Finite Group Databasebeta

← C23×#18276 #19165902

Subgroups — C23×#18276 #19165902

Order 5888.

966 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3559C211yesmaximal, characteristic, complemented
2C23×#3109C211yesmaximal, characteristic, complemented
3C23×#3105C211yesmaximal, characteristic, complemented
4C23×#2562C211yesmaximal, characteristic, complemented
5C23×#1616C212yesmaximal, complemented
6C23×#1615C212yesmaximal, complemented
7C23×#1598C212yesmaximal, complemented
8C23×#1597C212yesmaximal, complemented
9C23×#1509C211yesmaximal, characteristic, complemented
10C23×#1502C211yesmaximal, characteristic, complemented
11C23×#1470C211yesmaximal, characteristic, complemented
12C23×#563—21no—
13C23×#562—21no—
14C23×#554—21no—
15C23×#553—21no—
16C23×#553—21no—
17C23×#547—21no—
18C23×#547—22no—
19C23×#544—21no—
20C23×#540—21no—
21C23×#539—22no—
22C23×#539C2212yes—
23C23×#537C2212yes—
24C23×#532C2211yescharacteristic
25C23×#529C2211yescharacteristic
26C2×C23×#125C2211yescharacteristic
27C2×C4×D4×C23C2211yescharacteristic
28C23×#484C2211yescharacteristic
29C23×#481C2211yescharacteristic
30C23×#481C2212yes—
31C23×#478—21no—
32C23×#476—21no—
33C23×#463—21no—
34C23×#459—21no—
35C23×#450C2211yescharacteristic
36C23×#447C2211yescharacteristic
37C23×#447C2212yes—
38C23×#442—22no—
39C4×SD16×C23—21no—
40C4×D8×C23—21no—
41C23×#432C2211yescharacteristic
42C23×#431C2211yescharacteristic
43C23×#400C2212yes—
44C23×#394C2212yes—
45C23×#388—22no—
46C23×#380—21no—
47C23×#380—21no—
48C4×C4⋊C4×C23C2211yescharacteristic
49C23×#328C2212yes—
50C23×#327C2212yes—
51C23×#327C2212yes—
52C23×#327C2212yes—
53C23×#324C2212yes—
54C23×#324C2212yes—
55C2×#41×C23—21no—
56C2×#41×C23—22no—
57C22×D4×C23C2311yescharacteristic
58C23×C4×C23—21no—
59C2×SD16×C23—21no—
60C2×D8×C23—21no—
61C2×D8×C23—22no—
62C23×C4⋊Q8—21no—
63C23×C4⋊Q8—22no—
64C23×C4⋊Q8D411yescharacteristic
65C23×#127D411yescharacteristic
66C23×#126—22no—
67C23×#126—22no—
68C23×#126—21no—
69C23×#125—21no—
70C23×#125—22no—
71C23×#125—21no—
72C23×#125—21no—
73C23×#125—21no—
74C23×#125—21no—
75C23×#125D411yescharacteristic
76C23×#124—21no—
77C23×#124D411yescharacteristic
78C23×#123—22no—
79C23×#123—22no—
80C23×#123—22no—
81C23×#123—22no—
82C23×#123—22no—
83C23×#123—21no—
84C23×#123—22no—
85C23×C22⋊Q8—22no—
86C23×C22⋊Q8—22no—
87C23×C22⋊Q8—22no—
88C23×C22⋊Q8—22no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8C2311yescharacteristic
92C23×#121—21no—
93C23×#121—22no—
94C23×#121—21no—
95C23×#121—22no—
96C23×#121—22no—
97C23×#121—21no—
98C23×#121—22no—
99C23×#121C2311yescharacteristic
100C23×#121C2312yes—
101C4×Q8×C23D411yescharacteristic
102C4×D4×C23—22no—
103C4×D4×C23—21no—
104C4×D4×C23—21no—
105C4×D4×C23—21no—
106C4×D4×C23—22no—
107C4×D4×C23—21no—
108C4×D4×C23—21no—
109C4×D4×C23—21no—
110C4×D4×C23—21no—
111C4×D4×C23—21no—
112C4×D4×C23D411yescharacteristic
113C4×D4×C23D412yes—
114C23×#117—21no—
115C23×#117—21no—
