Finite Group Databasebeta

← C3×#24156 #10642763

Subgroups — C3×#24156 #10642763

Order 768.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3C2×C3×#460C211yesmaximal, characteristic, complemented
4C3×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
5C3×#2033C212yesmaximal, complemented
6C3×#2000C212yesmaximal, complemented
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1982C211yesmaximal, characteristic, complemented
9C3×#1877C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#24156C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×SD16C2212yes—
15C3×#532—24no—
16C3×#532—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121—21no—
22C2×C3×#121C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2212yes—
26C22×C3×C42C2211yescharacteristic
27C3×#477—24no—
28C3×#460—24no—
29C2×C3×#107C2212yes—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C3×#391C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#387—21no—
40C3×#385—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C4⋊C4—21no—
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C22⋊C4—21no—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#532C2×C311yescharacteristic
49C2×#477C2×C311yescharacteristic, complemented
50C2×#460C2×C311yescharacteristic, complemented
51C4×C22⋊Q8C2×C311yescharacteristic, complemented
52#2033C2×C312yescomplemented
53#2000C2×C312yescomplemented
54#1987C2×C311yescharacteristic, complemented
55#1982C2×C311yescharacteristic, complemented
56#1877C2×C312yescomplemented
57#1862C2×C312yescomplemented
58#1797C2×C311yescharacteristic, complemented
59C2×C3×#41—28no—
60C2×C3×#41—44no—
61C2×C3×#41—44no—
62C22×C3×Q8—21no—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4—22no—
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4C2312yes—
68C23×C3×C4—22no—
69C23×C3×C4D411yescharacteristic
70C23×C3×C4D411yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C2×C3×SD16—24no—
73C2×C3×SD16—24no—
74C2×C3×SD16—42no—
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8D414yes—
79C3×#127—22no—
80C3×#127D414yes—
81C3×#124—28no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×#121—28no—
85C3×#121—24no—
86C3×#121—24no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×D4—28no—
90C3×C4×D4—24no—
91C3×C4×D4—24no—
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⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C42—22no—
105C2×C3×C42—22no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D411yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#107—24no—
113C3×C4⋊C8—22no—
114C3×C4⋊C8—22no—
115C3×#103—24no—
116C3×#102—24no—
117C3×#102—24no—
118C3×#102—42no—
119C3×C22⋊C8—42no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C22×#41—21no—
126C22×SD16C22×C312yes—
127#532—24no—
128#532—24no—
129C2×C4⋊Q8C22×C311yescharacteristic
130C2×#127C22×C311yescharacteristic
131C2×#124—21no—
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×#121—21no—
134C2×#121C22×C312yes—
135C2×C4×Q8C22×C311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4C22×C312yes—
138C22×C42C22×C311yescharacteristic
139#477—24no—
140#460—24no—
141C2×#107C22×C312yes—
142C2×C4⋊C8C22×C311yescharacteristic
143C2×C4⋊C8C22×C311yescharacteristic
144C2×#103C22×C312yes—
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×#102C22×C312yes—
148C2×C22⋊C8C22×C312yes—
149#391C22×C311yescharacteristic
150#389C22×C311yescharacteristic
151#387—21no—
152#385—21no—
153#380C22×C311yescharacteristic
154C4×C4⋊C4—21no—
155C4×C4⋊C4C22×C311yescharacteristic
156C4×C22⋊C4—21no—
157#340C22×C312yes—
158#340C22×C312yes—
159#336C22×C312yes—
160C24×C3—22no—
161C24×C3C2×D411yescharacteristic
162C3×#41—432no—
163C2×C3×Q8—24no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—42no—
170C2×C3×Q8—42no—
171C2×C3×Q8#4112yescomplemented
172C2×C3×D4—28no—
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
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×D4C2×D414yes—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—28no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—44no—
206C22×C3×C4—44no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C3×SD16—48no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4C2×D414yes—
231C3×C4⋊C4#4112yescomplemented
232C3×C22⋊C4—28no—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—42no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—42no—
241C3×C42—42no—
242C3×C42C2×D412yes—
243C3×C42C2×D412yes—
244C2×#41—28no—
245C2×#41—44no—
246C2×#41—44no—
247C22×Q8—21no—
248C22×Q8C23×C311yescharacteristic
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C23×C312yes—
253C23×C4—22no—
254C23×C4C3×D411yescharacteristic
255C23×C4C3×D411yescharacteristic
256C23×C4C23×C311yescharacteristic
257C2×SD16—24no—
258C2×SD16—24no—
259C2×SD16—42no—
260C22×C8C23×C312yes—
261C22×C8C23×C312yes—
262C4⋊Q8—22no—
263C4⋊Q8C3×D414yes—
264#127—22no—
265#127C3×D414yes—
266#124—28no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269#121—28no—
270#121—24no—
271#121—24no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×D4—28no—
275C4×D4—24no—
276C4×D4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C312yes—
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C3×D411yescharacteristic
