Finite Group Databasebeta

← C3×#25586 #10644193

Subgroups — C3×#25586 #10644193

Order 768.

1196 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3505C211yesmaximal, characteristic
2C3×#2503C211yesmaximal, characteristic, complemented
3C3×#2501C211yesmaximal, characteristic, complemented
4C3×#2486C211yesmaximal, characteristic, complemented
5C3×#2022C214yesmaximal, complemented
6C3×#1996C211yesmaximal, characteristic, complemented
7C3×#1964C211yesmaximal, characteristic, complemented
8C3×#1890C214yesmaximal, complemented
9C3×C4×#99C211yesmaximal, characteristic, complemented
10#25586C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#553—21no—
12C3×#553—22no—
13C3×#552—21no—
14C3×#551—22no—
15C3×#548—21no—
16C3×#546—21no—
17C3×#539—21no—
18C3×#538—21no—
19C3×#534C2212yes—
20C3×#529—21no—
21C3×#525—21no—
22C2×C3×#123C2214yes—
23C2×C3×#123C2212yes—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×#121C2211yescharacteristic
26C3×#518C2212yes—
27C2×C3×C4×D4C2211yescharacteristic
28C22×C3×C4⋊C4C2211yescharacteristic
29C2×C3×#99C2212yes—
30C2×C3×#99C2214yes—
31C3×#400—22no—
32C3×#399—22no—
33C3×#398—21no—
34C3×#397—21no—
35C3×#391—21no—
36C3×#390—21no—
37C3×#389—21no—
38C3×#384—21no—
39C3×#382—21no—
40C3×#381C2214yes—
41C3×#381C2212yes—
42C3×#380—21no—
43C3×#380C2211yescharacteristic
44C3×C4×C22⋊C4C2211yescharacteristic
45C3×C4×C22⋊C4C2211yescharacteristic
46C3×#342C2214yes—
47C3×#342C2212yes—
48#3505C2×C311yescharacteristic
49#2503C2×C311yescharacteristic, complemented
50#2501C2×C311yescharacteristic, complemented
51#2486C2×C311yescharacteristic, complemented
52#2022C2×C314yescomplemented
53#1996C2×C311yescharacteristic, complemented
54#1964C2×C311yescharacteristic, complemented
55#1890C2×C314yescomplemented
56C4×#99C2×C311yescharacteristic, complemented
57C2×C3×#41—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4D411yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C22×C3×D4C2312yes—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C3×#126—22no—
66C3×#126—22no—
67C3×#126—22no—
68C3×#125—21no—
69C3×#125—22no—
70C3×#124—21no—
71C3×#124—22no—
72C3×#123—22no—
73C3×#123—22no—
74C3×#123—22no—
75C3×#123—22no—
76C3×#123—22no—
77C3×#123—28no—
78C3×#123—44no—
79C3×#123—44no—
80C3×C22⋊Q8—21no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8—21no—
86C3×#121—21no—
87C3×#121—22no—
88C3×#121—21no—
89C3×#121—21no—
90C3×#121—21no—
91C3×#121—21no—
92C3×#121—22no—
93C3×#121—22no—
94C3×#120—22no—
95C3×#120—22no—
96C3×C4×Q8—21no—
97C3×C4×D4—22no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—22no—
104C3×C4×D4—22no—
105C3×C4×D4—22no—
106C3×C4×D4—22no—
107C3×#117—21no—
108C3×#117—22no—
109C3×#117—22no—
110C2×C3×C4⋊C4—21no—
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4—24no—
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—21no—
117C2×C3×C4⋊C4—21no—
118C2×C3×C4⋊C4—21no—
119C2×C3×C4⋊C4D411yescharacteristic
120C2×C3×C4⋊C4D411yescharacteristic
121C2×C3×C4⋊C4D411yescharacteristic
122C2×C3×C4⋊C4D412yes—
123C2×C3×C22⋊C4—22no—
124C2×C3×C22⋊C4—22no—
125C2×C3×C22⋊C4—24no—
126C2×C3×C22⋊C4D411yescharacteristic
127C2×C3×C22⋊C4D412yes—
128C2×C3×C22⋊C4C2312yes—
129C2×C3×C22⋊C4C2312yes—
130C2×C3×C22⋊C4C2311yescharacteristic
131C2×C3×C22⋊C4C2312yes—
132C2×C3×C22⋊C4C2312yes—
133C2×C3×C42—21no—
134C2×C3×C42—21no—
135C2×C3×C42D411yescharacteristic
136C3×#99—24no—
137C3×#99—44no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—22no—
141C3×#95—21no—
142C3×#95—24no—
143C3×#95—22no—
144C3×#95—24no—
145#553—21no—
146#553—22no—
147#552—21no—
148#551—22no—
149#548—21no—
150#546—21no—
151#539—21no—
152#538—21no—
153#534C22×C312yes—
154#529—21no—
155#525—21no—
156C2×#123C22×C314yes—
157C2×#123C22×C312yes—
158C2×C22⋊Q8C22×C311yescharacteristic
159C2×#121C22×C311yescharacteristic
160#518C22×C312yes—
161C2×C4×D4C22×C311yescharacteristic
