Finite Group Databasebeta

← C3×#23915 #10642522

Subgroups — C3×#23915 #10642522

Order 768.

1886 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3487C211yesmaximal, characteristic, complemented
2C2×C3×#535C211yesmaximal, characteristic, complemented
3C2×C3×#454C212yesmaximal, complemented
4C3×#3003C212yesmaximal, complemented
5C3×#2002C214yesmaximal, complemented
6C3×#1897C212yesmaximal, complemented
7C3×#1888C212yesmaximal, complemented
8C3×#1785C211yesmaximal, characteristic, complemented
9#23915C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×2−1+4C2211yescharacteristic, complemented
11C22×C3×#41C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C2×C3×AGL(1,ℤ/8)C2214yes—
14C3×#550—24no—
15C3×#548—24no—
16C3×#535—22no—
17C3×#535—28no—
18C3×#535C2212yescomplemented
19C2×C3×#127C2212yes—
20C2×C3×#124C2212yes—
21C3×#525—22no—
22C3×#525C2212yes—
23C2×C3×#121—22no—
24C2×C3×#120—22no—
25C3×#519C2211yescharacteristic, complemented
26C2×C3×#117C2211yescharacteristic
27C3×#454—28no—
28C3×#450—24no—
29C3×#447—24no—
30C2×C3×#104C2214yescomplemented
31C3×#418C2214yes—
32C2×C3×#101C2212yescomplemented
33C2×C3×#100C2212yescomplemented
34C3×#407C2212yes—
35C3×#339C2214yescomplemented
36#3487C2×C311yescharacteristic, complemented
37C2×#535C2×C311yescharacteristic, complemented
38C2×#454C2×C312yescomplemented
39#3003C2×C312yescomplemented
40#2002C2×C314yescomplemented
41#1897C2×C312yescomplemented
42#1888C2×C312yescomplemented
43#1785C2×C311yescharacteristic, complemented
44C25×C3—21no—
45C3×2−1+4—24no—
46C3×2−1+4—42no—
47C2×C3×#41—24no—
48C2×C3×#41—22no—
49C2×C3×#41—24no—
50C2×C3×#41—22no—
51C2×C3×#41—42no—
52C2×C3×#41C2312yes—
53C22×C3×Q8—22no—
54C22×C3×Q8C2311yescharacteristic
55C22×C3×D4—24no—
56C22×C3×D4—24no—
57C22×C3×D4—21no—
58C22×C3×D4—24no—
59C22×C3×D4—42no—
60C22×C3×D4D412yes—
61C22×C3×D4C2311yescharacteristic
62C22×C3×D4C2312yes—
63C23×C3×C4D412yes—
64C23×C3×C4C2311yescharacteristic
65C3×AGL(1,ℤ/8)—28no—
66C3×AGL(1,ℤ/8)—44no—
67C2×C3×SD16—24no—
68C2×C3×D8—24no—
69C2×C3×M16C2314yes—
70C3×#127—24no—
71C3×#127—24no—
72C3×#127—22no—
73C3×#127—24no—
74C3×#124—24no—
75C3×#124—24no—
76C3×#124—22no—
77C3×#124D418yescomplemented
78C3×#123—22no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8D414yescomplemented
81C3×#121—216no—
82C3×#121—24no—
83C3×#121—28no—
84C3×#121—24no—
85C3×#121D414yescomplemented
86C3×#120—216no—
87C3×#120—24no—
88C3×#120—22no—
89C3×C4×Q8—24no—
90C3×C4×D4—28no—
91C3×C4×D4—24no—
92C3×#117—22no—
93C3×#117—24no—
94C3×#117C2312yes—
95C2×C3×C4⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—24no—
98C2×C3×C42C2312yes—
99C3×#104—28no—
100C3×#102—28no—
101C3×#101—24no—
102C3×#100—42no—
103C3×C22⋊C8—24no—
104C2×2−1+4C22×C311yescharacteristic, complemented
105C22×#41C22×C311yescharacteristic
106C23×D4C22×C311yescharacteristic
107C2×AGL(1,ℤ/8)C22×C314yes—
108#550—24no—
109#548—24no—
110#535—22no—
111#535—28no—
112#535C22×C312yescomplemented
113C2×#127C22×C312yes—
114C2×#124C22×C312yes—
115#525—22no—
116#525C22×C312yes—
117C2×#121—22no—
118C2×#120—22no—
119#519C22×C311yescharacteristic, complemented
120C2×#117C22×C311yescharacteristic
121#454—28no—
122#450—24no—
123#447—24no—
124C2×#104C22×C314yescomplemented
125#418C22×C314yes—
126C2×#101C22×C312yescomplemented
127C2×#100C22×C312yescomplemented
128#407C22×C312yes—
129#339C22×C314yescomplemented
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—42no—
133C24×C3—42no—
134C24×C3—84no—
135C24×C3C2×D412yes—
