Finite Group Databasebeta

← C3×#23932 #10642539

Subgroups — C3×#23932 #10642539

Order 768.

1556 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×#532C211yesmaximal, characteristic, complemented
2C2×C3×#449C212yesmaximal, complemented
3C3×#2297C211yesmaximal, characteristic, complemented
4C3×#2024C211yesmaximal, characteristic, complemented
5C3×#2023C211yesmaximal, characteristic, complemented
6C3×#2014C211yesmaximal, characteristic, complemented
7C3×#2011C211yesmaximal, characteristic, complemented
8C3×#1879C212yesmaximal, complemented
9C3×#1877C212yesmaximal, complemented
10C3×#1875C212yesmaximal, complemented
11C3×#1817C211yesmaximal, characteristic, complemented
12#23932C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×#41C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C22×C3×D8C2212yes—
17C3×#532—28no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124—21no—
21C2×C3×#121—21no—
22C2×C3×#121C2212yes—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2212yes—
25C2×C3×#117C2211yescharacteristic
26C22×C3×C42C2211yescharacteristic
27C3×#449—28no—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2212yes—
33C3×#391C2211yescharacteristic
34C3×#391C2211yescharacteristic
35C3×#390C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#386—21no—
38C3×#386—21no—
39C3×C4×C22⋊C4—21no—
40C3×C4×C22⋊C4—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C3×#336C2212yes—
46C2×#532C2×C311yescharacteristic, complemented
47C2×#449C2×C312yescomplemented
48#2297C2×C311yescharacteristic, complemented
49#2024C2×C311yescharacteristic, complemented
50#2023C2×C311yescharacteristic, complemented
51#2014C2×C311yescharacteristic, complemented
52#2011C2×C311yescharacteristic, complemented
53#1879C2×C312yescomplemented
54#1877C2×C312yescomplemented
55#1875C2×C312yescomplemented
56#1817C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41—24no—
61C2×C3×#41—42no—
62C2×C3×#41—44no—
63C2×C3×#41—44no—
64C22×C3×Q8—21no—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—21no—
69C22×C3×D4—21no—
70C22×C3×D4C2312yes—
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—22no—
73C23×C3×C4—21no—
74C23×C3×C4D411yescharacteristic
75C23×C3×C4D411yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—42no—
80C2×C3×D8—24no—
81C2×C3×D8—24no—
82C2×C3×D8—42no—
83C22×C3×C8C2312yes—
84C22×C3×C8C2312yes—
85C3×C4⋊Q8—24no—
86C3×#127—24no—
87C3×#124—28no—
88C3×#121—28no—
89C3×#121—24no—
90C3×#121D418yes—
91C3×C4×D4—28no—
92C3×C4×D4—24no—
93C3×C4×D4—24no—
94C3×#117—42no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2312yes—
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×C42—24no—
107C2×C3×C42—21no—
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D411yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#103—24no—
113C3×#102—24no—
114C3×#102—42no—
115C3×#102—42no—
116C3×C22⋊C8—24no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C22×#41—21no—
122C22×#41C22×C311yescharacteristic
123C22×SD16C22×C312yes—
124C22×D8C22×C312yes—
125#532—28no—
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#127C22×C311yescharacteristic
128C2×#124—21no—
129C2×#121—21no—
130C2×#121C22×C312yes—
131C2×C4×D4—21no—
132C2×C4×D4C22×C312yes—
133C2×#117C22×C311yescharacteristic
134C22×C42C22×C311yescharacteristic
135#449—28no—
136C2×#103C22×C312yes—
137C2×#102C22×C312yes—
138C2×#102C22×C312yes—
139C2×#102C22×C312yes—
140C2×C22⋊C8C22×C312yes—
141#391C22×C311yescharacteristic
142#391C22×C311yescharacteristic
143#390C22×C311yescharacteristic
144#390C22×C311yescharacteristic
145#386—21no—
146#386—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149#340C22×C312yes—
150#340C22×C312yes—
151#336C22×C312yes—
152#336C22×C311yescharacteristic
153#336C22×C311yescharacteristic
154C24×C3—21no—
155C24×C3—22no—
156C24×C3C2×D411yescharacteristic
157C3×#41—48no—
158C3×#41—48no—
159C3×#41—432no—
160C2×C3×Q8—24no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—42no—
165C2×C3×Q8—42no—
166C2×C3×Q8#4112yescomplemented
167C2×C3×D4—24no—
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—21no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—42no—
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×D4C2×D414yes—
188C2×C3×D4#4112yescomplemented
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—28no—
196C22×C3×C4—24no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—41no—
204C22×C3×C4—42no—
205C22×C3×C4—44no—
206C22×C3×C4—42no—
207C22×C3×C4—44no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D414yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C3×SD16—48no—
216C3×D8—48no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4C2×D414yes—
228C3×C22⋊C4—28no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—42no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42—42no—
240C2×#41—28no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—42no—
245C2×#41—44no—
246C2×#41—44no—
247C22×Q8—21no—
248C22×Q8C23×C311yescharacteristic
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C23×C312yes—
254C22×D4C23×C311yescharacteristic
255C23×C4—22no—
256C23×C4—21no—
257C23×C4C3×D411yescharacteristic
258C23×C4C3×D411yescharacteristic
259C23×C4C23×C311yescharacteristic
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—42no—
263C2×D8—24no—
264C2×D8—24no—
265C2×D8—42no—
266C22×C8C23×C312yes—
267C22×C8C23×C312yes—
268C4⋊Q8—24no—
269#127—24no—
270#124—28no—
271#121—28no—
272#121—24no—
273#121C3×D418yes—
274C4×D4—28no—
