Finite Group Databasebeta

← C2×C3×#1465 #10635965

Subgroups — C2×C3×#1465 #10635965

Order 768.

1738 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
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×#407C212yesmaximal, complemented
3C3×#1465C218yesmaximal, direct product factor
4C2×C3×#324C212yesmaximal, complemented
5C2×C3×#323C212yesmaximal, complemented
6C2×#1465C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×D4C411yescharacteristic
8C24×C3×C4C411yescharacteristic
9C24×C3×C4C2211yescharacteristic
10C22×C3×M16—22no—
11C2×C3×C4×D4—22no—
12C2×C3×C4×D4—22no—
13C2×C3×C4×D4—24no—
14C2×C3×C4×D4C414yes—
15C2×C3×C4×D4C2214yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C411yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C42C411yescharacteristic
20C3×#407—24no—
21C3×#407C2218yescomplemented
22C2×C3×C22⋊C8—22no—
23C2×C3×C22⋊C8C2214yes—
24C3×#324C2218yescomplemented
25C3×#323C2218yescomplemented
26C22×C4×D4C2×C311yescharacteristic
27C2×#407C2×C312yescomplemented
28#1465C2×C318yesdirect product factor
29C2×#324C2×C312yescomplemented
30C2×#323C2×C312yescomplemented
31C25×C3C2×C411yescharacteristic
32C25×C3C2×C411yescharacteristic
33C22×C3×D4—22no—
34C22×C3×D4—22no—
35C22×C3×D4—24no—
36C22×C3×D4—24no—
37C22×C3×D4C2×C414yes—
38C23×C3×C4—22no—
39C23×C3×C4—21no—
40C23×C3×C4—21no—
41C23×C3×C4—44no—
42C23×C3×C4—44no—
43C23×C3×C4C2×C411yescharacteristic
44C23×C3×C4C2×C411yescharacteristic
45C23×C3×C4C2×C411yescharacteristic
46C23×C3×C4C2×C414yes—
47C23×C3×C4C2×C411yescharacteristic
48C23×C3×C4C2×C412yes—
49C23×C3×C4D412yes—
50C23×C3×C4D412yes—
51C23×C3×C4D414yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C2×C3×M16—28no—
55C2×C3×M16—48no—
56C22×C3×C8—24no—
57C3×C4×D4—28no—
58C3×C4×D4—28no—
59C3×C4×D4—28no—
60C3×C4×D4—28no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4C2×C412yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4C2×C414yes—
70C2×C3×C22⋊C4C2×C412yes—
71C2×C3×C22⋊C4C2312yes—
72C2×C3×C42—22no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42C2×C414yes—
76C3×C22⋊C8—28no—
77C3×C22⋊C8—24no—
78C3×C22⋊C8C2318yes—
79C23×D4C3×C411yescharacteristic
80C24×C4C3×C411yescharacteristic
81C24×C4C22×C311yescharacteristic
82C22×M16—22no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—24no—
86C2×C4×D4C3×C414yes—
87C2×C4×D4C22×C314yes—
88C22×C4⋊C4C22×C311yescharacteristic
89C22×C22⋊C4C3×C411yescharacteristic
90C22×C22⋊C4C22×C311yescharacteristic
91C22×C42C3×C411yescharacteristic
92#407—24no—
93#407C22×C318yescomplemented
94C2×C22⋊C8—22no—
95C2×C22⋊C8C22×C314yes—
96#324C22×C318yescomplemented
97#323C22×C318yescomplemented
98C24×C3—21no—
99C24×C3—21no—
100C24×C3—22no—
101C24×C3—22no—
102C24×C3—22no—
103C24×C3—42no—
104C24×C3—42no—
105C24×C3—42no—
106C24×C3—42no—
107C24×C3C22⋊C412yes—
108C24×C3C22⋊C412yes—
109C24×C3M1614yescomplemented
110C24×C3C22×C411yescharacteristic
111C24×C3C22×C412yes—
112C24×C3C22×C412yes—
113C2×C3×D4—216no—
114C2×C3×D4—216no—
115C2×C3×D4—28no—
116C2×C3×D4—28no—
117C2×C3×D4—28no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—28no—
122C22×C3×C4—24no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—44no—
130C22×C3×C4—48no—
131C22×C3×C4—44no—
132C22×C3×C4—44no—
133C22×C3×C4—42no—
134C22×C3×C4—44no—
135C22×C3×C4—48no—
136C22×C3×C4—48no—
137C22×C3×C4—44no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4C22⋊C412yes—
141C22×C3×C4C22⋊C414yes—
