Finite Group Databasebeta

← C3×#23945 #10642552

Subgroups — C3×#23945 #10642552

Order 768.

1382 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#542C211yesmaximal, characteristic, complemented
2C2×C3×#450C211yesmaximal, characteristic, complemented
3C2×C3×#447C211yesmaximal, characteristic
4C3×#2309C211yesmaximal, characteristic, complemented
5C3×#2023C212yesmaximal, complemented
6C3×#2012C212yesmaximal, complemented
7C3×#1880C212yesmaximal, complemented
8C3×#1879C212yesmaximal, complemented
9C3×#1875C211yesmaximal, characteristic, complemented
10C3×#1873C211yesmaximal, characteristic, complemented
11C3×#1818C211yesmaximal, characteristic, complemented
12#23945C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C3×#542—24no—
17C3×#542—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic, complemented
19C2×C3×#126—21no—
20C2×C3×#124—21no—
21C2×C3×#124C2211yescharacteristic, complemented
22C2×C3×#123—21no—
23C2×C3×C22⋊Q8—21no—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×#121C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic, complemented
27C2×C3×C4×D4C2211yescharacteristic, complemented
28C2×C3×#117C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C3×#450—24no—
31C3×#447—24no—
32C2×C3×#103C2212yescomplemented
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yescomplemented
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2211yescharacteristic
38C3×#390—21no—
39C3×#390—21no—
40C3×#390C2212yes—
41C3×#390C2212yes—
42C3×#379—21no—
43C3×#379—21no—
44C3×#340C2212yescomplemented
45C3×#340C2212yescomplemented
46C3×#336C2212yescomplemented
47C3×#336C2212yescomplemented
48C2×#542C2×C311yescharacteristic, complemented
49C2×#450C2×C311yescharacteristic, complemented
50C2×#447C2×C311yescharacteristic
51#2309C2×C311yescharacteristic, complemented
52#2023C2×C312yescomplemented
53#2012C2×C312yescomplemented
54#1880C2×C312yescomplemented
55#1879C2×C312yescomplemented
56#1875C2×C311yescharacteristic, complemented
57#1873C2×C311yescharacteristic, complemented
58#1818C2×C311yescharacteristic, complemented
59C25×C3—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4—24no—
64C22×C3×D4—21no—
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4—42no—
68C22×C3×D4C2311yescharacteristic
69C22×C3×D4C2312yes—
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C2×C3×SD16—24no—
73C2×C3×SD16—24no—
74C2×C3×SD16—42no—
75C2×C3×D8—24no—
76C2×C3×D8—24no—
77C2×C3×D8—42no—
78C22×C3×C8C2312yes—
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—22no—
82C3×#126—28no—
83C3×#124—28no—
84C3×#124—22no—
85C3×#124—22no—
86C3×#123—28no—
87C3×C22⋊Q8—28no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8D414yes—
90C3×#121—22no—
91C3×#121D414yes—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×#117—22no—
97C3×#117—22no—
98C3×#117—42no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
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—21no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4D411yescharacteristic
118C2×C3×C22⋊C4D411yescharacteristic
119C2×C3×C42—21no—
120C2×C3×C42—21no—
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C3×#103—42no—
124C3×#102—24no—
125C3×#102—24no—
126C3×#102—42no—
127C3×C22⋊C8—22no—
128C3×C22⋊C8—22no—
129C23×D4C22×C311yescharacteristic
130C22×SD16C22×C312yes—
131C22×D8C22×C312yes—
132#542—24no—
133#542—24no—
134C2×C4⋊Q8C22×C311yescharacteristic, complemented
135C2×#126—21no—
136C2×#124—21no—
137C2×#124C22×C311yescharacteristic, complemented
138C2×#123—21no—
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×#121C22×C311yescharacteristic
142C2×C4×Q8C22×C311yescharacteristic, complemented
143C2×C4×D4C22×C311yescharacteristic, complemented
144C2×#117C22×C311yescharacteristic
