Finite Group Databasebeta

← C23×#18265 #19165891

Subgroups — C23×#18265 #19165891

Order 5888.

1124 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3537C211yesmaximal, characteristic
2C23×#3092C211yesmaximal, characteristic, complemented
3C2×C23×#481C211yesmaximal, characteristic, complemented
4C23×#2501C211yesmaximal, characteristic, complemented
5C23×#1618C214yesmaximal, complemented
6C23×#1613C214yesmaximal, complemented
7C23×#1507C211yesmaximal, characteristic, complemented
8C23×#1501C211yesmaximal, characteristic, complemented
9C23×#1471C211yesmaximal, characteristic, complemented
10C23×#560—22no—
11C23×#553—21no—
12C23×#553—21no—
13D42×C23—22no—
14D42×C23—21no—
15D42×C23—21no—
16C23×#540—21no—
17C23×#538—21no—
18C23×#538C2212yes—
19C23×#535—21no—
20C23×#534—21no—
21C23×#534C2212yes—
22C23×#525C2211yescharacteristic
23C2×C23×#123—22no—
24C2×C23×#121C2211yescharacteristic
25C2×C23×#121C2211yescharacteristic
26C22×C4⋊C4×C23C2211yescharacteristic, complemented
27C23×#484C2212yescomplemented
28C23×#483—21no—
29C23×#481—22no—
30C23×#481C2214yescomplemented
31C23×#481C2212yescomplemented
32C23×#480—21no—
33C23×#428—21no—
34C2×C23×#106—21no—
35C23×#417—21no—
36C2×C23×#102—21no—
37C23×#407C2211yescharacteristic, complemented
38C2×C23×C22⋊C8C2211yescharacteristic, complemented
39C23×#399C2214yescomplemented
40C23×#390—21no—
41C23×#384—21no—
42C4×C22⋊C4×C23C2211yescharacteristic, complemented
43C23×#328C2212yescomplemented
44C23×#327C2212yescomplemented
45C23×#327C2214yescomplemented
46C23×#323C2214yescomplemented
47C2×#41×C23D411yescharacteristic
48C22×D4×C23—22no—
49C22×D4×C23—21no—
50C22×D4×C23—21no—
51C22×D4×C23—22no—
52C22×D4×C23D411yescharacteristic
53C22×D4×C23D412yes—
54C22×D4×C23D411yescharacteristic
55C23×C4×C23—21no—
56C23×C4×C23C2311yescharacteristic
57C2×M16×C23—21no—
58C22×C8×C23—21no—
59C23×#127—22no—
60C23×#127—21no—
61C23×#126—22no—
62C23×#126—22no—
63C23×#125—21no—
64C23×#124—22no—
65C23×#123—22no—
66C23×#123—21no—
67C23×#123—22no—
68C23×#123—22no—
69C23×#123—22no—
70C23×#123—48no—
71C23×C22⋊Q8D411yescharacteristic
72C23×C22⋊Q8C2311yescharacteristic
73C23×#121—22no—
74C23×#121—21no—
75C23×#121—21no—
76C23×#121—22no—
77C23×#121—22no—
78C23×#121—22no—
79C23×#121—22no—
80C23×#121—21no—
81C23×#121—21no—
82C23×#121—22no—
83C23×#121—21no—
84C23×#121—21no—
85C23×#121—21no—
86C23×#121—21no—
87C23×#121D412yes—
88C23×#121D411yescharacteristic
89C23×#121C2312yes—
90C23×#121C2311yescharacteristic
91C23×#120—22no—
92C23×#120—22no—
93C23×#120—22no—
94C23×#120—21no—
95C23×#120—22no—
96C23×#120—22no—
97C23×#120—22no—
98C4×D4×C23—22no—
99C4×D4×C23—21no—
100C4×D4×C23—21no—
101C4×D4×C23—21no—
102C4×D4×C23—21no—
103C4×D4×C23—22no—
104C23×#117—21no—
105C23×#117D411yescharacteristic
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23—21no—
108C2×C4⋊C4×C23—21no—
109C2×C4⋊C4×C23—21no—
110C2×C4⋊C4×C23—21no—
111C2×C4⋊C4×C23—24no—
112C2×C4⋊C4×C23D411yescharacteristic
113C2×C4⋊C4×C23D411yescharacteristic
114C2×C4⋊C4×C23C2312yes—
115C2×C22⋊C4×C23—21no—
116C2×C22⋊C4×C23—21no—
117C2×C22⋊C4×C23—24no—
118C2×C22⋊C4×C23C2312yes—
119C2×C42×C23—21no—
120C23×#107—22no—
121C23×#106—24no—
122C23×#106—22no—
123C23×#103—22no—
124C23×#102—24no—
125C23×#102—22no—
126C23×C22⋊C8—21no—
127C23×C22⋊C8—21no—
128C23×C22⋊C8C2312yes—
129C23×C22⋊C8C2312yes—
130C23×#95C2312yes—
131C24×C23—22no—
132C24×C23—21no—
133C24×C23C2×D411yescharacteristic
134C24×C23C2×D412yes—
