Finite Group Databasebeta

← C3×#23910 #10642517

Subgroups — C3×#23910 #10642517

Order 768.

1866 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3486C211yesmaximal, characteristic, complemented
2C22×C3×#124C211yesmaximal, characteristic, complemented
3C2×C3×#454C212yesmaximal, complemented
4C3×#3003C212yesmaximal, complemented
5C3×#2003C214yesmaximal, complemented
6C3×#1896C212yesmaximal, complemented
7C3×#1888C212yesmaximal, complemented
8C3×#1784C211yesmaximal, characteristic, complemented
9#23910C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×2−1+4C2211yescharacteristic, complemented
11C23×C3×Q8C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C2×C3×AGL(1,ℤ/8)C2214yes—
14C3×D4×Q8—24no—
15C3×#548—24no—
16C3×#536—22no—
17C3×#536C2212yescomplemented
18C2×C3×#124—22no—
19C2×C3×#124—28no—
20C2×C3×#124C2212yes—
21C2×C3×#124C2212yes—
22C3×#525—22no—
23C3×#525C2212yes—
24C3×#519C2211yescharacteristic, complemented
25C22×C3×C22⋊C4—22no—
26C22×C3×C42C2211yescharacteristic
27C3×#454—28no—
28C3×#450—24no—
29C3×#447—24no—
30C2×C3×#104C2214yescomplemented
31C3×#418C2214yes—
32C2×C3×#101C2212yescomplemented
33C2×C3×#101C2212yescomplemented
34C3×#407C2212yes—
35C3×#339C2214yescomplemented
36#3486C2×C311yescharacteristic, complemented
37C22×#124C2×C311yescharacteristic, complemented
38C2×#454C2×C312yescomplemented
39#3003C2×C312yescomplemented
40#2003C2×C314yescomplemented
41#1896C2×C312yescomplemented
42#1888C2×C312yescomplemented
43#1784C2×C311yescharacteristic, complemented
44C25×C3—21no—
45C3×2−1+4—24no—
46C3×2−1+4—42no—
47C2×C3×#41—22no—
48C2×C3×#41—22no—
49C2×C3×#41C2312yes—
50C22×C3×Q8—24no—
51C22×C3×Q8—22no—
52C22×C3×Q8—24no—
53C22×C3×Q8—42no—
54C22×C3×Q8D412yes—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—24no—
58C22×C3×D4—21no—
59C22×C3×D4—24no—
60C22×C3×D4—42no—
61C22×C3×D4C2312yes—
62C23×C3×C4—21no—
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×C4⋊Q8—24no—
71C3×C4⋊Q8—22no—
72C3×#124—216no—
73C3×#124—24no—
74C3×#124—24no—
75C3×#124—24no—
76C3×#124—22no—
77C3×#124—24no—
78C3×#124D418yescomplemented
79C3×#123—24no—
80C3×#123—22no—
81C3×C22⋊Q8—24no—
82C3×C22⋊Q8—28no—
83C3×C22⋊Q8—24no—
84C3×C22⋊Q8D414yes—
85C3×#121—24no—
86C3×#121D414yes—
87C3×#120—22no—
88C3×C4×Q8—24no—
89C3×C4×D4—28no—
90C3×C4×D4—24no—
91C3×#117—22no—
92C3×#117C2312yes—
93C2×C3×C22⋊C4—216no—
94C2×C3×C22⋊C4—24no—
95C2×C3×C22⋊C4—24no—
96C2×C3×C42—21no—
97C2×C3×C42—24no—
98C2×C3×C42C2312yes—
99C3×#104—28no—
100C3×#102—28no—
101C3×#101—24no—
102C3×#101—42no—
103C3×C22⋊C8—24no—
104C2×2−1+4C22×C311yescharacteristic, complemented
105C23×Q8C22×C311yescharacteristic
106C23×D4C22×C311yescharacteristic
107C2×AGL(1,ℤ/8)C22×C314yes—
108D4×Q8—24no—
109#548—24no—
110#536—22no—
111#536C22×C312yescomplemented
112C2×#124—22no—
113C2×#124—28no—
114C2×#124C22×C312yes—
115C2×#124C22×C312yes—
116#525—22no—
117#525C22×C312yes—
118#519C22×C311yescharacteristic, complemented
119C22×C22⋊C4—22no—
120C22×C42C22×C311yescharacteristic
121#454—28no—
122#450—24no—
123#447—24no—
124C2×#104C22×C314yescomplemented
125#418C22×C314yes—
126C2×#101C22×C312yescomplemented
127C2×#101C22×C312yescomplemented
128#407C22×C312yes—
129#339C22×C314yescomplemented
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—48no—
133C24×C3—42no—
