Finite Group Databasebeta

← C3×#25664 #10644271

Subgroups — C3×#25664 #10644271

Order 768.

1136 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3505C211yesmaximal, characteristic
2C3×#2504C211yesmaximal, characteristic, complemented
3C3×#2502C211yesmaximal, characteristic, complemented
4C3×#2487C211yesmaximal, characteristic, complemented
5C3×#2044C214yesmaximal, complemented
6C3×#1996C211yesmaximal, characteristic, complemented
7C3×#1964C211yesmaximal, characteristic, complemented
8C3×#1867C214yesmaximal, complemented
9C3×C4×#99C211yesmaximal, characteristic, complemented
10#25664C311yesmaximal, 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×C22⋊Q8C2211yescharacteristic
23C2×C3×#121C2211yescharacteristic
24C3×#518C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C2×C3×#99C2212yes—
28C2×C3×#99C2214yes—
29C3×#400—21no—
30C3×#399—21no—
31C3×#398—22no—
32C3×#397—22no—
33C3×#391—21no—
34C3×#390—21no—
35C3×#389—21no—
36C3×#386C2212yes—
37C3×#386C2214yes—
38C3×#385C2214yes—
39C3×#385C2212yes—
40C3×#384—21no—
41C3×#382—21no—
42C3×#380—21no—
43C3×#380C2211yescharacteristic
44C3×C4×C22⋊C4C2211yescharacteristic
45C3×C4×C22⋊C4C2211yescharacteristic
46C3×#342C2214yes—
47C3×#342C2212yes—
48#3505C2×C311yescharacteristic
49#2504C2×C311yescharacteristic, complemented
50#2502C2×C311yescharacteristic, complemented
51#2487C2×C311yescharacteristic, complemented
52#2044C2×C314yescomplemented
53#1996C2×C311yescharacteristic, complemented
54#1964C2×C311yescharacteristic, complemented
55#1867C2×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×C22⋊Q8—21no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—21no—
80C3×C22⋊Q8—21no—
81C3×C22⋊Q8—21no—
82C3×C22⋊Q8—21no—
83C3×#121—21no—
84C3×#121—22no—
85C3×#121—21no—
86C3×#121—21no—
87C3×#121—21no—
88C3×#121—21no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#120—22no—
92C3×#120—22no—
93C3×C4×Q8—21no—
94C3×C4×D4—22no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×#117—21no—
105C3×#117—22no—
106C3×#117—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—21no—
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⋊C4D411yescharacteristic
117C2×C3×C4⋊C4D411yescharacteristic
118C2×C3×C4⋊C4D412yes—
119C2×C3×C4⋊C4D411yescharacteristic
120C2×C3×C22⋊C4—22no—
121C2×C3×C22⋊C4—22no—
122C2×C3×C22⋊C4—24no—
123C2×C3×C22⋊C4D412yes—
124C2×C3×C22⋊C4D411yescharacteristic
125C2×C3×C22⋊C4C2312yes—
126C2×C3×C22⋊C4C2312yes—
127C2×C3×C22⋊C4C2311yescharacteristic
128C2×C3×C22⋊C4C2312yes—
129C2×C3×C22⋊C4C2312yes—
130C2×C3×C42—21no—
131C2×C3×C42—21no—
132C2×C3×C42D411yescharacteristic
133C3×#99—24no—
134C3×#99—28no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—22no—
138C3×#95—21no—
139C3×#95—24no—
140C3×#95—24no—
141C3×#95—22no—
142#553—21no—
143#553—22no—
144#552—21no—
145#551—22no—
146#548—21no—
147#546—21no—
148#539—21no—
149#538—21no—
150#534C22×C312yes—
151#529—21no—
152#525—21no—
153C2×C22⋊Q8C22×C311yescharacteristic
154C2×#121C22×C311yescharacteristic
155#518C22×C312yes—
156C2×C4×D4C22×C311yescharacteristic
157C22×C4⋊C4C22×C311yescharacteristic
158C2×#99C22×C314yes—
159C2×#99C22×C312yes—
160#400—21no—
161#399—21no—
162#398—22no—
163#397—22no—
164#391—21no—
165#390—21no—
166#389—21no—
167#386C22×C314yes—
168#386C22×C312yes—
169#385C22×C312yes—
170#385C22×C314yes—
171#384—21no—
172#382—21no—
173#380—21no—
174#380C22×C311yescharacteristic
175C4×C22⋊C4C22×C311yescharacteristic
176C4×C22⋊C4C22×C311yescharacteristic
