Finite Group Databasebeta

← C2×C3×#1510 #10636158

Subgroups — C2×C3×#1510 #10636158

Order 768.

1790 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C2×C3×#407C211yesmaximal, characteristic, complemented
3C2×C3×#381C211yesmaximal, characteristic, complemented
4C3×#1510C218yesmaximal, direct product factor
5C2×C3×#327C214yesmaximal, complemented
6C2×#1510C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×D4—21no—
8C23×C3×D4C411yescharacteristic, complemented
9C24×C3×C4C2211yescharacteristic
10C22×C3×M16—21no—
11C2×C3×#121—21no—
12C2×C3×#121—22no—
13C2×C3×#121—21no—
14C2×C3×#121—22no—
15C2×C3×#121C412yescomplemented
16C2×C3×#121C414yescomplemented
17C2×C3×#121C2212yes—
18C2×C3×#121C2214yes—
19C22×C3×C4⋊C4C411yescharacteristic, complemented
20C22×C3×C22⋊C4—21no—
21C22×C3×C22⋊C4—21no—
22C3×#407—22no—
23C3×#407C2214yescomplemented
24C2×C3×C22⋊C8—21no—
25C2×C3×C22⋊C8C2212yescomplemented
26C3×#381—22no—
27C3×#381C2214yescomplemented
28C2×C3×#95—21no—
29C2×C3×#95C2212yescomplemented
30C3×#327C22116yescomplemented
31C22×#121C2×C311yescharacteristic
32C2×#407C2×C311yescharacteristic, complemented
33C2×#381C2×C311yescharacteristic, complemented
34#1510C2×C318yesdirect product factor
35C2×#327C2×C314yescomplemented
36C25×C3—21no—
37C25×C3D411yescharacteristic
38C22×C3×D4—24no—
39C22×C3×D4—22no—
40C22×C3×D4—24no—
41C22×C3×D4—22no—
42C22×C3×D4—22no—
43C22×C3×D4—21no—
44C22×C3×D4—44no—
45C22×C3×D4—48no—
46C22×C3×D4C2×C412yes—
47C22×C3×D4C2×C414yescomplemented
48C23×C3×C4—21no—
49C23×C3×C4—22no—
50C23×C3×C4—44no—
51C23×C3×C4C2×C411yescharacteristic
52C23×C3×C4C2×C412yes—
53C23×C3×C4C2×C411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D412yes—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D412yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C2×C3×M16—24no—
62C2×C3×M16—44no—
63C22×C3×C8—22no—
64C3×#121—24no—
65C3×#121—24no—
66C3×#121—24no—
67C3×#121—22no—
68C3×#121—24no—
69C3×#121—22no—
70C3×#121—24no—
71C3×#121C2×C414yes—
72C3×#121C2×C418yescomplemented
73C3×#121C2314yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2×C412yes—
77C2×C3×C4⋊C4C2×C414yescomplemented
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—44no—
84C3×C22⋊C8—24no—
85C3×C22⋊C8—22no—
86C3×C22⋊C8C2314yescomplemented
87C3×#95—24no—
88C3×#95—22no—
89C3×#95C2314yescomplemented
90C23×D4—21no—
91C23×D4C3×C411yescharacteristic, complemented
92C24×C4C22×C311yescharacteristic
93C22×M16—21no—
94C2×#121—21no—
95C2×#121—22no—
96C2×#121—21no—
97C2×#121—22no—
98C2×#121C3×C412yescomplemented
99C2×#121C3×C414yescomplemented
100C2×#121C22×C312yes—
101C2×#121C22×C314yes—
102C22×C4⋊C4C3×C411yescharacteristic, complemented
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105#407—22no—
106#407C22×C314yescomplemented
107C2×C22⋊C8—21no—
108C2×C22⋊C8C22×C312yescomplemented
109#381—22no—
110#381C22×C314yescomplemented
111C2×#95—21no—
112C2×#95C22×C312yescomplemented
113#327C22×C3116yescomplemented
114C24×C3—24no—
115C24×C3—24no—
116C24×C3—24no—
117C24×C3—22no—
118C24×C3—42no—
119C24×C3—44no—
120C24×C3—44no—
121C24×C3—44no—
122C24×C3C22⋊C411yescharacteristic
123C24×C3C22⋊C412yes—
124C24×C3C22⋊C411yescharacteristic
125C24×C3C2×D412yes—
126C24×C3C2×D411yescharacteristic
127C2×C3×D4—24no—
128C2×C3×D4—28no—
129C2×C3×D4—24no—
130C2×C3×D4—22no—
131C2×C3×D4—24no—
132C2×C3×D4—44no—
133C2×C3×D4—48no—
134C2×C3×D4—48no—
135C2×C3×D4—44no—
136C2×C3×D4—416no—
137C2×C3×D4—416no—
138C2×C3×D4—48no—
139C2×C3×D4—48no—
140C2×C3×D4C22×C414yes—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—44no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—44no—
150C22×C3×C4—42no—
151C22×C3×C4—44no—
152C22×C3×C4—42no—
153C22×C3×C4—44no—
154C22×C3×C4—44no—
155C22×C3×C4—44no—
156C22×C3×C4—44no—
157C22×C3×C4—44no—
158C22×C3×C4C22⋊C412yes—
