Finite Group Databasebeta

← C3×#33084 #10651691

Subgroups — C3×#33084 #10651691

Order 768.

852 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2863C211yesmaximal, characteristic, complemented
2C3×C42×D4C211yesmaximal, characteristic
3C3×#1928C211yesmaximal, characteristic, complemented
4C3×#1928C211yesmaximal, characteristic, complemented
5C3×#1927C211yesmaximal, characteristic, complemented
6C3×#1921C211yesmaximal, characteristic, complemented
7C3×#1919C211yesmaximal, characteristic, complemented
8C3×#1918C211yesmaximal, characteristic, complemented
9C3×#1877C211yesmaximal, characteristic
10C3×#1877C211yesmaximal, characteristic
11C3×#1877C211yesmaximal, characteristic
12C3×#1864C211yesmaximal, characteristic
13C3×#1862C211yesmaximal, characteristic
14C3×#1862C211yesmaximal, characteristic
15C3×#1839C211yesmaximal, characteristic, complemented
16#33084C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C22×C3×C42—21no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C3×#391C2211yescharacteristic
32C3×#391C2211yescharacteristic
33C3×#391C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#387—21no—
38C3×#387—21no—
39C3×#387—21no—
40C3×#383—21no—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C22⋊C4—21no—
45C3×C4×C22⋊C4—21no—
46C3×C4×C22⋊C4—21no—
47C3×C43C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2863C2×C311yescharacteristic, complemented
61C42×D4C2×C311yescharacteristic
62#1928C2×C311yescharacteristic, complemented
63#1928C2×C311yescharacteristic, complemented
64#1927C2×C311yescharacteristic, complemented
65#1921C2×C311yescharacteristic, complemented
66#1919C2×C311yescharacteristic, complemented
67#1918C2×C311yescharacteristic, complemented
68#1877C2×C311yescharacteristic
69#1877C2×C311yescharacteristic
70#1877C2×C311yescharacteristic
71#1864C2×C311yescharacteristic
72#1862C2×C311yescharacteristic
73#1862C2×C311yescharacteristic
74#1839C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4—21no—
80C2×C3×SD16—22no—
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42—44no—
109C2×C3×C42D411yescharacteristic
110C2×C3×C42D411yescharacteristic
111C2×C3×C42D411yescharacteristic
112C2×C3×C42D411yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C3×#107—22no—
117C3×#107—22no—
118C3×#106—22no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C22×SD16C22×C311yescharacteristic
138C2×C4×D4C22×C311yescharacteristic
139C2×C4×D4C22×C311yescharacteristic
140C2×C4×D4C22×C311yescharacteristic
141C22×C42—21no—
142C2×#107C22×C311yescharacteristic
143C2×#107C22×C311yescharacteristic
144C2×#106C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#103C22×C311yescharacteristic
147C2×#103C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149C2×#102C22×C311yescharacteristic
150C2×#102C22×C311yescharacteristic
151#391C22×C311yescharacteristic
152#391C22×C311yescharacteristic
153#391C22×C311yescharacteristic
154#389C22×C311yescharacteristic
155#389C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#387—21no—
158#387—21no—
159#387—21no—
160#383—21no—
161C4×C4⋊C4C22×C311yescharacteristic
162C4×C4⋊C4C22×C311yescharacteristic
163C4×C4⋊C4C22×C311yescharacteristic
164C4×C22⋊C4—21no—
165C4×C22⋊C4—21no—
166C4×C22⋊C4—21no—
167C43C22×C311yescharacteristic
168#340C22×C311yescharacteristic
169#340C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#336C22×C311yescharacteristic
175#336C22×C311yescharacteristic
176#336C22×C311yescharacteristic
177#336C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179#336C22×C311yescharacteristic
180C24×C3—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4#4112yes—
191C2×C3×D4#4112yes—
192C2×C3×D4#4112yes—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
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—21no—
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×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C3×SD16—44no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4#4112yes—
242C3×C4⋊C4#4112yes—
243C3×C4⋊C4#4112yes—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—24no—
247C3×C42—22no—
248C3×C42—22no—
249C3×C42—22no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—41no—
257C3×C42—42no—
258C3×C42—42no—
259C3×C42—42no—
260C22×Q8C23×C311yescharacteristic
261C22×D4C23×C311yescharacteristic
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C2×SD16—22no—
266C2×SD16—22no—
267C2×SD16—22no—
268C22×C8C23×C311yescharacteristic
269C22×C8C23×C311yescharacteristic