162C22×C4⋊C4C22×C311yescharacteristic
163C2×#99C22×C312yes—
164C2×#99C22×C314yes—
165#400—22no—
166#399—22no—
167#398—21no—
168#397—21no—
169#391—21no—
170#390—21no—
171#389—21no—
172#384—21no—
173#382—21no—
174#381C22×C312yes—
175#381C22×C314yes—
176#380—21no—
177#380C22×C311yescharacteristic
178C4×C22⋊C4C22×C311yescharacteristic
179C4×C22⋊C4C22×C311yescharacteristic
180#342C22×C312yes—
181#342C22×C314yes—
182C24×C3C2×D412yes—
183C24×C3C2×D411yescharacteristic
184C24×C3C2411yescharacteristic
185C3×#41—82no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—42no—
189C2×C3×D4—21no—
190C2×C3×D4—22no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—24no—
195C2×C3×D4—24no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—42no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—24no—
213C22×C3×C4—24no—
214C22×C3×C4—42no—
215C22×C3×C4—42no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D412yes—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D412yes—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—48no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—44no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—41no—
246C3×C4⋊C4—41no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—21no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—21no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—21no—
260C3×C22⋊C4—21no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—48no—
266C3×C22⋊C4—44no—
267C3×C22⋊C4#4114yescomplemented
268C3×C42—22no—
269C3×C42—22no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42—42no—
273C3×C42—42no—
274C2×#41—21no—
275C22×Q8C3×D411yescharacteristic
276C22×D4C3×D411yescharacteristic
277C22×D4C23×C311yescharacteristic
278C22×D4C23×C312yes—
279C23×C4C3×D411yescharacteristic
280C23×C4C23×C312yes—
281C23×C4C23×C311yescharacteristic
282#126—22no—
283#126—22no—
284#126—22no—
285#125—21no—
286#125—22no—
287#124—21no—
288#124—22no—
289#123—22no—
290#123—22no—
291#123—22no—
292#123—22no—
293#123—22no—
294#123—28no—
295#123—44no—
296#123—44no—
297C22⋊Q8—21no—
298C22⋊Q8—22no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302C22⋊Q8—21no—
303#121—21no—
304#121—22no—
305#121—21no—
306#121—21no—
307#121—21no—
308#121—21no—
309#121—22no—
310#121—22no—
311#120—22no—
312#120—22no—
313C4×Q8—21no—
314C4×D4—22no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324#117—21no—
325#117—22no—
326#117—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4C3×D411yescharacteristic
337C2×C4⋊C4C3×D411yescharacteristic
338C2×C4⋊C4C3×D411yescharacteristic
339C2×C4⋊C4C3×D412yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4C3×D411yescharacteristic
344C2×C22⋊C4C3×D412yes—
345C2×C22⋊C4C23×C312yes—
346C2×C22⋊C4C23×C312yes—
347C2×C22⋊C4C23×C311yescharacteristic
348C2×C22⋊C4C23×C312yes—
349C2×C22⋊C4C23×C312yes—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42C3×D411yescharacteristic
353#99—24no—
354#99—44no—
355#95—21no—
356#95—21no—
357#95—22no—
358#95—21no—
359#95—24no—
360#95—22no—
361#95—24no—
362C23×C3—21no—
363C23×C3—22no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—24no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—42no—
372C23×C3—42no—
373C23×C3C22×D411yescharacteristic
374C23×C3C22×D411yescharacteristic
375C23×C3C22×D411yescharacteristic
376C23×C3C2×#4112yes—
377C23×C3H(5,𝔽2)12yes—
378C3×Q8—42no—
379C3×Q8—81no—
380C3×D4—42no—
381C3×D4—42no—
382C3×D4—42no—
383C3×D4—42no—
384C3×D4—44no—
385C3×D4—42no—
386C3×D4—81no—
387C3×D4—82no—
388C3×D4—82no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—24no—
401C2×C3×C4—24no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—42no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—42no—
411C2×C3×C4—42no—
412C2×C3×C4—42no—
413C2×C3×C4—42no—
414C2×C3×C4—42no—
415C2×C3×C4—42no—
416C2×C3×C4—42no—
417C2×C3×C4—42no—
418C2×C3×C4—42no—
419C2×C3×C4—42no—
