Finite Group Databasebeta

← C3×#29628 #10648235

Subgroups — C3×#29628 #10648235

Order 768.

1440 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×C3×#449C212yesmaximal, complemented
3C3×#2854C211yesmaximal, characteristic, complemented
4C3×#2043C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2041C211yesmaximal, characteristic, complemented
7C3×#2013C211yesmaximal, characteristic, complemented
8C3×#2012C212yesmaximal, complemented
9C3×#2011C211yesmaximal, characteristic, complemented
10C3×#1817C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal, complemented
12#29628C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C3×#532—28no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×#123—21no—
21C2×C3×#121—21no—
22C2×C3×#121—21no—
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4—21no—
26C22×C3×C42C2211yescharacteristic
27C3×#449—28no—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×C22⋊C8C2212yes—
35C3×#390C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#388—21no—
40C3×#388—21no—
41C3×#386—21no—
42C3×#380C2211yescharacteristic
43C3×#336C2211yescharacteristic
44C3×#336C2212yes—
45C3×#336C2211yescharacteristic
46C2×#532C2×C311yescharacteristic, complemented
47C2×#449C2×C312yescomplemented
48#2854C2×C311yescharacteristic, complemented
49#2043C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2041C2×C311yescharacteristic, complemented
52#2013C2×C311yescharacteristic, complemented
53#2012C2×C312yescomplemented
54#2011C2×C311yescharacteristic, complemented
55#1817C2×C311yescharacteristic, complemented
56#1788C2×C312yescomplemented
57C2×C3×#41—24no—
58C2×C3×#41—42no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C2×C3×#41—42no—
62C2×C3×#41D418yes—
63C22×C3×Q8C2312yes—
64C22×C3×D4—22no—
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C22×C3×D4C2312yes—
70C23×C3×C4—22no—
71C23×C3×C4D411yescharacteristic
72C23×C3×C4D411yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C2×C3×SD16—24no—
75C2×C3×SD16—42no—
76C2×C3×SD16—42no—
77C2×C3×D8—24no—
78C2×C3×D8—42no—
79C2×C3×D8—42no—
80C22×C3×C8C2312yes—
81C22×C3×C8C2312yes—
82C3×C4⋊Q8—24no—
83C3×#127—24no—
84C3×#124—24no—
85C3×#124—24no—
86C3×#123—44no—
87C3×#121—28no—
88C3×#121—28no—
89C3×#121—24no—
90C3×C4×D4—28no—
91C3×C4×D4—28no—
92C2×C3×C4⋊C4—21no—
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×C22⋊C4—22no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C42—24no—
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D411yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#103—24no—
113C3×#103—42no—
114C3×#103—42no—
115C3×#102—24no—
116C3×#102—42no—
117C3×#102—42no—
118C3×C22⋊C8—24no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C22×#41C22×C312yes—
123C22×SD16C22×C312yes—
124C22×D8C22×C312yes—
125#532—28no—
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#127C22×C311yescharacteristic
128C2×#124C22×C312yes—
129C2×#123—21no—
130C2×#121—21no—
131C2×#121—21no—
132C2×#121C22×C311yescharacteristic
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C22×C42C22×C311yescharacteristic
136#449—28no—
137C2×#103C22×C312yes—
138C2×#103C22×C312yes—
139C2×#103C22×C312yes—
140C2×#102C22×C312yes—
141C2×#102C22×C312yes—
142C2×#102C22×C312yes—
143C2×C22⋊C8C22×C312yes—
144#390C22×C311yescharacteristic
145#390C22×C311yescharacteristic
146#389C22×C311yescharacteristic
147#389C22×C311yescharacteristic
148#388—21no—
149#388—21no—
150#386—21no—
151#380C22×C311yescharacteristic
152#336C22×C311yescharacteristic
153#336C22×C312yes—
154#336C22×C311yescharacteristic
155C24×C3—22no—
156C24×C3—21no—
157C24×C3C2×D411yescharacteristic
158C3×#41—48no—
159C3×#41—48no—
160C3×#41—48no—
161C3×#41—48no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8C2×D414yes—
168C2×C3×D4—24no—
169C2×C3×D4—24no—
170C2×C3×D4—28no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—44no—
183C2×C3×D4—44no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4C2×D414yes—
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—28no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—41no—
203C22×C3×C4—41no—
204C22×C3×C4—44no—
205C22×C3×C4—44no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D414yes—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C3×SD16—48no—
214C3×D8—48no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4—42no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—42no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42—22no—
240C3×C42—22no—
241C2×#41—24no—
242C2×#41—42no—
243C2×#41—42no—
244C2×#41—42no—
245C2×#41—42no—
246C2×#41C3×D418yes—
247C22×Q8C23×C312yes—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C23×C311yescharacteristic
253C22×D4C23×C312yes—
254C23×C4—22no—
255C23×C4C3×D411yescharacteristic
256C23×C4C3×D411yescharacteristic
257C23×C4C23×C311yescharacteristic
258C2×SD16—24no—
259C2×SD16—42no—
