Finite Group Databasebeta

← C3×#25753 #10644360

Subgroups — C3×#25753 #10644360

Order 768.

1998 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D42C211yesmaximal, characteristic
2C2×C3×#492C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C2×C3×#450C214yesmaximal, complemented
5C3×#2542C211yesmaximal, characteristic, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1982C211yesmaximal, characteristic, complemented
8C3×#1954C211yesmaximal, characteristic, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25753C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4—21no—
12C23×C3×D4C2212yes—
13C22×C3×SD16C2212yes—
14C22×C3×SD16C2214yescomplemented
15C3×D42—24no—
16C3×D42—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic, complemented
18C2×C3×#127C2211yescharacteristic
19C2×C3×C22⋊Q8C2214yescomplemented
20C2×C3×#121—21no—
21C2×C3×#121C2212yes—
22C2×C3×#120—22no—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C3×#492—24no—
26C3×#461—24no—
27C3×#450—216no—
28C2×C3×C4⋊C8C2211yescharacteristic, complemented
29C2×C3×#103C2212yescomplemented
30C2×C3×#102C2212yes—
31C2×C3×#102C2214yescomplemented
32C2×C3×C22⋊C8C2214yescomplemented
33C2×C3×C4×C8C2211yescharacteristic, complemented
34C3×#399—22no—
35C3×#390—21no—
36C3×#384—21no—
37C3×C4×C4⋊C4C2211yescharacteristic, complemented
38C3×#340C2212yescomplemented
39C2×D42C2×C311yescharacteristic
40C2×#492C2×C311yescharacteristic, complemented
41C2×#461C2×C311yescharacteristic, complemented
42C2×#450C2×C314yescomplemented
43#2542C2×C311yescharacteristic, complemented
44#2000C2×C314yescomplemented
45#1982C2×C311yescharacteristic, complemented
46#1954C2×C311yescharacteristic, complemented
47C4×#102C2×C311yescharacteristic, complemented
48C25×C3—22no—
49C22×C3×Q8C2312yes—
50C22×C3×D4—28no—
51C22×C3×D4—22no—
52C22×C3×D4—24no—
53C22×C3×D4—24no—
54C22×C3×D4—28no—
55C22×C3×D4—21no—
56C22×C3×D4—22no—
57C22×C3×D4—21no—
58C22×C3×D4—44no—
59C22×C3×D4—44no—
60C22×C3×D4—44no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2312yes—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2312yes—
66C2×C3×SD16—28no—
67C2×C3×SD16—24no—
68C2×C3×SD16—28no—
69C2×C3×SD16—28no—
70C2×C3×SD16—44no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2312yes—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×#127—22no—
76C3×#127D414yes—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—44no—
79C3×#121—28no—
80C3×#121—24no—
81C3×#121—24no—
82C3×#120—216no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4D414yes—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—24no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—24no—
96C2×C3×C22⋊C4D412yes—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C3×C4⋊C8—22no—
100C3×#103—24no—
101C3×#102—28no—
102C3×#102D418yescomplemented
103C3×C22⋊C8—28no—
104C3×C4×C8—22no—
105C3×#95—21no—
106C23×D4—21no—
107C23×D4C22×C312yes—
108C22×SD16C22×C312yes—
109C22×SD16C22×C314yescomplemented
110D42—24no—
111D42—24no—
112C2×C4⋊Q8C22×C311yescharacteristic, complemented
113C2×#127C22×C311yescharacteristic
114C2×C22⋊Q8C22×C314yescomplemented
115C2×#121—21no—
116C2×#121C22×C312yes—
117C2×#120—22no—
118C2×C4×D4C22×C311yescharacteristic
119C2×C4×D4C22×C311yescharacteristic
120#492—24no—
121#461—24no—
122#450—216no—
123C2×C4⋊C8C22×C311yescharacteristic, complemented
124C2×#103C22×C312yescomplemented
125C2×#102C22×C312yes—
126C2×#102C22×C314yescomplemented
127C2×C22⋊C8C22×C314yescomplemented
128C2×C4×C8C22×C311yescharacteristic, complemented
129#399—22no—
130#390—21no—
131#384—21no—
132C4×C4⋊C4C22×C311yescharacteristic, complemented
133#340C22×C312yescomplemented
134C24×C3—21no—
135C24×C3—28no—
136C24×C3—22no—
137C24×C3—48no—
138C24×C3—44no—
139C24×C3—44no—
140C24×C3—84no—
141C24×C3C2×D412yes—
142C2×C3×Q8—22no—
143C2×C3×Q8—24no—
144C2×C3×Q8—22no—
145C2×C3×Q8—24no—
