Finite Group Databasebeta

← C23×#19072 #19166698

Subgroups — C23×#19072 #19166698

Order 5888.

992 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3545C211yesmaximal, characteristic, complemented
2C23×#3180C212yesmaximal, complemented
3C23×#3153C211yesmaximal, characteristic, complemented
4C23×#1661C212yesmaximal, complemented
5C23×#1660C211yesmaximal, characteristic, complemented
6C23×#1659C211yesmaximal, characteristic, complemented
7C23×#1657C211yesmaximal, characteristic, complemented
8C23×#1656C212yesmaximal, complemented
9C23×#1655C211yesmaximal, characteristic, complemented
10C23×#1524C212yesmaximal, complemented
11C2×C23×#329C211yesmaximal, characteristic, complemented
12C2×C23×#137—21no—
13C2×C23×AGL(1,ℤ/8)—21no—
14C23×#563C2211yescharacteristic
15C23×#562C2212yes—
16C23×H(3,𝔽4)C2211yescharacteristic
17C23×#555—21no—
18C23×#551—21no—
19C23×#551—22no—
20C23×#546—22no—
21C23×#542—22no—
22C23×#538—22no—
23C23×#536—21no—
24C23×#534—21no—
25C23×#532C2211yescharacteristic, complemented
26C23×#529C2212yes—
27C2×C23×#124C2211yescharacteristic
28C23×#501C2212yescomplemented
29C23×#497C2211yescharacteristic
30C23×#497C2211yescharacteristic, complemented
31C23×#496C2211yescharacteristic
32C23×#496C2212yescomplemented
33C23×#496C2211yescharacteristic, complemented
34C23×#495—21no—
35C23×#495—21no—
36C23×#490C2212yescomplemented
37C23×#489C2212yescomplemented
38C23×#466—22no—
39C23×#465—22no—
40C23×#442—22no—
41C23×#440—22no—
42C4×M16×C23C2211yescharacteristic, complemented
43C2×C23×#97C2212yescomplemented
44C23×#330C2212yescomplemented
45C23×#330C2212yescomplemented
46C23×#329C2211yescharacteristic
47C23×#329C2211yescharacteristic
48C23×#329C2212yescomplemented
49C23×#329C2212yescomplemented
50C23×#329C2212yescomplemented
51C2×#41×C23—21no—
52C2×#41×C23—21no—
53C22×Q8×C23D411yescharacteristic
54C22×D4×C23—21no—
55C22×D4×C23D411yescharacteristic
56C23×#137—44no—
57C23×AGL(1,ℤ/8)—44no—
58C2×Q16×C23—21no—
59C2×Q16×C23—21no—
60C2×SD16×C23—21no—
61C2×SD16×C23—21no—
62C2×SD16×C23—22no—
63C2×SD16×C23—21no—
64C2×SD16×C23—21no—
65C2×D8×C23—21no—
66C2×D8×C23—21no—
67C2×D8×C23—22no—
68C2×M16×C23—21no—
69C22×C8×C23—22no—
70C23×C4⋊Q8—21no—
71C23×C4⋊Q8—21no—
72C23×C4⋊Q8C2311yescharacteristic
73C23×#127C2311yescharacteristic
74C23×#126—22no—
75C23×#126—22no—
76C23×#126—22no—
77C23×#125C2312yes—
78C23×#124—22no—
79C23×#124—21no—
80C23×#124—22no—
81C23×#124—21no—
82C23×#124—22no—
83C23×#124—21no—
84C23×#124D411yescharacteristic
85C23×#124D411yescharacteristic
86C23×#124C2312yes—
87C23×#124C2311yescharacteristic
88C23×#124C2311yescharacteristic
89C23×#123—22no—
90C23×#123—22no—
91C23×#123—21no—
92C23×#123—21no—
93C23×#123—22no—
94C23×#123—22no—
95C23×#123—22no—
96C23×C22⋊Q8—22no—
97C23×C22⋊Q8—22no—
98C23×C22⋊Q8—22no—
99C23×C22⋊Q8—22no—
100C23×C22⋊Q8D412yes—
101C23×#121—22no—
102C23×#121—22no—
103C23×#121—22no—
104C23×#121—22no—
105C23×#121—22no—
106C23×#121D412yes—
107C23×#120—22no—
108C23×#120—21no—
109C23×#120—21no—
110C23×#120—22no—
111C4×Q8×C23D412yes—
112C4×D4×C23—22no—
113C4×D4×C23—22no—
114C4×D4×C23—22no—
115C4×D4×C23—22no—
116C4×D4×C23D412yes—
117C23×#117—21no—
118C23×#117—22no—
119C23×#117—22no—
120C2×C22⋊C4×C23—21no—
121C2×C22⋊C4×C23—22no—
122C2×C22⋊C4×C23—22no—
123C2×C42×C23C2311yescharacteristic
124C23×#107—22no—
125C23×#106—22no—
126C23×#103—22no—
