Finite Group Databasebeta

← C3×#32888 #10651495

Subgroups — C3×#32888 #10651495

Order 768.

4070 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×C3×#453C214yesmaximal, complemented
3C3×#3006C212yesmaximal, complemented
4C3×#2433C211yesmaximal, characteristic, complemented
5C3×#1792C214yesmaximal, complemented
6C3×#1789C212yesmaximal, complemented
7C3×#1783C211yesmaximal, characteristic, complemented
8#32888C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×H(5,𝔽2)—216no—
10C2×C3×H(5,𝔽2)—24no—
11C2×C3×H(5,𝔽2)—22no—
12C2×C3×H(5,𝔽2)C2214yes—
13C22×C3×#41—22no—
14C22×C3×#41C2212yes—
15C23×C3×D4—22no—
16C23×C3×D4—22no—
17C23×C3×D4C2211yescharacteristic
18C2×C3×AGL(1,ℤ/8)C2214yescomplemented
19C2×C3×#127C2212yescomplemented
20C2×C3×#121C2211yescharacteristic, complemented
21C2×C3×#120—22no—
22C3×#453—216no—
23C3×#449—24no—
24C3×#448—22no—
25C3×#447—22no—
26C2×C3×#104C2218yescomplemented
27C3×#417C2214yescomplemented
28C2×C3×#100C2214yescomplemented
29C3×#407C2212yescomplemented
30C3×#392—22no—
31C3×#390C2212yescomplemented
32C3×C4×C22⋊C4C2211yescharacteristic, complemented
33C22×H(5,𝔽2)C2×C311yescharacteristic, complemented
34C2×#453C2×C314yescomplemented
35#3006C2×C312yescomplemented
36#2433C2×C311yescharacteristic, complemented
37#1792C2×C314yescomplemented
38#1789C2×C312yescomplemented
39#1783C2×C311yescharacteristic, complemented
40C25×C3—22no—
41C25×C3D412yes—
42C3×H(5,𝔽2)—232no—
43C3×H(5,𝔽2)—232no—
44C3×H(5,𝔽2)—232no—
45C3×H(5,𝔽2)—216no—
46C3×H(5,𝔽2)—28no—
47C3×H(5,𝔽2)D4116yescomplemented
48C2×C3×#41—232no—
49C2×C3×#41—28no—
50C2×C3×#41—216no—
51C2×C3×#41—28no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41D414yescomplemented
55C2×C3×#41C2314yes—
56C22×C3×Q8D412yescomplemented
57C22×C3×D4—24no—
58C22×C3×D4—232no—
59C22×C3×D4—232no—
60C22×C3×D4—28no—
61C22×C3×D4—28no—
62C22×C3×D4—22no—
63C22×C3×D4—28no—
64C22×C3×D4—28no—
65C22×C3×D4—28no—
66C22×C3×D4—21no—
67C22×C3×D4—24no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4D412yes—
71C22×C3×D4D412yescomplemented
72C22×C3×D4C2312yes—
73C22×C3×D4C2311yescharacteristic, complemented
74C23×C3×C4—22no—
75C23×C3×C4—22no—
76C23×C3×C4C2311yescharacteristic
77C3×AGL(1,ℤ/8)—216no—
78C2×C3×SD16—24no—
79C2×C3×D8—24no—
80C2×C3×M16C2314yescomplemented
81C3×#127—28no—
82C3×#121—22no—
83C3×#121—22no—
84C3×#120—48no—
85C2×C3×C4⋊C4C2311yescharacteristic, complemented
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2312yescomplemented
90C3×#104—216no—
91C3×#103—24no—
92C3×#102—24no—
93C3×#100—28no—
94C3×C22⋊C8—22no—
95C3×C22⋊C8—22no—
96C3×#95—21no—
97C2×H(5,𝔽2)—216no—
98C2×H(5,𝔽2)—24no—
99C2×H(5,𝔽2)—22no—
100C2×H(5,𝔽2)C22×C314yes—
101C22×#41—22no—
102C22×#41C22×C312yes—
103C23×D4—22no—
104C23×D4—22no—
105C23×D4C22×C311yescharacteristic
106C2×AGL(1,ℤ/8)C22×C314yescomplemented
107C2×#127C22×C312yescomplemented
108C2×#121C22×C311yescharacteristic, complemented
109C2×#120—22no—
110#453—216no—