145C2×#117C22×C311yescharacteristic
146#450—24no—
147#447—24no—
148C2×#103C22×C312yescomplemented
149C2×#102C22×C312yes—
150C2×#102C22×C312yes—
151C2×#102C22×C312yescomplemented
152C2×C22⋊C8C22×C311yescharacteristic
153C2×C22⋊C8C22×C311yescharacteristic
154#390—21no—
155#390—21no—
156#390C22×C312yes—
157#390C22×C312yes—
158#379—21no—
159#379—21no—
160#340C22×C312yescomplemented
161#340C22×C312yescomplemented
162#336C22×C312yescomplemented
163#336C22×C312yescomplemented
164C24×C3—24no—
165C24×C3—22no—
166C24×C3—21no—
167C24×C3—42no—
168C24×C3—42no—
169C24×C3—82no—
170C24×C3—82no—
171C24×C3C2×D411yescharacteristic
172C2×C3×Q8—24no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—82no—
177C2×C3×Q8C2×D412yes—
178C2×C3×D4—24no—
179C2×C3×D4—24no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—48no—
189C2×C3×D4—48no—
190C2×C3×D4—42no—
191C2×C3×D4—42no—
192C2×C3×D4—82no—
193C2×C3×D4C2×D412yes—
194C2×C3×D4#4114yes—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—24no—
201C22×C3×C4—22no—
202C22×C3×C4—82no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2411yescharacteristic
211C3×SD16—48no—
212C3×D8—48no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4C2×D412yes—
229C3×C4⋊C4C2×D412yes—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—82no—
239C3×C22⋊C4—82no—
240C3×C22⋊C4—82no—
241C3×C22⋊C4—82no—
242C3×C42—24no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—42no—
248C25—21no—
249C22×Q8—21no—
250C22×Q8C23×C311yescharacteristic
251C22×D4—21no—
252C22×D4—24no—
253C22×D4—21no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—42no—
257C22×D4C23×C311yescharacteristic
258C22×D4C23×C312yes—
259C23×C4—21no—
260C23×C4C23×C311yescharacteristic
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—42no—
264C2×D8—24no—
265C2×D8—24no—
266C2×D8—42no—
267C22×C8C23×C312yes—
268C22×C8C23×C312yes—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271#126—28no—
272#124—28no—
273#124—22no—
274#124—22no—
275#123—28no—
276C22⋊Q8—28no—
277C22⋊Q8—22no—
278C22⋊Q8C3×D414yes—
279#121—22no—
280#121C3×D414yes—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×D4—22no—
284C4×D4—22no—
285#117—22no—
286#117—22no—
287#117—42no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C3×D411yescharacteristic
292C2×C4⋊C4C3×D411yescharacteristic
293C2×C4⋊C4C23×C312yes—
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C22⋊C4—21no—
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×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4C3×D411yescharacteristic
307C2×C22⋊C4C3×D411yescharacteristic
308C2×C42—21no—
309C2×C42—21no—
310C2×C42C23×C311yescharacteristic
311C2×C42C23×C311yescharacteristic
312#103—42no—
313#102—24no—
314#102—24no—
315#102—42no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—24no—
324C23×C3—48no—
325C23×C3—42no—
326C23×C3—41no—
327C23×C3—42no—
328C23×C3—41no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—84no—
336C23×C3—84no—
337C23×C3—88no—
338C23×C3—88no—
339C23×C3#12012yes—
340C23×C3C22×D411yescharacteristic
341C3×Q8—24no—
342C3×Q8—42no—
343C3×Q8—84no—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—24no—
347C3×D4—48no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—84no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—84no—
375C2×C3×C4#12012yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4112yes—
379C2×C3×C4H(5,𝔽2)12yes—
380C3×C8—42no—
381C3×C8—42no—
382C24—24no—
383C24—21no—
384C24—22no—
385C24—42no—
386C24—42no—
387C24—82no—
388C24—82no—
389C24C2×C3×D411yescharacteristic
390C2×Q8—24no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—82no—
395C2×Q8C2×C3×D412yes—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—82no—
411C2×D4C2×C3×D412yes—