142C22×C3×C4C22⋊C411yescharacteristic
143C22×C3×C4C22⋊C414yes—
144C22×C3×C4C22⋊C411yescharacteristic
145C22×C3×C4M1614yescomplemented
146C22×C3×C4C22×C412yes—
147C22×C3×C4C22×C412yes—
148C22×C3×C4C22×C412yes—
149C22×C3×C4C22×C412yes—
150C22×C3×C4C22×C411yescharacteristic
151C22×C3×C4C2×D414yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D414yes—
155C22×C3×C4C2411yescharacteristic
156C3×M16—416no—
157C2×C3×C8—28no—
158C2×C3×C8—48no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—24no—
161C3×C22⋊C4—28no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—24no—
164C3×C42—24no—
165C3×C42—24no—
166C25C2×C3×C411yescharacteristic
167C25C2×C3×C411yescharacteristic
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4C2×C3×C414yes—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4C2×C3×C411yescharacteristic
179C23×C4C2×C3×C411yescharacteristic
180C23×C4C2×C3×C411yescharacteristic
181C23×C4C2×C3×C411yescharacteristic
182C23×C4C2×C3×C414yes—
183C23×C4C2×C3×C412yes—
184C23×C4C3×D412yes—
185C23×C4C3×D412yes—
186C23×C4C3×D414yes—
187C23×C4C23×C311yescharacteristic
188C23×C4C23×C312yes—
189C2×M16—28no—
190C2×M16—48no—
191C22×C8—24no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4C2×C3×C412yes—
199C2×C4⋊C4C23×C312yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4C2×C3×C414yes—
205C2×C22⋊C4C2×C3×C412yes—
206C2×C22⋊C4C23×C312yes—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42C2×C3×C414yes—
211C22⋊C8—28no—
212C22⋊C8—24no—
213C22⋊C8C23×C318yes—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—24no—
221C23×C3—24no—
222C23×C3—24no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—24no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3—48no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—48no—
236C23×C3—48no—
237C23×C3—48no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3C2×C22⋊C414yes—
241C23×C3C2×C22⋊C411yescharacteristic
242C23×C3C2×C22⋊C411yescharacteristic
243C23×C3C2×M1612yes—
244C23×C3C2×M1612yes—
245C23×C3C2×M1614yescomplemented
246C23×C3C23×C411yescharacteristic
247C3×D4—232no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—28no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—28no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—44no—
261C2×C3×C4—41no—
262C2×C3×C4—44no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—48no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—48no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4#9914yes—
274C2×C3×C4C2×C22⋊C412yes—
275C2×C3×C4C2×C22⋊C412yes—
276C2×C3×C4C2×C22⋊C412yes—
277C2×C3×C4C2×M1614yescomplemented
278C2×C3×C4C22×D412yes—
279C3×C8—48no—
280C24—21no—
281C24—21no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24C3×C22⋊C412yes—
290C24C3×C22⋊C412yes—
291C24C3×M1614yescomplemented
292C24C22×C3×C411yescharacteristic
293C24C22×C3×C412yes—
294C24C22×C3×C412yes—
295C2×D4—216no—
296C2×D4—216no—
297C2×D4—28no—
298C2×D4—28no—
299C2×D4—28no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—44no—
312C22×C4—48no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4C3×C22⋊C411yescharacteristic
323C22×C4C3×C22⋊C412yes—
324C22×C4C3×C22⋊C414yes—
325C22×C4C3×C22⋊C414yes—
326C22×C4C3×C22⋊C411yescharacteristic
327C22×C4C3×M1614yescomplemented
328C22×C4C22×C3×C412yes—
329C22×C4C22×C3×C412yes—
330C22×C4C22×C3×C412yes—
331C22×C4C22×C3×C412yes—
332C22×C4C22×C3×C411yescharacteristic
333C22×C4C2×C3×D412yes—
334C22×C4C2×C3×D414yes—
