Finite Group Databasebeta

← C23×#24082 #19171708

Subgroups — C23×#24082 #19171708

Order 5888.

1130 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3494C211yesmaximal, characteristic
2C23×#3092C211yesmaximal, characteristic, complemented
3C2×C23×#428C211yesmaximal, characteristic, complemented
4C23×#2501C211yesmaximal, characteristic, complemented
5C23×#2060C212yesmaximal, complemented
6C23×#2037C212yesmaximal, complemented
7C23×#1900C211yesmaximal, characteristic, complemented
8C23×#1871C211yesmaximal, characteristic, complemented
9C23×#1859C212yesmaximal, complemented
10C23×#1856C212yesmaximal, complemented
11C23×#1826C211yesmaximal, characteristic, complemented
12C22×M16×C23C2211yescharacteristic
13C23×#555—21no—
14C23×#553—21no—
15C23×#553—21no—
16C23×#551—22no—
17C23×#547—21no—
18C23×#546—22no—
19C23×#544C2211yescharacteristic
20C23×#542—21no—
21C23×#540—21no—
22C23×#535C2211yescharacteristic
23C23×#532—21no—
24C23×#529—21no—
25C23×#525C2211yescharacteristic
26C2×C23×#123—22no—
27C2×C23×#121C2211yescharacteristic
28C23×#520C2211yescharacteristic
29C23×#518C2211yescharacteristic
30C2×C23×#117C2211yescharacteristic
31C22×C4⋊C4×C23C2211yescharacteristic
32C23×#484—21no—
33C23×#483—21no—
34C23×#481—21no—
35C23×#480—21no—
36C23×#428—21no—
37C23×#428—21no—
38C23×#428C2212yes—
39C23×#428C2212yes—
40C23×#428C2212yes—
41C2×C23×#107—21no—
42C2×C23×#106—21no—
43C2×C23×#104C2212yes—
44C23×#417C2212yes—
45C2×C23×#100C2212yes—
46C23×#407C2211yescharacteristic
47C23×#399—22no—
48C23×#390C2212yes—
49C23×#384C2212yes—
50C4×C22⋊C4×C23C2211yescharacteristic
51C23×#343C2212yes—
52C23×#339C2212yes—
53C23×#339C2212yes—
54C23×#339C2212yes—
55C2×#41×C23—22no—
56C2×#41×C23C2311yescharacteristic
57C22×D4×C23D412yes—
58C22×D4×C23C2311yescharacteristic
59C23×C4×C23—21no—
60C23×C4×C23C2311yescharacteristic
61C2×M16×C23—21no—
62C2×M16×C23—24no—
63C2×M16×C23C2312yes—
64C2×M16×C23C2312yes—
65C22×C8×C23—21no—
66C23×C4⋊Q8—21no—
67C23×C4⋊Q8—21no—
68C23×#127—21no—
69C23×#126—22no—
70C23×#126—22no—
71C23×#125—21no—
72C23×#125—21no—
73C23×#124—22no—
74C23×#124—22no—
75C23×#124—21no—
76C23×#123—22no—
77C23×#123—22no—
78C23×#123—22no—
79C23×#123—22no—
80C23×#123—21no—
81C23×#123—48no—
82C23×C22⋊Q8—22no—
83C23×C22⋊Q8—22no—
84C23×C22⋊Q8D411yescharacteristic
85C23×C22⋊Q8D411yescharacteristic
86C23×#121—22no—
87C23×#121—22no—
88C23×#121—22no—
89C23×#121—21no—
90C23×#121—21no—
91C23×#121—21no—
92C23×#121—21no—
93C23×#121—21no—
94C23×#121—21no—
95C23×#121D411yescharacteristic
96C23×#121D411yescharacteristic
97C23×#120—22no—
98C23×#120—21no—
99C4×Q8×C23—21no—
100C4×D4×C23—22no—
101C4×D4×C23—22no—
102C4×D4×C23—21no—
103C4×D4×C23—21no—
104C4×D4×C23—22no—
105C4×D4×C23—21no—
106C4×D4×C23—21no—
107C4×D4×C23—21no—
108C4×D4×C23—21no—
109C4×D4×C23—21no—
110C23×#117—22no—
111C23×#117—21no—
112C23×#117—21no—
113C23×#117—21no—
114C23×#117D411yescharacteristic
115C23×#117D411yescharacteristic
116C23×#117C2311yescharacteristic
117C23×#117C2311yescharacteristic
118C23×#117C2311yescharacteristic