177#342C22×C312yes—
178#342C22×C314yes—
179C24×C3C2×D412yes—
180C24×C3C2×D411yescharacteristic
181C24×C3C2411yescharacteristic
182C3×#41—82no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—42no—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—24no—
192C2×C3×D4—22no—
193C2×C3×D4—24no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—42no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—82no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—41no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4—84no—
242C3×C4⋊C4—82no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—21no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—84no—
262C3×C22⋊C4—82no—
263C3×C22⋊C4#4114yescomplemented
264C3×C42—22no—
265C3×C42—22no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—42no—
269C3×C42—42no—
270C2×#41—21no—
271C22×Q8C3×D411yescharacteristic
272C22×D4C3×D411yescharacteristic
273C22×D4C23×C311yescharacteristic
274C22×D4C23×C312yes—
275C23×C4C3×D411yescharacteristic
276C23×C4C23×C312yes—
277C23×C4C23×C311yescharacteristic
278#126—22no—
279#126—22no—
280#126—22no—
281#125—21no—
282#125—22no—
283#124—21no—
284#124—22no—
285#123—22no—
286#123—22no—
287#123—22no—
288#123—22no—
289#123—22no—
290C22⋊Q8—21no—
291C22⋊Q8—22no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296#121—21no—
297#121—22no—
298#121—21no—
299#121—21no—
300#121—21no—
301#121—21no—
302#121—22no—
303#121—22no—
304#120—22no—
305#120—22no—
306C4×Q8—21no—
307C4×D4—22no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317#117—21no—
318#117—22no—
319#117—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4C3×D411yescharacteristic
330C2×C4⋊C4C3×D411yescharacteristic
331C2×C4⋊C4C3×D411yescharacteristic
332C2×C4⋊C4C3×D412yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4C3×D411yescharacteristic
337C2×C22⋊C4C3×D412yes—
338C2×C22⋊C4C23×C312yes—
339C2×C22⋊C4C23×C312yes—
340C2×C22⋊C4C23×C311yescharacteristic
341C2×C22⋊C4C23×C312yes—
342C2×C22⋊C4C23×C312yes—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42C3×D411yescharacteristic
346#99—28no—
347#99—24no—
348#95—21no—
349#95—21no—
350#95—22no—
351#95—21no—
352#95—24no—
353#95—24no—
354#95—22no—
355C23×C3—21no—
356C23×C3—22no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—24no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—42no—
365C23×C3—42no—
366C23×C3C22×D411yescharacteristic
367C23×C3C22×D411yescharacteristic
368C23×C3C22×D411yescharacteristic
369C23×C3C2×#4112yes—
370C23×C32−1+412yes—
371C3×Q8—42no—
372C3×Q8—81no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—44no—
378C3×D4—42no—
379C3×D4—81no—
380C3×D4—82no—
381C3×D4—82no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—24no—
394C2×C3×C4—24no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—42no—
399C2×C3×C4—42no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—42no—
403C2×C3×C4—41no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—42no—
408C2×C3×C4—42no—
409C2×C3×C4—44no—
410C2×C3×C4—42no—
411C2×C3×C4—44no—
412C2×C3×C4—42no—
413C2×C3×C4—42no—
414C2×C3×C4—42no—
415C2×C3×C4—42no—
416C2×C3×C4—81no—
417C2×C3×C4—82no—
418C2×C3×C4—82no—
419C2×C3×C4#12011yescharacteristic
420C2×C3×C4#12011yescharacteristic
421C2×C3×C4#12011yescharacteristic
422C2×C3×C4#12011yescharacteristic
423C24C2×C3×D412yes—
424C24C2×C3×D411yescharacteristic
425C24C24×C311yescharacteristic