134C24×C3—42no—
135C24×C3C2×D411yescharacteristic
136C3×#41—24no—
137C3×#41—24no—
138C3×#41—44no—
139C3×#41—48no—
140C3×#41—44no—
141C2×C3×Q8—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8—24no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—42no—
150C2×C3×Q8—416no—
151C2×C3×Q8—48no—
152C2×C3×Q8C2×D412yes—
153C2×C3×Q8C2×D414yes—
154C2×C3×D4—22no—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—22no—
158C2×C3×D4—24no—
159C2×C3×D4—22no—
160C2×C3×D4—24no—
161C2×C3×D4—416no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4C2×D412yes—
165C2×C3×D4C2×D414yes—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—42no—
175C22×C3×C4—44no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2411yescharacteristic
181C3×SD16—48no—
182C3×D8—48no—
183C3×M16—24no—
184C3×M16—24no—
185C2×C3×C8—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4C2×D414yes—
190C3×C22⋊C4—216no—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—216no—
195C3×C22⋊C4—24no—
196C3×C42—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—22no—
205C2×#41—22no—
206C2×#41C23×C312yes—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—42no—
211C22×Q8C3×D412yes—
212C22×Q8C23×C311yescharacteristic
213C22×Q8C23×C311yescharacteristic
214C22×D4—24no—
215C22×D4—21no—
216C22×D4—24no—
217C22×D4—42no—
218C22×D4C23×C312yes—
219C23×C4—21no—
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—
227C4⋊Q8—24no—
228C4⋊Q8—22no—
229#124—216no—
230#124—24no—
231#124—24no—
232#124—24no—
233#124—22no—
234#124—24no—
235#124C3×D418yescomplemented
236#123—24no—
237#123—22no—
238C22⋊Q8—24no—
239C22⋊Q8—28no—
240C22⋊Q8—24no—
241C22⋊Q8C3×D414yes—
242#121—24no—
243#121C3×D414yes—
244#120—22no—
245C4×Q8—24no—
246C4×D4—28no—
247C4×D4—24no—
248#117—22no—
249#117C23×C312yes—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C42—21no—
254C2×C42—24no—
255C2×C42C23×C312yes—
256#104—28no—
257#102—28no—
258#101—24no—
259#101—42no—
260C22⋊C8—24no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—24no—
264C23×C3—24no—
265C23×C3—22no—
266C23×C3—48no—
267C23×C3—416no—
268C23×C3—48no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—88no—
272C23×C3—84no—
273C23×C3—84no—
274C23×C3#12012yes—
275C23×C3C22×D411yescharacteristic
276C3×Q8—28no—
277C3×Q8—24no—
278C3×Q8—48no—
279C3×Q8—44no—
280C3×Q8—44no—
281C3×D4—24no—
282C3×D4—28no—
283C3×D4—48no—
284C3×D4—44no—
285C3×D4—42no—
286C3×D4—42no—
287C3×D4—48no—
288C3×D4—44no—
289C2×C3×C4—21no—
290C2×C3×C4—22no—
291C2×C3×C4—21no—
292C2×C3×C4—22no—
293C2×C3×C4—28no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—48no—
301C2×C3×C4—48no—
302C2×C3×C4—48no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4#12014yes—
306C2×C3×C4#12012yes—
307C2×C3×C4C22×D412yes—
308C2×C3×C4C22×D412yes—
309C2×C3×C4H(5,𝔽2)12yes—
310C3×C8—44no—
311C24—24no—
312C24—22no—
313C24—42no—
314C24—48no—
315C24—42no—
316C24C2×C3×D411yescharacteristic
317#41—24no—
318#41—24no—
319#41—44no—
320#41—48no—
321#41—44no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—42no—
331C2×Q8—416no—
332C2×Q8—48no—
333C2×Q8C2×C3×D412yes—
334C2×Q8C2×C3×D414yes—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—416no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4C2×C3×D412yes—
346C2×D4C2×C3×D414yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4C2×C3×D412yes—