119C2×C4⋊C4×C23—21no—
120C2×C4⋊C4×C23—21no—
121C2×C4⋊C4×C23—21no—
122C2×C4⋊C4×C23—21no—
123C2×C4⋊C4×C23—21no—
124C2×C4⋊C4×C23—24no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23D411yescharacteristic
127C2×C4⋊C4×C23D411yescharacteristic
128C2×C4⋊C4×C23C2312yes—
129C2×C4⋊C4×C23C2311yescharacteristic
130C2×C22⋊C4×C23—22no—
131C2×C22⋊C4×C23—21no—
132C2×C22⋊C4×C23—21no—
133C2×C22⋊C4×C23—24no—
134C2×C22⋊C4×C23D412yes—
135C2×C42×C23—21no—
136C2×C42×C23C2312yes—
137C23×#107—22no—
138C23×#107—22no—
139C23×#107—22no—
140C23×#106—22no—
141C23×#106—22no—
142C23×#106—22no—
143C23×#104—24no—
144C23×#103—22no—
145C23×#102—22no—
146C23×#100—24no—
147C23×C22⋊C8—21no—
148C23×C22⋊C8—21no—
149C23×#95—21no—
150C24×C23C2×D411yescharacteristic
151C24×C23C2×D412yes—
152#41×C23—22no—
153#41×C23—22no—
154#41×C23—84no—
155C2×Q8×C23—22no—
156C2×Q8×C23C2×D411yescharacteristic
157C2×D4×C23—21no—
158C2×D4×C23—22no—
159C2×D4×C23—21no—
160C2×D4×C23—22no—
161C2×D4×C23—22no—
162C2×D4×C23—21no—
163C2×D4×C23—22no—
164C2×D4×C23—21no—
165C2×D4×C23—21no—
166C2×D4×C23—22no—
167C2×D4×C23—22no—
168C2×D4×C23—44no—
169C2×D4×C23C2×D411yescharacteristic
170C22×C4×C23—22no—
171C22×C4×C23—22no—
172C22×C4×C23—21no—
173C22×C4×C23—22no—
174C22×C4×C23—22no—
175C22×C4×C23—22no—
176C22×C4×C23—22no—
177C22×C4×C23—21no—
178C22×C4×C23—21no—
179C22×C4×C23—22no—
180C22×C4×C23—44no—
181C22×C4×C23—41no—
182C22×C4×C23—41no—
183C22×C4×C23C2×D411yescharacteristic
184C22×C4×C23C2×D411yescharacteristic
185C22×C4×C23C2×D411yescharacteristic
186C22×C4×C23C2×D411yescharacteristic
187C22×C4×C23C2×D411yescharacteristic
188C22×C4×C23C2×D412yes—
189C22×C4×C23C2411yescharacteristic
190M16×C23—24no—
191M16×C23—22no—
192M16×C23—22no—
193M16×C23—22no—
194M16×C23#4114yescomplemented
195C2×C8×C23—24no—
196C2×C8×C23—22no—
197C2×C8×C23—22no—
198C4⋊C4×C23—22no—
199C4⋊C4×C23—22no—
200C4⋊C4×C23—22no—
201C4⋊C4×C23—22no—
202C4⋊C4×C23—21no—
203C4⋊C4×C23—21no—
204C4⋊C4×C23—21no—
205C4⋊C4×C23—21no—
206C4⋊C4×C23—21no—
207C4⋊C4×C23—21no—
208C4⋊C4×C23—22no—
209C4⋊C4×C23—22no—
210C4⋊C4×C23—48no—
211C4⋊C4×C23—41no—
212C4⋊C4×C23—41no—
213C4⋊C4×C23C2×D411yescharacteristic
214C4⋊C4×C23C2×D411yescharacteristic
215C4⋊C4×C23C2×D411yescharacteristic
216C4⋊C4×C23C2×D411yescharacteristic
217C4⋊C4×C23C2×D411yescharacteristic
218C4⋊C4×C23C2×D411yescharacteristic
219C22⋊C4×C23—22no—
220C22⋊C4×C23—22no—
221C22⋊C4×C23—24no—
222C22⋊C4×C23—22no—
223C22⋊C4×C23—22no—
224C22⋊C4×C23—22no—
225C22⋊C4×C23—21no—
226C22⋊C4×C23—21no—
227C22⋊C4×C23—21no—
228C22⋊C4×C23—21no—
229C22⋊C4×C23—21no—
230C22⋊C4×C23—21no—
231C22⋊C4×C23—21no—
232C22⋊C4×C23—22no—
233C22⋊C4×C23—22no—
234C22⋊C4×C23—21no—
235C22⋊C4×C23—21no—
236C22⋊C4×C23—48no—
237C42×C23—22no—
238C42×C23—21no—
239C42×C23—21no—
240C42×C23—21no—
241C42×C23—21no—
242C42×C23—21no—
243C42×C23—22no—
244C42×C23—22no—
245#24082C2311yesmaximal, characteristic, direct product factor, Sylow
246C23×C23—22no—
247C23×C23—22no—
248C23×C23—22no—
249C23×C23—21no—
250C23×C23—24no—
251C23×C23—41no—
252C23×C23—42no—