135#41×C23—42no—
136C2×Q8×C23C2×D411yescharacteristic
137C2×D4×C23—22no—
138C2×D4×C23—21no—
139C2×D4×C23—21no—
140C2×D4×C23—22no—
141C2×D4×C23—22no—
142C2×D4×C23—22no—
143C2×D4×C23—22no—
144C2×D4×C23—22no—
145C2×D4×C23—22no—
146C2×D4×C23—21no—
147C2×D4×C23—22no—
148C2×D4×C23—22no—
149C2×D4×C23—21no—
150C2×D4×C23—21no—
151C2×D4×C23—21no—
152C2×D4×C23—42no—
153C2×D4×C23—42no—
154C2×D4×C23—44no—
155C2×D4×C23—82no—
156C2×D4×C23—82no—
157C2×D4×C23—84no—
158C2×D4×C23—84no—
159C2×D4×C23C2×D411yescharacteristic
160C2×D4×C23C2×D412yes—
161C22×C4×C23—22no—
162C22×C4×C23—22no—
163C22×C4×C23—22no—
164C22×C4×C23—21no—
165C22×C4×C23—21no—
166C22×C4×C23—22no—
167C22×C4×C23—42no—
168C22×C4×C23—44no—
169C22×C4×C23—41no—
170C22×C4×C23—41no—
171C22×C4×C23C2×D411yescharacteristic
172C22×C4×C23C2×D411yescharacteristic
173C22×C4×C23C2×D411yescharacteristic
174C22×C4×C23C2×D412yes—
175C22×C4×C23C2×D411yescharacteristic
176C22×C4×C23C2×D411yescharacteristic
177C22×C4×C23C2411yescharacteristic
178M16×C23—42no—
179C2×C8×C23—22no—
180C2×C8×C23—22no—
181C2×C8×C23—42no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—21no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—21no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—48no—
192C4⋊C4×C23—41no—
193C4⋊C4×C23—41no—
194C4⋊C4×C23C2×D412yes—
195C4⋊C4×C23C2×D411yescharacteristic
196C4⋊C4×C23C2×D411yescharacteristic
197C22⋊C4×C23—24no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—21no—
204C22⋊C4×C23—21no—
205C22⋊C4×C23—21no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—21no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—48no—
213C42×C23—22no—
214C42×C23—21no—
215C42×C23—42no—
216#18265C2311yesmaximal, characteristic, direct product factor, Sylow
217C23×C23—22no—
218C23×C23—22no—
219C23×C23—22no—
220C23×C23—22no—
221C23×C23—21no—
222C23×C23—24no—
223C23×C23—41no—
224C23×C23—42no—
225C23×C23—42no—
226C23×C23—42no—
227C23×C23—81no—
228C23×C23—81no—
229C23×C23—82no—
230C23×C23—82no—
231C23×C23—82no—
232C23×C23#12011yescharacteristic
233C23×C23#12011yescharacteristic
234C23×C23C22×D411yescharacteristic
235Q8×C23—41no—
236D4×C23—42no—
237D4×C23—41no—
238D4×C23—44no—
239D4×C23—42no—
240D4×C23—44no—
241D4×C23—42no—
242D4×C23—42no—
243D4×C23—42no—
244D4×C23—81no—
245D4×C23—82no—
246D4×C23—82no—
247D4×C23—81no—
248D4×C23—82no—
249D4×C23—82no—
250D4×C23—82no—
251D4×C23—82no—
252D4×C23—82no—
253C2×C4×C23—22no—
254C2×C4×C23—22no—
255C2×C4×C23—22no—
256C2×C4×C23—24no—
257C2×C4×C23—21no—
258C2×C4×C23—22no—
259C2×C4×C23—22no—
260C2×C4×C23—21no—
261C2×C4×C23—21no—
262C2×C4×C23—41no—
263C2×C4×C23—41no—
264C2×C4×C23—41no—
265C2×C4×C23—42no—
266C2×C4×C23—42no—
267C2×C4×C23—42no—
268C2×C4×C23—42no—
269C2×C4×C23—44no—
270C2×C4×C23—42no—
271C2×C4×C23—81no—
272C2×C4×C23—82no—
273C2×C4×C23—82no—
274C2×C4×C23—82no—
275C2×C4×C23—82no—
276C2×C4×C23#12011yescharacteristic
277C2×C4×C23#12011yescharacteristic
278C2×C4×C23C22×D411yescharacteristic
279C2×C4×C23C22×D411yescharacteristic
280C8×C23—42no—
281C8×C23—42no—
282#3537C2×C2311yescharacteristic
283#3092C2×C2311yescharacteristic, complemented
284C2×#481C2×C2311yescharacteristic, complemented
285#2501C2×C2311yescharacteristic, complemented
286#1618C2×C2314yescomplemented
287#1613C2×C2314yescomplemented
288#1507C2×C2311yescharacteristic, complemented
289#1501C2×C2311yescharacteristic, complemented
290#1471C2×C2311yescharacteristic, complemented
291C22×C23—21no—
292C22×C23—21no—