111#449—24no—
112#448—22no—
113#447—22no—
114C2×#104C22×C318yescomplemented
115#417C22×C314yescomplemented
116C2×#100C22×C314yescomplemented
117#407C22×C312yescomplemented
118#392—22no—
119#390C22×C312yescomplemented
120C4×C22⋊C4C22×C311yescharacteristic, complemented
121C24×C3—216no—
122C24×C3—28no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—28no—
126C24×C3—22no—
127C24×C3—21no—
128C24×C3—44no—
129C24×C3—82no—
130C24×C3—84no—
131C24×C3—84no—
132C24×C3C2×D411yescharacteristic
133C24×C3C2×D414yes—
134C3×#41—264no—
135C3×#41—216no—
136C3×#41—232no—
137C3×#41—216no—
138C3×#41—232no—
139C3×#41—28no—
140C3×#41—28no—
141C3×#41—28no—
142C3×#41C2×D4116yes—
143C2×C3×Q8—28no—
144C2×C3×Q8—22no—
145C2×C3×Q8C2×D414yes—
146C2×C3×D4—232no—
147C2×C3×D4—216no—
148C2×C3×D4—216no—
149C2×C3×D4—216no—
150C2×C3×D4—264no—
151C2×C3×D4—264no—
152C2×C3×D4—232no—
153C2×C3×D4—232no—
154C2×C3×D4—216no—
155C2×C3×D4—24no—
156C2×C3×D4—232no—
157C2×C3×D4—28no—
158C2×C3×D4—24no—
159C2×C3×D4—28no—
160C2×C3×D4—232no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—22no—
166C2×C3×D4—24no—
167C2×C3×D4—48no—
168C2×C3×D4—44no—
169C2×C3×D4—41no—
170C2×C3×D4—41no—
171C2×C3×D4—48no—
172C2×C3×D4—84no—
173C2×C3×D4C2×D418yes—
174C2×C3×D4C2×D414yes—
175C22×C3×C4—216no—
176C22×C3×C4—216no—
177C22×C3×C4—28no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2411yescharacteristic
188C3×SD16—48no—
189C3×D8—48no—
190C3×M16—28no—
191C2×C3×C8—24no—
192C3×C4⋊C4—22no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—48no—
195C3×C22⋊C4—42no—
196C3×C42—24no—
197C25—22no—
198C25C3×D412yes—
199H(5,𝔽2)—232no—
200H(5,𝔽2)—232no—
201H(5,𝔽2)—232no—
202H(5,𝔽2)—216no—
203H(5,𝔽2)—28no—
204H(5,𝔽2)C3×D4116yescomplemented
205C2×#41—232no—
206C2×#41—28no—
207C2×#41—216no—
208C2×#41—28no—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41C3×D414yescomplemented
212C2×#41C23×C314yes—
213C22×Q8C3×D412yescomplemented
214C22×D4—24no—
215C22×D4—232no—
216C22×D4—232no—
217C22×D4—28no—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—28no—
221C22×D4—28no—
222C22×D4—28no—
223C22×D4—21no—
224C22×D4—24no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4C3×D412yes—
228C22×D4C3×D412yescomplemented
229C22×D4C23×C312yes—
230C22×D4C23×C311yescharacteristic, complemented
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C23×C311yescharacteristic
234AGL(1,ℤ/8)—216no—
235C2×SD16—24no—
236C2×D8—24no—
237C2×M16C23×C314yescomplemented
238#127—28no—
239#121—22no—
240#121—22no—
241#120—48no—
242C2×C4⋊C4C23×C311yescharacteristic, complemented
243C2×C22⋊C4—21no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—21no—
246C2×C42C23×C312yescomplemented
247#104—216no—
248#103—24no—
249#102—24no—
250#100—28no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253#95—21no—
254C23×C3—28no—
255C23×C3—28no—
256C23×C3—28no—
257C23×C3—232no—
258C23×C3—22no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—24no—