136C24×C3C2×D411yescharacteristic
137C3×#41—24no—
138C3×#41—24no—
139C3×#41—416no—
140C3×#41—44no—
141C3×#41—48no—
142C3×#41—44no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—48no—
147C2×C3×Q8C2×D412yes—
148C2×C3×Q8C2×D414yes—
149C2×C3×D4—24no—
150C2×C3×D4—22no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—28no—
154C2×C3×D4—216no—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—416no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—88no—
168C2×C3×D4C2×D412yes—
169C2×C3×D4C2×D414yes—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—42no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2411yescharacteristic
182C3×SD16—48no—
183C3×D8—48no—
184C3×M16—24no—
185C3×M16—24no—
186C2×C3×C8—24no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4C2×D414yes—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—28no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—216no—
196C3×C22⋊C4—24no—
197C3×C42—24no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42C2×D414yes—
201C25—21no—
2022−1+4—24no—
2032−1+4—42no—
204C2×#41—24no—
205C2×#41—22no—
206C2×#41—22no—
207C2×#41—24no—
208C2×#41—42no—
209C2×#41C23×C312yes—
210C22×Q8—22no—
211C22×Q8C23×C311yescharacteristic
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—21no—
215C22×D4—24no—
216C22×D4—42no—
217C22×D4C3×D412yes—
218C22×D4C23×C311yescharacteristic
219C22×D4C23×C312yes—
220C23×C4C3×D412yes—
221C23×C4C23×C311yescharacteristic
222AGL(1,ℤ/8)—28no—
223AGL(1,ℤ/8)—44no—
224C2×SD16—24no—
225C2×D8—24no—
226C2×M16C23×C314yes—
227#127—24no—
228#127—24no—
229#127—22no—
230#127—24no—
231#124—24no—
232#124—24no—
233#124—22no—
234#124C3×D418yescomplemented
235#123—22no—
236C22⋊Q8—24no—
237C22⋊Q8C3×D414yescomplemented
238#121—216no—
239#121—24no—
240#121—28no—
241#121—24no—
242#121C3×D414yescomplemented
243#120—216no—
244#120—24no—
245#120—22no—
246C4×Q8—24no—
247C4×D4—28no—
248C4×D4—24no—
249#117—22no—
250#117—24no—
251#117C23×C312yes—
252C2×C4⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—24no—
255C2×C42C23×C312yes—
256#104—28no—
257#102—28no—
258#101—24no—
259#100—42no—
260C22⋊C8—24no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—24no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—42no—
269C23×C3—44no—
270C23×C3—48no—
271C23×C3—48no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—88no—
275C23×C3—88no—
276C23×C3—88no—
277C23×C3#12012yes—
278C23×C3C22×D411yescharacteristic
279C3×Q8—28no—
280C3×Q8—24no—
281C3×Q8—44no—
282C3×D4—24no—
283C3×D4—28no—
284C3×D4—48no—
285C3×D4—48no—
286C3×D4—44no—
287C3×D4—42no—
288C3×D4—42no—
289C3×D4—44no—
290C3×D4—44no—
291C3×D4—84no—
292C3×D4—88no—
293C3×D4—816no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—21no—
297C2×C3×C4—22no—
298C2×C3×C4—28no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—48no—
303C2×C3×C4—48no—
304C2×C3×C4—44no—
305C2×C3×C4—44no—
306C2×C3×C4#12014yes—
307C2×C3×C4#12012yes—
308C2×C3×C4C22×D412yes—
309C2×C3×C4C22×D412yes—
310C2×C3×C4H(5,𝔽2)12yes—
311C3×C8—44no—
312C24—24no—
313C24—22no—
314C24—42no—
315C24—42no—
316C24—84no—
317C24C2×C3×D412yes—
318C24C2×C3×D411yescharacteristic
319#41—24no—
320#41—24no—
321#41—416no—
322#41—44no—
323#41—48no—
324#41—44no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—48no—
329C2×Q8C2×C3×D412yes—
330C2×Q8C2×C3×D414yes—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—28no—
336C2×D4—216no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—416no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—88no—