127C23×#102—22no—
128C23×#102—22no—
129C23×#102—22no—
130C23×#97—22no—
131C23×#97C2312yes—
132C23×#97C2312yes—
133C23×#97C2311yescharacteristic
134C23×#97C2311yescharacteristic
135C4×C8×C23—21no—
136C24×C23—21no—
137C24×C23—21no—
138#41×C23—42no—
139#41×C23—42no—
140#41×C23—82no—
141C2×Q8×C23—21no—
142C2×Q8×C23—21no—
143C2×Q8×C23—21no—
144C2×Q8×C23—42no—
145C2×Q8×C23C2×D412yes—
146C2×Q8×C23C2×D411yescharacteristic
147C2×Q8×C23C2×D411yescharacteristic
148C2×D4×C23—22no—
149C2×D4×C23—22no—
150C2×D4×C23—22no—
151C2×D4×C23—21no—
152C2×D4×C23—22no—
153C2×D4×C23—21no—
154C2×D4×C23—22no—
155C2×D4×C23—21no—
156C2×D4×C23—42no—
157C2×D4×C23—41no—
158C2×D4×C23—82no—
159C2×D4×C23C2×D412yes—
160C2×D4×C23C2×D411yescharacteristic
161C2×D4×C23C2×D411yescharacteristic
162C22×C4×C23—21no—
163C22×C4×C23—22no—
164C22×C4×C23—22no—
165C22×C4×C23—21no—
166C22×C4×C23—22no—
167C22×C4×C23—41no—
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D412yes—
170Q16×C23—42no—
171Q16×C23—42no—
172SD16×C23—44no—
173SD16×C23—42no—
174SD16×C23—42no—
175SD16×C23—42no—
176SD16×C23—42no—
177D8×C23—44no—
178D8×C23—42no—
179D8×C23—42no—
180M16×C23—24no—
181C2×C8×C23—22no—
182C2×C8×C23—21no—
183C2×C8×C23—21no—
184C2×C8×C23—22no—
185C2×C8×C23—44no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23C2×D412yes—
195C4⋊C4×C23C2×D412yes—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—21no—
198C22⋊C4×C23—21no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—22no—
204C22⋊C4×C23—22no—
205C22⋊C4×C23—22no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—42no—
212C42×C23—21no—
213C42×C23—21no—
214C42×C23—22no—
215C42×C23C2×D411yescharacteristic
216C42×C23C2×D412yes—
217C42×C23C2411yescharacteristic
218#19072C2311yesmaximal, characteristic, direct product factor, Sylow
219C23×C23—21no—
220C23×C23—22no—
221C23×C23—21no—
222C23×C23—42no—
223C23×C23—41no—
224C23×C23—82no—
225C23×C23—81no—
226C23×C23—82no—
227C23×C23#12011yescharacteristic
228Q8×C23—41no—
229Q8×C23—41no—
230Q8×C23—42no—
231Q8×C23—41no—
232Q8×C23—41no—
233Q8×C23—81no—
234Q8×C23—81no—
235D4×C23—41no—
236D4×C23—41no—
237D4×C23—42no—
238D4×C23—41no—
239D4×C23—42no—
240D4×C23—41no—
241D4×C23—42no—
242D4×C23—44no—
243D4×C23—81no—
244D4×C23—82no—
245D4×C23—82no—
246D4×C23—82no—
247D4×C23—81no—
248D4×C23—82no—
249C2×C4×C23—22no—
250C2×C4×C23—22no—
251C2×C4×C23—21no—
252C2×C4×C23—22no—
253C2×C4×C23—22no—
254C2×C4×C23—21no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
257C2×C4×C23—41no—
258C2×C4×C23—82no—
259C2×C4×C23—82no—
260C2×C4×C23—82no—
261C2×C4×C23—81no—
262C2×C4×C23—82no—
263C2×C4×C23#12012yes—
264C2×C4×C23#12011yescharacteristic
265C2×C4×C23C22×D412yes—
266C2×C4×C23C22×D411yescharacteristic
267C8×C23—24no—
268C8×C23—22no—
269C8×C23—22no—
270C8×C23—42no—
271#3545C2×C2311yescharacteristic, complemented
272#3180C2×C2312yescomplemented
273#3153C2×C2311yescharacteristic, complemented
274#1661C2×C2312yescomplemented
275#1660C2×C2311yescharacteristic, complemented
276#1659C2×C2311yescharacteristic, complemented
277#1657C2×C2311yescharacteristic, complemented
278#1656C2×C2312yescomplemented
279#1655C2×C2311yescharacteristic, complemented
280#1524C2×C2312yescomplemented
281C2×#329C2×C2311yescharacteristic, complemented
282C22×C23—22no—
283C22×C23—42no—
284C22×C23—41no—
285C22×C23—41no—
286C22×C23—42no—
287C22×C23—42no—