159C22×C3×C4C22⋊C412yes—
160C22×C3×C4C22⋊C411yescharacteristic
161C22×C3×C4C22⋊C411yescharacteristic
162C22×C3×C4C22⋊C412yes—
163C22×C3×C4C22⋊C412yes—
164C22×C3×C4C22⋊C412yes—
165C22×C3×C4C22×C412yes—
166C22×C3×C4C22×C411yescharacteristic
167C22×C3×C4C22×C411yescharacteristic
168C22×C3×C4C22×C412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C3×M16—48no—
177C2×C3×C8—24no—
178C2×C3×C8—44no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4C22×C414yes—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—28no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—48no—
186C25—21no—
187C25C3×D411yescharacteristic
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—44no—
195C22×D4—48no—
196C22×D4C2×C3×C412yes—
197C22×D4C2×C3×C414yescomplemented
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—44no—
201C23×C4C2×C3×C411yescharacteristic
202C23×C4C2×C3×C411yescharacteristic
203C23×C4C2×C3×C412yes—
204C23×C4C3×D411yescharacteristic
205C23×C4C3×D411yescharacteristic
206C23×C4C3×D412yes—
207C23×C4C3×D411yescharacteristic
208C23×C4C3×D412yes—
209C23×C4C23×C311yescharacteristic
210C23×C4C23×C312yes—
211C2×M16—24no—
212C2×M16—44no—
213C22×C8—22no—
214#121—24no—
215#121—24no—
216#121—24no—
217#121—22no—
218#121—24no—
219#121—22no—
220#121—24no—
221#121C2×C3×C414yes—
222#121C2×C3×C418yescomplemented
223#121C23×C314yes—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C2×C3×C412yes—
227C2×C4⋊C4C2×C3×C414yescomplemented
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—44no—
234C22⋊C8—24no—
235C22⋊C8—22no—
236C22⋊C8C23×C314yescomplemented
237#95—24no—
238#95—22no—
239#95C23×C314yescomplemented
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—24no—
245C23×C3—24no—
246C23×C3—24no—
247C23×C3—22no—
248C23×C3—24no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—24no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—416no—
255C23×C3—48no—
256C23×C3—48no—
257C23×C3—48no—
258C23×C3—44no—
259C23×C3—44no—
260C23×C3—44no—
261C23×C3—816no—
262C23×C3#9914yes—
263C23×C3C2×C22⋊C412yes—
264C23×C3C2×C22⋊C412yes—
265C23×C3C2×C22⋊C411yescharacteristic
266C23×C3C2×C22⋊C411yescharacteristic
267C23×C3AGL(1,ℤ/8)14yes—
268C23×C3C22×D411yescharacteristic
269C3×D4—44no—
270C3×D4—44no—
271C3×D4—48no—
272C3×D4—416no—
273C3×D4—48no—
274C3×D4—48no—
275C2×C3×C4—22no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—48no—
295C2×C3×C4—44no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4—44no—
299C2×C3×C4—88no—
300C2×C3×C4C2×C22⋊C411yescharacteristic
301C2×C3×C4C2×C22⋊C412yes—
302C2×C3×C4C2×C22⋊C411yescharacteristic
303C2×C3×C4C2×C22⋊C412yes—
304C2×C3×C4C23×C411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C3×C8—44no—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—24no—
311C24—42no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24C3×C22⋊C411yescharacteristic
316C24C3×C22⋊C412yes—
317C24C3×C22⋊C411yescharacteristic
318C24C2×C3×D412yes—
319C24C2×C3×D411yescharacteristic
320C2×D4—24no—
321C2×D4—28no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—44no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—44no—
329C2×D4—416no—
330C2×D4—416no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4C22×C3×C414yes—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4C3×C22⋊C412yes—
352C22×C4C3×C22⋊C411yescharacteristic
353C22×C4C3×C22⋊C412yes—
354C22×C4C3×C22⋊C411yescharacteristic
355C22×C4C3×C22⋊C412yes—
356C22×C4C3×C22⋊C412yes—
357C22×C4C3×C22⋊C412yes—
358C22×C4C22×C3×C411yescharacteristic
359C22×C4C22×C3×C412yes—
360C22×C4C22×C3×C411yescharacteristic
361C22×C4C22×C3×C412yes—
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D412yes—
364C22×C4C2×C3×D412yes—
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D412yes—
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C24×C311yescharacteristic