263C23×C3—41no—
264C23×C3—44no—
265C23×C3—44no—
266C23×C3—44no—
267C23×C3—44no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—44no—
271C23×C3—44no—
272C23×C3—42no—
273C23×C3—41no—
274C23×C3—82no—
275C23×C3—82no—
276C23×C3—816no—
277C23×C3—832no—
278C23×C3—88no—
279C23×C3—88no—
280C23×C3#12011yescharacteristic
281C23×C3#12011yescharacteristic
282C23×C3#12018yes—
283C23×C3C22×D411yescharacteristic
284C3×Q8—28no—
285C3×Q8—24no—
286C3×Q8#12018yescomplemented
287C3×D4—264no—
288C3×D4—216no—
289C3×D4—216no—
290C3×D4—28no—
291C3×D4—216no—
292C3×D4—216no—
293C3×D4—24no—
294C3×D4—44no—
295C3×D4—816no—
296C3×D4—82no—
297C3×D4—82no—
298C3×D4#12018yescomplemented
299C2×C3×C4—28no—
300C2×C3×C4—232no—
301C2×C3×C4—216no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—216no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4#12012yes—
313C2×C3×C4C22×D414yes—
314C2×C3×C4C22×D412yes—
315C3×C8—44no—
316C24—216no—
317C24—28no—
318C24—22no—
319C24—22no—
320C24—28no—
321C24—22no—
322C24—21no—
323C24—44no—
324C24—82no—
325C24—84no—
326C24—84no—
327C24C2×C3×D411yescharacteristic
328C24C2×C3×D414yes—
329#41—264no—
330#41—216no—
331#41—232no—
332#41—216no—
333#41—232no—
334#41—28no—
335#41—28no—
336#41—28no—
337#41C2×C3×D4116yes—
338C2×Q8—28no—
339C2×Q8—22no—
340C2×Q8C2×C3×D414yes—
341C2×D4—232no—
342C2×D4—216no—
343C2×D4—216no—
344C2×D4—216no—
345C2×D4—264no—
346C2×D4—264no—
347C2×D4—232no—
348C2×D4—232no—
349C2×D4—216no—
350C2×D4—24no—
351C2×D4—232no—
352C2×D4—28no—
353C2×D4—24no—
354C2×D4—28no—
355C2×D4—232no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—48no—
367C2×D4—84no—
368C2×D4C2×C3×D418yes—
369C2×D4C2×C3×D414yes—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D412yes—
381C22×C4C2×C3×D412yes—
382C22×C4C24×C311yescharacteristic
383SD16—48no—
384D8—48no—
385M16—28no—
386C2×C8—24no—
387C4⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—48no—
390C22⋊C4—42no—
391C42—24no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—28no—
398C22×C3—28no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—44no—
402C22×C3—48no—
403C22×C3—416no—
404C22×C3—416no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—84no—
410C22×C3—84no—
411C22×C3—88no—
412C22×C3—832no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3—81no—
416C22×C3#45312yescomplemented
417C22×C3C2×#12012yes—
418C22×C3C2×#12011yescharacteristic
419C3×C4—28no—
420C3×C4—22no—
421C3×C4—44no—
422C3×C4—81no—
423C3×C4C2×#12014yes—
424C23—28no—
425C23—28no—
426C23—28no—
427C23—232no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—41no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—41no—
443C23—42no—
444C23—82no—
445C23—82no—
446C23—816no—
447C23—832no—
448C23—88no—
449C23—88no—
450C23C3×#12011yescharacteristic
451C23C3×#12011yescharacteristic
452C23C3×#12018yes—
453C23C22×C3×D411yescharacteristic
454Q8—28no—
455Q8—24no—
456Q8C3×#12018yescomplemented
457D4—264no—
