Finite Group Databasebeta

← C2×C3×#1615 #10636713

Subgroups — C2×C3×#1615 #10636713

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#539C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic, complemented
3C2×C3×#481C211yesmaximal, characteristic
4C2×C3×#400C211yesmaximal, characteristic, complemented
5C3×#1615C218yesmaximal, direct product factor
6C2×C3×#328C211yesmaximal, characteristic, complemented
7C2×C3×#327C211yesmaximal, characteristic
8C2×C3×#324C211yesmaximal, characteristic, complemented
9C2×#1615C311yesmaximal, characteristic, direct product factor, Sylow
10C3×#539—22no—
11C3×#539—22no—
12C3×#539—22no—
13C3×#539C2214yes—
14C2×C3×#126—21no—
15C2×C3×#125—21no—
16C2×C3×#123—21no—
17C2×C3×#123—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C4⋊C4C2211yescharacteristic
24C3×#484—22no—
25C3×#484C2214yescomplemented
26C3×#481—22no—
27C3×#481C2214yes—
28C2×C3×#107—21no—
29C2×C3×#106—21no—
30C2×C3×#103—21no—
31C2×C3×#102—21no—
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#400—22no—
35C3×#400C2214yescomplemented
36C2×C3×#95—21no—
37C2×C3×#95—21no—
38C2×C3×#95C2211yescharacteristic
39C3×#328C2214yescomplemented
40C3×#327C2214yes—
41C3×#324C2214yescomplemented
42C2×#539C2×C311yescharacteristic
43C2×#484C2×C311yescharacteristic, complemented
44C2×#481C2×C311yescharacteristic
45C2×#400C2×C311yescharacteristic, complemented
46#1615C2×C318yesdirect product factor
47C2×#328C2×C311yescharacteristic, complemented
48C2×#327C2×C311yescharacteristic
49C2×#324C2×C311yescharacteristic, complemented
50C22×C3×Q8D411yescharacteristic
51C22×C3×D4—21no—
52C22×C3×D4D411yescharacteristic
53C23×C3×C4—21no—
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×C3×C8—21no—
60C22×C3×C8—21no—
61C3×#126—24no—
62C3×#126—24no—
63C3×#125—24no—
64C3×#125—24no—
65C3×#123—24no—
66C3×#123—24no—
67C3×#123—24no—
68C3×#123—24no—
69C3×C22⋊Q8—24no—
70C3×C22⋊Q8—24no—
71C3×C22⋊Q8—21no—
72C3×C22⋊Q8—21no—
73C3×C22⋊Q8D412yes—
74C3×C22⋊Q8C2312yes—
75C3×#121—21no—
76C3×#121—21no—
77C3×#121D412yes—
78C3×#121C2312yes—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C42—21no—
105C3×#107—24no—
106C3×#106—24no—
107C3×#103—24no—
108C3×#102—24no—
109C3×C22⋊C8—21no—
110C3×C22⋊C8—21no—
111C3×C22⋊C8C2312yes—
112C3×C22⋊C8C2312yes—
113C3×#95—24no—
114C3×#95—24no—
115C3×#95—21no—
116C3×#95C2312yes—
117#539—22no—
118#539—22no—
119#539—22no—
120#539C22×C314yes—
121C2×#126—21no—
122C2×#125—21no—
123C2×#123—21no—
124C2×#123—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8C22×C311yescharacteristic
127C2×#121C22×C311yescharacteristic
128C2×C4×D4—21no—
129C22×C4⋊C4C22×C311yescharacteristic
130C22×C4⋊C4C22×C311yescharacteristic
131#484—22no—
132#484C22×C314yescomplemented
133#481—22no—
134#481C22×C314yes—
135C2×#107—21no—
136C2×#106—21no—
137C2×#103—21no—
138C2×#102—21no—
139C2×C22⋊C8C22×C311yescharacteristic
140C2×C22⋊C8C22×C311yescharacteristic
141#400—22no—
142#400C22×C314yescomplemented
143C2×#95—21no—
144C2×#95—21no—
145C2×#95C22×C311yescharacteristic
146#328C22×C314yescomplemented
147#327C22×C314yes—
148#324C22×C314yescomplemented
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C2×C3×Q8—21no—
152C2×C3×Q8—42no—
153C2×C3×Q8C2×D412yes—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4C2×D412yes—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4—44no—
178C22×C3×C4—41no—
179C22×C3×C4—41no—
180C22×C3×C4—42no—
181C22×C3×C4—82no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—42no—
193C2×C3×C8—42no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4—41no—
209C3×C4⋊C4C2×D412yes—
210C3×C4⋊C4C2×D412yes—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C42—24no—
219C22×Q8C3×D411yescharacteristic
220C22×D4—21no—
221C22×D4C3×D411yescharacteristic
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4C3×D411yescharacteristic
226C23×C4C3×D411yescharacteristic
227C23×C4C23×C311yescharacteristic
228C22×C8—21no—
229C22×C8—21no—
230#126—24no—
231#126—24no—
232#125—24no—
233#125—24no—
234#123—24no—
235#123—24no—
236#123—24no—
237#123—24no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8C3×D412yes—
243C22⋊Q8C23×C312yes—
244#121—21no—
245#121—21no—
