Finite Group Databasebeta

← C3×#24153 #10642760

Subgroups — C3×#24153 #10642760

Order 768.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C3×C4×#121C211yesmaximal, characteristic
5C3×#2036C212yesmaximal, complemented
6C3×#2000C212yesmaximal
7C3×#1988C211yesmaximal, characteristic, complemented
8C3×#1981C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1863C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#24153C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C3×#533—24no—
15C3×#533—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121C2211yescharacteristic
22C2×C3×C4×Q8—21no—
23C2×C3×C4×Q8C2212yes—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117—21no—
27C22×C3×C42C2211yescharacteristic
28C3×#474—24no—
29C3×#461—24no—
30C2×C3×#107C2212yes—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C3×#390C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#388—21no—
41C3×#386—21no—
42C3×#380C2211yescharacteristic
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C22⋊C4—21no—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#533C2×C311yescharacteristic, complemented
49C2×#474C2×C311yescharacteristic, complemented
50C2×#461C2×C311yescharacteristic, complemented
51C4×#121C2×C311yescharacteristic
52#2036C2×C312yescomplemented
53#2000C2×C312yes—
54#1988C2×C311yescharacteristic, complemented
55#1981C2×C311yescharacteristic, complemented
56#1880C2×C312yescomplemented
57#1863C2×C312yescomplemented
58#1797C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8C2312yes—
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C2×C3×SD16—24no—
68C2×C3×SD16—24no—
69C2×C3×SD16—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8D414yescomplemented
75C3×C4⋊Q8D414yescomplemented
76C3×#125—28no—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—24no—
80C3×#121—22no—
81C3×#121—22no—
82C3×C4×Q8—28no—
83C3×C4×Q8—24no—
84C3×C4×Q8—24no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—44no—
88C3×#117—28no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—22no—
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—22no—
106C2×C3×C42—22no—
107C2×C3×C42—22no—
108C2×C3×C42—21no—
109C2×C3×C42D411yescharacteristic
110C2×C3×C42D411yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C3×#107—24no—
114C3×C4⋊C8—22no—
115C3×C4⋊C8—22no—
116C3×#103—24no—
117C3×#103—24no—
118C3×#103—42no—
119C3×#102—24no—
120C3×C22⋊C8—42no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C22×SD16C22×C312yes—
125#533—24no—
126#533—24no—
127C2×C4⋊Q8C22×C311yescharacteristic
128C2×C4⋊Q8C22×C311yescharacteristic
129C2×#125—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8C22×C312yes—
132C2×#121C22×C311yescharacteristic
133C2×C4×Q8—21no—
134C2×C4×Q8C22×C312yes—
135C2×C4×D4—21no—
136C2×C4×D4C22×C311yescharacteristic
137C2×#117—21no—
138C22×C42C22×C311yescharacteristic
139#474—24no—
140#461—24no—
141C2×#107C22×C312yes—
142C2×C4⋊C8C22×C311yescharacteristic
143C2×C4⋊C8C22×C311yescharacteristic
144C2×#103C22×C312yes—
145C2×#103C22×C312yes—
146C2×#103C22×C312yes—
147C2×#102C22×C312yes—
148C2×C22⋊C8C22×C312yes—
149#390C22×C311yescharacteristic
150#389C22×C311yescharacteristic
151#388—21no—
152#386—21no—
153#380C22×C311yescharacteristic
154C4×C4⋊C4C22×C311yescharacteristic
155C4×C22⋊C4—21no—
156#340C22×C312yes—
157#340C22×C312yes—
158#336C22×C312yes—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—24no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—42no—
168C2×C3×Q8—42no—
169C2×C3×Q8C2×D414yes—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—42no—
177C2×C3×D4—42no—
178C2×C3×D4#4112yes—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
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—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4—42no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C3×SD16—48no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—28no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4C2×D414yes—
225C3×C4⋊C4#4112yes—
226C3×C22⋊C4—28no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—42no—
232C3×C42—28no—
233C3×C42—22no—
234C3×C42—22no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—42no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240C22×Q8—21no—
241C22×Q8C23×C312yes—
242C22×D4—21no—
243C22×D4C23×C311yescharacteristic
244C23×C4—21no—
245C23×C4C3×D411yescharacteristic
246C23×C4C3×D411yescharacteristic
247C23×C4C23×C311yescharacteristic
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—42no—
251C22×C8C23×C312yes—
252C22×C8C23×C312yes—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8C3×D414yescomplemented
256C4⋊Q8C3×D414yescomplemented
257#125—28no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261#121—22no—
262#121—22no—
263C4×Q8—28no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—44no—
269#117—28no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C4⋊C4C23×C312yes—
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42C3×D411yescharacteristic