369M16—48no—
370C2×C8—24no—
371C2×C8—44no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4C22×C3×C414yes—
375C22⋊C4—24no—
376C22⋊C4—28no—
377C22⋊C4—24no—
378C22⋊C4—48no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—22no—
384C22×C3—21no—
385C22×C3—28no—
386C22×C3—28no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—48no—
391C22×C3—48no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3—816no—
397C22×C3—88no—
398C22×C3C2×#9914yes—
399C22×C3C2×#9912yes—
400C22×C3#41812yes—
401C22×C3#41812yes—
402C22×C3#42112yes—
403C22×C3C22×C22⋊C411yescharacteristic
404C22×C3C2×AGL(1,ℤ/8)12yes—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—22no—
408C3×C4—21no—
409C3×C4—22no—
410C3×C4—21no—
411C3×C4—44no—
412C3×C4—84no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—22no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—416no—
428C23—48no—
429C23—48no—
430C23—48no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—816no—
435C23C3×#9914yes—
436C23C2×C3×C22⋊C412yes—
437C23C2×C3×C22⋊C412yes—
438C23C2×C3×C22⋊C411yescharacteristic
439C23C2×C3×C22⋊C411yescharacteristic
440C23C3×AGL(1,ℤ/8)14yes—
441C23C22×C3×D411yescharacteristic
442D4—44no—
443D4—44no—
444D4—48no—
445D4—416no—
446D4—48no—
447D4—48no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—88no—
473C2×C4C2×C3×C22⋊C411yescharacteristic
474C2×C4C2×C3×C22⋊C412yes—
475C2×C4C2×C3×C22⋊C411yescharacteristic
476C2×C4C2×C3×C22⋊C412yes—
477C2×C4C23×C3×C411yescharacteristic
478C2×C4C22×C3×D411yescharacteristic
479C8—44no—
480C2×C3—22no—
481C2×C3—24no—
482C2×C3—22no—
483C2×C3—42no—
484C2×C3—84no—
485C2×C3#151014yesdirect product factor
486C2×C3C22×#9911yescharacteristic
487C2×C3C2×#41811yescharacteristic
488C2×C3C2×#42111yescharacteristic
489C22—21no—
490C22—21no—
491C22—22no—
492C22—21no—
493C22—22no—
494C22—21no—
495C22—28no—
496C22—28no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—48no—
501C22—48no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22—816no—
507C22—88no—
508C22C2×C3×#9914yes—
509C22C2×C3×#9912yes—
510C22C3×#41812yes—
511C22C3×#41812yes—
512C22C3×#42112yes—
513C22C22×C3×C22⋊C411yescharacteristic
514C22C2×C3×AGL(1,ℤ/8)12yes—
515C4—21no—
516C4—21no—
517C4—22no—
518C4—21no—
519C4—22no—
520C4—21no—
521C4—44no—
522C4—84no—
523C3C2×#151011yescharacteristic, direct product factor, Sylow
524C2—22no—
525C2—24no—
526C2—22no—
527C2—42no—
528C2—84no—
529C2C3×#151014yesdirect product factor
530C2C22×C3×#9911yescharacteristic
531C2C2×C3×#41811yescharacteristic
532C2C2×C3×#42111yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#407C2semidirect
3. C2×C3×#381C2semidirect
4. C3×#1510C2direct (N × H)
5. C2×C3×#327C2semidirect
6. C2×#1510C3direct (N × H)
8. C23×C3×D4C4semidirect
15. C2×C3×#121C4semidirect
16. C2×C3×#121C4semidirect
19. C22×C3×C4⋊C4C4semidirect
23. C3×#407C22semidirect
25. C2×C3×C22⋊C8C22semidirect
27. C3×#381C22semidirect
29. C2×C3×#95C22semidirect
30. C3×#327C22semidirect
32. C2×#407C2×C3semidirect
33. C2×#381C2×C3semidirect
34. #1510C2×C3direct (N × H)
35. C2×#327C2×C3semidirect
47. C22×C3×D4C2×C4semidirect
72. C3×#121C2×C4semidirect
77. C2×C3×C4⋊C4C2×C4semidirect
86. C3×C22⋊C8C23semidirect
89. C3×#95C23semidirect
91. C23×D4C3×C4semidirect
98. C2×#121C3×C4semidirect
99. C2×#121C3×C4semidirect
102. C22×C4⋊C4C3×C4semidirect
106. #407C22×C3semidirect
108. C2×C22⋊C8C22×C3semidirect
110. #381C22×C3semidirect
112. C2×#95C22×C3semidirect
113. #327C22×C3semidirect
197. C22×D4C2×C3×C4semidirect
222. #121C2×C3×C4semidirect
227. C2×C4⋊C4C2×C3×C4semidirect
236. C22⋊C8C23×C3semidirect
239. #95C23×C3semidirect
485. C2×C3#1510direct (N × H)
523. C3C2×#1510direct (N × H)
529. C2C3×#1510direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C2×C3×#4072
C2×C3×#3812
C3×#15102
C2×C3×#3272
C2×#15103