288C22×C23—42no—
289C22×C23—42no—
290C22×C23—81no—
291C22×C23—82no—
292C22×C23—162no—
293C22×C23—162no—
294C22×C23#45312yes—
295C22×C23C2×#12011yescharacteristic
296C4×C23—21no—
297C4×C23—21no—
298C4×C23—22no—
299C4×C23—22no—
300C4×C23—41no—
301C4×C23—41no—
302C4×C23—42no—
303C4×C23—42no—
304C4×C23—42no—
305C4×C23—81no—
306C2×#137—21no—
307C2×AGL(1,ℤ/8)—21no—
308#563C22×C2311yescharacteristic
309#562C22×C2312yes—
310H(3,𝔽4)C22×C2311yescharacteristic
311#555—21no—
312#551—21no—
313#551—22no—
314#546—22no—
315#542—22no—
316#538—22no—
317#536—21no—
318#534—21no—
319#532C22×C2311yescharacteristic, complemented
320#529C22×C2312yes—
321C2×#124C22×C2311yescharacteristic
322#501C22×C2312yescomplemented
323#497C22×C2311yescharacteristic
324#497C22×C2311yescharacteristic, complemented
325#496C22×C2311yescharacteristic
326#496C22×C2311yescharacteristic, complemented
327#496C22×C2312yescomplemented
328#495—21no—
329#495—21no—
330#490C22×C2312yescomplemented
331#489C22×C2312yescomplemented
332#466—22no—
333#465—22no—
334#442—22no—
335#440—22no—
336C4×M16C22×C2311yescharacteristic, complemented
337C2×#97C22×C2312yescomplemented
338#330C22×C2312yescomplemented
339#330C22×C2312yescomplemented
340#329C22×C2311yescharacteristic
341#329C22×C2311yescharacteristic
342#329C22×C2312yescomplemented
343#329C22×C2312yescomplemented
344#329C22×C2312yescomplemented
345C2×C23—22no—
346C2×C23—82no—
347C2×C23—81no—
348C2×C23—81no—
349C2×C23—161no—
350C2×C23C2×#45311yescharacteristic
351C2×C23#301812yes—
352C2×#41—21no—
353C2×#41—21no—
354C22×Q8D4×C2311yescharacteristic
355C22×D4—21no—
356C22×D4D4×C2311yescharacteristic
357#137—44no—
358AGL(1,ℤ/8)—44no—
359C2×Q16—21no—
360C2×Q16—21no—
361C2×SD16—21no—
362C2×SD16—21no—
363C2×SD16—22no—
364C2×SD16—21no—
365C2×SD16—21no—
366C2×D8—21no—
367C2×D8—21no—
368C2×D8—22no—
369C2×M16—21no—
370C22×C8—22no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8C23×C2311yescharacteristic
374#127C23×C2311yescharacteristic
375#126—22no—
376#126—22no—
377#126—22no—
378#125C23×C2312yes—
379#124—21no—
380#124—22no—
381#124—21no—
382#124—22no—
383#124—22no—
384#124—21no—
385#124D4×C2311yescharacteristic
386#124D4×C2311yescharacteristic
387#124C23×C2311yescharacteristic
388#124C23×C2311yescharacteristic
389#124C23×C2312yes—
390#123—21no—
391#123—22no—
392#123—22no—
393#123—21no—
394#123—22no—
395#123—22no—
396#123—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8D4×C2312yes—
402#121—22no—
403#121—22no—
404#121—22no—
405#121—22no—
406#121—22no—
407#121D4×C2312yes—
408#120—21no—
409#120—22no—
410#120—21no—
411#120—22no—
412C4×Q8D4×C2312yes—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4D4×C2312yes—
418#117—21no—
419#117—22no—
420#117—22no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—22no—
424C2×C42C23×C2311yescharacteristic
425#107—22no—
426#106—22no—
427#103—22no—
428#102—22no—
429#102—22no—
430#102—22no—
431#97—22no—
432#97C23×C2312yes—
433#97C23×C2312yes—
434#97C23×C2311yescharacteristic
435#97C23×C2311yescharacteristic
436C4×C8—21no—
437C23#1907211yescharacteristic, direct product factor, Sylow
438C24—21no—
439C24—21no—
440#41—42no—
441#41—42no—
442#41—82no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—42no—
447C2×Q8C2×D4×C2312yes—
448C2×Q8C2×D4×C2311yescharacteristic
449C2×Q8C2×D4×C2311yescharacteristic
