Finite Group Databasebeta

← C3×#29665 #10648272

Subgroups — C3×#29665 #10648272

Order 768.

1164 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C3×#2857C211yesmaximal, characteristic, complemented
5C3×#2026C212yesmaximal, complemented
6C3×#2015C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#2000C212yesmaximal
9C3×#1803C211yesmaximal, characteristic, complemented
10C3×#1802C211yesmaximal, characteristic, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29665C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C3×#529—24no—
17C3×#529—24no—
18C2×C3×#126—21no—
19C2×C3×#125C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×#123—21no—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C22⋊Q8C2212yes—
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4—21no—
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×#117—21no—
29C22×C3×C42C2211yescharacteristic
30C3×#462—24no—
31C3×#461—24no—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#103C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C3×#391C2212yes—
40C3×#389—21no—
41C3×#389—21no—
42C3×#384C2211yescharacteristic
43C3×#380—21no—
44C3×#380—21no—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#529C2×C311yescharacteristic
49C2×#462C2×C311yescharacteristic, complemented
50C2×#461C2×C311yescharacteristic, complemented
51#2857C2×C311yescharacteristic, complemented
52#2026C2×C312yescomplemented
53#2015C2×C312yescomplemented
54#2001C2×C312yescomplemented
55#2000C2×C312yes—
56#1803C2×C311yescharacteristic, complemented
57#1802C2×C311yescharacteristic, complemented
58#1797C2×C311yescharacteristic, complemented
59C22×C3×Q8C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C2×C3×Q16—24no—
68C2×C3×Q16—42no—
69C2×C3×Q16—42no—
70C2×C3×SD16—24no—
71C2×C3×SD16—42no—
72C2×C3×SD16—42no—
73C22×C3×C8C2312yes—
74C22×C3×C8C2312yes—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8—22no—
77C3×#126—28no—
78C3×#125—22no—
79C3×#125—22no—
80C3×#124—22no—
81C3×#124—22no—
82C3×#123—28no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—42no—
86C3×C22⋊Q8—42no—
87C3×#121—22no—
88C3×#121—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8D414yes—
91C3×C4×D4—28no—
92C3×C4×D4—22no—
93C3×C4×D4D414yes—
94C3×#117—28no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C42—21no—
114C2×C3×C42—22no—
115C2×C3×C42—22no—
116C2×C3×C42D411yescharacteristic
117C2×C3×C42D411yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×C4⋊C8—22no—
121C3×C4⋊C8—22no—
122C3×#103—24no—
123C3×#103—24no—
124C3×#103—42no—
125C3×#102—42no—
126C3×C22⋊C8—42no—
127C3×#95—22no—
128C3×#95—22no—
129C22×Q16C22×C312yes—
130C22×SD16C22×C312yes—
131C2×C4⋊Q8C22×C311yescharacteristic
132#529—24no—
133#529—24no—
134C2×#126—21no—
135C2×#125C22×C311yescharacteristic
136C2×#124C22×C311yescharacteristic
137C2×#123—21no—
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×C22⋊Q8C22×C312yes—
140C2×#121C22×C311yescharacteristic
141C2×C4×Q8C22×C311yescharacteristic
142C2×C4×D4—21no—
143C2×C4×D4C22×C311yescharacteristic
144C2×#117—21no—
145C22×C42C22×C311yescharacteristic
146#462—24no—
147#461—24no—
148C2×C4⋊C8C22×C311yescharacteristic
149C2×C4⋊C8C22×C311yescharacteristic
150C2×#103C22×C312yes—
151C2×#103C22×C312yes—
152C2×#103C22×C312yes—
153C2×#102C22×C312yes—
154C2×C22⋊C8C22×C312yes—
155#391C22×C312yes—
156#389—21no—
157#389—21no—
158#384C22×C311yescharacteristic
159#380—21no—
160#380—21no—
161#340C22×C312yes—
162#340C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—21no—
176C2×C3×Q8—22no—
177C2×C3×Q8C2×D412yes—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4C2×D412yes—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—24no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—41no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C3×Q16—48no—
214C3×SD16—48no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—44no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4C2×D412yes—
235C3×C4⋊C4C2×D412yes—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—44no—
242C3×C42—22no—
243C3×C42—22no—
244C3×C42—24no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42C2×D412yes—
248C3×C42C2×D412yes—
249C22×Q8C23×C312yes—
250C22×Q8C23×C311yescharacteristic
251C22×D4—21no—
252C22×D4C23×C311yescharacteristic
253C23×C4—21no—
254C23×C4C3×D411yescharacteristic
255C23×C4C3×D411yescharacteristic
256C23×C4C23×C311yescharacteristic
257C2×Q16—24no—
258C2×Q16—42no—
259C2×Q16—42no—
260C2×SD16—24no—
261C2×SD16—42no—
262C2×SD16—42no—
263C22×C8C23×C312yes—
264C22×C8C23×C312yes—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267#126—28no—
268#125—22no—
269#125—22no—
270#124—22no—
271#124—22no—
272#123—28no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275C22⋊Q8—42no—
276C22⋊Q8—42no—
277#121—22no—
278#121—22no—
279C4×Q8—22no—
280C4×Q8C3×D414yes—
281C4×D4—28no—