293C22×C23—21no—
294C22×C23—21no—
295C22×C23—21no—
296C22×C23—21no—
297C22×C23—21no—
298C22×C23—44no—
299C22×C23—42no—
300C22×C23—42no—
301C22×C23—41no—
302C22×C23—42no—
303C22×C23—42no—
304C22×C23—41no—
305C22×C23—41no—
306C22×C23—42no—
307C22×C23—42no—
308C22×C23—84no—
309C22×C23—162no—
310C22×C23—164no—
311C22×C23#45312yes—
312C22×C23C2×#12011yescharacteristic
313C22×C23#57712yes—
314C4×C23—21no—
315C4×C23—21no—
316C4×C23—21no—
317C4×C23—44no—
318C4×C23—42no—
319C4×C23—41no—
320C4×C23—42no—
321C4×C23—41no—
322C4×C23—41no—
323C4×C23—82no—
324C4×C23#45812yes—
325#560—22no—
326#553—21no—
327#553—21no—
328D42—22no—
329D42—21no—
330D42—21no—
331#540—21no—
332#538—21no—
333#538C22×C2312yes—
334#535—21no—
335#534—21no—
336#534C22×C2312yes—
337#525C22×C2311yescharacteristic
338C2×#123—22no—
339C2×#121C22×C2311yescharacteristic
340C2×#121C22×C2311yescharacteristic
341C22×C4⋊C4C22×C2311yescharacteristic, complemented
342#484C22×C2312yescomplemented
343#483—21no—
344#481—22no—
345#481C22×C2314yescomplemented
346#481C22×C2312yescomplemented
347#480—21no—
348#428—21no—
349C2×#106—21no—
350#417—21no—
351C2×#102—21no—
352#407C22×C2311yescharacteristic, complemented
353C2×C22⋊C8C22×C2311yescharacteristic, complemented
354#399C22×C2314yescomplemented
355#390—21no—
356#384—21no—
357C4×C22⋊C4C22×C2311yescharacteristic, complemented
358#328C22×C2312yescomplemented
359#327C22×C2312yescomplemented
360#327C22×C2314yescomplemented
361#323C22×C2314yescomplemented
362C2×C23—22no—
363C2×C23—22no—
364C2×C23—41no—
365C2×C23—82no—
366C2×C23—82no—
367C2×C23—81no—
368C2×C23#300911yescharacteristic
369C2×C23C2×#45311yescharacteristic
370C2×C23C2×#45811yescharacteristic
371C2×#41D4×C2311yescharacteristic
372C22×D4—22no—
373C22×D4—21no—
374C22×D4—21no—
375C22×D4—22no—
376C22×D4D4×C2312yes—
377C22×D4D4×C2311yescharacteristic
378C22×D4D4×C2311yescharacteristic
379C23×C4—21no—
380C23×C4C23×C2311yescharacteristic
381C2×M16—21no—
382C22×C8—21no—
383#127—22no—
384#127—21no—
385#126—22no—
386#126—22no—
387#125—21no—
388#124—22no—
389#123—22no—
390#123—21no—
391#123—22no—
392#123—22no—
393#123—22no—
394#123—48no—
395C22⋊Q8D4×C2311yescharacteristic
396C22⋊Q8C23×C2311yescharacteristic
397#121—22no—
398#121—21no—
399#121—21no—
400#121—22no—
401#121—22no—
402#121—22no—
403#121—22no—
404#121—21no—
405#121—21no—
406#121—21no—
407#121—21no—
408#121—21no—
409#121—21no—
410#121—22no—
411#121D4×C2312yes—
412#121D4×C2311yescharacteristic
413#121C23×C2312yes—
414#121C23×C2311yescharacteristic
415#120—22no—
416#120—22no—
417#120—22no—
418#120—21no—
419#120—22no—
420#120—22no—
421#120—22no—
422C4×D4—22no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4—22no—
428#117—21no—
429#117D4×C2311yescharacteristic
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—24no—
436C2×C4⋊C4D4×C2311yescharacteristic
437C2×C4⋊C4D4×C2311yescharacteristic
438C2×C4⋊C4C23×C2312yes—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—24no—
442C2×C22⋊C4C23×C2312yes—
443C2×C42—21no—
444#107—22no—
445#106—24no—
446#106—22no—
447#103—22no—
448#102—24no—
449#102—22no—
450C22⋊C8—21no—
451C22⋊C8—21no—
452C22⋊C8C23×C2312yes—
453C22⋊C8C23×C2312yes—
454#95C23×C2312yes—
455C23#1826511yescharacteristic, direct product factor, Sylow
456C24—22no—
457C24—21no—
458C24C2×D4×C2311yescharacteristic
459C24C2×D4×C2312yes—
460#41—42no—
461C2×Q8C2×D4×C2311yescharacteristic
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—44no—