350C2×D4C2×C3×D412yes—
351C2×D4C2×C3×D414yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4C2×C3×D412yes—
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C2×C3×D414yes—
363C22×C4C24×C311yescharacteristic
364SD16—48no—
365D8—48no—
366M16—24no—
367M16—24no—
368C2×C8—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4C2×C3×D414yes—
373C22⋊C4—24no—
374C22⋊C4—28no—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—216no—
378C22⋊C4—24no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42C2×C3×D414yes—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—24no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—44no—
389C22×C3—48no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—82no—
396C22×C3—84no—
397C22×C3—816no—
398C22×C3—84no—
399C22×C3C2×#12011yescharacteristic
400C22×C3C2×#12011yescharacteristic
401C22×C3#53411yescharacteristic
402C3×C4—22no—
403C3×C4—24no—
404C3×C4—44no—
405C3×C4—42no—
406C3×C4—44no—
407C3×C4D4214yes—
408C23—21no—
409C23—21no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—42no—
416C23—44no—
417C23—48no—
418C23—48no—
419C23—42no—
420C23—42no—
421C23—88no—
422C23—88no—
423C23—88no—
424C23C3×#12012yes—
425C23C22×C3×D411yescharacteristic
426Q8—28no—
427Q8—24no—
428Q8—44no—
429D4—24no—
430D4—28no—
431D4—48no—
432D4—48no—
433D4—44no—
434D4—42no—
435D4—42no—
436D4—44no—
437D4—44no—
438D4—84no—
439D4—816no—
440D4—88no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4C3×#12014yes—
454C2×C4C3×#12012yes—
455C2×C4C22×C3×D412yes—
456C2×C4C22×C3×D412yes—
457C2×C4C3×H(5,𝔽2)12yes—
458C8—44no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—42no—
462C2×C3—82no—
463C2×C3—82no—
464C2×C3—84no—
465C2×C3#240311yescharacteristic
466C2×C3#302112yes—
467C22—22no—
468C22—22no—
469C22—24no—
470C22—42no—
471C22—42no—
472C22—44no—
473C22—48no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—82no—
480C22—84no—
481C22—816no—
482C22—84no—
483C22C2×C3×#12011yescharacteristic
484C22C2×C3×#12011yescharacteristic
485C22C3×#53411yescharacteristic
486C4—22no—
487C4—24no—
488C4—44no—
489C4—42no—
490C4—44no—
491C4C3×D4214yes—
492C3#2391511yescharacteristic, direct product factor, Sylow
493C2—21no—
494C2—21no—
495C2—42no—
496C2—82no—
497C2—84no—
498C2—82no—
499C2C3×#240311yescharacteristic
500C2C3×#302112yes—
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. C3×#3487C2semidirect
2. C2×C3×#535C2semidirect
3. C2×C3×#454C2semidirect
4. C3×#3003C2semidirect
5. C3×#2002C2semidirect
6. C3×#1897C2semidirect
7. C3×#1888C2semidirect
8. C3×#1785C2semidirect
9. #23915C3direct (N × H)
10. C2×C3×2−1+4C22semidirect
18. C3×#535C22semidirect
25. C3×#519C22semidirect
30. C2×C3×#104C22semidirect
32. C2×C3×#101C22semidirect
33. C2×C3×#100C22semidirect
35. C3×#339C22semidirect
36. #3487C2×C3semidirect
37. C2×#535C2×C3semidirect
38. C2×#454C2×C3semidirect
39. #3003C2×C3semidirect
40. #2002C2×C3semidirect
41. #1897C2×C3semidirect
42. #1888C2×C3semidirect
43. #1785C2×C3semidirect
77. C3×#124D4semidirect
80. C3×C22⋊Q8D4semidirect
85. C3×#121D4semidirect
104. C2×2−1+4C22×C3semidirect
112. #535C22×C3semidirect
119. #519C22×C3semidirect
124. C2×#104C22×C3semidirect
126. C2×#101C22×C3semidirect
127. C2×#100C22×C3semidirect
129. #339C22×C3semidirect
234. #124C3×D4semidirect
237. C22⋊Q8C3×D4semidirect
242. #121C3×D4semidirect
492. C3#23915direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34872
C2×C3×#5352
C2×C3×#4542
C3×#30032
C3×#20022
C3×#18972
C3×#18882
C3×#17852
#239153