335C22×C4C2×C3×D412yes—
336C22×C4C2×C3×D414yes—
337C22×C4C24×C311yescharacteristic
338M16—416no—
339C2×C8—28no—
340C2×C8—48no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C22⋊C4—28no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C42—24no—
347C42—24no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—24no—
355C22×C3—24no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3—44no—
359C22×C3—44no—
360C22×C3—44no—
361C22×C3—44no—
362C22×C3—48no—
363C22×C3—48no—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—48no—
367C22×C3—48no—
368C22×C3#40712yes—
369C22×C3#40712yes—
370C22×C3C2×#9912yes—
371C22×C3#41312yes—
372C22×C3C22×C22⋊C411yescharacteristic
373C22×C3C22×M1611yescharacteristic
374C22×C3C22×M1611yescharacteristic
375C3×C4—24no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4C2×#9914yes—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—22no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—48no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—48no—
404C23—48no—
405C23—48no—
406C23—42no—
407C23—42no—
408C23C2×C3×C22⋊C414yes—
409C23C2×C3×C22⋊C411yescharacteristic
410C23C2×C3×C22⋊C411yescharacteristic
411C23C2×C3×M1612yes—
412C23C2×C3×M1612yes—
413C23C2×C3×M1614yescomplemented
414C23C23×C3×C411yescharacteristic
415D4—232no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—28no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4C3×#9914yes—
442C2×C4C2×C3×C22⋊C412yes—
443C2×C4C2×C3×C22⋊C412yes—
444C2×C4C2×C3×C22⋊C412yes—
445C2×C4C2×C3×M1614yescomplemented
446C2×C4C22×C3×D412yes—
447C8—48no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—22no—
451C2×C3—44no—
452C2×C3—44no—
453C2×C3#146514yesdirect product factor
454C2×C3C2×#40711yescharacteristic
455C2×C3C22×#9911yescharacteristic
456C2×C3C2×#41311yescharacteristic
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—44no—
471C22—48no—
472C22—48no—
473C22—42no—
474C22—42no—
475C22—48no—
476C22—48no—
477C22C3×#40712yes—
478C22C3×#40712yes—
479C22C2×C3×#9912yes—
480C22C3×#41312yes—
481C22C22×C3×C22⋊C411yescharacteristic
482C22C22×C3×M1611yescharacteristic
483C22C22×C3×M1611yescharacteristic
484C4—24no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—42no—
489C4—42no—
490C4C2×C3×#9914yes—
491C3C2×#146511yescharacteristic, direct product factor, Sylow
492C2—21no—
493C2—21no—
494C2—22no—
495C2—44no—
496C2—44no—
497C2C3×#146514yesdirect product factor
498C2C2×C3×#40711yescharacteristic
499C2C22×C3×#9911yescharacteristic
500C2C2×C3×#41311yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C4
LCSG ▷ C23 ▷ C2 ▷ 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. C3×#1465C2direct (N × H)
4. C2×C3×#324C2semidirect
5. C2×C3×#323C2semidirect
6. C2×#1465C3direct (N × H)
21. C3×#407C22semidirect
24. C3×#324C22semidirect
25. C3×#323C22semidirect
27. C2×#407C2×C3semidirect
28. #1465C2×C3direct (N × H)
29. C2×#324C2×C3semidirect
30. C2×#323C2×C3semidirect
93. #407C22×C3semidirect
96. #324C22×C3semidirect
97. #323C22×C3semidirect
109. C24×C3M16semidirect
145. C22×C3×C4M16semidirect
245. C23×C3C2×M16semidirect
277. C2×C3×C4C2×M16semidirect
291. C24C3×M16semidirect
327. C22×C4C3×M16semidirect
413. C23C2×C3×M16semidirect
445. C2×C4C2×C3×M16semidirect
453. C2×C3#1465direct (N × H)
491. C3C2×#1465direct (N × H)
497. C2C3×#1465direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×C3×#4072
C3×#14652
C2×C3×#3242
C2×C3×#3232
C2×#14653