480C2×D4—82no—
481C2×D4—82no—
482C2×D4—84no—
483C2×D4—84no—
484C2×D4C2×D4×C2311yescharacteristic
485C2×D4C2×D4×C2312yes—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4C2×D4×C2311yescharacteristic
497C22×C4C2×D4×C2311yescharacteristic
498C22×C4C2×D4×C2311yescharacteristic
499C22×C4C2×D4×C2312yes—
500C22×C4C2×D4×C2311yescharacteristic
501C22×C4C2×D4×C2311yescharacteristic
502C22×C4C24×C2311yescharacteristic
503M16—42no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—42no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—48no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4C2×D4×C2312yes—
520C4⋊C4C2×D4×C2311yescharacteristic
521C4⋊C4C2×D4×C2311yescharacteristic
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—22no—
532C22⋊C4—21no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—48no—
538C42—22no—
539C42—21no—
540C42—42no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—21no—
546C23—24no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—81no—
552C23—81no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23C23×#12011yescharacteristic
557C23C23×#12011yescharacteristic
558C23C22×D4×C2311yescharacteristic
559Q8—41no—
560D4—42no—
561D4—41no—
562D4—44no—
563D4—42no—
564D4—44no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—81no—
569D4—82no—
570D4—82no—
571D4—81no—
572D4—82no—
573D4—82no—
574D4—82no—
575D4—82no—
576D4—82no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4C23×#12011yescharacteristic
601C2×C4C23×#12011yescharacteristic
602C2×C4C22×D4×C2311yescharacteristic
603C2×C4C22×D4×C2311yescharacteristic
604C8—42no—
605C8—42no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—84no—
624C22—162no—
625C22—164no—
626C22C23×#45312yes—
627C22C2×C23×#12011yescharacteristic
628C22C23×#57712yes—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—44no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—82no—
639C4C23×#45812yes—
640C2—22no—
641C2—22no—
642C2—41no—
643C2—82no—
644C2—82no—
645C2—81no—
646C2C23×#300911yescharacteristic
647C2C2×C23×#45311yescharacteristic
648C2C2×C23×#45811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3092C2semidirect
3. C2×C23×#481C2semidirect
4. C23×#2501C2semidirect
5. C23×#1618C2semidirect
6. C23×#1613C2semidirect
7. C23×#1507C2semidirect
8. C23×#1501C2semidirect
9. C23×#1471C2semidirect
26. C22×C4⋊C4×C23C22semidirect
27. C23×#484C22semidirect
30. C23×#481C22semidirect
31. C23×#481C22semidirect
37. C23×#407C22semidirect
38. C2×C23×C22⋊C8C22semidirect
39. C23×#399C22semidirect
42. C4×C22⋊C4×C23C22semidirect
43. C23×#328C22semidirect
44. C23×#327C22semidirect
45. C23×#327C22semidirect
46. C23×#323C22semidirect
216. #18265C23direct (N × H)
283. #3092C2×C23semidirect
284. C2×#481C2×C23semidirect
285. #2501C2×C23semidirect
286. #1618C2×C23semidirect
287. #1613C2×C23semidirect
288. #1507C2×C23semidirect
289. #1501C2×C23semidirect
290. #1471C2×C23semidirect
341. C22×C4⋊C4C22×C23semidirect
342. #484C22×C23semidirect
345. #481C22×C23semidirect
346. #481C22×C23semidirect
352. #407C22×C23semidirect
353. C2×C22⋊C8C22×C23semidirect
354. #399C22×C23semidirect
357. C4×C22⋊C4C22×C23semidirect
358. #328C22×C23semidirect
359. #327C22×C23semidirect
360. #327C22×C23semidirect
361. #323C22×C23semidirect
455. C23#18265direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35372
C23×#30922
C2×C23×#4812
C23×#25012
C23×#16182
C23×#16132
C23×#15072
C23×#15012
C23×#14712
#1826523