Finite Group Databasebeta

← C3×#33081 #10651688

Subgroups — C3×#33081 #10651688

Order 768.

786 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2860C211yesmaximal, characteristic
2C3×C42×Q8C211yesmaximal, characteristic, complemented
3C3×#1926C211yesmaximal, characteristic
4C3×#1925C211yesmaximal, characteristic
5C3×#1925C211yesmaximal, characteristic
6C3×#1921C211yesmaximal, characteristic, complemented
7C3×#1920C211yesmaximal, characteristic, complemented
8C3×#1918C211yesmaximal, characteristic, complemented
9C3×#1880C211yesmaximal, characteristic
10C3×#1880C211yesmaximal, characteristic
11C3×#1880C211yesmaximal, characteristic
12C3×#1865C211yesmaximal, characteristic, complemented
13C3×#1863C211yesmaximal, characteristic, complemented
14C3×#1863C211yesmaximal, characteristic, complemented
15C3×#1839C211yesmaximal, characteristic, complemented
16#33081C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×#126—21no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
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×#390C2211yescharacteristic
32C3×#390C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#388—21no—
38C3×#388—21no—
39C3×#388—21no—
40C3×C4×C4⋊C4—21no—
41C3×C4×C4⋊C4—21no—
42C3×C4×C4⋊C4—21no—
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×C4×C4⋊C4C2211yescharacteristic
46C3×C43—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#2860C2×C311yescharacteristic
61C42×Q8C2×C311yescharacteristic, complemented
62#1926C2×C311yescharacteristic
63#1925C2×C311yescharacteristic
64#1925C2×C311yescharacteristic
65#1921C2×C311yescharacteristic, complemented
66#1920C2×C311yescharacteristic, complemented
67#1918C2×C311yescharacteristic, complemented
68#1880C2×C311yescharacteristic
69#1880C2×C311yescharacteristic
70#1880C2×C311yescharacteristic
71#1865C2×C311yescharacteristic, complemented
72#1863C2×C311yescharacteristic, complemented
73#1863C2×C311yescharacteristic, complemented
74#1839C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C2×C3×SD16—22no—
78C2×C3×SD16—22no—
79C2×C3×SD16—22no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#126—44no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42—21no—
115C2×C3×C42D411yescharacteristic
116C2×C3×C42D411yescharacteristic
117C2×C3×C42D411yescharacteristic
118C2×C3×C42D411yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2311yescharacteristic
122C3×#107—22no—
123C3×#107—22no—
124C3×#106—22no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#102—22no—
129C3×#102—22no—
130C3×#102—22no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C22×SD16C22×C311yescharacteristic
138C2×#126—21no—
139C2×C4×Q8C22×C311yescharacteristic
140C2×C4×Q8C22×C311yescharacteristic
141C2×C4×Q8C22×C311yescharacteristic
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#390C22×C311yescharacteristic
152#390C22×C311yescharacteristic
153#390C22×C311yescharacteristic
154#389C22×C311yescharacteristic
155#389C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#388—21no—
158#388—21no—
159#388—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4C22×C311yescharacteristic
164C4×C4⋊C4C22×C311yescharacteristic
165C4×C4⋊C4C22×C311yescharacteristic
166C43—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#4112yes—
185C2×C3×Q8#4112yes—
186C2×C3×Q8#4112yes—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×SD16—44no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4#4112yes—
236C3×C4⋊C4#4112yes—
237C3×C4⋊C4#4112yes—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C42—24no—
245C3×C42—24no—
246C3×C42—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
262C2×SD16—22no—
263C2×SD16—22no—
264C2×SD16—22no—
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269#126—44no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42C3×D411yescharacteristic
301C2×C42C3×D411yescharacteristic
302C2×C42C3×D411yescharacteristic
303C2×C42C3×D411yescharacteristic
304C2×C42C23×C311yescharacteristic
305C2×C42C23×C311yescharacteristic
306C2×C42C23×C311yescharacteristic
307#107—22no—
308#107—22no—
309#106—22no—
