Finite Group Databasebeta

← C3×#23929 #10642536

Subgroups — C3×#23929 #10642536

Order 768.

1824 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#537C212yesmaximal, complemented
2C2×C3×#452C212yesmaximal, complemented
3C2×C3×#449C212yesmaximal, complemented
4C3×#2001C211yesmaximal, characteristic, complemented
5C3×#2000C212yesmaximal, complemented
6C3×#1999C211yesmaximal, characteristic, complemented
7C3×#1874C212yesmaximal, complemented
8C3×#1872C212yesmaximal, complemented
9C3×#1780C211yesmaximal, characteristic, complemented
10#23929C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C22×C3×#41C2211yescharacteristic
13C22×C3×Q16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×D8C2211yescharacteristic
16C3×#537—28no—
17C3×#537—28no—
18C2×C3×C22⋊Q8—22no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121—22no—
22C2×C3×#121—22no—
23C2×C3×#121—22no—
24C2×C3×#121C2212yes—
25C2×C3×#121C2212yes—
26C22×C3×C4⋊C4C2212yes—
27C3×#452—28no—
28C3×#449—28no—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2212yes—
34C2×C3×C22⋊C8C2212yes—
35C2×C3×C22⋊C8C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C3×#340C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2211yescharacteristic
40C2×#537C2×C312yescomplemented
41C2×#452C2×C312yescomplemented
42C2×#449C2×C312yescomplemented
43#2001C2×C311yescharacteristic, complemented
44#2000C2×C312yescomplemented
45#1999C2×C311yescharacteristic, complemented
46#1874C2×C312yescomplemented
47#1872C2×C312yescomplemented
48#1780C2×C311yescharacteristic, complemented
49C2×C3×#41—24no—
50C2×C3×#41—24no—
51C2×C3×#41—24no—
52C2×C3×#41—24no—
53C2×C3×#41—24no—
54C2×C3×#41—24no—
55C2×C3×#41—41no—
56C2×C3×#41—41no—
57C2×C3×#41—42no—
58C2×C3×#41—42no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2312yes—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4D411yescharacteristic
65C22×C3×D4D412yes—
66C22×C3×D4C2312yes—
67C23×C3×C4—22no—
68C23×C3×C4—22no—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2312yes—
71C2×C3×Q16—24no—
72C2×C3×Q16—41no—
73C2×C3×SD16—24no—
74C2×C3×SD16—24no—
75C2×C3×SD16—42no—
76C2×C3×D8—24no—
77C2×C3×D8—41no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2312yes—
80C22×C3×C8C2311yescharacteristic
81C3×C22⋊Q8—216no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—24no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8D418yescomplemented
86C3×#121—216no—
87C3×#121—216no—
88C3×#121—216no—
89C3×#121—24no—
90C3×#121—24no—
91C3×#121—24no—
92C3×#121D418yescomplemented
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—24no—
97C2×C3×C4⋊C4—24no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C3×#103—24no—
105C3×#103—24no—
106C3×#102—24no—
107C3×#102—24no—
108C3×C22⋊C8—24no—
109C3×C22⋊C8—24no—
110C3×C22⋊C8—41no—
111C3×C22⋊C8—41no—
112C22×#41C22×C312yes—
113C22×#41C22×C311yescharacteristic
114C22×Q16C22×C311yescharacteristic
115C22×SD16C22×C312yes—
116C22×D8C22×C311yescharacteristic
117#537—28no—
118#537—28no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8C22×C312yes—
121C2×C22⋊Q8C22×C312yes—
122C2×#121—22no—
123C2×#121—22no—
124C2×#121—22no—
125C2×#121C22×C312yes—
126C2×#121C22×C312yes—
127C22×C4⋊C4C22×C312yes—
128#452—28no—
129#449—28no—
130C2×#103C22×C312yes—
131C2×#103C22×C312yes—
132C2×#102C22×C312yes—
133C2×#102C22×C312yes—
134C2×C22⋊C8C22×C312yes—
135C2×C22⋊C8C22×C312yes—
136C2×C22⋊C8C22×C311yescharacteristic
137C2×C22⋊C8C22×C311yescharacteristic
