Finite Group Databasebeta

← C2×C3×#3117 #10679147

Subgroups — C2×C3×#3117 #10679147

Order 768.

2150 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
3C3×#3117C2116yesmaximal, direct product factor
4C2×C3×#484C211yesmaximal, characteristic, complemented
5C2×C3×#481C211yesmaximal, characteristic
6C2×C3×#479C212yesmaximal, complemented
7C2×C3×#464C212yesmaximal, complemented
8C2×C3×#451C211yesmaximal, characteristic, complemented
9C2×C3×#448C211yesmaximal, characteristic
10C2×C3×#441C212yesmaximal, complemented
11C2×C3×#440C212yesmaximal, complemented
12C2×C3×#432C211yesmaximal, characteristic, complemented
13C2×#3117C311yesmaximal, characteristic, direct product factor, Sylow
14C23×C3×Q8C2211yescharacteristic
15C22×C3×Q16C2212yes—
16C22×C3×SD16C2212yes—
17C3×#529—24no—
18C3×#529C2218yes—
19C2×C3×#126—21no—
20C2×C3×#125C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic
22C2×C3×#123—21no—
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×Q8—21no—
26C2×C3×C4×Q8—21no—
27C2×C3×C4×Q8—24no—
28C2×C3×C4×Q8C2212yes—
29C2×C3×C4×Q8C2212yes—
30C2×C3×C4×Q8C2211yescharacteristic
31C2×C3×C4×Q8C2218yescomplemented
32C2×C3×C4×D4—21no—
33C2×C3×C4×D4C2211yescharacteristic
34C2×C3×#117—21no—
35C22×C3×C4⋊C4—21no—
36C22×C3×C4⋊C4C2211yescharacteristic
37C22×C3×C42—21no—
38C22×C3×C42C2211yescharacteristic
39C3×#484C2218yescomplemented
40C3×#481C2218yes—
41C3×#479C22116yescomplemented
42C3×#464C22116yescomplemented
43C3×#451C2218yescomplemented
44C3×#448C2218yes—
45C3×#441C22116yescomplemented
46C3×#440C22116yescomplemented
47C3×#432C2218yescomplemented
48C2×C3×#107C2212yes—
49C2×C3×#106C2212yes—
50C2×C3×C4⋊C8C2212yes—
51C2×C3×#103C2212yes—
52C2×C3×#103C2212yes—
53C2×C3×#103C2212yes—
54C2×C3×#102C2212yes—
55C2×C3×C22⋊C8C2211yescharacteristic
56C2×C3×C22⋊C8C2211yescharacteristic
57C2×C3×#97C2212yes—
58C2×#529C2×C311yescharacteristic
59C22×C4×Q8C2×C311yescharacteristic, complemented
60#3117C2×C3116yesdirect product factor
61C2×#484C2×C311yescharacteristic, complemented
62C2×#481C2×C311yescharacteristic
63C2×#479C2×C312yescomplemented
64C2×#464C2×C312yescomplemented
65C2×#451C2×C311yescharacteristic, complemented
66C2×#448C2×C311yescharacteristic
67C2×#441C2×C312yescomplemented
68C2×#440C2×C312yescomplemented
69C2×#432C2×C311yescharacteristic, complemented
70C22×C3×Q8—28no—
71C22×C3×Q8—22no—
72C22×C3×Q8—21no—
73C22×C3×Q8C2311yescharacteristic
74C22×C3×Q8C2314yes—
75C22×C3×Q8C2312yes—
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4—21no—
80C23×C3×C4—21no—
81C23×C3×C4D411yescharacteristic
82C23×C3×C4D411yescharacteristic
83C23×C3×C4C2311yescharacteristic
84C2×C3×Q16—28no—
85C2×C3×Q16C2318yes—
86C2×C3×SD16—28no—
87C2×C3×SD16C2318yes—
88C22×C3×C8C2312yes—
89C22×C3×C8C2312yes—
90C3×#126—28no—
91C3×#125—22no—
92C3×#125C2314yes—
93C3×#124—22no—
94C3×#124C2314yes—
95C3×#123—28no—
96C3×C22⋊Q8—22no—
97C3×C22⋊Q8C2314yes—
98C3×#121—22no—
99C3×#121C2314yes—
100C3×C4×Q8—24no—
101C3×C4×Q8—24no—
102C3×C4×Q8—28no—
103C3×C4×Q8—28no—
104C3×C4×Q8—22no—
105C3×C4×Q8C2318yes—
106C3×C4×Q8C2314yes—
107C3×C4×Q8C2318yes—
108C3×C4×D4—28no—
109C3×C4×D4—22no—
110C3×C4×D4C2314yes—
111C3×#117—28no—