146C2×C3×D4—28no—
147C2×C3×D4—28no—
148C2×C3×D4—22no—
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4—22no—
152C2×C3×D4—21no—
153C2×C3×D4—24no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—24no—
157C2×C3×D4—48no—
158C2×C3×D4—416no—
159C2×C3×D4—48no—
160C2×C3×D4—432no—
161C2×C3×D4—44no—
162C2×C3×D4—44no—
163C2×C3×D4—48no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—82no—
167C2×C3×D4SD1614yescomplemented
168C2×C3×D4C2×D414yes—
169C2×C3×D4C2×D414yes—
170C2×C3×D4C2×D412yes—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C3×SD16—416no—
186C3×SD16—48no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—24no—
190C2×C3×C8—22no—
191C2×C3×C8C2×D414yescomplemented
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—48no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4C2×D412yes—
199C3×C4⋊C4C2×D414yescomplemented
200C3×C22⋊C4—28no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—48no—
204C3×C42—42no—
205C3×C42C2×D412yes—
206C3×C42C2×D412yes—
207C25—22no—
208C22×Q8C23×C312yes—
209C22×D4—28no—
210C22×D4—22no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—28no—
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4C3×D411yescharacteristic
221C22×D4C23×C312yes—
222C22×D4C23×C311yescharacteristic
223C23×C4—21no—
224C23×C4C23×C312yes—
225C2×SD16—28no—
226C2×SD16—24no—
227C2×SD16—28no—
228C2×SD16—28no—
229C2×SD16—44no—
230C22×C8C23×C312yes—
231C22×C8C23×C312yes—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234#127—22no—
235#127C3×D414yes—
236C22⋊Q8—28no—
237C22⋊Q8—44no—
238#121—28no—
239#121—24no—
240#121—24no—
241#120—216no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4C3×D414yes—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4C3×D411yescharacteristic
250C2×C4⋊C4C23×C311yescharacteristic
251C2×C4⋊C4C23×C312yes—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4C3×D412yes—
256C2×C42—21no—
257C2×C42C23×C311yescharacteristic
258C4⋊C8—22no—
259#103—24no—
260#102—28no—
261#102C3×D418yescomplemented
262C22⋊C8—28no—
263C4×C8—22no—
264#95—21no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—24no—
268C23×C3—24no—
269C23×C3—22no—
270C23×C3—24no—
271C23×C3—24no—
272C23×C3—416no—
273C23×C3—416no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—42no—
277C23×C3—44no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—44no—
281C23×C3—816no—
282C23×C3—88no—
283C23×C3—816no—
284C23×C3C2×SD1614yes—
285C23×C3C22×D411yescharacteristic
286C3×Q8—42no—
287C3×Q8—42no—
288C3×Q8—44no—
289C3×D4—22no—
290C3×D4—24no—
291C3×D4—28no—
292C3×D4—28no—
293C3×D4—416no—
294C3×D4—416no—
295C3×D4—42no—
296C3×D4—42no—
297C3×D4—48no—
298C3×D4—84no—
299C3×D4#12714yes—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—44no—
312C2×C3×C4—48no—
313C2×C3×C4—42no—
314C2×C3×C4—44no—
315C2×C3×C4#12012yes—
316C2×C3×C4#12012yes—
317C2×C3×C4C2×SD1612yes—
318C2×C3×C4AGL(1,ℤ/8)12yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D412yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4H(5,𝔽2)12yes—
323C3×C8—24no—
324C3×C8—42no—
325C24—21no—
326C24—28no—
327C24—22no—
328C24—48no—
329C24—44no—
330C24—44no—
331C24—84no—
332C24C2×C3×D412yes—
333C2×Q8—22no—
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—24no—
348C2×D4—48no—
349C2×D4—416no—
350C2×D4—432no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—82no—
358C2×D4C3×SD1614yescomplemented
359C2×D4C2×C3×D414yes—
360C2×D4C2×C3×D414yes—
361C2×D4C2×C3×D412yes—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4C2×C3×D412yes—
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C24×C311yescharacteristic
376SD16—416no—
377SD16—48no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8C2×C3×D414yescomplemented
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—48no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4C2×C3×D412yes—
390C4⋊C4C2×C3×D414yescomplemented