138#340C22×C311yescharacteristic
139#340C22×C312yes—
140#340C22×C311yescharacteristic
141C24×C3—22no—
142C24×C3C2×D411yescharacteristic
143C24×C3C2×D412yes—
144C3×#41—44no—
145C3×#41—48no—
146C3×#41—44no—
147C3×#41—48no—
148C3×#41—48no—
149C3×#41—48no—
150C3×#41—48no—
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—41no—
158C2×C3×Q8C2×D414yes—
159C2×C3×D4—28no—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—28no—
163C2×C3×D4—28no—
164C2×C3×D4—24no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—42no—
171C2×C3×D4—41no—
172C2×C3×D4—44no—
173C2×C3×D4—44no—
174C2×C3×D4—82no—
175C2×C3×D4—82no—
176C2×C3×D4—82no—
177C2×C3×D4—82no—
178C2×C3×D4C2×D414yes—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—28no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—28no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—44no—
193C22×C3×C4—44no—
194C22×C3×C4—82no—
195C22×C3×C4—82no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D414yes—
199C22×C3×C4C2×D414yes—
200C22×C3×C4C2411yescharacteristic
201C3×Q16—44no—
202C3×SD16—48no—
203C3×D8—44no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—21no—
209C2×C3×C8—22no—
210C2×C3×C8—21no—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—28no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4C2×D414yes—
217C3×C4⋊C4C2×D414yes—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—28no—
222C2×#41—24no—
223C2×#41—24no—
224C2×#41—24no—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—41no—
229C2×#41—41no—
230C2×#41—42no—
231C2×#41—42no—
232C22×Q8C3×D411yescharacteristic
233C22×Q8C23×C312yes—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4C3×D411yescharacteristic
238C22×D4C3×D412yes—
239C22×D4C23×C312yes—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4C23×C311yescharacteristic
243C23×C4C23×C312yes—
244C2×Q16—24no—
245C2×Q16—41no—
246C2×SD16—24no—
247C2×SD16—24no—
248C2×SD16—42no—
249C2×D8—24no—
250C2×D8—41no—
251C22×C8C23×C311yescharacteristic
252C22×C8C23×C312yes—
253C22×C8C23×C311yescharacteristic
254C22⋊Q8—216no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8C3×D418yescomplemented
259#121—216no—
260#121—216no—
261#121—216no—
262#121—24no—
263#121—24no—
264#121—24no—
265#121C3×D418yescomplemented
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C312yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277#103—24no—
278#103—24no—
279#102—24no—
280#102—24no—
281C22⋊C8—24no—
282C22⋊C8—24no—
283C22⋊C8—41no—
284C22⋊C8—41no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—24no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3#12014yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—24no—
306C3×Q8—24no—
307C3×Q8—44no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×D4—24no—
311C3×D4—24no—
312C3×D4—48no—
313C3×D4—44no—
314C3×D4—416no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—88no—
318C3×D4—88no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—416no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—88no—
341C2×C3×C4#12014yes—
342C2×C3×C4C22×D412yes—
343C2×C3×C4C22×D412yes—
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345C2×C3×C4H(5,𝔽2)11yescharacteristic
346C3×C8—41no—
347C3×C8—42no—
348C3×C8—41no—
349C24—22no—
350C24C2×C3×D411yescharacteristic
351C24C2×C3×D412yes—
352#41—44no—
353#41—48no—
354#41—44no—
355#41—48no—
356#41—48no—
357#41—48no—
358#41—48no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—41no—