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4—22no—
114C2×C3×C4⋊C4—28no—
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—21no—
117C2×C3×C4⋊C4—21no—
118C2×C3×C4⋊C4—21no—
119C2×C3×C4⋊C4—21no—
120C2×C3×C4⋊C4—21no—
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C4⋊C4C2311yescharacteristic
123C2×C3×C4⋊C4C2311yescharacteristic
124C2×C3×C4⋊C4C2314yes—
125C2×C3×C22⋊C4—21no—
126C2×C3×C22⋊C4—21no—
127C2×C3×C22⋊C4—21no—
128C2×C3×C22⋊C4—21no—
129C2×C3×C22⋊C4—21no—
130C2×C3×C42—22no—
131C2×C3×C42—22no—
132C2×C3×C42—28no—
133C2×C3×C42—21no—
134C2×C3×C42D411yescharacteristic
135C2×C3×C42D411yescharacteristic
136C2×C3×C42D414yes—
137C2×C3×C42C2311yescharacteristic
138C2×C3×C42C2311yescharacteristic
139C2×C3×C42C2314yes—
140C3×#107C2318yes—
141C3×#106C2318yes—
142C3×C4⋊C8C2318yes—
143C3×#103C2318yes—
144C3×#103C2318yes—
145C3×#103C2318yes—
146C3×#102C2318yes—
147C3×C22⋊C8C2314yes—
148C3×C22⋊C8C2314yes—
149C3×#97C2318yes—
150C23×Q8C22×C311yescharacteristic
151C22×Q16C22×C312yes—
152C22×SD16C22×C312yes—
153#529—24no—
154#529C22×C318yes—
155C2×#126—21no—
156C2×#125C22×C311yescharacteristic
157C2×#124C22×C311yescharacteristic
158C2×#123—21no—
159C2×C22⋊Q8C22×C311yescharacteristic
160C2×#121C22×C311yescharacteristic
161C2×C4×Q8—21no—
162C2×C4×Q8—21no—
163C2×C4×Q8—24no—
164C2×C4×Q8C22×C312yes—
165C2×C4×Q8C22×C312yes—
166C2×C4×Q8C22×C311yescharacteristic
167C2×C4×Q8C22×C318yescomplemented
168C2×C4×D4—21no—
169C2×C4×D4C22×C311yescharacteristic
170C2×#117—21no—
171C22×C4⋊C4—21no—
172C22×C4⋊C4C22×C311yescharacteristic
173C22×C42—21no—
174C22×C42C22×C311yescharacteristic
175#484C22×C318yescomplemented
176#481C22×C318yes—
177#479C22×C3116yescomplemented
178#464C22×C3116yescomplemented
179#451C22×C318yescomplemented
180#448C22×C318yes—
181#441C22×C3116yescomplemented
182#440C22×C3116yescomplemented
183#432C22×C318yescomplemented
184C2×#107C22×C312yes—
185C2×#106C22×C312yes—
186C2×C4⋊C8C22×C312yes—
187C2×#103C22×C312yes—
188C2×#103C22×C312yes—
189C2×#103C22×C312yes—
190C2×#102C22×C312yes—
191C2×C22⋊C8C22×C311yescharacteristic
192C2×C22⋊C8C22×C311yescharacteristic
193C2×#97C22×C312yes—
194C24×C3—21no—
195C24×C3C2×D411yescharacteristic
196C2×C3×Q8—216no—
197C2×C3×Q8—28no—
198C2×C3×Q8—216no—
199C2×C3×Q8—24no—
200C2×C3×Q8—22no—
201C2×C3×Q8—21no—
202C2×C3×Q8—24no—
203C2×C3×Q8—24no—
204C2×C3×Q8#4118yes—
205C2×C3×Q8C2412yes—
206C2×C3×Q8C2414yes—
207C2×C3×D4—24no—
208C2×C3×D4—21no—
209C2×C3×D4—24no—
210C2×C3×D4—44no—
211C2×C3×D4C2412yes—
212C22×C3×C4—28no—
213C22×C3×C4—24no—
214C22×C3×C4—24no—
215C22×C3×C4—22no—
216C22×C3×C4—24no—
217C22×C3×C4—24no—
218C22×C3×C4—22no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—24no—
223C22×C3×C4—42no—
224C22×C3×C4—44no—
225C22×C3×C4—44no—
226C22×C3×C4C2×D414yes—
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D414yes—
234C22×C3×C4C2412yes—
235C22×C3×C4C2411yescharacteristic
236C3×Q16—216no—
237C3×SD16—216no—
238C2×C3×C8—24no—
239C2×C3×C8—24no—
240C2×C3×C8C2414yes—
241C2×C3×C8C2414yes—
242C3×C4⋊C4—28no—
243C3×C4⋊C4—28no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4C2414yes—
253C3×C4⋊C4C2412yes—
254C3×C4⋊C4C2412yes—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—24no—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—24no—