291C2×C42C3×D411yescharacteristic
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C311yescharacteristic
294#107—24no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297#103—24no—
298#103—24no—
299#103—42no—
300#102—24no—
301C22⋊C8—42no—
302#95—21no—
303#95—21no—
304#95—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—22no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3#13712yes—
319C23×C3C22×D411yescharacteristic
320C3×Q8—24no—
321C3×Q8—24no—
322C3×Q8—42no—
323C3×Q8—42no—
324C3×Q8—48no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—48no—
328C3×D4—41no—
329C3×D4—41no—
330C2×C3×C4—28no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—44no—
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—84no—
362C2×C3×C4#12012yes—
363C2×C3×C4#12012yes—
364C2×C3×C4#13512yes—
365C2×C3×C4AGL(1,ℤ/8)12yes—
366C2×C3×C4#13712yes—
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372C2×C3×C42−1+411yescharacteristic
373C3×C8—42no—
374C3×C8—42no—
375C24—21no—
376C24C2×C3×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8C2×C3×D414yes—
386C2×D4—24no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4C3×#4112yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C24×C311yescharacteristic
421SD16—48no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—42no—
440C4⋊C4C2×C3×D414yes—
441C4⋊C4C3×#4112yes—
442C22⋊C4—28no—
443C22⋊C4—24no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C42—28no—
449C42—22no—
450C42—22no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42C2×C3×D412yes—
455C42C2×C3×D412yes—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3C2×#12011yescharacteristic
471C22×C3#52511yescharacteristic
472C22×C3#53711yescharacteristic
473C22×C3C2×#13511yescharacteristic
474C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3C2×#13711yescharacteristic
477C3×C4—22no—
478C3×C4—22no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4—44no—
482C3×C4—41no—
483C3×C4—44no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4#44914yes—
487C3×C4#54611yescharacteristic
488C3×C4#54711yescharacteristic
489C3×C4#54811yescharacteristic
490C3×C4#55011yescharacteristic
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23C3×#13712yes—
505C23C22×C3×D411yescharacteristic
506Q8—24no—
507Q8—24no—
508Q8—48no—
509Q8—42no—
510Q8—42no—
511D4—22no—
512D4—22no—
513D4—48no—
514D4—41no—
515D4—41no—
516C2×C4—28no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
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—84no—
548C2×C4C3×#12012yes—
549C2×C4C3×#12012yes—
550C2×C4C3×#13512yes—
551C2×C4C3×AGL(1,ℤ/8)12yes—
552C2×C4C3×#13712yes—
553C2×C4C22×C3×D411yescharacteristic
554C2×C4C22×C3×D411yescharacteristic
555C2×C4C2×C3×#4111yescharacteristic
556C2×C4C2×C3×#4111yescharacteristic
557C2×C4C3×H(5,𝔽2)11yescharacteristic
558C2×C4C3×2−1+411yescharacteristic
559C8—42no—
560C8—42no—
561C2×C3—42no—
562C2×C3—82no—
563C2×C3#240711yescharacteristic
564C2×C3C2×#44911yescharacteristic
565C2×C3#300411yescharacteristic
566C2×C3#310711yescharacteristic
567C2×C3#311011yescharacteristic
568C2×C3#324811yescharacteristic
569C2×C3#325411yescharacteristic
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22C2×C3×#12011yescharacteristic
585C22C3×#52511yescharacteristic
586C22C3×#53711yescharacteristic
587C22C2×C3×#13511yescharacteristic
588C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
589C22C2×C3×#13711yescharacteristic
590C22C2×C3×#13711yescharacteristic
591C4—22no—
592C4—22no—
593C4—41no—
594C4—44no—
595C4—41no—
596C4—44no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4C3×#44914yes—
601C4C3×#54611yescharacteristic
602C4C3×#54711yescharacteristic
603C4C3×#54811yescharacteristic
604C4C3×#55011yescharacteristic
605C3#2415311yescharacteristic, direct product factor, Sylow
606C2—42no—
607C2—82no—
608C2C3×#240711yescharacteristic
609C2C2×C3×#44911yescharacteristic
610C2C3×#300411yescharacteristic
611C2C3×#310711yescharacteristic
612C2C3×#311011yescharacteristic
613C2C3×#324811yescharacteristic
614C2C3×#325411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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×#533C2semidirect
2. C2×C3×#474C2semidirect
3. C2×C3×#461C2semidirect
5. C3×#2036C2semidirect
7. C3×#1988C2semidirect
8. C3×#1981C2semidirect
9. C3×#1880C2semidirect
10. C3×#1863C2semidirect
11. C3×#1797C2semidirect
12. #24153C3direct (N × H)
48. C2×#533C2×C3semidirect
49. C2×#474C2×C3semidirect
50. C2×#461C2×C3semidirect
52. #2036C2×C3semidirect
54. #1988C2×C3semidirect
55. #1981C2×C3semidirect
56. #1880C2×C3semidirect
57. #1863C2×C3semidirect
58. #1797C2×C3semidirect
74. C3×C4⋊Q8D4semidirect
75. C3×C4⋊Q8D4semidirect
255. C4⋊Q8C3×D4semidirect
256. C4⋊Q8C3×D4semidirect
605. C3#24153direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4742
C2×C3×#4612
C3×C4×#1212
C3×#20362
C3×#20002
C3×#19882
C3×#19812
C3×#18802
C3×#18632
C3×#17972
#241533