420C2×C3×C4—44no—
421C2×C3×C4—42no—
422C2×C3×C4—44no—
423C2×C3×C4—81no—
424C2×C3×C4—82no—
425C2×C3×C4—82no—
426C2×C3×C4#12011yescharacteristic
427C2×C3×C4#12011yescharacteristic
428C2×C3×C4#12011yescharacteristic
429C2×C3×C4#12011yescharacteristic
430C24C2×C3×D412yes—
431C24C2×C3×D411yescharacteristic
432C24C24×C311yescharacteristic
433#41—82no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—42no—
437C2×D4—21no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4C2×C3×D411yescharacteristic
465C22×C4C2×C3×D412yes—
466C22×C4C2×C3×D411yescharacteristic
467C22×C4C2×C3×D411yescharacteristic
468C22×C4C2×C3×D411yescharacteristic
469C22×C4C2×C3×D411yescharacteristic
470C22×C4C2×C3×D412yes—
471C22×C4C2×C3×D411yescharacteristic
472C22×C4C2×C3×D412yes—
473C22×C4C2×C3×D411yescharacteristic
474C22×C4C2×C3×D411yescharacteristic
475C22×C4C2×C3×D411yescharacteristic
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—24no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4C3×#4114yescomplemented
516C42—22no—
517C42—22no—
518C42—21no—
519C42—21no—
520C42—42no—
521C42—42no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—21no—
525C22×C3—21no—
526C22×C3—21no—
527C22×C3—21no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—41no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—41no—
537C22×C3—84no—
538C22×C3—81no—
539C22×C3C2×#12011yescharacteristic
540C22×C3#53211yescharacteristic
541C22×C3#53511yescharacteristic
542C22×C3#54612yes—
543C22×C3#55012yes—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—44no—
548C3×C4—41no—
549C3×C4—42no—
550C3×C4—41no—
551C3×C4—42no—
552C3×C4—41no—
553C3×C4—42no—
554C3×C4—82no—
555C3×C4—82no—
556C3×C4—82no—
557C3×C4#45812yes—
558C23—21no—
559C23—22no—
560C23—21no—
561C23—21no—
562C23—24no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23C22×C3×D411yescharacteristic
570C23C22×C3×D411yescharacteristic
571C23C22×C3×D411yescharacteristic
572C23C2×C3×#4112yes—
573C23C3×H(5,𝔽2)12yes—
574Q8—42no—
575Q8—81no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—42no—
582D4—81no—
583D4—82no—
584D4—82no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—81no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4C3×#12011yescharacteristic
623C2×C4C3×#12011yescharacteristic
624C2×C4C3×#12011yescharacteristic
625C2×C4C3×#12011yescharacteristic
626C2×C3—22no—
627C2×C3—21no—
628C2×C3—22no—
629C2×C3—21no—
630C2×C3—81no—
631C2×C3—82no—
632C2×C3#243111yescharacteristic
633C2×C3C2×#45811yescharacteristic
634C2×C3#302811yescharacteristic
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—42no—
649C22—41no—
650C22—84no—
651C22—81no—
652C22C2×C3×#12011yescharacteristic
653C22C3×#53211yescharacteristic
654C22C3×#53511yescharacteristic
655C22C3×#54612yes—
656C22C3×#55012yes—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—44no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4C3×#45812yes—
671C3#2558611yescharacteristic, direct product factor, Sylow
672C2—22no—
673C2—21no—
674C2—22no—
675C2—21no—
676C2—81no—
677C2—82no—
678C2C3×#243111yescharacteristic
679C2C2×C3×#45811yescharacteristic
680C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC3×#3505C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2503C2semidirect
3. C3×#2501C2semidirect
4. C3×#2486C2semidirect
5. C3×#2022C2semidirect
6. C3×#1996C2semidirect
7. C3×#1964C2semidirect
8. C3×#1890C2semidirect
9. C3×C4×#99C2semidirect
10. #25586C3direct (N × H)
49. #2503C2×C3semidirect
50. #2501C2×C3semidirect
51. #2486C2×C3semidirect
52. #2022C2×C3semidirect
53. #1996C2×C3semidirect
54. #1964C2×C3semidirect
55. #1890C2×C3semidirect
56. C4×#99C2×C3semidirect
267. C3×C22⋊C4#41semidirect
515. C22⋊C4C3×#41semidirect
671. C3#25586direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35052
C3×#25032
C3×#25012
C3×#24862
C3×#20222
C3×#19962
C3×#19642
C3×#18902
C3×C4×#992
#255863