246#121C3×D412yes—
247#121C23×C312yes—
248C4×D4—24no—
249C4×D4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C3×D412yes—
264C2×C4⋊C4C3×D411yescharacteristic
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C42—21no—
274#107—24no—
275#106—24no—
276#103—24no—
277#102—24no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8C23×C312yes—
281C22⋊C8C23×C312yes—
282#95—24no—
283#95—24no—
284#95—21no—
285#95C23×C312yes—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3#12011yescharacteristic
293C23×C3#12011yescharacteristic
294C23×C3#13512yes—
295C23×C3#13712yes—
296C23×C3C22×D411yescharacteristic
297C3×Q8—42no—
298C3×Q8—81no—
299C3×D4—42no—
300C3×D4—42no—
301C3×D4—42no—
302C3×D4—44no—
303C3×D4—81no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—81no—
330C2×C3×C4—84no—
331C2×C3×C4—84no—
332C2×C3×C4—82no—
333C2×C3×C4—82no—
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4#12011yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C3×C8—42no—
339C3×C8—42no—
340C24—21no—
341C24C2×C3×D411yescharacteristic
342C2×Q8—21no—
343C2×Q8—42no—
344C2×Q8C2×C3×D412yes—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—21no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4C2×C3×D412yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—82no—
373C22×C4C2×C3×D412yes—
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C2×C3×D411yescharacteristic
376C22×C4C2×C3×D412yes—
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C24×C311yescharacteristic
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4C2×C3×D412yes—
401C4⋊C4C2×C3×D412yes—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C42—24no—
410C22×C3—24no—
411C22×C3—24no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—84no—
420C22×C3#45212yes—
421C22×C3#45212yes—
422C22×C3#45412yes—
423C22×C3#45812yes—
424C22×C3C2×#12011yescharacteristic
425C22×C3C2×#13511yescharacteristic
426C22×C3C2×#13711yescharacteristic
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—82no—
437C3×C4—82no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23C3×#12011yescharacteristic
445C23C3×#12011yescharacteristic
446C23C3×#13512yes—
447C23C3×#13712yes—
448C23C22×C3×D411yescharacteristic
449Q8—42no—
450Q8—81no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—44no—
455D4—81no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—81no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4C3×#12011yescharacteristic
487C2×C4C3×#12011yescharacteristic
488C2×C4C22×C3×D411yescharacteristic
489C2×C4C22×C3×D411yescharacteristic
490C8—42no—
491C8—42no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—22no—
495C2×C3—82no—
496C2×C3#161514yesdirect product factor
497C2×C3C2×#45211yescharacteristic
498C2×C3C2×#45411yescharacteristic
499C2×C3C2×#45811yescharacteristic
500C22—24no—
501C22—24no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—84no—
510C22C3×#45212yes—
511C22C3×#45212yes—
512C22C3×#45412yes—
513C22C3×#45812yes—
514C22C2×C3×#12011yescharacteristic
515C22C2×C3×#13511yescharacteristic
516C22C2×C3×#13711yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—82no—
527C4—82no—
528C3C2×#161511yescharacteristic, direct product factor, Sylow
529C2—21no—
530C2—21no—
531C2—22no—
532C2—82no—
533C2C3×#161514yesdirect product factor
534C2C2×C3×#45211yescharacteristic
535C2C2×C3×#45411yescharacteristic
536C2C2×C3×#45811yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#484C2semidirect
4. C2×C3×#400C2semidirect
5. C3×#1615C2direct (N × H)
6. C2×C3×#328C2semidirect
8. C2×C3×#324C2semidirect
9. C2×#1615C3direct (N × H)
25. C3×#484C22semidirect
35. C3×#400C22semidirect
39. C3×#328C22semidirect
41. C3×#324C22semidirect
43. C2×#484C2×C3semidirect
45. C2×#400C2×C3semidirect
46. #1615C2×C3direct (N × H)
47. C2×#328C2×C3semidirect
49. C2×#324C2×C3semidirect
132. #484C22×C3semidirect
142. #400C22×C3semidirect
146. #328C22×C3semidirect
148. #324C22×C3semidirect
496. C2×C3#1615direct (N × H)
528. C3C2×#1615direct (N × H)
533. C2C3×#1615direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5392
C2×C3×#4842
C2×C3×#4812
C2×C3×#4002
C3×#16152
C2×C3×#3282
C2×C3×#3272
C2×C3×#3242
C2×#16153