253C23×C23—42no—
254C23×C23#12011yescharacteristic
255C23×C23#12011yescharacteristic
256C23×C23C22×D411yescharacteristic
257Q8×C23—22no—
258Q8×C23—82no—
259D4×C23—22no—
260D4×C23—42no—
261D4×C23—44no—
262D4×C23—42no—
263D4×C23—41no—
264D4×C23—41no—
265D4×C23—42no—
266D4×C23—81no—
267D4×C23—82no—
268D4×C23—82no—
269D4×C23—82no—
270C2×C4×C23—22no—
271C2×C4×C23—21no—
272C2×C4×C23—21no—
273C2×C4×C23—21no—
274C2×C4×C23—21no—
275C2×C4×C23—22no—
276C2×C4×C23—22no—
277C2×C4×C23—24no—
278C2×C4×C23—21no—
279C2×C4×C23—21no—
280C2×C4×C23—21no—
281C2×C4×C23—21no—
282C2×C4×C23—21no—
283C2×C4×C23—21no—
284C2×C4×C23—21no—
285C2×C4×C23—22no—
286C2×C4×C23—42no—
287C2×C4×C23—42no—
288C2×C4×C23—42no—
289C2×C4×C23—41no—
290C2×C4×C23—41no—
291C2×C4×C23—42no—
292C2×C4×C23—44no—
293C2×C4×C23—42no—
294C2×C4×C23—81no—
295C2×C4×C23—82no—
296C2×C4×C23—82no—
297C2×C4×C23—82no—
298C2×C4×C23—82no—
299C2×C4×C23—82no—
300C2×C4×C23#12011yescharacteristic
301C2×C4×C23#12011yescharacteristic
302C2×C4×C23C22×D411yescharacteristic
303C2×C4×C23C22×D411yescharacteristic
304C2×C4×C23C2×#4112yes—
305C2×C4×C232−1+412yes—
306C8×C23—24no—
307C8×C23—42no—
308#3494C2×C2311yescharacteristic
309#3092C2×C2311yescharacteristic, complemented
310C2×#428C2×C2311yescharacteristic, complemented
311#2501C2×C2311yescharacteristic, complemented
312#2060C2×C2312yescomplemented
313#2037C2×C2312yescomplemented
314#1900C2×C2311yescharacteristic, complemented
315#1871C2×C2311yescharacteristic, complemented
316#1859C2×C2312yescomplemented
317#1856C2×C2312yescomplemented
318#1826C2×C2311yescharacteristic, complemented
319C22×C23—21no—
320C22×C23—21no—
321C22×C23—21no—
322C22×C23—21no—
323C22×C23—21no—
324C22×C23—21no—
325C22×C23—21no—
326C22×C23—44no—
327C22×C23—41no—
328C22×C23—42no—
329C22×C23—41no—
330C22×C23—42no—
331C22×C23—42no—
332C22×C23—41no—
333C22×C23—84no—
334C22×C23#45312yes—
335C22×C23C2×#12011yescharacteristic
336C22×C23#54712yes—
337C4×C23—21no—
338C4×C23—21no—
339C4×C23—44no—
340C4×C23—41no—
341C4×C23—41no—
342C4×C23—41no—
343C4×C23—42no—
344C4×C23—41no—
345C4×C23—41no—
346C4×C23—41no—
347C4×C23—41no—
348C4×C23—82no—
349C4×C23#52511yescharacteristic
350C4×C23#53611yescharacteristic
351C4×C23#54712yes—
352C22×M16C22×C2311yescharacteristic
353#555—21no—
354#553—21no—
355#553—21no—
356#551—22no—
357#547—21no—
358#546—22no—
359#544C22×C2311yescharacteristic
360#542—21no—
361#540—21no—
362#535C22×C2311yescharacteristic
363#532—21no—
364#529—21no—
365#525C22×C2311yescharacteristic
366C2×#123—22no—
367C2×#121C22×C2311yescharacteristic
368#520C22×C2311yescharacteristic
369#518C22×C2311yescharacteristic
370C2×#117C22×C2311yescharacteristic
371C22×C4⋊C4C22×C2311yescharacteristic
372#484—21no—
373#483—21no—
374#481—21no—
375#480—21no—
376#428—21no—
377#428—21no—
378#428C22×C2312yes—
379#428C22×C2312yes—
380#428C22×C2312yes—
381C2×#107—21no—
382C2×#106—21no—
383C2×#104C22×C2312yes—
384#417C22×C2312yes—
385C2×#100C22×C2312yes—
386#407C22×C2311yescharacteristic
387#399—22no—
388#390C22×C2312yes—