458D4—216no—
459D4—216no—
460D4—28no—
461D4—216no—
462D4—216no—
463D4—24no—
464D4—44no—
465D4—816no—
466D4—82no—
467D4—82no—
468D4C3×#12018yescomplemented
469C2×C4—28no—
470C2×C4—232no—
471C2×C4—216no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—216no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4C3×#12012yes—
483C2×C4C22×C3×D414yes—
484C2×C4C22×C3×D412yes—
485C8—44no—
486C2×C3—22no—
487C2×C3—22no—
488C2×C3—41no—
489C2×C3—48no—
490C2×C3—48no—
491C2×C3—161no—
492C2×C3#284611yescharacteristic
493C2×C3C2×#45311yescharacteristic
494C2×C3#301911yescharacteristic
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—28no—
503C22—28no—
504C22—44no—
505C22—48no—
506C22—416no—
507C22—416no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—84no—
513C22—84no—
514C22—88no—
515C22—832no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22C3×#45312yescomplemented
520C22C2×C3×#12012yes—
521C22C2×C3×#12011yescharacteristic
522C4—28no—
523C4—22no—
524C4—44no—
525C4—81no—
526C4C2×C3×#12014yes—
527C3#3288811yescharacteristic, direct product factor, Sylow
528C2—22no—
529C2—22no—
530C2—41no—
531C2—48no—
532C2—48no—
533C2—161no—
534C2C3×#284611yescharacteristic
535C2C2×C3×#45311yescharacteristic
536C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×H(5,𝔽2)C2semidirect
2. C2×C3×#453C2semidirect
3. C3×#3006C2semidirect
4. C3×#2433C2semidirect
5. C3×#1792C2semidirect
6. C3×#1789C2semidirect
7. C3×#1783C2semidirect
8. #32888C3direct (N × H)
18. C2×C3×AGL(1,ℤ/8)C22semidirect
19. C2×C3×#127C22semidirect
20. C2×C3×#121C22semidirect
26. C2×C3×#104C22semidirect
27. C3×#417C22semidirect
28. C2×C3×#100C22semidirect
29. C3×#407C22semidirect
31. C3×#390C22semidirect
32. C3×C4×C22⋊C4C22semidirect
33. C22×H(5,𝔽2)C2×C3semidirect
34. C2×#453C2×C3semidirect
35. #3006C2×C3semidirect
36. #2433C2×C3semidirect
37. #1792C2×C3semidirect
38. #1789C2×C3semidirect
39. #1783C2×C3semidirect
47. C3×H(5,𝔽2)D4semidirect
54. C2×C3×#41D4semidirect
56. C22×C3×Q8D4semidirect
71. C22×C3×D4D4semidirect
73. C22×C3×D4C23semidirect
80. C2×C3×M16C23semidirect
85. C2×C3×C4⋊C4C23semidirect
89. C2×C3×C42C23semidirect
106. C2×AGL(1,ℤ/8)C22×C3semidirect
107. C2×#127C22×C3semidirect
108. C2×#121C22×C3semidirect
114. C2×#104C22×C3semidirect
115. #417C22×C3semidirect
116. C2×#100C22×C3semidirect
117. #407C22×C3semidirect
119. #390C22×C3semidirect
120. C4×C22⋊C4C22×C3semidirect
204. H(5,𝔽2)C3×D4semidirect
211. C2×#41C3×D4semidirect
213. C22×Q8C3×D4semidirect
228. C22×D4C3×D4semidirect
230. C22×D4C23×C3semidirect
237. C2×M16C23×C3semidirect
242. C2×C4⋊C4C23×C3semidirect
246. C2×C42C23×C3semidirect
286. C3×Q8#120semidirect
298. C3×D4#120semidirect
416. C22×C3#453semidirect
456. Q8C3×#120semidirect
468. D4C3×#120semidirect
519. C22C3×#453semidirect
527. C3#32888direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×H(5,𝔽2)2
C2×C3×#4532
C3×#30062
C3×#24332
C3×#17922
C3×#17892
C3×#17832
#328883