Finite Group Databasebeta

← C3×#24242 #10642849

Subgroups — C3×#24242 #10642849

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C2×C3×#452C212yesmaximal, complemented
4C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
5C3×#2538C211yesmaximal, characteristic, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1954C211yesmaximal, characteristic, complemented
10C3×#1803C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal, complemented
12#24242C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C3×#550—24no—
17C3×#550—24no—
18C2×C3×#127—21no—
19C2×C3×#127C2211yescharacteristic
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121—22no—
22C2×C3×#121C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C3×#460—24no—
28C3×#452—28no—
29C3×C4×Q16—24no—
30C2×C3×#107C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C2×C3×C22⋊C8C2212yes—
40C2×C3×C4×C8C2211yescharacteristic
41C3×#396—22no—
42C3×#390—21no—
43C3×#389—21no—
44C3×#384C2211yescharacteristic
45C3×#380C2212yes—
46C3×#340C2212yes—
47C2×#550C2×C311yescharacteristic
48C2×#460C2×C311yescharacteristic, complemented
49C2×#452C2×C312yescomplemented
50C2×C4×Q16C2×C311yescharacteristic, complemented
51#2538C2×C311yescharacteristic, complemented
52#2033C2×C312yescomplemented
53#2000C2×C312yescomplemented
54#1957C2×C311yescharacteristic, complemented
55#1954C2×C311yescharacteristic, complemented
56#1803C2×C311yescharacteristic, complemented
57#1788C2×C312yescomplemented
58C2×C3×#41—24no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41—42no—
63C2×C3×#41—42no—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4—21no—
69C22×C3×D4—21no—
70C22×C3×D4D411yescharacteristic
71C22×C3×D4C2312yes—
72C23×C3×C4—22no—
73C23×C3×C4C2312yes—
74C2×C3×Q16—22no—
75C2×C3×Q16—22no—
76C2×C3×Q16D414yescomplemented
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—42no—
80C22×C3×C8C2312yes—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#127—28no—
84C3×#127—22no—
85C3×#127D414yes—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—42no—
88C3×#121—216no—
89C3×#121—24no—
90C3×#121—24no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×Q8D414yes—
95C3×C4×D4—28no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—22no—
110C2×C3×C22⋊C4D412yes—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42C2311yescharacteristic
114C3×#107—22no—
115C3×C4⋊C8—22no—
116C3×#103—22no—
117C3×#103—24no—
118C3×#103—22no—
119C3×#102—24no—
120C3×#102—24no—
121C3×#102—42no—
122C3×C22⋊C8—24no—
123C3×C22⋊C8—42no—
124C3×C4×C8—22no—
125C3×#95—22no—
126C22×#41C22×C312yes—
127C22×Q16C22×C311yescharacteristic
128C22×SD16C22×C312yes—
129#550—24no—
130#550—24no—
131C2×#127—21no—
132C2×#127C22×C311yescharacteristic
133C2×C22⋊Q8C22×C312yes—
134C2×#121—22no—
135C2×#121C22×C312yes—
136C2×C4×Q8C22×C311yescharacteristic
137C2×C4×Q8C22×C311yescharacteristic
138C2×C4×D4—21no—
139C2×C4×D4C22×C311yescharacteristic
140#460—24no—
141#452—28no—
142C4×Q16—24no—
143C2×#107C22×C311yescharacteristic
144C2×C4⋊C8C22×C311yescharacteristic
145C2×#103C22×C312yes—
146C2×#103C22×C311yescharacteristic
147C2×#103C22×C311yescharacteristic
148C2×#102C22×C312yes—
149C2×#102C22×C312yes—
150C2×#102C22×C312yes—
151C2×C22⋊C8C22×C312yes—
152C2×C22⋊C8C22×C312yes—
153C2×C4×C8C22×C311yescharacteristic
154#396—22no—
155#390—21no—
156#389—21no—
157#384C22×C311yescharacteristic
158#380C22×C312yes—
159#340C22×C312yes—
160C24×C3—22no—
161C24×C3C2×D412yes—
162C3×#41—48no—
163C3×#41—48no—
164C3×#41—48no—
