Finite Group Databasebeta

← C3×#29718 #10648325

Subgroups — C3×#29718 #10648325

Order 768.

1254 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#450C211yesmaximal, characteristic, complemented
2C2×C3×#447C211yesmaximal, characteristic
3C3×#2850C211yesmaximal, characteristic, complemented
4C2×C3×#390C211yesmaximal, characteristic, complemented
5C3×#2073C212yesmaximal, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2054C212yesmaximal, complemented
8C3×#2012C212yesmaximal, complemented
9C3×#1874C211yesmaximal, characteristic, complemented
10C3×#1872C211yesmaximal, characteristic, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29718C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C2×C3×#124C2211yescharacteristic, complemented
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4—21no—
21C22×C3×C22⋊C4—21no—
22C22×C3×C42C2211yescharacteristic
23C3×#450—24no—
24C3×#447—24no—
25C2×C3×C4⋊C8C2212yescomplemented
26C2×C3×#103C2212yescomplemented
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yescomplemented
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C3×#391C2211yescharacteristic, complemented
33C3×#390—21no—
34C3×#390—21no—
35C3×#390C2212yes—
36C3×#390C2212yes—
37C3×#390C2211yescharacteristic, complemented
38C3×#388—21no—
39C3×#388—21no—
40C3×#386—21no—
41C3×#386—21no—
42C3×#382C2211yescharacteristic, complemented
43C3×#340C2212yescomplemented
44C3×#340C2212yescomplemented
45C3×#336C2212yescomplemented
46C2×#450C2×C311yescharacteristic, complemented
47C2×#447C2×C311yescharacteristic
48#2850C2×C311yescharacteristic, complemented
49C2×#390C2×C311yescharacteristic, complemented
50#2073C2×C312yescomplemented
51#2055C2×C312yescomplemented
52#2054C2×C312yescomplemented
53#2012C2×C312yescomplemented
54#1874C2×C311yescharacteristic, complemented
55#1872C2×C311yescharacteristic, complemented
56#1797C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—42no—
63C22×C3×D4—42no—
64C22×C3×D4D414yes—
65C22×C3×D4C2311yescharacteristic
66C22×C3×D4C2312yes—
67C23×C3×C4—21no—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D411yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C2×C3×SD16—42no—
74C2×C3×D8—24no—
75C2×C3×D8—42no—
76C2×C3×D8—42no—
77C22×C3×C8C2312yes—
78C22×C3×C8C2312yes—
79C3×#124—42no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C3×#121—22no—
83C3×#121—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—42no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—44no—
101C2×C3×C22⋊C4—44no—
102C2×C3×C42—42no—
103C2×C3×C42D411yescharacteristic
104C2×C3×C42D411yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C3×C4⋊C8—42no—
108C3×#103—42no—
109C3×#102—24no—
110C3×#102—24no—
111C3×#102—42no—
112C3×C22⋊C8—22no—
113C3×C22⋊C8—22no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C23×D4C22×C311yescharacteristic
119C22×SD16C22×C312yes—
120C22×D8C22×C312yes—
121C2×#124C22×C311yescharacteristic, complemented
122C2×C22⋊Q8C22×C311yescharacteristic
123C2×#121C22×C311yescharacteristic
124C22×C4⋊C4C22×C311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42C22×C311yescharacteristic
128#450—24no—
129#447—24no—
130C2×C4⋊C8C22×C312yescomplemented
131C2×#103C22×C312yescomplemented
132C2×#102C22×C312yes—
133C2×#102C22×C312yes—
134C2×#102C22×C312yescomplemented
135C2×C22⋊C8C22×C311yescharacteristic
136C2×C22⋊C8C22×C311yescharacteristic
137#391C22×C311yescharacteristic, complemented
138#390—21no—
139#390—21no—
140#390C22×C312yes—
141#390C22×C312yes—
142#390C22×C311yescharacteristic, complemented
143#388—21no—
144#388—21no—
145#386—21no—
146#386—21no—
147#382C22×C311yescharacteristic, complemented
148#340C22×C312yescomplemented
149#340C22×C312yescomplemented
150#336C22×C312yescomplemented
151C24×C3—24no—
152C24×C3—22no—
153C24×C3—21no—
154C24×C3—42no—
155C24×C3—42no—
156C24×C3—42no—
157C24×C3—42no—
158C24×C3—42no—
159C24×C3—42no—
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×D4—24no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—48no—
176C2×C3×D4—42no—
177C2×C3×D4—42no—
178C2×C3×D4—48no—
179C2×C3×D4—42no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4C2×D414yes—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—41no—
193C22×C3×C4—41no—
194C22×C3×C4—41no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4#4112yes—
205C22×C3×C4C2411yescharacteristic
206C3×SD16—48no—
207C3×D8—48no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—42no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—44no—
225C3×C22⋊C4—44no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—44no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42—41no—
235C3×C42—41no—
236C25—21no—
237C22×Q8C23×C311yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—42no—
242C22×D4—42no—
243C22×D4C3×D414yes—
244C22×D4C23×C311yescharacteristic
245C22×D4C23×C312yes—
246C23×C4—21no—
247C23×C4C3×D411yescharacteristic
248C23×C4C3×D411yescharacteristic
249C23×C4C23×C311yescharacteristic
250C2×SD16—24no—
251C2×SD16—42no—
252C2×SD16—42no—
253C2×D8—24no—
254C2×D8—42no—
255C2×D8—42no—
256C22×C8C23×C312yes—
257C22×C8C23×C312yes—
258#124—42no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261#121—22no—
262#121—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C42—42no—
282C2×C42C3×D411yescharacteristic
283C2×C42C3×D411yescharacteristic
284C2×C42C23×C311yescharacteristic