260C3×C42—28no—
261C3×C42—28no—
262C3×C42—24no—
263C3×C42C2×D414yes—
264C3×C42C2×D412yes—
265C3×C42C2×D412yes—
266C3×C42C2×D414yes—
267C3×C42C2412yes—
268C3×C42C2412yes—
269C22×Q8—28no—
270C22×Q8—21no—
271C22×Q8—22no—
272C22×Q8C23×C311yescharacteristic
273C22×Q8C23×C314yes—
274C22×Q8C23×C312yes—
275C22×D4—21no—
276C22×D4C23×C311yescharacteristic
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4C3×D411yescharacteristic
281C23×C4C3×D411yescharacteristic
282C23×C4C23×C311yescharacteristic
283C2×Q16—28no—
284C2×Q16C23×C318yes—
285C2×SD16—28no—
286C2×SD16C23×C318yes—
287C22×C8C23×C312yes—
288C22×C8C23×C312yes—
289#126—28no—
290#125—22no—
291#125C23×C314yes—
292#124—22no—
293#124C23×C314yes—
294#123—28no—
295C22⋊Q8—22no—
296C22⋊Q8C23×C314yes—
297#121—22no—
298#121C23×C314yes—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—28no—
302C4×Q8—28no—
303C4×Q8—22no—
304C4×Q8C23×C318yes—
305C4×Q8C23×C318yes—
306C4×Q8C23×C314yes—
307C4×D4—28no—
308C4×D4—22no—
309C4×D4C23×C314yes—
310#117—28no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C23×C312yes—
321C2×C4⋊C4C23×C314yes—
322C2×C4⋊C4C23×C311yescharacteristic
323C2×C4⋊C4C23×C311yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—28no—
333C2×C42C3×D411yescharacteristic
334C2×C42C3×D411yescharacteristic
335C2×C42C3×D414yes—
336C2×C42C23×C311yescharacteristic
337C2×C42C23×C311yescharacteristic
338C2×C42C23×C314yes—
339#107C23×C318yes—
340#106C23×C318yes—
341C4⋊C8C23×C318yes—
342#103C23×C318yes—
343#103C23×C318yes—
344#103C23×C318yes—
345#102C23×C318yes—
346C22⋊C8C23×C314yes—
347C22⋊C8C23×C314yes—
348#97C23×C318yes—
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—42no—
353C23×C3—42no—
354C23×C3—42no—
355C23×C3#13714yes—
356C23×C3C22×D412yes—
357C23×C3C22×D411yescharacteristic
358C3×Q8—216no—
359C3×Q8—28no—
360C3×Q8—24no—
361C3×Q8—24no—
362C3×Q8—42no—
363C3×Q8C2×#4118yes—
364C3×D4—24no—
365C3×D4—48no—
366C3×D4—42no—
367C2×C3×C4—216no—
368C2×C3×C4—28no—
369C2×C3×C4—22no—
370C2×C3×C4—28no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—44no—
383C2×C3×C4—48no—
384C2×C3×C4—48no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—44no—
388C2×C3×C4—42no—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D412yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4C22×D412yes—
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C2×#4114yes—
396C2×C3×C4C2511yescharacteristic
397C3×C8—24no—
398C3×C8—24no—
399C24—21no—
400C24C2×C3×D411yescharacteristic
401C2×Q8—216no—
402C2×Q8—28no—
403C2×Q8—216no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8C3×#4118yes—
410C2×Q8C24×C312yes—
411C2×Q8C24×C314yes—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—24no—
415C2×D4—44no—
416C2×D4C24×C312yes—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D414yes—
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D412yes—
437C22×C4C2×C3×D411yescharacteristic
438C22×C4C2×C3×D414yes—
439C22×C4C24×C312yes—
440C22×C4C24×C311yescharacteristic
441Q16—216no—
442SD16—216no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8C24×C314yes—
446C2×C8C24×C314yes—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4C24×C314yes—