165C3×#41—48no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8C2×D412yes—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8C2×D412yes—
177C2×C3×Q8#4112yes—
178C2×C3×D4—28no—
179C2×C3×D4—28no—
180C2×C3×D4—22no—
181C2×C3×D4—24no—
182C2×C3×D4—24no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—44no—
191C2×C3×D4—44no—
192C2×C3×D4—42no—
193C2×C3×D4—42no—
194C2×C3×D4—81no—
195C2×C3×D4—81no—
196C2×C3×D4—82no—
197C2×C3×D4—82no—
198C2×C3×D4C2×D414yes—
199C22×C3×C4—21no—
200C22×C3×C4—28no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—21no—
210C22×C3×C4—22no—
211C22×C3×C4—44no—
212C22×C3×C4—44no—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C3×Q16—28no—
218C3×SD16—48no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—21no—
225C2×C3×C8—22no—
226C2×C3×C8—21no—
227C2×C3×C8C2×D412yes—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—44no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—81no—
239C3×C4⋊C4C2×D412yes—
240C3×C22⋊C4—28no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—44no—
243C3×C22⋊C4—82no—
244C3×C42—24no—
245C3×C42—42no—
246C3×C42C2×D412yes—
247C3×C42C2×D412yes—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—42no—
253C2×#41—42no—
254C22×Q8C23×C311yescharacteristic
255C22×Q8C23×C311yescharacteristic
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4C3×D411yescharacteristic
261C22×D4C23×C312yes—
262C23×C4—22no—
263C23×C4C23×C312yes—
264C2×Q16—22no—
265C2×Q16—22no—
266C2×Q16C3×D414yescomplemented
267C2×SD16—24no—
268C2×SD16—24no—
269C2×SD16—42no—
270C22×C8C23×C312yes—
271C22×C8C23×C311yescharacteristic
272C22×C8C23×C311yescharacteristic
273#127—28no—
274#127—22no—
275#127C3×D414yes—
276C22⋊Q8—24no—
277C22⋊Q8—42no—
278#121—216no—
279#121—24no—
280#121—24no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8C3×D414yes—
285C4×D4—28no—
286C4×D4—22no—
287C4×D4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C3×D411yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C312yes—
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4C3×D412yes—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42C23×C311yescharacteristic
304#107—22no—
305C4⋊C8—22no—
306#103—22no—
307#103—24no—
308#103—22no—
309#102—24no—
310#102—24no—
311#102—42no—
312C22⋊C8—24no—
313C22⋊C8—42no—
314C4×C8—22no—
315#95—22no—
316C23×C3—22no—
317C23×C3—24no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3C22×D411yescharacteristic
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—41no—
337C3×Q8—41no—
338C3×Q8#12114yes—
339C3×D4—24no—
340C3×D4—24no—
341C3×D4—416no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—44no—
345C3×D4—48no—
346C3×D4—84no—
347C3×D4—88no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—416no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—42no—
372C2×C3×C4—41no—
373C2×C3×C4#12012yes—
374C2×C3×C4#12012yes—
375C2×C3×C4#13512yes—
376C2×C3×C4#13712yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C3×C8—22no—
384C3×C8—41no—
385C3×C8—42no—
386C24—22no—
387C24C2×C3×D412yes—
388#41—48no—
389#41—48no—
390#41—48no—
391#41—48no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8C2×C3×D412yes—
401C2×Q8C2×C3×D412yes—
402C2×Q8C2×C3×D412yes—
403C2×Q8C3×#4112yes—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4C2×C3×D414yes—
425C22×C4—21no—
426C22×C4—28no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4C2×C3×D412yes—
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C24×C311yescharacteristic
443Q16—28no—
444SD16—48no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—21no—
451C2×C8—22no—
452C2×C8—21no—