270C22×C8C23×C311yescharacteristic
271C22×C8C23×C311yescharacteristic
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—44no—
294C2×C42C3×D411yescharacteristic
295C2×C42C3×D411yescharacteristic
296C2×C42C3×D411yescharacteristic
297C2×C42C3×D411yescharacteristic
298C2×C42C23×C311yescharacteristic
299C2×C42C23×C311yescharacteristic
300C2×C42C23×C311yescharacteristic
301#107—22no—
302#107—22no—
303#106—22no—
304#103—22no—
305#103—22no—
306#103—22no—
307#102—22no—
308#102—22no—
309#102—22no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C3×Q8—41no—
331C3×Q8—41no—
332C3×Q8—41no—
333C3×Q8—41no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4#12614yes—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—48no—
357C2×C3×C4—48no—
358C2×C3×C4—48no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
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—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4#13512yes—
378C2×C3×C4AGL(1,ℤ/8)12yes—
379C2×C3×C4#13712yes—
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C3×C8—41no—
387C3×C8—41no—
388C3×C8—41no—
389C3×C8—41no—
390C24—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4C3×#4112yes—
401C2×D4C3×#4112yes—
402C2×D4C3×#4112yes—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C24×C311yescharacteristic
426SD16—44no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4C3×#4112yes—
452C4⋊C4C3×#4112yes—
453C4⋊C4C3×#4112yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—41no—
467C42—42no—
468C42—42no—
469C42—42no—
470C22×C3—24no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3#52511yescharacteristic
478C22×C3#52511yescharacteristic
479C22×C3#52511yescharacteristic
480C22×C3C2×#13511yescharacteristic
481C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
482C22×C3C2×#13711yescharacteristic
483C22×C3#57511yescharacteristic
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—41no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—41no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4C2×#12611yescharacteristic
498C3×C4#52911yescharacteristic
499C3×C4#52911yescharacteristic
500C3×C4#52911yescharacteristic
501C23—22no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C22×C3×D411yescharacteristic
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513D4—22no—
514D4—22no—
515D4—22no—
516D4C3×#12614yes—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4C3×#13512yes—
557C2×C4C3×AGL(1,ℤ/8)12yes—
558C2×C4C3×#13712yes—
559C2×C4C2×C3×#4111yescharacteristic
560C2×C4C2×C3×#4111yescharacteristic
561C2×C4C2×C3×#4111yescharacteristic
562C2×C4C2×C3×#4111yescharacteristic
563C2×C4C2×C3×#4111yescharacteristic
564C2×C4C2×C3×#4111yescharacteristic
565C8—41no—
566C8—41no—
567C8—41no—
568C8—41no—
569C2×C3—44no—
570C2×C3#285211yescharacteristic
571C2×C3#310211yescharacteristic
572C2×C3#310511yescharacteristic
573C2×C3#310711yescharacteristic
574C2×C3#311011yescharacteristic
575C2×C3#311411yescharacteristic
576C2×C3#311611yescharacteristic
577C22—24no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22C3×#52511yescharacteristic
585C22C3×#52511yescharacteristic
586C22C3×#52511yescharacteristic
587C22C2×C3×#13511yescharacteristic
588C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
589C22C2×C3×#13711yescharacteristic
590C22C3×#57511yescharacteristic
591C4—22no—
592C4—22no—
593C4—22no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4C2×C3×#12611yescharacteristic
605C4C3×#52911yescharacteristic
606C4C3×#52911yescharacteristic
607C4C3×#52911yescharacteristic
608C3#3308411yescharacteristic, direct product factor, Sylow
609C2—44no—
610C2C3×#285211yescharacteristic
611C2C3×#310211yescharacteristic
612C2C3×#310511yescharacteristic
613C2C3×#310711yescharacteristic
614C2C3×#311011yescharacteristic
615C2C3×#311411yescharacteristic
616C2C3×#311611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2863C2semidirect
3. C3×#1928C2semidirect
4. C3×#1928C2semidirect
5. C3×#1927C2semidirect
6. C3×#1921C2semidirect
7. C3×#1919C2semidirect
8. C3×#1918C2semidirect
15. C3×#1839C2semidirect
16. #33084C3direct (N × H)
60. #2863C2×C3semidirect
62. #1928C2×C3semidirect
63. #1928C2×C3semidirect
64. #1927C2×C3semidirect
65. #1921C2×C3semidirect
66. #1919C2×C3semidirect
67. #1918C2×C3semidirect
74. #1839C2×C3semidirect
608. C3#33084direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28632
C3×C42×D42
C3×#19282
C3×#19272
C3×#19212
C3×#19192
C3×#19182
C3×#18772
C3×#18642
C3×#18622
C3×#18392
#330843