389#384C22×C2312yes—
390C4×C22⋊C4C22×C2311yescharacteristic
391#343C22×C2312yes—
392#339C22×C2312yes—
393#339C22×C2312yes—
394#339C22×C2312yes—
395C2×C23—22no—
396C2×C23—22no—
397C2×C23—41no—
398C2×C23—81no—
399C2×C23—82no—
400C2×C23#240611yescharacteristic
401C2×C23C2×#45311yescharacteristic
402C2×C23#302211yescharacteristic
403C2×#41—22no—
404C2×#41C23×C2311yescharacteristic
405C22×D4D4×C2312yes—
406C22×D4C23×C2311yescharacteristic
407C23×C4—21no—
408C23×C4C23×C2311yescharacteristic
409C2×M16—21no—
410C2×M16—24no—
411C2×M16C23×C2312yes—
412C2×M16C23×C2312yes—
413C22×C8—21no—
414C4⋊Q8—21no—
415C4⋊Q8—21no—
416#127—21no—
417#126—22no—
418#126—22no—
419#125—21no—
420#125—21no—
421#124—22no—
422#124—22no—
423#124—21no—
424#123—22no—
425#123—22no—
426#123—22no—
427#123—22no—
428#123—21no—
429#123—48no—
430C22⋊Q8—22no—
431C22⋊Q8—22no—
432C22⋊Q8D4×C2311yescharacteristic
433C22⋊Q8D4×C2311yescharacteristic
434#121—22no—
435#121—22no—
436#121—22no—
437#121—21no—
438#121—21no—
439#121—21no—
440#121—21no—
441#121—21no—
442#121—21no—
443#121D4×C2311yescharacteristic
444#121D4×C2311yescharacteristic
445#120—21no—
446#120—22no—
447C4×Q8—21no—
448C4×D4—22no—
449C4×D4—22no—
450C4×D4—21no—
451C4×D4—22no—
452C4×D4—21no—
453C4×D4—21no—
454C4×D4—21no—
455C4×D4—21no—
456C4×D4—21no—
457C4×D4—21no—
458#117—22no—
459#117—21no—
460#117—21no—
461#117—21no—
462#117D4×C2311yescharacteristic
463#117D4×C2311yescharacteristic
464#117C23×C2311yescharacteristic
465#117C23×C2311yescharacteristic
466#117C23×C2311yescharacteristic
467C2×C4⋊C4—21no—
468C2×C4⋊C4—21no—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—21no—
473C2×C4⋊C4—24no—
474C2×C4⋊C4D4×C2311yescharacteristic
475C2×C4⋊C4D4×C2311yescharacteristic
476C2×C4⋊C4C23×C2312yes—
477C2×C4⋊C4C23×C2311yescharacteristic
478C2×C22⋊C4—21no—
479C2×C22⋊C4—22no—
480C2×C22⋊C4—21no—
481C2×C22⋊C4—24no—
482C2×C22⋊C4D4×C2312yes—
483C2×C42—21no—
484C2×C42C23×C2312yes—
485#107—22no—
486#107—22no—
487#107—22no—
488#106—22no—
489#106—22no—
490#106—22no—
491#104—24no—
492#103—22no—
493#102—22no—
494#100—24no—
495C22⋊C8—21no—
496C22⋊C8—21no—
497#95—21no—
498C23#2408211yescharacteristic, direct product factor, Sylow
499C24C2×D4×C2311yescharacteristic
500C24C2×D4×C2312yes—
501#41—22no—
502#41—22no—
503#41—84no—
504C2×Q8—22no—
505C2×Q8C2×D4×C2311yescharacteristic
506C2×D4—21no—
507C2×D4—22no—
508C2×D4—21no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—21no—
512C2×D4—22no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4—22no—
516C2×D4—22no—
517C2×D4—44no—
518C2×D4C2×D4×C2311yescharacteristic
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—44no—
530C22×C4—41no—
531C22×C4—41no—
532C22×C4C2×D4×C2311yescharacteristic
533C22×C4C2×D4×C2311yescharacteristic
534C22×C4C2×D4×C2311yescharacteristic
535C22×C4C2×D4×C2311yescharacteristic
536C22×C4C2×D4×C2311yescharacteristic
537C22×C4C2×D4×C2312yes—
538C22×C4C24×C2311yescharacteristic
539M16—24no—
540M16—22no—
541M16—22no—
542M16—22no—
543M16#41×C2314yescomplemented
544C2×C8—24no—