310#103—22no—
311#103—22no—
312#103—22no—
313#102—22no—
314#102—22no—
315#102—22no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C3×Q8—22no—
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8#12614yescomplemented
334C3×D4—41no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—41no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
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#13512yes—
375C2×C3×C4AGL(1,ℤ/8)12yes—
376C2×C3×C4#13712yes—
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C3×C8—41no—
384C3×C8—41no—
385C3×C8—41no—
386C3×C8—41no—
387C24—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8C3×#4112yes—
392C2×Q8C3×#4112yes—
393C2×Q8C3×#4112yes—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C24×C311yescharacteristic
414SD16—44no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4C3×#4112yes—
443C4⋊C4C3×#4112yes—
444C4⋊C4C3×#4112yes—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—41no—
464C42—42no—
465C42—42no—
466C42—42no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3#52511yescharacteristic
475C22×C3#52511yescharacteristic
476C22×C3#52511yescharacteristic
477C22×C3C2×#13511yescharacteristic
478C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
479C22×C3C2×#13711yescharacteristic
480C22×C3#57511yescharacteristic
481C3×C4—24no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—41no—
486C3×C4—42no—
487C3×C4—41no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4C2×#12611yescharacteristic
495C3×C4#52911yescharacteristic
496C3×C4#52911yescharacteristic
497C3×C4#52911yescharacteristic
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C22×C3×D411yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8C3×#12614yescomplemented
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
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—41no—
536C2×C4—41no—
537C2×C4—41no—
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×C4C3×#13512yes—
551C2×C4C3×AGL(1,ℤ/8)12yes—
552C2×C4C3×#13712yes—
553C2×C4C2×C3×#4111yescharacteristic
554C2×C4C2×C3×#4111yescharacteristic
555C2×C4C2×C3×#4111yescharacteristic
556C2×C4C2×C3×#4111yescharacteristic
557C2×C4C2×C3×#4111yescharacteristic
558C2×C4C2×C3×#4111yescharacteristic
559C8—41no—
560C8—41no—
561C8—41no—
562C8—41no—
563C2×C3—161no—
564C2×C3#285211yescharacteristic
565C2×C3#310211yescharacteristic
566C2×C3#310611yescharacteristic
567C2×C3#310811yescharacteristic
568C2×C3#311211yescharacteristic
569C2×C3#311511yescharacteristic
570C2×C3#311711yescharacteristic
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22C3×#52511yescharacteristic
579C22C3×#52511yescharacteristic
580C22C3×#52511yescharacteristic
581C22C2×C3×#13511yescharacteristic
582C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
583C22C2×C3×#13711yescharacteristic
584C22C3×#57511yescharacteristic
585C4—24no—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4C2×C3×#12611yescharacteristic
599C4C3×#52911yescharacteristic
600C4C3×#52911yescharacteristic
601C4C3×#52911yescharacteristic
602C3#3308111yescharacteristic, direct product factor, Sylow
603C2—161no—
604C2C3×#285211yescharacteristic
605C2C3×#310211yescharacteristic
606C2C3×#310611yescharacteristic
607C2C3×#310811yescharacteristic
608C2C3×#311211yescharacteristic
609C2C3×#311511yescharacteristic
610C2C3×#311711yescharacteristic
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
2. C3×C42×Q8C2semidirect
6. C3×#1921C2semidirect
7. C3×#1920C2semidirect
8. C3×#1918C2semidirect
12. C3×#1865C2semidirect
13. C3×#1863C2semidirect
14. C3×#1863C2semidirect
15. C3×#1839C2semidirect
16. #33081C3direct (N × H)
61. C42×Q8C2×C3semidirect
65. #1921C2×C3semidirect
66. #1920C2×C3semidirect
67. #1918C2×C3semidirect
71. #1865C2×C3semidirect
72. #1863C2×C3semidirect
73. #1863C2×C3semidirect
74. #1839C2×C3semidirect
333. C3×Q8#126semidirect
509. Q8C3×#126semidirect
602. C3#33081direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28602
C3×C42×Q82
C3×#19262
C3×#19252
C3×#19212
C3×#19202
C3×#19182
C3×#18802
C3×#18652
C3×#18632
C3×#18392
#330813