Finite Group Databasebeta

← C3×#23952 #10642559

Subgroups — C3×#23952 #10642559

Order 768.

1326 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3493C211yesmaximal, characteristic, complemented
2C3×#3003C212yesmaximal, complemented
3C3×#2309C211yesmaximal, characteristic, complemented
4C3×#2027C212yesmaximal, complemented
5C3×#2006C212yesmaximal, complemented
6C3×#1897C212yesmaximal, complemented
7C3×#1883C214yesmaximal, complemented
8C3×#1822C211yesmaximal, characteristic, complemented
9#23952C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C2×C3×AGL(1,ℤ/8)C2214yes—
12C3×#556—22no—
13C3×#555—22no—
14C3×#553—22no—
15C3×#552—22no—
16C3×#544C2211yescharacteristic, complemented
17C3×#538—22no—
18C3×#537C2211yescharacteristic, complemented
19C3×#529—22no—
20C3×#525C2212yes—
21C3×#520C2212yescomplemented
22C2×C3×#117C2211yescharacteristic
23C2×C3×#117C2211yescharacteristic
24C3×#450—22no—
25C3×#450—22no—
26C3×#447—22no—
27C3×#447—22no—
28C2×C3×#104C2214yescomplemented
29C3×#418C2214yes—
30C3×#407C2212yes—
31C3×#390—22no—
32C3×#390C2212yes—
33C3×#390C2212yes—
34C3×#379—22no—
35C3×#343C2212yescomplemented
36C3×#343C2212yescomplemented
37C3×#339C2214yescomplemented
38#3493C2×C311yescharacteristic, complemented
39#3003C2×C312yescomplemented
40#2309C2×C311yescharacteristic, complemented
41#2027C2×C312yescomplemented
42#2006C2×C312yescomplemented
43#1897C2×C312yescomplemented
44#1883C2×C314yescomplemented
45#1822C2×C311yescharacteristic, complemented
46C25×C3—21no—
47C2×C3×#41C2312yes—
48C22×C3×D4—24no—
49C22×C3×D4—21no—
50C22×C3×D4—22no—
51C22×C3×D4—22no—
52C22×C3×D4—42no—
53C22×C3×D4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C3×AGL(1,ℤ/8)—28no—
56C3×AGL(1,ℤ/8)—44no—
57C2×C3×SD16—24no—
58C2×C3×D8—24no—
59C2×C3×M16C2314yes—
60C3×C4⋊Q8—22no—
61C3×#126—24no—
62C3×#126—24no—
63C3×#125—22no—
64C3×#125—22no—
65C3×#124—22no—
66C3×#124—24no—
67C3×#123—24no—
68C3×#123—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8D412yes—
72C3×C22⋊Q8D412yes—
73C3×#121—24no—
74C3×#121—22no—
75C3×#121—22no—
76C3×#121D412yes—
77C3×#121D412yes—
78C3×#120—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—24no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×#117—24no—
87C3×#117—22no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117—42no—
91C3×#117C2312yes—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—24no—
98C2×C3×C22⋊C4—24no—
99C2×C3×C22⋊C4D412yes—
100C2×C3×C42D412yes—
101C2×C3×C42C2312yes—
102C3×#104—44no—
103C3×#102—24no—
104C3×#102—24no—
105C3×C22⋊C8—22no—
106C3×C22⋊C8—22no—
107C23×D4C22×C311yescharacteristic
108C2×AGL(1,ℤ/8)C22×C314yes—
109#556—22no—
110#555—22no—
111#553—22no—
112#552—22no—
113#544C22×C311yescharacteristic, complemented
114#538—22no—
115#537C22×C311yescharacteristic, complemented
116#529—22no—
117#525C22×C312yes—
118#520C22×C312yescomplemented
119C2×#117C22×C311yescharacteristic
120C2×#117C22×C311yescharacteristic
121#450—22no—
122#450—22no—
123#447—22no—
124#447—22no—
125C2×#104C22×C314yescomplemented
126#418C22×C314yes—
127#407C22×C312yes—
128#390—22no—
129#390C22×C312yes—
130#390C22×C312yes—
131#379—22no—
132#343C22×C312yescomplemented
133#343C22×C312yescomplemented
134#339C22×C314yescomplemented
135C24×C3—24no—
136C24×C3—22no—
137C24×C3—42no—
138C24×C3—42no—
139C24×C3—82no—
140C24×C3—82no—
141C24×C3C2×D411yescharacteristic
142C3×#41—24no—
143C3×#41—24no—
144C2×C3×Q8—22no—
145C2×C3×Q8C2×D412yes—
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—24no—
154C2×C3×D4—48no—
155C2×C3×D4—48no—
156C2×C3×D4—42no—
157C2×C3×D4—41no—
158C2×C3×D4—41no—
159C2×C3×D4C2×D412yes—
160C2×C3×D4#4114yes—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2411yescharacteristic
172C3×SD16—48no—
173C3×D8—48no—
174C3×M16—24no—
175C3×M16—24no—
176C2×C3×C8—24no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4C2×D412yes—
187C3×C4⋊C4C2×D412yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—84no—
198C3×C22⋊C4—84no—
199C3×C42—24no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—24no—
204C25—21no—
205C2×#41C23×C312yes—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—42no—
211C22×D4C23×C312yes—
212C23×C4C23×C311yescharacteristic
213AGL(1,ℤ/8)—28no—
214AGL(1,ℤ/8)—44no—
215C2×SD16—24no—
216C2×D8—24no—
217C2×M16C23×C314yes—
218C4⋊Q8—22no—
219#126—24no—
220#126—24no—
221#125—22no—
222#125—22no—
223#124—22no—
224#124—24no—
225#123—24no—
226#123—22no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C22⋊Q8C3×D412yes—
230C22⋊Q8C3×D412yes—
231#121—24no—
232#121—22no—
233#121—22no—
234#121C3×D412yes—
235#121C3×D412yes—
236#120—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—24no—
242C4×D4—22no—
243C4×D4—22no—
244#117—24no—