545C2×C8—22no—
546C2×C8—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—48no—
560C4⋊C4—41no—
561C4⋊C4—41no—
562C4⋊C4C2×D4×C2311yescharacteristic
563C4⋊C4C2×D4×C2311yescharacteristic
564C4⋊C4C2×D4×C2311yescharacteristic
565C4⋊C4C2×D4×C2311yescharacteristic
566C4⋊C4C2×D4×C2311yescharacteristic
567C4⋊C4C2×D4×C2311yescharacteristic
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—24no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—21no—
575C22⋊C4—21no—
576C22⋊C4—21no—
577C22⋊C4—21no—
578C22⋊C4—21no—
579C22⋊C4—21no—
580C22⋊C4—21no—
581C22⋊C4—22no—
582C22⋊C4—22no—
583C22⋊C4—21no—
584C22⋊C4—21no—
585C22⋊C4—48no—
586C42—22no—
587C42—21no—
588C42—21no—
589C42—21no—
590C42—21no—
591C42—21no—
592C42—22no—
593C42—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—21no—
598C23—24no—
599C23—41no—
600C23—42no—
601C23—42no—
602C23C23×#12011yescharacteristic
603C23C23×#12011yescharacteristic
604C23C22×D4×C2311yescharacteristic
605Q8—22no—
606Q8—82no—
607D4—22no—
608D4—42no—
609D4—44no—
610D4—42no—
611D4—42no—
612D4—41no—
613D4—41no—
614D4—82no—
615D4—81no—
616D4—82no—
617D4—82no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—24no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—81no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4C23×#12011yescharacteristic
649C2×C4C23×#12011yescharacteristic
650C2×C4C22×D4×C2311yescharacteristic
651C2×C4C22×D4×C2311yescharacteristic
652C2×C4C2×#41×C2312yes—
653C2×C4C23×2−1+412yes—
654C8—24no—
655C8—42no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—44no—
664C22—41no—
665C22—42no—
666C22—41no—
667C22—42no—
668C22—42no—
669C22—41no—
670C22—84no—
671C22C23×#45312yes—
672C22C2×C23×#12011yescharacteristic
673C22C23×#54712yes—
674C4—21no—
675C4—21no—
676C4—44no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—42no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—82no—
686C4C23×#52511yescharacteristic
687C4C23×#53611yescharacteristic
688C4C23×#54712yes—
689C2—22no—
690C2—22no—
691C2—41no—
692C2—82no—
693C2—81no—
694C2C23×#240611yescharacteristic
695C2C2×C23×#45311yescharacteristic
696C2C23×#302211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3494C2
LCSG ▷ C22×C4 ▷ C2 ▷ 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×#428C2semidirect
4. C23×#2501C2semidirect
5. C23×#2060C2semidirect
6. C23×#2037C2semidirect
7. C23×#1900C2semidirect
8. C23×#1871C2semidirect
9. C23×#1859C2semidirect
10. C23×#1856C2semidirect
11. C23×#1826C2semidirect
194. M16×C23#41semidirect
245. #24082C23direct (N × H)
309. #3092C2×C23semidirect
310. C2×#428C2×C23semidirect
311. #2501C2×C23semidirect
312. #2060C2×C23semidirect
313. #2037C2×C23semidirect
314. #1900C2×C23semidirect
315. #1871C2×C23semidirect
316. #1859C2×C23semidirect
317. #1856C2×C23semidirect
318. #1826C2×C23semidirect
498. C23#24082direct (N × H)
543. M16#41×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#34942
C23×#30922
C2×C23×#4282
C23×#25012
C23×#20602
C23×#20372
C23×#19002
C23×#18712
C23×#18592
C23×#18562
C23×#18262
#2408223