426#41—82no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—42no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—42no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—82no—
456C22×C4C2×C3×D411yescharacteristic
457C22×C4C2×C3×D412yes—
458C22×C4C2×C3×D411yescharacteristic
459C22×C4C2×C3×D411yescharacteristic
460C22×C4C2×C3×D411yescharacteristic
461C22×C4C2×C3×D411yescharacteristic
462C22×C4C2×C3×D412yes—
463C22×C4C2×C3×D411yescharacteristic
464C22×C4C2×C3×D412yes—
465C22×C4C2×C3×D411yescharacteristic
466C22×C4C2×C3×D411yescharacteristic
467C22×C4C2×C3×D411yescharacteristic
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—84no—
506C22⋊C4—82no—
507C22⋊C4C3×#4114yescomplemented
508C42—22no—
509C42—22no—
510C42—21no—
511C42—21no—
512C42—42no—
513C42—42no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3—41no—
524C22×C3—42no—
525C22×C3—41no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—41no—
529C22×C3—81no—
530C22×C3—84no—
531C22×C3C2×#12011yescharacteristic
532C22×C3#53211yescharacteristic
533C22×C3#53611yescharacteristic
534C22×C3#54712yes—
535C22×C3#54812yes—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—44no—
540C3×C4—41no—
541C3×C4—42no—
542C3×C4—41no—
543C3×C4—42no—
544C3×C4—41no—
545C3×C4—42no—
546C3×C4—82no—
547C3×C4—82no—
548C3×C4—82no—
549C3×C4#45812yes—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—21no—
554C23—24no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23C22×C3×D411yescharacteristic
562C23C22×C3×D411yescharacteristic
563C23C22×C3×D411yescharacteristic
564C23C2×C3×#4112yes—
565C23C3×2−1+412yes—
566Q8—42no—
567Q8—81no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—44no—
573D4—42no—
574D4—81no—
575D4—82no—
576D4—82no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—81no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4C3×#12011yescharacteristic
615C2×C4C3×#12011yescharacteristic
616C2×C4C3×#12011yescharacteristic
617C2×C4C3×#12011yescharacteristic
618C2×C3—22no—
619C2×C3—21no—
620C2×C3—22no—
621C2×C3—21no—
622C2×C3—81no—
623C2×C3—82no—
624C2×C3#243211yescharacteristic
625C2×C3C2×#45811yescharacteristic
626C2×C3#302711yescharacteristic
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22—41no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—81no—
643C22—84no—
644C22C2×C3×#12011yescharacteristic
645C22C3×#53211yescharacteristic
646C22C3×#53611yescharacteristic
647C22C3×#54712yes—
648C22C3×#54812yes—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—44no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4C3×#45812yes—
663C3#2566411yescharacteristic, direct product factor, Sylow
664C2—22no—
665C2—21no—
666C2—22no—
667C2—21no—
668C2—81no—
669C2—82no—
670C2C3×#243211yescharacteristic
671C2C2×C3×#45811yescharacteristic
672C2C3×#302711yescharacteristic
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×#2504C2semidirect
3. C3×#2502C2semidirect
4. C3×#2487C2semidirect
5. C3×#2044C2semidirect
6. C3×#1996C2semidirect
7. C3×#1964C2semidirect
8. C3×#1867C2semidirect
9. C3×C4×#99C2semidirect
10. #25664C3direct (N × H)
49. #2504C2×C3semidirect
50. #2502C2×C3semidirect
51. #2487C2×C3semidirect
52. #2044C2×C3semidirect
53. #1996C2×C3semidirect
54. #1964C2×C3semidirect
55. #1867C2×C3semidirect
56. C4×#99C2×C3semidirect
263. C3×C22⋊C4#41semidirect
507. C22⋊C4C3×#41semidirect
663. C3#25664direct (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×#25042
C3×#25022
C3×#24872
C3×#20442
C3×#19962
C3×#19642
C3×#18672
C3×C4×#992
#256643