412C2×D4C3×#4114yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—82no—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C2×C3×D412yes—
427C22×C4C2×C3×D412yes—
428C22×C4C24×C311yescharacteristic
429SD16—48no—
430D8—48no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4C2×C3×D412yes—
447C4⋊C4C2×C3×D412yes—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—24no—
455C22⋊C4—42no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C42—24no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—42no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—44no—
472C22×C3—44no—
473C22×C3—44no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—816no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—82no—
481C22×C3—82no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3C2×#12011yescharacteristic
485C22×C3#52511yescharacteristic
486C22×C3#53511yescharacteristic
487C22×C3#57612yes—
488C22×C3#57712yes—
489C3×C4—22no—
490C3×C4—42no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—82no—
498C3×C4#54612yes—
499C3×C4#54612yes—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—24no—
506C23—48no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—84no—
518C23—84no—
519C23—88no—
520C23—88no—
521C23C3×#12012yes—
522C23C22×C3×D411yescharacteristic
523Q8—24no—
524Q8—42no—
525Q8—84no—
526D4—24no—
527D4—24no—
528D4—24no—
529D4—48no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—84no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4C3×#12012yes—
558C2×C4C22×C3×D411yescharacteristic
559C2×C4C22×C3×D411yescharacteristic
560C2×C4C2×C3×#4112yes—
561C2×C4C3×H(5,𝔽2)12yes—
562C8—42no—
563C8—42no—
564C2×C3—42no—
565C2×C3—82no—
566C2×C3—84no—
567C2×C3#240411yescharacteristic
568C2×C3#300811yescharacteristic
569C2×C3#301011yescharacteristic
570C2×C3#311812yes—
571C2×C3#322412yes—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—816no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22C2×C3×#12011yescharacteristic
591C22C3×#52511yescharacteristic
592C22C3×#53511yescharacteristic
593C22C3×#57612yes—
594C22C3×#57712yes—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—82no—
604C4C3×#54612yes—
605C4C3×#54612yes—
606C3#2394511yescharacteristic, direct product factor, Sylow
607C2—42no—
608C2—84no—
609C2—82no—
610C2C3×#240411yescharacteristic
611C2C3×#300811yescharacteristic
612C2C3×#301011yescharacteristic
613C2C3×#311812yes—
614C2C3×#322412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#542C2semidirect
2. C2×C3×#450C2semidirect
4. C3×#2309C2semidirect
5. C3×#2023C2semidirect
6. C3×#2012C2semidirect
7. C3×#1880C2semidirect
8. C3×#1879C2semidirect
9. C3×#1875C2semidirect
10. C3×#1873C2semidirect
11. C3×#1818C2semidirect
12. #23945C3direct (N × H)
18. C2×C3×C4⋊Q8C22semidirect
21. C2×C3×#124C22semidirect
26. C2×C3×C4×Q8C22semidirect
27. C2×C3×C4×D4C22semidirect
32. C2×C3×#103C22semidirect
35. C2×C3×#102C22semidirect
44. C3×#340C22semidirect
45. C3×#340C22semidirect
46. C3×#336C22semidirect
47. C3×#336C22semidirect
48. C2×#542C2×C3semidirect
49. C2×#450C2×C3semidirect
51. #2309C2×C3semidirect
52. #2023C2×C3semidirect
53. #2012C2×C3semidirect
54. #1880C2×C3semidirect
55. #1879C2×C3semidirect
56. #1875C2×C3semidirect
57. #1873C2×C3semidirect
58. #1818C2×C3semidirect
134. C2×C4⋊Q8C22×C3semidirect
137. C2×#124C22×C3semidirect
142. C2×C4×Q8C22×C3semidirect
143. C2×C4×D4C22×C3semidirect
148. C2×#103C22×C3semidirect
151. C2×#102C22×C3semidirect
160. #340C22×C3semidirect
161. #340C22×C3semidirect
162. #336C22×C3semidirect
163. #336C22×C3semidirect
606. C3#23945direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5422
C2×C3×#4502
C2×C3×#4472
C3×#23092
C3×#20232
C3×#20122
C3×#18802
C3×#18792
C3×#18752
C3×#18732
C3×#18182
#239453