453C2×C8C2×C3×D412yes—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4C2×C3×D412yes—
466C22⋊C4—28no—
467C22⋊C4—24no—
468C22⋊C4—44no—
469C22⋊C4—82no—
470C42—24no—
471C42—42no—
472C42C2×C3×D412yes—
473C42C2×C3×D412yes—
474C22×C3—42no—
475C22×C3—44no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—44no—
479C22×C3—42no—
480C22×C3—44no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3C2×#12011yescharacteristic
489C22×C3D4211yescharacteristic
490C22×C3#54611yescharacteristic
491C22×C3C2×#13511yescharacteristic
492C22×C3C2×#13711yescharacteristic
493C22×C3#57611yescharacteristic
494C22×C3#57711yescharacteristic
495C3×C4—24no—
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—81no—
501C3×C4—82no—
502C3×C4#45214yescomplemented
503C3×C4C2×#12111yescharacteristic
504C3×C4#53511yescharacteristic
505C3×C4D4211yescharacteristic
506C3×C4#54611yescharacteristic
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C22×C3×D411yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8C3×#12114yes—
530D4—24no—
531D4—24no—
532D4—416no—
533D4—44no—
534D4—48no—
535D4—42no—
536D4—42no—
537D4—84no—
538D4—88no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—416no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4C3×#12012yes—
565C2×C4C3×#12012yes—
566C2×C4C3×#13512yes—
567C2×C4C3×#13712yes—
568C2×C4C22×C3×D411yescharacteristic
569C2×C4C22×C3×D411yescharacteristic
570C2×C4C22×C3×D411yescharacteristic
571C2×C4C2×C3×#4111yescharacteristic
572C2×C4C3×H(5,𝔽2)11yescharacteristic
573C2×C4C3×H(5,𝔽2)11yescharacteristic
574C8—22no—
575C8—41no—
576C8—42no—
577C2×C3—82no—
578C2×C3—84no—
579C2×C3#240811yescharacteristic
580C2×C3C2×#45211yescharacteristic
581C2×C3#301211yescharacteristic
582C2×C3#327811yescharacteristic
583C2×C3#328711yescharacteristic
584C2×C3#330411yescharacteristic
585C2×C3#333311yescharacteristic
586C22—44no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22C2×C3×#12011yescharacteristic
601C22C3×D4211yescharacteristic
602C22C3×#54611yescharacteristic
603C22C2×C3×#13511yescharacteristic
604C22C2×C3×#13711yescharacteristic
605C22C3×#57611yescharacteristic
606C22C3×#57711yescharacteristic
607C4—24no—
608C4—22no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—81no—
613C4—82no—
614C4C3×#45214yescomplemented
615C4C2×C3×#12111yescharacteristic
616C4C3×#53511yescharacteristic
617C4C3×D4211yescharacteristic
618C4C3×#54611yescharacteristic
619C3#2424211yescharacteristic, direct product factor, Sylow
620C2—82no—
621C2—84no—
622C2C3×#240811yescharacteristic
623C2C2×C3×#45211yescharacteristic
624C2C3×#301211yescharacteristic
625C2C3×#327811yescharacteristic
626C2C3×#328711yescharacteristic
627C2C3×#330411yescharacteristic
628C2C3×#333311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#460C2semidirect
3. C2×C3×#452C2semidirect
4. C2×C3×C4×Q16C2semidirect
5. C3×#2538C2semidirect
6. C3×#2033C2semidirect
7. C3×#2000C2semidirect
8. C3×#1957C2semidirect
9. C3×#1954C2semidirect
10. C3×#1803C2semidirect
11. C3×#1788C2semidirect
12. #24242C3direct (N × H)
48. C2×#460C2×C3semidirect
49. C2×#452C2×C3semidirect
50. C2×C4×Q16C2×C3semidirect
51. #2538C2×C3semidirect
52. #2033C2×C3semidirect
53. #2000C2×C3semidirect
54. #1957C2×C3semidirect
55. #1954C2×C3semidirect
56. #1803C2×C3semidirect
57. #1788C2×C3semidirect
76. C2×C3×Q16D4semidirect
266. C2×Q16C3×D4semidirect
502. C3×C4#452semidirect
614. C4C3×#452semidirect
619. C3#24242direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4602
C2×C3×#4522
C2×C3×C4×Q162
C3×#25382
C3×#20332
C3×#20002
C3×#19572
C3×#19542
C3×#18032
C3×#17882
#242423