260C2×SD16—42no—
261C2×D8—24no—
262C2×D8—42no—
263C2×D8—42no—
264C22×C8C23×C312yes—
265C22×C8C23×C312yes—
266C4⋊Q8—24no—
267#127—24no—
268#124—24no—
269#124—24no—
270#123—44no—
271#121—28no—
272#121—28no—
273#121—24no—
274C4×D4—28no—
275C4×D4—28no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C42—24no—
292C2×C42C3×D411yescharacteristic
293C2×C42C3×D411yescharacteristic
294C2×C42C23×C311yescharacteristic
295C2×C42C23×C311yescharacteristic
296#103—24no—
297#103—42no—
298#103—42no—
299#102—24no—
300#102—42no—
301#102—42no—
302C22⋊C8—24no—
303#95—21no—
304#95—21no—
305#95—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—24no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3#12012yes—
327C23×C3C22×D411yescharacteristic
328C3×Q8—24no—
329C3×Q8—24no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—48no—
335C3×D4—48no—
336C3×D4—416no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—84no—
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—24no—
349C2×C3×C4—24no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—24no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—44no—
358C2×C3×C4—44no—
359C2×C3×C4—416no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
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#12012yes—
372C2×C3×C4#13512yes—
373C2×C3×C4AGL(1,ℤ/8)12yes—
374C2×C3×C4C22×D412yes—
375C2×C3×C4H(5,𝔽2)12yes—
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C3×C8—42no—
379C3×C8—42no—
380C24—22no—
381C24—21no—
382C24C2×C3×D411yescharacteristic
383#41—48no—
384#41—48no—
385#41—48no—
386#41—48no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8C2×C3×D414yes—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×C3×D414yes—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D414yes—
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C24×C311yescharacteristic
438SD16—48no—
439D8—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3C2×#12011yescharacteristic
488C22×C3#53511yescharacteristic
489C22×C3#53511yescharacteristic
490C22×C3C2×#13511yescharacteristic
491C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
492C22×C3#57511yescharacteristic
493C22×C3#57511yescharacteristic
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—44no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4#53412yes—
504C3×C4#53412yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—24no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C3×#12012yes—
526C23C22×C3×D411yescharacteristic
527Q8—24no—
528Q8—24no—
529Q8—42no—
530Q8—42no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—48no—
535D4—416no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—84no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—416no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4C3×#12012yes—
571C2×C4C3×#13512yes—
572C2×C4C3×AGL(1,ℤ/8)12yes—
573C2×C4C22×C3×D412yes—
574C2×C4C3×H(5,𝔽2)12yes—
575C2×C4C3×H(5,𝔽2)11yescharacteristic
576C2×C4C3×H(5,𝔽2)11yescharacteristic
577C8—42no—
578C8—42no—
579C2×C3—42no—
580C2×C3—84no—
581C2×C3—161no—
582C2×C3#261311yescharacteristic
583C2×C3#300211yescharacteristic
584C2×C3#300611yescharacteristic
585C2×C3#320711yescharacteristic
586C2×C3#321211yescharacteristic
587C2×C3#321411yescharacteristic
588C2×C3#321911yescharacteristic
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22C2×C3×#12011yescharacteristic
611C22C3×#53511yescharacteristic
612C22C3×#53511yescharacteristic
613C22C2×C3×#13511yescharacteristic
614C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
615C22C3×#57511yescharacteristic
616C22C3×#57511yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—44no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4C3×#53412yes—
627C4C3×#53412yes—
628C3#2962811yescharacteristic, direct product factor, Sylow
629C2—42no—
630C2—84no—
631C2—161no—
632C2C3×#261311yescharacteristic
633C2C3×#300211yescharacteristic
634C2C3×#300611yescharacteristic
635C2C3×#320711yescharacteristic
636C2C3×#321211yescharacteristic
637C2C3×#321411yescharacteristic
638C2C3×#321911yescharacteristic
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×#449C2semidirect
3. C3×#2854C2semidirect
4. C3×#2043C2semidirect
5. C3×#2042C2semidirect
6. C3×#2041C2semidirect
7. C3×#2013C2semidirect
8. C3×#2012C2semidirect
9. C3×#2011C2semidirect
10. C3×#1817C2semidirect
11. C3×#1788C2semidirect
12. #29628C3direct (N × H)
46. C2×#532C2×C3semidirect
47. C2×#449C2×C3semidirect
48. #2854C2×C3semidirect
49. #2043C2×C3semidirect
50. #2042C2×C3semidirect
51. #2041C2×C3semidirect
52. #2013C2×C3semidirect
53. #2012C2×C3semidirect
54. #2011C2×C3semidirect
55. #1817C2×C3semidirect
56. #1788C2×C3semidirect
628. C3#29628direct (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×#4492
C3×#28542
C3×#20432
C3×#20422
C3×#20412
C3×#20132
C3×#20122
C3×#20112
C3×#18172
C3×#17882
#296283