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—82no—
461C2×D4C2×D4×C2312yes—
462C2×D4C2×D4×C2311yescharacteristic
463C2×D4C2×D4×C2311yescharacteristic
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4C2×D4×C2311yescharacteristic
471C22×C4C2×D4×C2312yes—
472Q16—42no—
473Q16—42no—
474SD16—42no—
475SD16—42no—
476SD16—42no—
477SD16—42no—
478SD16—44no—
479D8—42no—
480D8—42no—
481D8—44no—
482M16—24no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—44no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4C2×D4×C2312yes—
497C4⋊C4C2×D4×C2312yes—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—42no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42C2×D4×C2311yescharacteristic
518C42C2×D4×C2312yes—
519C42C24×C2311yescharacteristic
520C23—21no—
521C23—22no—
522C23—21no—
523C23—42no—
524C23—41no—
525C23—82no—
526C23—81no—
527C23—82no—
528C23C23×#12011yescharacteristic
529Q8—41no—
530Q8—41no—
531Q8—42no—
532Q8—41no—
533Q8—41no—
534Q8—81no—
535Q8—81no—
536D4—41no—
537D4—41no—
538D4—42no—
539D4—41no—
540D4—42no—
541D4—41no—
542D4—42no—
543D4—44no—
544D4—81no—
545D4—82no—
546D4—81no—
547D4—82no—
548D4—82no—
549D4—82no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4C23×#12012yes—
565C2×C4C23×#12011yescharacteristic
566C2×C4C22×D4×C2312yes—
567C2×C4C22×D4×C2311yescharacteristic
568C8—24no—
569C8—22no—
570C8—22no—
571C8—42no—
572C22—22no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—82no—
582C22—162no—
583C22—162no—
584C22C23×#45312yes—
585C22C2×C23×#12011yescharacteristic
586C4—21no—
587C4—21no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—81no—
596C2—22no—
597C2—81no—
598C2—82no—
599C2—81no—
600C2—161no—
601C2C2×C23×#45311yescharacteristic
602C2C23×#301812yes—
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3545C2semidirect
2. C23×#3180C2semidirect
3. C23×#3153C2semidirect
4. C23×#1661C2semidirect
5. C23×#1660C2semidirect
6. C23×#1659C2semidirect
7. C23×#1657C2semidirect
8. C23×#1656C2semidirect
9. C23×#1655C2semidirect
10. C23×#1524C2semidirect
11. C2×C23×#329C2semidirect
25. C23×#532C22semidirect
28. C23×#501C22semidirect
30. C23×#497C22semidirect
32. C23×#496C22semidirect
33. C23×#496C22semidirect
36. C23×#490C22semidirect
37. C23×#489C22semidirect
42. C4×M16×C23C22semidirect
43. C2×C23×#97C22semidirect
44. C23×#330C22semidirect
45. C23×#330C22semidirect
48. C23×#329C22semidirect
49. C23×#329C22semidirect
50. C23×#329C22semidirect
218. #19072C23direct (N × H)
271. #3545C2×C23semidirect
272. #3180C2×C23semidirect
273. #3153C2×C23semidirect
274. #1661C2×C23semidirect
275. #1660C2×C23semidirect
276. #1659C2×C23semidirect
277. #1657C2×C23semidirect
278. #1656C2×C23semidirect
279. #1655C2×C23semidirect
280. #1524C2×C23semidirect
281. C2×#329C2×C23semidirect
319. #532C22×C23semidirect
322. #501C22×C23semidirect
324. #497C22×C23semidirect
326. #496C22×C23semidirect
327. #496C22×C23semidirect
330. #490C22×C23semidirect
331. #489C22×C23semidirect
336. C4×M16C22×C23semidirect
337. C2×#97C22×C23semidirect
338. #330C22×C23semidirect
339. #330C22×C23semidirect
342. #329C22×C23semidirect
343. #329C22×C23semidirect
344. #329C22×C23semidirect
437. C23#19072direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35452
C23×#31802
C23×#31532
C23×#16612
C23×#16602
C23×#16592
C23×#16572
C23×#16562
C23×#16552
C23×#15242
C2×C23×#3292
#1907223