—24no—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—216no—
142C2×C3×D4—216no—
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4C2×D414yes—
148C2×C3×D4C2×D414yes—
149C2×C3×D4#4114yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—24no—
155C22×C3×C4—28no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—28no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—41no—
163C22×C3×C4—44no—
164C22×C3×C4—432no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—44no—
168C22×C3×C4—44no—
169C22×C3×C4—44no—
170C22×C3×C4SD1614yescomplemented
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4#4112yes—
178C22×C3×C4C2411yescharacteristic
179C3×SD16—416no—
180C2×C3×C8—24no—
181C2×C3×C8—24no—
182C2×C3×C8—24no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—44no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4C2×D414yes—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—28no—
195C3×C22⋊C4—44no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42C2×D414yes—
200C25—21no—
201C22×Q8C23×C312yes—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4C3×D414yes—
207C22×D4C23×C312yes—
208C23×C4—28no—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C3×D411yescharacteristic
215C23×C4C3×D411yescharacteristic
216C23×C4C23×C311yescharacteristic
217C2×SD16—28no—
218C2×SD16—28no—
219C2×SD16—28no—
220C22×C8C23×C314yes—
221C4⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C4×D4—28no—
225C4×D4—28no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4C3×D418yes—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C23×C312yes—
238C2×C4⋊C4C23×C312yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—24no—
244C2×C42C3×D411yescharacteristic
245C2×C42C3×D411yescharacteristic
246C2×C42C23×C311yescharacteristic
247C2×C42C23×C311yescharacteristic
248C4⋊C8—24no—
249#103—28no—
250#102—28no—
251C22⋊C8—24no—
252#95—22no—
253#95—22no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—216no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—48no—
263C23×C3—48no—
264C23×C3—48no—
265C23×C3—48no—
266C23×C3—48no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—48no—
272C23×C3#12112yes—
273C23×C3C2×SD1612yes—
274C23×C3#13712yes—
275C23×C3C22×D411yescharacteristic
276C3×Q8—44no—
277C3×Q8—44no—
278C3×D4—216no—
279C3×D4—24no—
280C3×D4—24no—
281C3×D4—24no—
282C3×D4#12118yes—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—28no—
292C2×C3×C4—21no—
293C2×C3×C4—24no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—44no—
297C2×C3×C4—416no—
298C2×C3×C4—48no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—416no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12012yes—
310C2×C3×C4#12112yes—
311C2×C3×C4C2×SD1612yes—
312C2×C3×C4C2×SD1612yes—
313C2×C3×C4AGL(1,ℤ/8)12yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D412yes—
316C2×C3×C4C2×#4112yes—
317C2×C3×C4C2×#4111yescharacteristic
318C3×C8—44no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24C2×C3×D411yescharacteristic
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—216no—
335C2×D4—216no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4C2×C3×D414yes—
341C2×D4C2×C3×D414yes—
342C2×D4C3×#4114yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—41no—
356C22×C4—44no—
357C22×C4—432no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4C3×SD1614yescomplemented
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D412yes—
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C3×#4112yes—
371C22×C4C24×C311yescharacteristic
372SD16—416no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4C2×C3×D414yes—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4—44no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42C2×C3×D414yes—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—28no—
400C22×C3—432no—
401C22×C3—44no—
402C22×C3—44no—
403C22×C3—44no—
404C22×C3—44no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3#46114yescomplemented
408C22×C3C2×#12011yescharacteristic
409C22×C3C2×#12111yescharacteristic
410C22×C3#52511yescharacteristic
411C22×C3C22×SD1611yescharacteristic
412C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
413C22×C3C2×#13711yescharacteristic
414C22×C3#57511yescharacteristic
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—44no—
419C3×C4—41no—
420C3×C4—82no—
421C3×C4—82no—
422C3×C4#45014yescomplemented
423C3×C4C2×#12112yes—
424C3×C4#52512yes—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—216no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—48no—
434C23—48no—
435C23—48no—
436C23—48no—
437C23—48no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—48no—
443C23C3×#12112yes—
444C23C2×C3×SD1612yes—
445C23C3×#13712yes—
446C23C22×C3×D411yescharacteristic
447Q8—44no—
448Q8—44no—
449D4—216no—
450D4—24no—
451D4—24no—
452D4—24no—
453D4C3×#12118yes—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—44no—
468C2×C4—416no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—416no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4C3×#12012yes—
480C2×C4C3×#12012yes—
481C2×C4C3×#12112yes—
482C2×C4C2×C3×SD1612yes—
483C2×C4C2×C3×SD1612yes—
484C2×C4C3×AGL(1,ℤ/8)12yes—
485C2×C4C22×C3×D411yescharacteristic
486C2×C4C22×C3×D412yes—
487C2×C4C2×C3×#4112yes—
488C2×C4C2×C3×#4111yescharacteristic
489C8—44no—
490C2×C3—24no—
491C2×C3—48no—
492C2×C3#284711yescharacteristic
493C2×C3C2×#45011yescharacteristic
494C2×C3#300711yescharacteristic
495C2×C3C2×#46111yescharacteristic
496C2×C3#303611yescharacteristic
497C2×C3#310111yescharacteristic
498C2×C3#311011yescharacteristic
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—28no—
506C22—432no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22C3×#46114yescomplemented
514C22C2×C3×#12011yescharacteristic
515C22C2×C3×#12111yescharacteristic
516C22C3×#52511yescharacteristic
517C22C22×C3×SD1611yescharacteristic
518C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
519C22C2×C3×#13711yescharacteristic
520C22C3×#57511yescharacteristic
521C4—22no—
522C4—22no—
523C4—22no—
524C4—44no—
525C4—41no—
526C4—82no—
527C4—82no—
528C4C3×#45014yescomplemented
529C4C2×C3×#12112yes—
530C4C3×#52512yes—
531C3#3290511yescharacteristic, direct product factor, Sylow
532C2—24no—
533C2—48no—
534C2C3×#284711yescharacteristic
535C2C2×C3×#45011yescharacteristic
536C2C3×#300711yescharacteristic
537C2C2×C3×#46111yescharacteristic
538C2C3×#303611yescharacteristic
539C2C3×#310111yescharacteristic
540C2C3×#311011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#461C2semidirect
4. C3×#2857C2semidirect
5. C3×#1877C2semidirect
6. C3×#1872C2semidirect
7. C3×#1802C2semidirect
8. C3×#1797C2semidirect
9. #32905C3direct (N × H)
13. C2×C3×C4⋊Q8C22semidirect
25. C2×C3×C4⋊C8C22semidirect
26. C2×C3×#103C22semidirect
29. C3×#391C22semidirect
32. C3×#384C22semidirect
35. C3×#340C22semidirect
36. C3×#336C22semidirect
38. C2×#461C2×C3semidirect
40. #2857C2×C3semidirect
41. #1877C2×C3semidirect
42. #1872C2×C3semidirect
43. #1802C2×C3semidirect
44. #1797C2×C3semidirect
102. C2×C4⋊Q8C22×C3semidirect
114. C2×C4⋊C8C22×C3semidirect
115. C2×#103C22×C3semidirect
118. #391C22×C3semidirect
121. #384C22×C3semidirect
124. #340C22×C3semidirect
125. #336C22×C3semidirect
170. C22×C3×C4SD16semidirect
363. C22×C4C3×SD16semidirect
407. C22×C3#461semidirect
422. C3×C4#450semidirect
513. C22C3×#461semidirect
528. C4C3×#450semidirect
531. C3#32905direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×C3×#4612
C2×C3×#4502
C3×#28572
C3×#18772
C3×#18722
C3×#18022
C3×#17972
#329053