245#117—22no—
246#117—22no—
247#117—22no—
248#117—42no—
249#117C23×C312yes—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C4⋊C4C23×C311yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4C3×D412yes—
258C2×C42C3×D412yes—
259C2×C42C23×C312yes—
260#104—44no—
261#102—24no—
262#102—24no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—22no—
268C23×C3—24no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—44no—
272C23×C3—44no—
273C23×C3—48no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—84no—
277C23×C3—84no—
278C23×C3—88no—
279C23×C3—88no—
280C23×C3#12012yes—
281C23×C3C22×D411yescharacteristic
282C3×Q8—24no—
283C3×Q8—82no—
284C3×D4—24no—
285C3×D4—28no—
286C3×D4—44no—
287C3×D4—44no—
288C3×D4—42no—
289C3×D4—42no—
290C3×D4—44no—
291C3×D4—82no—
292C3×D4—82no—
293C3×D4—82no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—44no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—84no—
310C2×C3×C4#12011yescharacteristic
311C2×C3×C4#12011yescharacteristic
312C2×C3×C4C22×D412yes—
313C2×C3×C4C2×#4112yes—
314C2×C3×C4H(5,𝔽2)12yes—
315C3×C8—44no—
316C24—24no—
317C24—22no—
318C24—42no—
319C24—42no—
320C24—82no—
321C24—82no—
322C24C2×C3×D411yescharacteristic
323#41—24no—
324#41—24no—
325C2×Q8—22no—
326C2×Q8C2×C3×D412yes—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4C2×C3×D412yes—
341C2×D4C3×#4114yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D412yes—
349C22×C4C2×C3×D412yes—
350C22×C4C2×C3×D412yes—
351C22×C4C2×C3×D412yes—
352C22×C4C24×C311yescharacteristic
353SD16—48no—
354D8—48no—
355M16—24no—
356M16—24no—
357C2×C8—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4C2×C3×D412yes—
368C4⋊C4C2×C3×D412yes—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—42no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C42—24no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—24no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—48no—
393C22×C3—42no—
394C22×C3—44no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—84no—
398C22×C3—816no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3C2×#12011yescharacteristic
402C22×C3#52511yescharacteristic
403C22×C3#53511yescharacteristic
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—82no—
412C3×C4#54614yes—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—24no—
417C23—41no—
418C23—41no—
419C23—44no—
420C23—44no—
421C23—48no—
422C23—42no—
423C23—42no—
424C23—84no—
425C23—84no—
426C23—88no—
427C23—88no—
428C23C3×#12012yes—
429C23C22×C3×D411yescharacteristic
430Q8—24no—
431Q8—82no—
432D4—24no—
433D4—28no—
434D4—44no—
435D4—44no—
436D4—42no—
437D4—42no—
438D4—44no—
439D4—82no—
440D4—82no—
441D4—82no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—84no—
458C2×C4C3×#12011yescharacteristic
459C2×C4C3×#12011yescharacteristic
460C2×C4C22×C3×D412yes—
461C2×C4C2×C3×#4112yes—
462C2×C4C3×H(5,𝔽2)12yes—
463C8—44no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—42no—
467C2×C3—82no—
468C2×C3—84no—
469C2×C3#240411yescharacteristic
470C2×C3#302111yescharacteristic
471C2×C3#302111yescharacteristic
472C22—21no—
473C22—21no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—42no—
478C22—42no—
479C22—48no—
480C22—42no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—84no—
485C22—816no—
486C22—82no—
487C22—82no—
488C22C2×C3×#12011yescharacteristic
489C22C3×#52511yescharacteristic
490C22C3×#53511yescharacteristic
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—82no—
499C4C3×#54614yes—
500C3#2395211yescharacteristic, direct product factor, Sylow
501C2—21no—
502C2—21no—
503C2—42no—
504C2—84no—
505C2—82no—
506C2C3×#240411yescharacteristic
507C2C3×#302111yescharacteristic
508C2C3×#302111yescharacteristic
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×#3493C2semidirect
2. C3×#3003C2semidirect
3. C3×#2309C2semidirect
4. C3×#2027C2semidirect
5. C3×#2006C2semidirect
6. C3×#1897C2semidirect
7. C3×#1883C2semidirect
8. C3×#1822C2semidirect
9. #23952C3direct (N × H)
16. C3×#544C22semidirect
18. C3×#537C22semidirect
21. C3×#520C22semidirect
28. C2×C3×#104C22semidirect
35. C3×#343C22semidirect
36. C3×#343C22semidirect
37. C3×#339C22semidirect
38. #3493C2×C3semidirect
39. #3003C2×C3semidirect
40. #2309C2×C3semidirect
41. #2027C2×C3semidirect
42. #2006C2×C3semidirect
43. #1897C2×C3semidirect
44. #1883C2×C3semidirect
45. #1822C2×C3semidirect
113. #544C22×C3semidirect
115. #537C22×C3semidirect
118. #520C22×C3semidirect
125. C2×#104C22×C3semidirect
132. #343C22×C3semidirect
133. #343C22×C3semidirect
134. #339C22×C3semidirect
500. C3#23952direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34932
C3×#30032
C3×#23092
C3×#20272
C3×#20062
C3×#18972
C3×#18832
C3×#18222
#239523