275C4×D4—24no—
276C4×D4—24no—
277#117—42no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C312yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C42—24no—
290C2×C42—21no—
291C2×C42C3×D411yescharacteristic
292C2×C42C3×D411yescharacteristic
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#103—24no—
296#102—24no—
297#102—42no—
298#102—42no—
299C22⋊C8—24no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—24no—
311C23×C3—22no—
312C23×C3—41no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—42no—
316C23×C3—41no—
317C23×C3—42no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—42no—
324C23×C3#12012yes—
325C23×C3C22×D411yescharacteristic
326C3×Q8—24no—
327C3×Q8—48no—
328C3×Q8—42no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—24no—
332C3×D4—48no—
333C3×D4—48no—
334C3×D4—48no—
335C3×D4—416no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—42no—
339C2×C3×C4—28no—
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—21no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—21no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—44no—
356C2×C3×C4—48no—
357C2×C3×C4—416no—
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—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4#12012yes—
369C2×C3×C4#13512yes—
370C2×C3×C4AGL(1,ℤ/8)12yes—
371C2×C3×C4AGL(1,ℤ/8)12yes—
372C2×C3×C4C22×D412yes—
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)12yes—
376C3×C8—42no—
377C3×C8—42no—
378C24—21no—
379C24—22no—
380C24C2×C3×D411yescharacteristic
381#41—48no—
382#41—48no—
383#41—432no—
384C2×Q8—24no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8C3×#4112yescomplemented
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C2×C3×D414yes—
412C2×D4C3×#4112yescomplemented
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—41no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D414yes—
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C2×C3×D411yescharacteristic
438C22×C4C24×C311yescharacteristic
439SD16—48no—
440D8—48no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4C2×C3×D414yes—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—42no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—82no—
478C22×C3—82no—
479C22×C3—81no—
480C22×C3—82no—
481C22×C3—81no—
482C22×C3—82no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3C2×#12011yescharacteristic
486C22×C3#52511yescharacteristic
487C22×C3#53511yescharacteristic
488C22×C3C2×#13511yescharacteristic
489C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
490C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
491C22×C3#57511yescharacteristic
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—42no—
497C3×C4—41no—
498C3×C4—44no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4#54612yes—
502C3×C4#54612yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—24no—
510C23—22no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23C3×#12012yes—
524C23C22×C3×D411yescharacteristic
525Q8—24no—
526Q8—48no—
527Q8—42no—
528D4—24no—
529D4—24no—
530D4—24no—
531D4—48no—
532D4—48no—
533D4—48no—
534D4—416no—
535D4—42no—
536D4—42no—
537D4—42no—
538C2×C4—28no—
539C2×C4—21no—
540C2×C4—21no—
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—24no—
553C2×C4—22no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—416no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4C3×#12012yes—
568C2×C4C3×#13512yes—
569C2×C4C3×AGL(1,ℤ/8)12yes—
570C2×C4C3×AGL(1,ℤ/8)12yes—
571C2×C4C22×C3×D412yes—
572C2×C4C2×C3×#4111yescharacteristic
573C2×C4C2×C3×#4111yescharacteristic
574C2×C4C3×H(5,𝔽2)12yes—
575C8—42no—
576C8—42no—
577C2×C3—42no—
578C2×C3—82no—
579C2×C3—84no—
580C2×C3#240411yescharacteristic
581C2×C3#300211yescharacteristic
582C2×C3#300311yescharacteristic
583C2×C3#310611yescharacteristic
584C2×C3#311511yescharacteristic
585C2×C3#320711yescharacteristic
586C2×C3#321411yescharacteristic
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—81no—
601C22—82no—
602C22—81no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22—82no—
607C22—82no—
608C22C2×C3×#12011yescharacteristic
609C22C3×#52511yescharacteristic
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—22no—
618C4—22no—
619C4—42no—
620C4—41no—
621C4—44no—
622C4—81no—
623C4—81no—
624C4C3×#54612yes—
625C4C3×#54612yes—
626C3#2393211yescharacteristic, direct product factor, Sylow
627C2—42no—
628C2—82no—
629C2—84no—
630C2C3×#240411yescharacteristic
631C2C3×#300211yescharacteristic
632C2C3×#300311yescharacteristic
633C2C3×#310611yescharacteristic
634C2C3×#311511yescharacteristic
635C2C3×#320711yescharacteristic
636C2C3×#321411yescharacteristic
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×#2297C2semidirect
4. C3×#2024C2semidirect
5. C3×#2023C2semidirect
6. C3×#2014C2semidirect
7. C3×#2011C2semidirect
8. C3×#1879C2semidirect
9. C3×#1877C2semidirect
10. C3×#1875C2semidirect
11. C3×#1817C2semidirect
12. #23932C3direct (N × H)
46. C2×#532C2×C3semidirect
47. C2×#449C2×C3semidirect
48. #2297C2×C3semidirect
49. #2024C2×C3semidirect
50. #2023C2×C3semidirect
51. #2014C2×C3semidirect
52. #2011C2×C3semidirect
53. #1879C2×C3semidirect
54. #1877C2×C3semidirect
55. #1875C2×C3semidirect
56. #1817C2×C3semidirect
166. C2×C3×Q8#41semidirect
188. C2×C3×D4#41semidirect
390. C2×Q8C3×#41semidirect
412. C2×D4C3×#41semidirect
626. C3#23932direct (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×#22972
C3×#20242
C3×#20232
C3×#20142
C3×#20112
C3×#18792
C3×#18772
C3×#18752
C3×#18172
#239323