294C2×C42C3×D411yescharacteristic
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297#107—24no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300#103—24no—
301#102—24no—
302#102—24no—
303#102—42no—
304C22⋊C8—42no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—24no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3AGL(1,ℤ/8)12yes—
324C23×C3C22×D411yescharacteristic
325C3×Q8—22no—
326C3×Q8—22no—
327C3×Q8—48no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×D4—24no—
331C3×D4—24no—
332C3×D4—48no—
333C3×D4—416no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—48no—
337C2×C3×C4—28no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—416no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—44no—
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#12012yes—
370C2×C3×C4#12012yes—
371C2×C3×C4#13512yes—
372C2×C3×C4AGL(1,ℤ/8)12yes—
373C2×C3×C4#13712yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C42−1+411yescharacteristic
380C3×C8—42no—
381C3×C8—42no—
382C24—22no—
383C24C2×C3×D411yescharacteristic
384#41—432no—
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8C3×#4112yescomplemented
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4C2×C3×D414yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C24×C311yescharacteristic
435SD16—48no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4C2×C3×D414yes—
453C4⋊C4C3×#4112yescomplemented
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—24no—
457C22⋊C4—42no—
458C42—22no—
459C42—22no—
460C42—21no—
461C42—21no—
462C42—42no—
463C42—42no—
464C42C2×C3×D412yes—
465C42C2×C3×D412yes—
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×C3C2×#12011yescharacteristic
481C22×C3#52511yescharacteristic
482C22×C3#53711yescharacteristic
483C22×C3C2×#13511yescharacteristic
484C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
485C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
486C22×C3C2×#13711yescharacteristic
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—42no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—41no—
493C3×C4—44no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4#45214yes—
497C3×C4#54611yescharacteristic
498C3×C4#54711yescharacteristic
499C3×C4#54811yescharacteristic
500C3×C4#55011yescharacteristic
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23C3×AGL(1,ℤ/8)12yes—
515C23C22×C3×D411yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—48no—
519Q8—41no—
520Q8—41no—
521D4—24no—
522D4—24no—
523D4—48no—
524D4—416no—
525D4—48no—
526D4—42no—
527D4—42no—
528C2×C4—28no—
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—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—416no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4C3×#12012yes—
561C2×C4C3×#12012yes—
562C2×C4C3×#13512yes—
563C2×C4C3×AGL(1,ℤ/8)12yes—
564C2×C4C3×#13712yes—
565C2×C4C22×C3×D411yescharacteristic
566C2×C4C22×C3×D411yescharacteristic
567C2×C4C2×C3×#4111yescharacteristic
568C2×C4C2×C3×#4111yescharacteristic
569C2×C4C3×H(5,𝔽2)11yescharacteristic
570C2×C4C3×2−1+411yescharacteristic
571C8—42no—
572C8—42no—
573C2×C3—42no—
574C2×C3—84no—
575C2×C3#240711yescharacteristic
576C2×C3C2×#45211yescharacteristic
577C2×C3#300311yescharacteristic
578C2×C3#310611yescharacteristic
579C2×C3#311211yescharacteristic
580C2×C3#324711yescharacteristic
581C2×C3#325411yescharacteristic
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C2×C3×#12011yescharacteristic
597C22C3×#52511yescharacteristic
598C22C3×#53711yescharacteristic
599C22C2×C3×#13511yescharacteristic
600C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
601C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
602C22C2×C3×#13711yescharacteristic
603C4—22no—
604C4—22no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—44no—
609C4—42no—
610C4—81no—
611C4—81no—
612C4C3×#45214yes—
613C4C3×#54611yescharacteristic
614C4C3×#54711yescharacteristic
615C4C3×#54811yescharacteristic
616C4C3×#55011yescharacteristic
617C3#2415611yescharacteristic, direct product factor, Sylow
618C2—42no—
619C2—84no—
620C2C3×#240711yescharacteristic
621C2C2×C3×#45211yescharacteristic
622C2C3×#300311yescharacteristic
623C2C3×#310611yescharacteristic
624C2C3×#311211yescharacteristic
625C2C3×#324711yescharacteristic
626C2C3×#325411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
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
2. C2×C3×#477C2semidirect
3. C2×C3×#460C2semidirect
4. C3×C4×C22⋊Q8C2semidirect
5. C3×#2033C2semidirect
6. C3×#2000C2semidirect
7. C3×#1987C2semidirect
8. C3×#1982C2semidirect
9. C3×#1877C2semidirect
10. C3×#1862C2semidirect
11. C3×#1797C2semidirect
12. #24156C3direct (N × H)
49. C2×#477C2×C3semidirect
50. C2×#460C2×C3semidirect
51. C4×C22⋊Q8C2×C3semidirect
52. #2033C2×C3semidirect
53. #2000C2×C3semidirect
54. #1987C2×C3semidirect
55. #1982C2×C3semidirect
56. #1877C2×C3semidirect
57. #1862C2×C3semidirect
58. #1797C2×C3semidirect
171. C2×C3×Q8#41semidirect
231. C3×C4⋊C4#41semidirect
393. C2×Q8C3×#41semidirect
453. C4⋊C4C3×#41semidirect
617. C3#24156direct (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×#4772
C2×C3×#4602
C3×C4×C22⋊Q82
C3×#20332
C3×#20002
C3×#19872
C3×#19822
C3×#18772
C3×#18622
C3×#17972
#241563