458C4⋊C4C24×C312yes—
459C4⋊C4C24×C312yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C42—28no—
466C42—28no—
467C42—24no—
468C42C2×C3×D414yes—
469C42C2×C3×D412yes—
470C42C2×C3×D412yes—
471C42C2×C3×D414yes—
472C42C24×C312yes—
473C42C24×C312yes—
474C22×C3—28no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—84no—
481C22×C3#52512yes—
482C22×C3C2×#13714yes—
483C22×C3C2×#13712yes—
484C22×C3#57512yes—
485C22×C3C23×D411yescharacteristic
486C3×C4—28no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—42no—
493C3×C4—42no—
494C3×C4—44no—
495C3×C4—42no—
496C3×C4#52512yes—
497C3×C4C22×#4112yes—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23C3×#13714yes—
505C23C22×C3×D412yes—
506C23C22×C3×D411yescharacteristic
507Q8—216no—
508Q8—24no—
509Q8—28no—
510Q8—24no—
511Q8—42no—
512Q8C2×C3×#4118yes—
513D4—24no—
514D4—48no—
515D4—42no—
516C2×C4—216no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4C22×C3×D412yes—
539C2×C4C22×C3×D412yes—
540C2×C4C22×C3×D412yes—
541C2×C4C22×C3×D412yes—
542C2×C4C22×C3×D412yes—
543C2×C4C22×C3×D411yescharacteristic
544C2×C4C2×C3×#4114yes—
545C2×C4C25×C311yescharacteristic
546C8—24no—
547C8—24no—
548C2×C3—24no—
549C2×C3—82no—
550C2×C3#311714yesdirect product factor
551C2×C3C2×#52511yescharacteristic
552C2×C3C22×#13711yescharacteristic
553C2×C3C2×#57511yescharacteristic
554C22—22no—
555C22—22no—
556C22—28no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—84no—
561C22C3×#52512yes—
562C22C2×C3×#13714yes—
563C22C2×C3×#13712yes—
564C22C3×#57512yes—
565C22C23×C3×D411yescharacteristic
566C4—28no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—44no—
575C4—42no—
576C4C3×#52512yes—
577C4C22×C3×#4112yes—
578C3C2×#311711yescharacteristic, direct product factor, Sylow
579C2—24no—
580C2—82no—
581C2C3×#311714yesdirect product factor
582C2C2×C3×#52511yescharacteristic
583C2C22×C3×#13711yescharacteristic
584C2C2×C3×#57511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4×Q8C2semidirect
3. C3×#3117C2direct (N × H)
4. C2×C3×#484C2semidirect
6. C2×C3×#479C2semidirect
7. C2×C3×#464C2semidirect
8. C2×C3×#451C2semidirect
10. C2×C3×#441C2semidirect
11. C2×C3×#440C2semidirect
12. C2×C3×#432C2semidirect
13. C2×#3117C3direct (N × H)
31. C2×C3×C4×Q8C22semidirect
39. C3×#484C22semidirect
41. C3×#479C22semidirect
42. C3×#464C22semidirect
43. C3×#451C22semidirect
45. C3×#441C22semidirect
46. C3×#440C22semidirect
47. C3×#432C22semidirect
59. C22×C4×Q8C2×C3semidirect
60. #3117C2×C3direct (N × H)
61. C2×#484C2×C3semidirect
63. C2×#479C2×C3semidirect
64. C2×#464C2×C3semidirect
65. C2×#451C2×C3semidirect
67. C2×#441C2×C3semidirect
68. C2×#440C2×C3semidirect
69. C2×#432C2×C3semidirect
167. C2×C4×Q8C22×C3semidirect
175. #484C22×C3semidirect
177. #479C22×C3semidirect
178. #464C22×C3semidirect
179. #451C22×C3semidirect
181. #441C22×C3semidirect
182. #440C22×C3semidirect
183. #432C22×C3semidirect
550. C2×C3#3117direct (N × H)
578. C3C2×#3117direct (N × H)
581. C2C3×#3117direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C22×C3×C4×Q82
C3×#31172
C2×C3×#4842
C2×C3×#4812
C2×C3×#4792
C2×C3×#4642
C2×C3×#4512
C2×C3×#4482
C2×C3×#4412
C2×C3×#4402
C2×C3×#4322
C2×#31173