Finite Group Databasebeta

← C3×#29667 #10648274

Subgroups — C3×#29667 #10648274

Order 768.

1438 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×C3×#463C212yesmaximal, complemented
3C3×#2850C211yesmaximal, characteristic, complemented
4C3×#2024C211yesmaximal, characteristic, complemented
5C3×#2023C211yesmaximal, characteristic, complemented
6C3×#2014C211yesmaximal, characteristic, complemented
7C3×#2011C211yesmaximal, characteristic, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1999C212yesmaximal, complemented
10C3×#1804C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29667C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C3×#532—28no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124—21no—
20C2×C3×#124C2211yescharacteristic
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121—21no—
23C2×C3×#121C2212yes—
24C2×C3×#121C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2212yes—
27C22×C3×C42C2211yescharacteristic
28C3×#463—28no—
29C2×C3×C4⋊C8C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×C22⋊C8C2212yes—
35C3×#391C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#388—21no—
38C3×#388—21no—
39C3×#386—21no—
40C3×#386—21no—
41C3×#382C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2211yescharacteristic
45C3×#336C2211yescharacteristic
46C2×#532C2×C311yescharacteristic, complemented
47C2×#463C2×C312yescomplemented
48#2850C2×C311yescharacteristic, complemented
49#2024C2×C311yescharacteristic, complemented
50#2023C2×C311yescharacteristic, complemented
51#2014C2×C311yescharacteristic, complemented
52#2011C2×C311yescharacteristic, complemented
53#2000C2×C312yescomplemented
54#1999C2×C312yescomplemented
55#1804C2×C312yescomplemented
56#1797C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4—21no—
66C22×C3×D4C2311yescharacteristic
67C22×C3×D4C2312yes—
68C23×C3×C4—22no—
69C23×C3×C4D411yescharacteristic
70C23×C3×C4D411yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C2×C3×SD16—24no—
73C2×C3×SD16—42no—
74C2×C3×SD16—42no—
75C2×C3×D8—24no—
76C2×C3×D8—42no—
77C2×C3×D8—42no—
78C22×C3×C8C2312yes—
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—24no—
81C3×#127—24no—
82C3×#124—28no—
83C3×#124—24no—
84C3×C22⋊Q8—42no—
85C3×#121—28no—
86C3×#121—24no—
87C3×#121—24no—
88C3×#121—42no—
89C3×C4×D4—28no—
90C3×C4×D4—24no—
91C3×C4×D4D418yescomplemented
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C42—24no—
106C2×C3×C42D411yescharacteristic
107C2×C3×C42D411yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×C4⋊C8—24no—
111C3×#103—24no—
112C3×#102—24no—
113C3×#102—42no—
114C3×#102—42no—
115C3×C22⋊C8—42no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C22×#41—21no—
121C22×SD16C22×C312yes—
122C22×D8C22×C312yes—
123#532—28no—
124C2×C4⋊Q8C22×C311yescharacteristic
125C2×#127C22×C311yescharacteristic
126C2×#124—21no—
127C2×#124C22×C311yescharacteristic
128C2×C22⋊Q8C22×C311yescharacteristic
129C2×#121—21no—
130C2×#121C22×C312yes—
131C2×#121C22×C311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4C22×C312yes—
134C22×C42C22×C311yescharacteristic
135#463—28no—
136C2×C4⋊C8C22×C312yes—
137C2×#103C22×C312yes—
138C2×#102C22×C312yes—
139C2×#102C22×C312yes—
140C2×#102C22×C312yes—
141C2×C22⋊C8C22×C312yes—
142#391C22×C311yescharacteristic
143#390C22×C311yescharacteristic
144#388—21no—
145#388—21no—
146#386—21no—
147#386—21no—
148#382C22×C311yescharacteristic
149#340C22×C312yes—
150#340C22×C312yes—
151#336C22×C311yescharacteristic
152#336C22×C311yescharacteristic
153C24×C3—22no—
154C24×C3—21no—
155C24×C3C2×D411yescharacteristic
156C3×#41—432no—
157C2×C3×Q8—24no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—42no—
163C2×C3×Q8—42no—
164C2×C3×D4—24no—
165C2×C3×D4—28no—
166C2×C3×D4—24no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—42no—
177C2×C3×D4—42no—
178C2×C3×D4—44no—
179C2×C3×D4—44no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4C2×D414yes—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—28no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—41no—
201C22×C3×C4—44no—
202C22×C3×C4—44no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C3×SD16—48no—
210C3×D8—48no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4C2×D414yes—
223C3×C22⋊C4—28no—
224C3×C22⋊C4—28no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C42—22no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42C2×D414yes—
235C2×#41—28no—
236C2×#41—44no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8C23×C311yescharacteristic
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C23×C311yescharacteristic
245C22×D4C23×C312yes—
246C23×C4—22no—
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—
258C4⋊Q8—24no—
259#127—24no—
260#124—28no—
261#124—24no—
262C22⋊Q8—42no—
263#121—28no—
264#121—24no—
265#121—24no—
266#121—42no—
267C4×D4—28no—
268C4×D4—24no—