391C22⋊C4—28no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—48no—
395C42—42no—
396C42C2×C3×D412yes—
397C42C2×C3×D412yes—
398C22×C3—24no—
399C22×C3—24no—
400C22×C3—28no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—44no—
410C22×C3—44no—
411C22×C3—832no—
412C22×C3—84no—
413C22×C3—84no—
414C22×C3C2×#12011yescharacteristic
415C22×C3D4212yes—
416C22×C3C22×SD1611yescharacteristic
417C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
418C22×C3#57712yes—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4—44no—
423C3×C4—82no—
424C3×C4#45014yescomplemented
425C3×C4C2×#12711yescharacteristic
426C3×C4#53511yescharacteristic
427C3×C4D4212yes—
428C23—22no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—416no—
436C23—416no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23—816no—
445C23—88no—
446C23—816no—
447C23C2×C3×SD1614yes—
448C23C22×C3×D411yescharacteristic
449Q8—42no—
450Q8—42no—
451Q8—44no—
452D4—28no—
453D4—22no—
454D4—24no—
455D4—28no—
456D4—416no—
457D4—416no—
458D4—48no—
459D4—42no—
460D4—42no—
461D4—84no—
462D4C3×#12714yes—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4C3×#12012yes—
479C2×C4C3×#12012yes—
480C2×C4C2×C3×SD1612yes—
481C2×C4C3×AGL(1,ℤ/8)12yes—
482C2×C4C22×C3×D411yescharacteristic
483C2×C4C22×C3×D412yes—
484C2×C4C22×C3×D411yescharacteristic
485C2×C4C3×H(5,𝔽2)12yes—
486C8—24no—
487C8—42no—
488C2×C3—44no—
489C2×C3—44no—
490C2×C3—84no—
491C2×C3#243311yescharacteristic
492C2×C3C2×#45011yescharacteristic
493C2×C3#300911yescharacteristic
494C2×C3#327212yes—
495C2×C3D4×SD1612yes—
496C22—24no—
497C22—24no—
498C22—28no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—832no—
510C22—84no—
511C22—84no—
512C22C2×C3×#12011yescharacteristic
513C22C3×D4212yes—
514C22C22×C3×SD1611yescharacteristic
515C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
516C22C3×#57712yes—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—44no—
521C4—82no—
522C4C3×#45014yescomplemented
523C4C2×C3×#12711yescharacteristic
524C4C3×#53511yescharacteristic
525C4C3×D4212yes—
526C3#2575311yescharacteristic, direct product factor, Sylow
527C2—44no—
528C2—44no—
529C2—84no—
530C2C3×#243311yescharacteristic
531C2C2×C3×#45011yescharacteristic
532C2C3×#300911yescharacteristic
533C2C3×#327212yes—
534C2C3×D4×SD1612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×D42C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#492C2semidirect
3. C2×C3×#461C2semidirect
4. C2×C3×#450C2semidirect
5. C3×#2542C2semidirect
6. C3×#2000C2semidirect
7. C3×#1982C2semidirect
8. C3×#1954C2semidirect
9. C3×C4×#102C2semidirect
10. #25753C3direct (N × H)
14. C22×C3×SD16C22semidirect
17. C2×C3×C4⋊Q8C22semidirect
19. C2×C3×C22⋊Q8C22semidirect
28. C2×C3×C4⋊C8C22semidirect
29. C2×C3×#103C22semidirect
31. C2×C3×#102C22semidirect
32. C2×C3×C22⋊C8C22semidirect
33. C2×C3×C4×C8C22semidirect
37. C3×C4×C4⋊C4C22semidirect
38. C3×#340C22semidirect
40. C2×#492C2×C3semidirect
41. C2×#461C2×C3semidirect
42. C2×#450C2×C3semidirect
43. #2542C2×C3semidirect
44. #2000C2×C3semidirect
45. #1982C2×C3semidirect
46. #1954C2×C3semidirect
47. C4×#102C2×C3semidirect
102. C3×#102D4semidirect
109. C22×SD16C22×C3semidirect
112. C2×C4⋊Q8C22×C3semidirect
114. C2×C22⋊Q8C22×C3semidirect
123. C2×C4⋊C8C22×C3semidirect
124. C2×#103C22×C3semidirect
126. C2×#102C22×C3semidirect
127. C2×C22⋊C8C22×C3semidirect
128. C2×C4×C8C22×C3semidirect
132. C4×C4⋊C4C22×C3semidirect
133. #340C22×C3semidirect
167. C2×C3×D4SD16semidirect
191. C2×C3×C8C2×D4semidirect
199. C3×C4⋊C4C2×D4semidirect
261. #102C3×D4semidirect
358. C2×D4C3×SD16semidirect
382. C2×C8C2×C3×D4semidirect
390. C4⋊C4C2×C3×D4semidirect
424. C3×C4#450semidirect
522. C4C3×#450semidirect
526. C3#25753direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D422
C2×C3×#4922
C2×C3×#4612
C2×C3×#4502
C3×#25422
C3×#20002
C3×#19822
C3×#19542
C3×C4×#1022
#257533