285C2×C42C23×C311yescharacteristic
286C4⋊C8—42no—
287#103—42no—
288#102—24no—
289#102—24no—
290#102—42no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—24no—
302C23×C3—48no—
303C23×C3—48no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—48no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—84no—
319C23×C3—84no—
320C23×C3—84no—
321C23×C3—84no—
322C23×C3#12112yes—
323C23×C3#13512yes—
324C23×C3AGL(1,ℤ/8)12yes—
325C23×C3C22×D411yescharacteristic
326C3×Q8—42no—
327C3×Q8—42no—
328C3×D4—24no—
329C3×D4—24no—
330C3×D4—48no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—84no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
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—21no—
351C2×C3×C4—44no—
352C2×C3×C4—44no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—84no—
366C2×C3×C4#12112yes—
367C2×C3×C4AGL(1,ℤ/8)12yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4H(5,𝔽2)12yes—
371C2×C3×C4H(5,𝔽2)12yes—
372C3×C8—42no—
373C3×C8—42no—
374C24—24no—
375C24—22no—
376C24—21no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24C2×C3×D411yescharacteristic
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—48no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—48no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4C2×C3×D414yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D412yes—
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C3×#4112yes—
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—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C42—22no—
456C42—22no—
457C42—41no—
458C42—41no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—816no—
469C22×C3—82no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3#46314yes—
481C22×C3C2×#12111yescharacteristic
482C22×C3#53411yescharacteristic
483C22×C3#53511yescharacteristic
484C22×C3C2×#13511yescharacteristic
485C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
486C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
487C22×C3#57511yescharacteristic
488C3×C4—22no—
489C3×C4—22no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—81no—
494C3×C4—82no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4#53412yes—
498C3×C4#53812yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—24no—
504C23—48no—
505C23—48no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—48no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23C3×#12112yes—
525C23C3×#13512yes—
526C23C3×AGL(1,ℤ/8)12yes—
527C23C22×C3×D411yescharacteristic
528Q8—42no—
529Q8—42no—
530D4—24no—
531D4—24no—
532D4—48no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—84no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4C3×#12112yes—
569C2×C4C3×AGL(1,ℤ/8)12yes—
570C2×C4C22×C3×D411yescharacteristic
571C2×C4C2×C3×#4111yescharacteristic
572C2×C4C3×H(5,𝔽2)12yes—
573C2×C4C3×H(5,𝔽2)12yes—
574C8—42no—
575C8—42no—
576C2×C3—24no—
577C2×C3—84no—
578C2×C3—161no—
579C2×C3#261711yescharacteristic
580C2×C3C2×#46311yescharacteristic
581C2×C3#303611yescharacteristic
582C2×C3#318811yescharacteristic
583C2×C3#319411yescharacteristic
584C2×C3#321111yescharacteristic
585C2×C3#321711yescharacteristic
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—816no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22C3×#46314yes—
608C22C2×C3×#12111yescharacteristic
609C22C3×#53411yescharacteristic
610C22C3×#53511yescharacteristic
611C22C2×C3×#13511yescharacteristic
612C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
613C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
614C22C3×#57511yescharacteristic
615C4—22no—
616C4—22no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—81no—
621C4—82no—
622C4—81no—
623C4—81no—
624C4C3×#53412yes—
625C4C3×#53812yes—
626C3#2971811yescharacteristic, direct product factor, Sylow
627C2—24no—
628C2—84no—
629C2—161no—
630C2C3×#261711yescharacteristic
631C2C2×C3×#46311yescharacteristic
632C2C3×#303611yescharacteristic
633C2C3×#318811yescharacteristic
634C2C3×#319411yescharacteristic
635C2C3×#321111yescharacteristic
636C2C3×#321711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#450C2semidirect
3. C3×#2850C2semidirect
4. C2×C3×#390C2semidirect
5. C3×#2073C2semidirect
6. C3×#2055C2semidirect
7. C3×#2054C2semidirect
8. C3×#2012C2semidirect
9. C3×#1874C2semidirect
10. C3×#1872C2semidirect
11. C3×#1797C2semidirect
12. #29718C3direct (N × H)
16. C2×C3×#124C22semidirect
25. C2×C3×C4⋊C8C22semidirect
26. C2×C3×#103C22semidirect
29. C2×C3×#102C22semidirect
32. C3×#391C22semidirect
37. C3×#390C22semidirect
42. C3×#382C22semidirect
43. C3×#340C22semidirect
44. C3×#340C22semidirect
45. C3×#336C22semidirect
46. C2×#450C2×C3semidirect
48. #2850C2×C3semidirect
49. C2×#390C2×C3semidirect
50. #2073C2×C3semidirect
51. #2055C2×C3semidirect
52. #2054C2×C3semidirect
53. #2012C2×C3semidirect
54. #1874C2×C3semidirect
55. #1872C2×C3semidirect
56. #1797C2×C3semidirect
121. C2×#124C22×C3semidirect
130. C2×C4⋊C8C22×C3semidirect
131. C2×#103C22×C3semidirect
134. C2×#102C22×C3semidirect
137. #391C22×C3semidirect
142. #390C22×C3semidirect
147. #382C22×C3semidirect
148. #340C22×C3semidirect
149. #340C22×C3semidirect
150. #336C22×C3semidirect
626. C3#29718direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4502
C2×C3×#4472
C3×#28502
C2×C3×#3902
C3×#20732
C3×#20552
C3×#20542
C3×#20122
C3×#18742
C3×#18722
C3×#17972
#297183