Finite Group Databasebeta

← C3×#23937 #10642544

Subgroups — C3×#23937 #10642544

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×C3×#451C211yesmaximal, characteristic, complemented
3C2×C3×#448C211yesmaximal, characteristic
4C2×C3×#391C211yesmaximal, characteristic, complemented
5C3×#2026C212yesmaximal, complemented
6C3×#2015C212yesmaximal, complemented
7C3×#1878C212yesmaximal, complemented
8C3×#1877C212yesmaximal, complemented
9C3×#1874C211yesmaximal, characteristic
10C3×#1872C211yesmaximal, characteristic, complemented
11C3×#1817C211yesmaximal, characteristic, complemented
12#23937C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
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×#121C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×#117—21no—
28C22×C3×C4⋊C4C2211yescharacteristic
29C22×C3×C42C2211yescharacteristic
30C3×#451—24no—
31C3×#448—24no—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2211yescharacteristic
38C3×#391—21no—
39C3×#391—21no—
40C3×#391C2212yes—
41C3×#391C2212yes—
42C2×C3×#95—21no—
43C2×C3×#95—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#529C2×C311yescharacteristic
49C2×#451C2×C311yescharacteristic, complemented
50C2×#448C2×C311yescharacteristic
51C2×#391C2×C311yescharacteristic, complemented
52#2026C2×C312yescomplemented
53#2015C2×C312yescomplemented
54#1878C2×C312yescomplemented
55#1877C2×C312yescomplemented
56#1874C2×C311yescharacteristic
57#1872C2×C311yescharacteristic, complemented
58#1817C2×C311yescharacteristic, complemented
59C22×C3×Q8—24no—
60C22×C3×Q8—21no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2312yes—
66C22×C3×D4—21no—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4D411yescharacteristic
72C23×C3×C4D411yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C2×C3×Q16—24no—
75C2×C3×Q16—24no—
76C2×C3×Q16—42no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—42no—
80C22×C3×C8C2312yes—
81C22×C3×C8C2312yes—
82C3×#126—28no—
83C3×#125—22no—
84C3×#125—22no—
85C3×#124—22no—
86C3×#124—22no—
87C3×#123—28no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8D414yes—
90C3×#121—22no—
91C3×#121D414yes—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×D4—28no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×#117—28no—
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—42no—
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×C42—22no—
113C2×C3×C42—22no—
114C2×C3×C42—21no—
115C2×C3×C42D411yescharacteristic
116C2×C3×C42D411yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C3×#103—24no—
120C3×#103—24no—
121C3×#103—42no—
122C3×#102—42no—
123C3×C22⋊C8—22no—
124C3×C22⋊C8—22no—
125C3×#95—22no—
126C3×#95—22no—
127C3×#95—22no—
128C3×#95—22no—
129C23×Q8C22×C311yescharacteristic
130C22×Q16C22×C312yes—
131C22×SD16C22×C312yes—
132#529—24no—
133#529—24no—
134C2×#126—21no—
135C2×#125C22×C311yescharacteristic
136C2×#124C22×C311yescharacteristic
137C2×#123—21no—
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×#121C22×C311yescharacteristic
140C2×C4×Q8C22×C311yescharacteristic
141C2×C4×D4—21no—
142C2×C4×D4C22×C311yescharacteristic
143C2×#117—21no—
144C22×C4⋊C4C22×C311yescharacteristic
145C22×C42C22×C311yescharacteristic
146#451—24no—
147#448—24no—
148C2×#103C22×C312yes—
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#102C22×C312yes—
152C2×C22⋊C8C22×C311yescharacteristic
153C2×C22⋊C8C22×C311yescharacteristic
154#391—21no—
155#391—21no—
156#391C22×C312yes—
157#391C22×C312yes—
158C2×#95—21no—
159C2×#95—21no—
160#340C22×C312yes—
161#340C22×C312yes—
162#336C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—24no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—48no—
176C2×C3×Q8—48no—
177C2×C3×Q8—42no—
178C2×C3×Q8—42no—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8#4114yes—
181C2×C3×D4—24no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4C2×D412yes—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—41no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C3×Q16—48no—
219C3×SD16—48no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4C2×D412yes—
236C3×C4⋊C4C2×D412yes—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—24no—
242C3×C42—24no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C22×Q8—24no—
250C22×Q8—21no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—42no—
254C22×Q8C23×C311yescharacteristic
255C22×Q8C23×C312yes—
256C22×D4—21no—
257C22×D4C23×C311yescharacteristic
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C3×D411yescharacteristic
262C23×C4C3×D411yescharacteristic
263C23×C4C23×C311yescharacteristic
264C2×Q16—24no—
265C2×Q16—24no—
266C2×Q16—42no—
267C2×SD16—24no—
