Finite Group Databasebeta

← C2×C3×#1513 #10636162

Subgroups — C2×C3×#1513 #10636162

Order 768.

1510 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#525C211yesmaximal, characteristic
2C2×C3×#407C211yesmaximal, characteristic, complemented
3C2×C3×#380C211yesmaximal, characteristic, complemented
4C3×#1513C218yesmaximal, direct product factor
5C2×C3×#328C212yesmaximal, complemented
6C2×C3×#327C212yesmaximal, complemented
7C2×#1513C311yesmaximal, characteristic, direct product factor, Sylow
8C22×C3×#41C411yescharacteristic, complemented
9C24×C3×C4C2211yescharacteristic
10C22×C3×M16—21no—
11C3×#525—22no—
12C3×#525—22no—
13C3×#525C414yescomplemented
14C3×#525C2214yes—
15C2×C3×#123—21no—
16C2×C3×C22⋊Q8C411yescharacteristic, complemented
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C411yescharacteristic, complemented
19C2×C3×#121C2211yescharacteristic
20C2×C3×#120—21no—
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4—21no—
23C2×C3×#117C411yescharacteristic, complemented
24C22×C3×C4⋊C4—21no—
25C22×C3×C22⋊C4—21no—
26C3×#407—22no—
27C3×#407C2214yescomplemented
28C2×C3×C22⋊C8—21no—
29C2×C3×C22⋊C8C2212yes—
30C3×#380—22no—
31C3×#380C2214yescomplemented
32C2×C3×#95C2212yes—
33C3×#328C2218yescomplemented
34C3×#327C2218yescomplemented
35C2×#525C2×C311yescharacteristic
36C2×#407C2×C311yescharacteristic, complemented
37C2×#380C2×C311yescharacteristic, complemented
38#1513C2×C318yesdirect product factor
39C2×#328C2×C312yescomplemented
40C2×#327C2×C312yescomplemented
41C25×C3D411yescharacteristic
42C2×C3×#41—22no—
43C2×C3×#41—44no—
44C2×C3×#41C2×C414yescomplemented
45C22×C3×Q8C2×C411yescharacteristic
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4C2×C411yescharacteristic
50C23×C3×C4—21no—
51C23×C3×C4—22no—
52C23×C3×C4—44no—
53C23×C3×C4C2×C411yescharacteristic
54C23×C3×C4C2×C411yescharacteristic
55C23×C3×C4C2×C412yes—
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D412yes—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D412yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C2×C3×M16—24no—
64C2×C3×M16—44no—
65C22×C3×C8—22no—
66C3×#123—24no—
67C3×#123—24no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—21no—
70C3×C22⋊Q8—21no—
71C3×C22⋊Q8C2×C412yes—
72C3×C22⋊Q8C2×C414yescomplemented
73C3×C22⋊Q8C2312yes—
74C3×#121—22no—
75C3×#121—21no—
76C3×#121—21no—
77C3×#121C2×C412yes—
78C3×#121C2×C414yescomplemented
79C3×#121C2312yes—
80C3×#120—24no—
81C3×#120—24no—
82C3×C4×D4—24no—
83C3×C4×D4—24no—
84C3×C4×D4—24no—
85C3×C4×D4—24no—
86C3×#117—22no—
87C3×#117C2×C414yescomplemented
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4—44no—
92C2×C3×C4⋊C4C2×C411yescharacteristic
93C2×C3×C4⋊C4C2×C411yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—24no—
100C2×C3×C22⋊C4—44no—
101C2×C3×C42—21no—
102C3×C22⋊C8—24no—
103C3×C22⋊C8—22no—
104C3×C22⋊C8C2314yes—
105C3×#95—22no—
106C3×#95C2314yes—
107C22×#41C3×C411yescharacteristic, complemented
108C24×C4C22×C311yescharacteristic
109C22×M16—21no—
110#525—22no—
111#525—22no—
112#525C3×C414yescomplemented
113#525C22×C314yes—
114C2×#123—21no—
115C2×C22⋊Q8C3×C411yescharacteristic, complemented
116C2×C22⋊Q8C22×C311yescharacteristic
117C2×#121C3×C411yescharacteristic, complemented
118C2×#121C22×C311yescharacteristic
119C2×#120—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×#117C3×C411yescharacteristic, complemented
123C22×C4⋊C4—21no—
124C22×C22⋊C4—21no—
125#407—22no—
126#407C22×C314yescomplemented
127C2×C22⋊C8—21no—
128C2×C22⋊C8C22×C312yes—
129#380—22no—
130#380C22×C314yescomplemented
131C2×#95C22×C312yes—
132#328C22×C318yescomplemented
133#327C22×C318yescomplemented
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—21no—
137C24×C3—42no—
138C24×C3—42no—
139C24×C3C22⋊C411yescharacteristic
140C24×C3C22⋊C411yescharacteristic
141C24×C3C22⋊C412yes—
142C24×C3C2×D411yescharacteristic
143C24×C3C2×D412yes—
144C3×#41—48no—
145C3×#41—48no—
146C2×C3×Q8—21no—
147C2×C3×Q8—42no—
148C2×C3×Q8C22×C412yes—
149C2×C3×D4—22no—