366C2×Q8C2×C3×D414yes—
367C2×D4—28no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—42no—
379C2×D4—41no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4C2×C3×D414yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D412yes—
406C22×C4C2×C3×D414yes—
407C22×C4C2×C3×D414yes—
408C22×C4C24×C311yescharacteristic
409Q16—44no—
410SD16—48no—
411D8—44no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—21no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4C2×C3×D414yes—
425C4⋊C4C2×C3×D414yes—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—28no—
430C22×C3—24no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—44no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—44no—
443C22×C3—42no—
444C22×C3—44no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3C2×#12012yes—
450C22×C3#53411yescharacteristic
451C22×C3#57611yescharacteristic
452C22×C3#57712yes—
453C22×C3#57811yescharacteristic
454C3×C4—24no—
455C3×C4—41no—
456C3×C4—44no—
457C3×C4—44no—
458C3×C4—44no—
459C3×C4D4211yescharacteristic
460C3×C4D4212yes—
461C3×C4D4211yescharacteristic
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—22no—
470C23—21no—
471C23—22no—
472C23—24no—
473C23—21no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23C3×#12014yes—
481C23C22×C3×D411yescharacteristic
482Q8—24no—
483Q8—24no—
484Q8—44no—
485Q8—42no—
486Q8—42no—
487D4—24no—
488D4—24no—
489D4—48no—
490D4—44no—
491D4—416no—
492D4—42no—
493D4—42no—
494D4—88no—
495D4—88no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—416no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—88no—
518C2×C4C3×#12014yes—
519C2×C4C22×C3×D412yes—
520C2×C4C22×C3×D412yes—
521C2×C4C3×H(5,𝔽2)11yescharacteristic
522C2×C4C3×H(5,𝔽2)11yescharacteristic
523C8—41no—
524C8—42no—
525C8—41no—
526C2×C3—44no—
527C2×C3—81no—
528C2×C3—84no—
529C2×C3#240311yescharacteristic
530C2×C3#301212yes—
531C2×C3#301312yes—
532C2×C3#320311yescharacteristic
533C2×C3#320411yescharacteristic
534C22—24no—
535C22—24no—
536C22—24no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—44no—
547C22—42no—
548C22—44no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22C2×C3×#12012yes—
554C22C3×#53411yescharacteristic
555C22C3×#57611yescharacteristic
556C22C3×#57712yes—
557C22C3×#57811yescharacteristic
558C4—24no—
559C4—41no—
560C4—44no—
561C4—44no—
562C4—44no—
563C4C3×D4211yescharacteristic
564C4C3×D4212yes—
565C4C3×D4211yescharacteristic
566C3#2392911yescharacteristic, direct product factor, Sylow
567C2—44no—
568C2—81no—
569C2—84no—
570C2C3×#240311yescharacteristic
571C2C3×#301212yes—
572C2C3×#301312yes—
573C2C3×#320311yescharacteristic
574C2C3×#320411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#537C2semidirect
2. C2×C3×#452C2semidirect
3. C2×C3×#449C2semidirect
4. C3×#2001C2semidirect
5. C3×#2000C2semidirect
6. C3×#1999C2semidirect
7. C3×#1874C2semidirect
8. C3×#1872C2semidirect
9. C3×#1780C2semidirect
10. #23929C3direct (N × H)
40. C2×#537C2×C3semidirect
41. C2×#452C2×C3semidirect
42. C2×#449C2×C3semidirect
43. #2001C2×C3semidirect
44. #2000C2×C3semidirect
45. #1999C2×C3semidirect
46. #1874C2×C3semidirect
47. #1872C2×C3semidirect
48. #1780C2×C3semidirect
85. C3×C22⋊Q8D4semidirect
92. C3×#121D4semidirect
258. C22⋊Q8C3×D4semidirect
265. #121C3×D4semidirect
566. C3#23929direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5372
C2×C3×#4522
C2×C3×#4492
C3×#20012
C3×#20002
C3×#19992
C3×#18742
C3×#18722
C3×#17802
#239293