268C2×SD16—24no—
269C2×SD16—42no—
270C22×C8C23×C312yes—
271C22×C8C23×C312yes—
272#126—28no—
273#125—22no—
274#125—22no—
275#124—22no—
276#124—22no—
277#123—28no—
278C22⋊Q8—22no—
279C22⋊Q8C3×D414yes—
280#121—22no—
281#121C3×D414yes—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×D4—28no—
285C4×D4—22no—
286C4×D4—22no—
287#117—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—42no—
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×C42—21no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42C3×D411yescharacteristic
306C2×C42C3×D411yescharacteristic
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309#103—24no—
310#103—24no—
311#103—42no—
312#102—42no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3#12012yes—
331C23×C3#13512yes—
332C23×C3#13712yes—
333C23×C3C22×D411yescharacteristic
334C3×Q8—24no—
335C3×Q8—24no—
336C3×Q8—24no—
337C3×Q8—48no—
338C3×Q8—42no—
339C3×Q8—42no—
340C3×Q8—42no—
341C3×D4—24no—
342C3×D4—48no—
343C3×D4—42no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—44no—
364C2×C3×C4—48no—
365C2×C3×C4—48no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—42no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—44no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—84no—
379C2×C3×C4#12012yes—
380C2×C3×C4#13712yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4112yes—
384C2×C3×C4H(5,𝔽2)12yes—
385C3×C8—42no—
386C3×C8—42no—
387C24—21no—
388C24C2×C3×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8C2×C3×D412yes—
403C2×Q8C3×#4114yes—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×C3×D412yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C2×C3×D412yes—
438C22×C4C2×C3×D412yes—
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—42no—
458C4⋊C4C2×C3×D412yes—
459C4⋊C4C2×C3×D412yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C42—24no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3#45214yes—
486C22×C3C2×#12011yescharacteristic
487C22×C3#52511yescharacteristic
488C22×C3#53511yescharacteristic
489C22×C3C2×#13511yescharacteristic
490C22×C3C2×#13711yescharacteristic
491C22×C3C2×#13711yescharacteristic
492C22×C3#57511yescharacteristic
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—44no—
496C3×C4—42no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—42no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—82no—
503C3×C4#54612yes—
504C3×C4#54612yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C3×#12012yes—
517C23C3×#13512yes—
518C23C3×#13712yes—
519C23C22×C3×D411yescharacteristic
520Q8—24no—
521Q8—24no—
522Q8—24no—
523Q8—48no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527D4—24no—
528D4—48no—
529D4—42no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—84no—
565C2×C4C3×#12012yes—
566C2×C4C3×#13712yes—
567C2×C4C22×C3×D411yescharacteristic
568C2×C4C22×C3×D411yescharacteristic
569C2×C4C2×C3×#4112yes—
570C2×C4C3×H(5,𝔽2)12yes—
571C8—42no—
572C8—42no—
573C2×C3—24no—
574C2×C3—82no—
575C2×C3#240411yescharacteristic
576C2×C3C2×#45211yescharacteristic
577C2×C3#300711yescharacteristic
578C2×C3#310811yescharacteristic
579C2×C3#311711yescharacteristic
580C2×C3#321111yescharacteristic
581C2×C3#321811yescharacteristic
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C3×#45214yes—
596C22C2×C3×#12011yescharacteristic
597C22C3×#52511yescharacteristic
598C22C3×#53511yescharacteristic
599C22C2×C3×#13511yescharacteristic
600C22C2×C3×#13711yescharacteristic
601C22C2×C3×#13711yescharacteristic
602C22C3×#57511yescharacteristic
603C4—22no—
604C4—22no—
605C4—44no—
606C4—42no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—82no—
613C4C3×#54612yes—
614C4C3×#54612yes—
615C3#2393711yescharacteristic, direct product factor, Sylow
616C2—24no—
617C2—82no—
618C2C3×#240411yescharacteristic
619C2C2×C3×#45211yescharacteristic
620C2C3×#300711yescharacteristic
621C2C3×#310811yescharacteristic
622C2C3×#311711yescharacteristic
623C2C3×#321111yescharacteristic
624C2C3×#321811yescharacteristic
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×#451C2semidirect
4. C2×C3×#391C2semidirect
5. C3×#2026C2semidirect
6. C3×#2015C2semidirect
7. C3×#1878C2semidirect
8. C3×#1877C2semidirect
10. C3×#1872C2semidirect
11. C3×#1817C2semidirect
12. #23937C3direct (N × H)
49. C2×#451C2×C3semidirect
51. C2×#391C2×C3semidirect
52. #2026C2×C3semidirect
53. #2015C2×C3semidirect
54. #1878C2×C3semidirect
55. #1877C2×C3semidirect
57. #1872C2×C3semidirect
58. #1817C2×C3semidirect
615. C3#23937direct (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×#4512
C2×C3×#4482
C2×C3×#3912
C3×#20262
C3×#20152
C3×#18782
C3×#18772
C3×#18742
C3×#18722
C3×#18172
#239373