269C4×D4C3×D418yescomplemented
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C312yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—24no—
284C2×C42C3×D411yescharacteristic
285C2×C42C3×D411yescharacteristic
286C2×C42C23×C311yescharacteristic
287C2×C42C23×C311yescharacteristic
288C4⋊C8—24no—
289#103—24no—
290#102—24no—
291#102—42no—
292#102—42no—
293C22⋊C8—42no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—24no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3AGL(1,ℤ/8)12yes—
319C23×C3C22×D411yescharacteristic
320C3×Q8—48no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×D4—24no—
324C3×D4—24no—
325C3×D4—48no—
326C3×D4—48no—
327C3×D4—416no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—84no—
333C2×C3×C4—28no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—24no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—44no—
347C2×C3×C4—416no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4#12012yes—
362C2×C3×C4#12012yes—
363C2×C3×C4#13512yes—
364C2×C3×C4AGL(1,ℤ/8)12yes—
365C2×C3×C4C22×D412yes—
366C2×C3×C4H(5,𝔽2)11yescharacteristic
367C2×C3×C4H(5,𝔽2)12yes—
368C2×C3×C42−1+411yescharacteristic
369C3×C8—42no—
370C3×C8—42no—
371C24—22no—
372C24—21no—
373C24C2×C3×D411yescharacteristic
374#41—432no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4C2×C3×D414yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C24×C311yescharacteristic
427SD16—48no—
428D8—48no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—42no—
440C4⋊C4C2×C3×D414yes—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42C2×C3×D414yes—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—44no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3—82no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3C2×#12011yescharacteristic
475C22×C3#53511yescharacteristic
476C22×C3#53711yescharacteristic
477C22×C3C2×#13511yescharacteristic
478C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
479C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
480C22×C3#57511yescharacteristic
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—41no—
486C3×C4—44no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4#53512yes—
491C3×C4#53712yes—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C3×AGL(1,ℤ/8)12yes—
513C23C22×C3×D411yescharacteristic
514Q8—48no—
515Q8—42no—
516Q8—42no—
517D4—24no—
518D4—24no—
519D4—48no—
520D4—48no—
521D4—416no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—84no—
527C2×C4—28no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
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×C4C3×#12012yes—
556C2×C4C3×#12012yes—
557C2×C4C3×#13512yes—
558C2×C4C3×AGL(1,ℤ/8)12yes—
559C2×C4C22×C3×D412yes—
560C2×C4C3×H(5,𝔽2)11yescharacteristic
561C2×C4C3×H(5,𝔽2)12yes—
562C2×C4C3×2−1+411yescharacteristic
563C8—42no—
564C8—42no—
565C2×C3—42no—
566C2×C3—84no—
567C2×C3—161no—
568C2×C3#261511yescharacteristic
569C2×C3#300211yescharacteristic
570C2×C3#300311yescharacteristic
571C2×C3#320711yescharacteristic
572C2×C3#321311yescharacteristic
573C2×C3#324711yescharacteristic
574C2×C3#325111yescharacteristic
575C22—22no—
576C22—22no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22C2×C3×#12011yescharacteristic
597C22C3×#53511yescharacteristic
598C22C3×#53711yescharacteristic
599C22C2×C3×#13511yescharacteristic
600C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
601C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
602C22C3×#57511yescharacteristic
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—44no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4C3×#53512yes—
613C4C3×#53712yes—
614C3#2966711yescharacteristic, direct product factor, Sylow
615C2—42no—
616C2—84no—
617C2—161no—
618C2C3×#261511yescharacteristic
619C2C3×#300211yescharacteristic
620C2C3×#300311yescharacteristic
621C2C3×#320711yescharacteristic
622C2C3×#321311yescharacteristic
623C2C3×#324711yescharacteristic
624C2C3×#325111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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×#532C2semidirect
2. C2×C3×#463C2semidirect
3. C3×#2850C2semidirect
4. C3×#2024C2semidirect
5. C3×#2023C2semidirect
6. C3×#2014C2semidirect
7. C3×#2011C2semidirect
8. C3×#2000C2semidirect
9. C3×#1999C2semidirect
10. C3×#1804C2semidirect
11. C3×#1797C2semidirect
12. #29667C3direct (N × H)
46. C2×#532C2×C3semidirect
47. C2×#463C2×C3semidirect
48. #2850C2×C3semidirect
49. #2024C2×C3semidirect
50. #2023C2×C3semidirect
51. #2014C2×C3semidirect
52. #2011C2×C3semidirect
53. #2000C2×C3semidirect
54. #1999C2×C3semidirect
55. #1804C2×C3semidirect
56. #1797C2×C3semidirect
91. C3×C4×D4D4semidirect
269. C4×D4C3×D4semidirect
614. C3#29667direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4632
C3×#28502
C3×#20242
C3×#20232
C3×#20142
C3×#20112
C3×#20002
C3×#19992
C3×#18042
C3×#17972
#296673