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4—42no—
155C2×C3×D4—44no—
156C2×C3×D4—44no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4C22×C412yes—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—24no—
168C22×C3×C4—44no—
169C22×C3×C4—44no—
170C22×C3×C4—44no—
171C22×C3×C4—44no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4—44no—
176C22×C3×C4—44no—
177C22×C3×C4—44no—
178C22×C3×C4—44no—
179C22×C3×C4C22⋊C412yes—
180C22×C3×C4C22⋊C412yes—
181C22×C3×C4C22⋊C412yes—
182C22×C3×C4C22⋊C411yescharacteristic
183C22×C3×C4C22⋊C412yes—
184C22×C3×C4C22⋊C412yes—
185C22×C3×C4C22⋊C411yescharacteristic
186C22×C3×C4C22×C412yes—
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C22×C411yescharacteristic
189C22×C3×C4C22×C412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2411yescharacteristic
197C3×M16—48no—
198C2×C3×C8—24no—
199C2×C3×C8—44no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—48no—
206C3×C4⋊C4C22×C412yes—
207C3×C4⋊C4C22×C412yes—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—48no—
216C3×C42—24no—
217C25C3×D411yescharacteristic
218C2×#41—22no—
219C2×#41—44no—
220C2×#41C2×C3×C414yescomplemented
221C22×Q8C2×C3×C411yescharacteristic
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2×C3×C411yescharacteristic
226C23×C4—21no—
227C23×C4—22no—
228C23×C4—44no—
229C23×C4C2×C3×C411yescharacteristic
230C23×C4C2×C3×C411yescharacteristic
231C23×C4C2×C3×C412yes—
232C23×C4C3×D411yescharacteristic
233C23×C4C3×D411yescharacteristic
234C23×C4C3×D412yes—
235C23×C4C3×D411yescharacteristic
236C23×C4C3×D412yes—
237C23×C4C23×C311yescharacteristic
238C23×C4C23×C312yes—
239C2×M16—24no—
240C2×M16—44no—
241C22×C8—22no—
242#123—24no—
243#123—24no—
244C22⋊Q8—22no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8C2×C3×C412yes—
248C22⋊Q8C2×C3×C414yescomplemented
249C22⋊Q8C23×C312yes—
250#121—22no—
251#121—21no—
252#121—21no—
253#121C2×C3×C412yes—
254#121C2×C3×C414yescomplemented
255#121C23×C312yes—
256#120—24no—
257#120—24no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—24no—
262#117—22no—
263#117C2×C3×C414yescomplemented
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×C3×C411yescharacteristic
269C2×C4⋊C4C2×C3×C411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—44no—
277C2×C42—21no—
278C22⋊C8—24no—
279C22⋊C8—22no—
280C22⋊C8C23×C314yes—
281#95—22no—
282#95C23×C314yes—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—24no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—48no—
300C23×C3—48no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3C2×C22⋊C412yes—
305C23×C3C2×C22⋊C412yes—
306C23×C3C2×C22⋊C411yescharacteristic
307C23×C3C2×C22⋊C411yescharacteristic
308C23×C3AGL(1,ℤ/8)12yes—
309C23×C3#13712yes—
310C23×C3C22×D411yescharacteristic
311C3×Q8—42no—
312C3×Q8—42no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—48no—
316C3×D4—48no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—44no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—48no—
344C2×C3×C4—44no—
345C2×C3×C4—44no—
346C2×C3×C4—44no—
347C2×C3×C4—88no—
348C2×C3×C4#9914yes—
349C2×C3×C4C2×C22⋊C411yescharacteristic
350C2×C3×C4C2×C22⋊C412yes—
351C2×C3×C4C2×C22⋊C411yescharacteristic
352C2×C3×C4C2×C22⋊C412yes—
353C2×C3×C4C23×C411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C3×C8—44no—
356C24—22no—
357C24—22no—
358C24—21no—
359C24—42no—
360C24—42no—
361C24C3×C22⋊C411yescharacteristic
362C24C3×C22⋊C411yescharacteristic
363C24C3×C22⋊C412yes—
364C24C2×C3×D411yescharacteristic
365C24C2×C3×D412yes—
366#41—48no—
367#41—48no—
368C2×Q8—21no—
369C2×Q8—42no—
370C2×Q8C22×C3×C412yes—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4C22×C3×C412yes—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4C3×C22⋊C411yescharacteristic
402C22×C4C3×C22⋊C412yes—
403C22×C4C3×C22⋊C412yes—
404C22×C4C3×C22⋊C412yes—
405C22×C4C3×C22⋊C412yes—
406C22×C4C3×C22⋊C412yes—
407C22×C4C3×C22⋊C411yescharacteristic
408C22×C4C22×C3×C411yescharacteristic
409C22×C4C22×C3×C412yes—
410C22×C4C22×C3×C411yescharacteristic
411C22×C4C22×C3×C412yes—