282C4×D4—22no—
283C4×D4C3×D414yes—
284#117—28no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4C23×C312yes—
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C3×D411yescharacteristic
307C2×C42C3×D411yescharacteristic
308C2×C42C23×C311yescharacteristic
309C2×C42C23×C311yescharacteristic
310C4⋊C8—22no—
311C4⋊C8—22no—
312#103—24no—
313#103—24no—
314#103—42no—
315#102—42no—
316C22⋊C8—42no—
317#95—22no—
318#95—22no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3#13712yes—
333C23×C3C22×D411yescharacteristic
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—41no—
337C3×Q8—41no—
338C3×Q8—42no—
339C3×Q8—42no—
340C3×Q8—42no—
341C3×Q8—42no—
342C3×D4—22no—
343C3×D4—22no—
344C3×D4—48no—
345C3×D4—41no—
346C3×D4—41no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—44no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—44no—
373C2×C3×C4—48no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4#12012yes—
381C2×C3×C4#12012yes—
382C2×C3×C4#13512yes—
383C2×C3×C4#13712yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C3×C8—42no—
391C3×C8—42no—
392C24—21no—
393C24C2×C3×D411yescharacteristic
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8C2×C3×D412yes—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×C3×D412yes—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C2×C3×D411yescharacteristic
438C22×C4C2×C3×D411yescharacteristic
439C22×C4C2×C3×D411yescharacteristic
440C22×C4C24×C311yescharacteristic
441Q16—48no—
442SD16—48no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4C2×C3×D412yes—
463C4⋊C4C2×C3×D412yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—44no—
470C42—22no—
471C42—22no—
472C42—24no—
473C42—21no—
474C42—21no—
475C42C2×C3×D412yes—
476C42C2×C3×D412yes—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3C2×#12011yescharacteristic
492C22×C3#53511yescharacteristic
493C22×C3#53711yescharacteristic
494C22×C3C2×#13511yescharacteristic
495C22×C3C2×#13711yescharacteristic
496C22×C3C2×#13711yescharacteristic
497C22×C3#57511yescharacteristic
498C3×C4—22no—
499C3×C4—41no—
500C3×C4—42no—
501C3×C4—41no—
502C3×C4—42no—
503C3×C4—41no—
504C3×C4—42no—
505C3×C4—82no—
506C3×C4—82no—
507C3×C4#45214yes—
508C3×C4#53511yescharacteristic
509C3×C4#53511yescharacteristic
510C3×C4#53711yescharacteristic
511C3×C4#53711yescharacteristic
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C3×#13712yes—
526C23C22×C3×D411yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535D4—22no—
536D4—22no—
537D4—48no—
538D4—41no—
539D4—41no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4C3×#12012yes—
574C2×C4C3×#12012yes—
575C2×C4C3×#13512yes—
576C2×C4C3×#13712yes—
577C2×C4C22×C3×D411yescharacteristic
578C2×C4C22×C3×D411yescharacteristic
579C2×C4C3×H(5,𝔽2)11yescharacteristic
580C2×C4C3×H(5,𝔽2)11yescharacteristic
581C2×C4C3×H(5,𝔽2)11yescharacteristic
582C2×C4C3×2−1+411yescharacteristic
583C8—42no—
584C8—42no—
585C2×C3—42no—
586C2×C3—82no—
587C2×C3#261511yescharacteristic
588C2×C3C2×#45211yescharacteristic
589C2×C3#300711yescharacteristic
590C2×C3#321211yescharacteristic
591C2×C3#321811yescharacteristic
592C2×C3#324811yescharacteristic
593C2×C3#325211yescharacteristic
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22C2×C3×#12011yescharacteristic
609C22C3×#53511yescharacteristic
610C22C3×#53711yescharacteristic
611C22C2×C3×#13511yescharacteristic
612C22C2×C3×#13711yescharacteristic
613C22C2×C3×#13711yescharacteristic
614C22C3×#57511yescharacteristic
615C4—22no—
616C4—41no—
617C4—42no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—42no—
622C4—82no—
623C4—82no—
624C4C3×#45214yes—
625C4C3×#53511yescharacteristic
626C4C3×#53511yescharacteristic
627C4C3×#53711yescharacteristic
628C4C3×#53711yescharacteristic
629C3#2966511yescharacteristic, direct product factor, Sylow
630C2—42no—
631C2—82no—
632C2C3×#261511yescharacteristic
633C2C2×C3×#45211yescharacteristic
634C2C3×#300711yescharacteristic
635C2C3×#321211yescharacteristic
636C2C3×#321811yescharacteristic
637C2C3×#324811yescharacteristic
638C2C3×#325211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×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. C2×C3×#462C2semidirect
3. C2×C3×#461C2semidirect
4. C3×#2857C2semidirect
5. C3×#2026C2semidirect
6. C3×#2015C2semidirect
7. C3×#2001C2semidirect
9. C3×#1803C2semidirect
10. C3×#1802C2semidirect
11. C3×#1797C2semidirect
12. #29665C3direct (N × H)
49. C2×#462C2×C3semidirect
50. C2×#461C2×C3semidirect
51. #2857C2×C3semidirect
52. #2026C2×C3semidirect
53. #2015C2×C3semidirect
54. #2001C2×C3semidirect
56. #1803C2×C3semidirect
57. #1802C2×C3semidirect
58. #1797C2×C3semidirect
629. C3#29665direct (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
C2×C3×#4622
C2×C3×#4612
C3×#28572
C3×#20262
C3×#20152
C3×#20012
C3×#20002
C3×#18032
C3×#18022
C3×#17972
#296653