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D414yes—
361C22×C4C24×C311yescharacteristic
362SD16—48no—
363D8—48no—
364M16—24no—
365M16—24no—
366C2×C8—24no—
367C4⋊C4—24no—
368C4⋊C4—28no—
369C4⋊C4—24no—
370C4⋊C4C2×C3×D414yes—
371C22⋊C4—216no—
372C22⋊C4—28no—
373C22⋊C4—24no—
374C22⋊C4—22no—
375C22⋊C4—216no—
376C22⋊C4—24no—
377C42—24no—
378C42—24no—
379C42—22no—
380C42—22no—
381C42C2×C3×D414yes—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—48no—
388C22×C3—42no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—84no—
392C22×C3—816no—
393C22×C3—84no—
394C22×C3#45414yescomplemented
395C22×C3C2×#12011yescharacteristic
396C22×C3C2×#12011yescharacteristic
397C22×C3#53411yescharacteristic
398C3×C4—22no—
399C3×C4—24no—
400C3×C4—44no—
401C3×C4—42no—
402C3×C4—42no—
403C3×C4—44no—
404C3×C4D4214yes—
405C23—24no—
406C23—21no—
407C23—21no—
408C23—24no—
409C23—22no—
410C23—48no—
411C23—416no—
412C23—48no—
413C23—42no—
414C23—42no—
415C23—88no—
416C23—84no—
417C23—84no—
418C23C3×#12012yes—
419C23C22×C3×D411yescharacteristic
420Q8—28no—
421Q8—24no—
422Q8—48no—
423Q8—44no—
424Q8—44no—
425D4—24no—
426D4—28no—
427D4—48no—
428D4—44no—
429D4—48no—
430D4—42no—
431D4—42no—
432D4—44no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4C3×#12014yes—
450C2×C4C3×#12012yes—
451C2×C4C22×C3×D412yes—
452C2×C4C22×C3×D412yes—
453C2×C4C3×H(5,𝔽2)12yes—
454C8—44no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—24no—
458C2×C3—82no—
459C2×C3—84no—
460C2×C3#240311yescharacteristic
461C2×C3C2×#45412yes—
462C22—24no—
463C22—24no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—48no—
468C22—42no—
469C22—44no—
470C22—44no—
471C22—816no—
472C22—84no—
473C22—84no—
474C22C3×#45414yescomplemented
475C22C2×C3×#12011yescharacteristic
476C22C2×C3×#12011yescharacteristic
477C22C3×#53411yescharacteristic
478C4—22no—
479C4—24no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—44no—
484C4C3×D4214yes—
485C3#2391011yescharacteristic, direct product factor, Sylow
486C2—21no—
487C2—24no—
488C2—21no—
489C2—84no—
490C2—82no—
491C2C3×#240311yescharacteristic
492C2C2×C3×#45412yes—
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×#3486C2semidirect
2. C22×C3×#124C2semidirect
3. C2×C3×#454C2semidirect
4. C3×#3003C2semidirect
5. C3×#2003C2semidirect
6. C3×#1896C2semidirect
7. C3×#1888C2semidirect
8. C3×#1784C2semidirect
9. #23910C3direct (N × H)
10. C2×C3×2−1+4C22semidirect
17. C3×#536C22semidirect
24. C3×#519C22semidirect
30. C2×C3×#104C22semidirect
32. C2×C3×#101C22semidirect
33. C2×C3×#101C22semidirect
35. C3×#339C22semidirect
36. #3486C2×C3semidirect
37. C22×#124C2×C3semidirect
38. C2×#454C2×C3semidirect
39. #3003C2×C3semidirect
40. #2003C2×C3semidirect
41. #1896C2×C3semidirect
42. #1888C2×C3semidirect
43. #1784C2×C3semidirect
78. C3×#124D4semidirect
104. C2×2−1+4C22×C3semidirect
111. #536C22×C3semidirect
118. #519C22×C3semidirect
124. C2×#104C22×C3semidirect
126. C2×#101C22×C3semidirect
127. C2×#101C22×C3semidirect
129. #339C22×C3semidirect
235. #124C3×D4semidirect
394. C22×C3#454semidirect
474. C22C3×#454semidirect
485. C3#23910direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34862
C22×C3×#1242
C2×C3×#4542
C3×#30032
C3×#20032
C3×#18962
C3×#18882
C3×#17842
#239103