412C22×C4C2×C3×D412yes—
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D412yes—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D412yes—
418C22×C4C24×C311yescharacteristic
419M16—48no—
420C2×C8—24no—
421C2×C8—44no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—48no—
428C4⋊C4C22×C3×C412yes—
429C4⋊C4C22×C3×C412yes—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—48no—
438C42—24no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—24no—
444C22×C3—24no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—48no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—84no—
460C22×C3C2×#9912yes—
461C22×C3#41712yes—
462C22×C3#41712yes—
463C22×C3#42112yes—
464C22×C3C22×C22⋊C411yescharacteristic
465C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
466C22×C3C2×#13711yescharacteristic
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—22no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—84no—
476C3×C4C2×#9914yes—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—48no—
494C23—48no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23C2×C3×C22⋊C412yes—
499C23C2×C3×C22⋊C412yes—
500C23C2×C3×C22⋊C411yescharacteristic
501C23C2×C3×C22⋊C411yescharacteristic
502C23C3×AGL(1,ℤ/8)12yes—
503C23C3×#13712yes—
504C23C22×C3×D411yescharacteristic
505Q8—42no—
506Q8—42no—
507D4—42no—
508D4—42no—
509D4—48no—
510D4—48no—
511D4—42no—
512D4—42no—
513D4—44no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—48no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—88no—
542C2×C4C3×#9914yes—
543C2×C4C2×C3×C22⋊C411yescharacteristic
544C2×C4C2×C3×C22⋊C412yes—
545C2×C4C2×C3×C22⋊C411yescharacteristic
546C2×C4C2×C3×C22⋊C412yes—
547C2×C4C23×C3×C411yescharacteristic
548C2×C4C22×C3×D411yescharacteristic
549C8—44no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—22no—
553C2×C3—42no—
554C2×C3—42no—
555C2×C3—82no—
556C2×C3#151314yesdirect product factor
557C2×C3C22×#9911yescharacteristic
558C2×C3C2×#41711yescharacteristic
559C2×C3C2×#42111yescharacteristic
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—84no—
581C22C2×C3×#9912yes—
582C22C3×#41712yes—
583C22C3×#41712yes—
584C22C3×#42112yes—
585C22C22×C3×C22⋊C411yescharacteristic
586C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
587C22C2×C3×#13711yescharacteristic
588C4—21no—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—21no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—84no—
597C4C2×C3×#9914yes—
598C3C2×#151311yescharacteristic, direct product factor, Sylow
599C2—21no—
600C2—21no—
601C2—22no—
602C2—42no—
603C2—42no—
604C2—82no—
605C2C3×#151314yesdirect product factor
606C2C22×C3×#9911yescharacteristic
607C2C2×C3×#41711yescharacteristic
608C2C2×C3×#42111yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#525C2
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×#380C2semidirect
4. C3×#1513C2direct (N × H)
5. C2×C3×#328C2semidirect
6. C2×C3×#327C2semidirect
7. C2×#1513C3direct (N × H)
8. C22×C3×#41C4semidirect
13. C3×#525C4semidirect
16. C2×C3×C22⋊Q8C4semidirect
18. C2×C3×#121C4semidirect
23. C2×C3×#117C4semidirect
27. C3×#407C22semidirect
31. C3×#380C22semidirect
33. C3×#328C22semidirect
34. C3×#327C22semidirect
36. C2×#407C2×C3semidirect
37. C2×#380C2×C3semidirect
38. #1513C2×C3direct (N × H)
39. C2×#328C2×C3semidirect
40. C2×#327C2×C3semidirect
44. C2×C3×#41C2×C4semidirect
72. C3×C22⋊Q8C2×C4semidirect
78. C3×#121C2×C4semidirect
87. C3×#117C2×C4semidirect
107. C22×#41C3×C4semidirect
112. #525C3×C4semidirect
115. C2×C22⋊Q8C3×C4semidirect
117. C2×#121C3×C4semidirect
122. C2×#117C3×C4semidirect
126. #407C22×C3semidirect
130. #380C22×C3semidirect
132. #328C22×C3semidirect
133. #327C22×C3semidirect
220. C2×#41C2×C3×C4semidirect
248. C22⋊Q8C2×C3×C4semidirect
254. #121C2×C3×C4semidirect
263. #117C2×C3×C4semidirect
556. C2×C3#1513direct (N × H)
598. C3C2×#1513direct (N × H)
605. C2C3×#1513direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5252
C2×C3×#4072
C2×C3×#3802
C3×#15132
C2×C3×#3282
C2×C3×#3272
C2×#15133