116C2×C4⋊C4×C23—21no—
117C2×C4⋊C4×C23—22no—
118C2×C4⋊C4×C23—21no—
119C2×C4⋊C4×C23D411yescharacteristic
120C2×C4⋊C4×C23D412yes—
121C2×C4⋊C4×C23D411yescharacteristic
122C2×C4⋊C4×C23C2311yescharacteristic
123C2×C4⋊C4×C23C2312yes—
124C2×C22⋊C4×C23—22no—
125C2×C22⋊C4×C23—21no—
126C2×C22⋊C4×C23—22no—
127C2×C42×C23—21no—
128C2×C42×C23C2311yescharacteristic
129C23×#107—21no—
130C23×#106—21no—
131C23×#106—22no—
132C23×C4⋊C8—21no—
133C23×C4⋊C8—21no—
134C23×#103—21no—
135C23×#102—21no—
136C23×#102—22no—
137C23×#102—21no—
138C23×#102—21no—
139C23×#102—22no—
140C23×C22⋊C8C2311yescharacteristic
141C23×C22⋊C8C2311yescharacteristic
142C23×C22⋊C8C2312yes—
143C23×#97—21no—
144C4×C8×C23—21no—
145C23×#95—22no—
146C23×#95—22no—
147C23×#95C2312yes—
148C24×C23—21no—
149#41×C23—81no—
150#41×C23—81no—
151#41×C23—82no—
152#41×C23—82no—
153C2×Q8×C23—22no—
154C2×Q8×C23—21no—
155C2×Q8×C23C2×D411yescharacteristic
156C2×D4×C23—21no—
157C2×D4×C23—22no—
158C2×D4×C23—22no—
159C2×D4×C23—22no—
160C2×D4×C23—21no—
161C2×D4×C23—21no—
162C2×D4×C23—21no—
163C2×D4×C23—21no—
164C2×D4×C23—42no—
165C2×D4×C23C2×D411yescharacteristic
166C2×D4×C23C2×D412yes—
167C22×C4×C23—22no—
168C22×C4×C23—22no—
169C22×C4×C23—22no—
170C22×C4×C23—21no—
171C22×C4×C23—22no—
172C22×C4×C23—22no—
173C22×C4×C23—21no—
174C22×C4×C23—42no—
175C22×C4×C23—42no—
176C22×C4×C23—41no—
177C22×C4×C23—41no—
178C22×C4×C23C2×D412yes—
179C22×C4×C23C2×D411yescharacteristic
180C22×C4×C23C2×D411yescharacteristic
181C22×C4×C23C2×D412yes—
182C22×C4×C23C2411yescharacteristic
183SD16×C23—42no—
184D8×C23—44no—
185D8×C23—42no—
186C2×C8×C23—22no—
187C2×C8×C23—21no—
188C2×C8×C23—21no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—22no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—21no—
198C4⋊C4×C23—22no—
199C4⋊C4×C23—21no—
200C4⋊C4×C23—21no—
201C4⋊C4×C23—21no—
202C4⋊C4×C23—21no—
203C4⋊C4×C23—22no—
204C4⋊C4×C23—22no—
205C4⋊C4×C23—21no—
206C4⋊C4×C23—21no—
207C4⋊C4×C23—21no—
208C4⋊C4×C23—42no—
209C4⋊C4×C23C2×D412yes—
210C4⋊C4×C23C2×D411yescharacteristic
211C4⋊C4×C23C2×D411yescharacteristic
212C22⋊C4×C23—22no—
213C22⋊C4×C23—22no—
214C22⋊C4×C23—22no—
215C22⋊C4×C23—22no—
216C22⋊C4×C23—22no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—21no—
219C22⋊C4×C23—21no—
220C22⋊C4×C23—21no—
221C22⋊C4×C23—22no—
222C22⋊C4×C23—21no—
223C22⋊C4×C23—22no—
224C22⋊C4×C23—44no—
225C22⋊C4×C23—41no—
226C22⋊C4×C23—41no—
227C22⋊C4×C23—82no—
228C42×C23—22no—
229C42×C23—21no—
230C42×C23—81no—
231C42×C23C2×D411yescharacteristic
232C42×C23C2×D411yescharacteristic
233C42×C23C2×D411yescharacteristic
234C42×C23C2×D411yescharacteristic
235#18276C2311yesmaximal, characteristic, direct product factor, Sylow
236C23×C23—22no—
237C23×C23—21no—
238C23×C23—22no—
239C23×C23—41no—
240C23×C23—41no—
241C23×C23—41no—
242C23×C23—41no—
243C23×C23—41no—
244C23×C23—41no—
245C23×C23C22×D411yescharacteristic
246Q8×C23—41no—
247Q8×C23—81no—
248Q8×C23—82no—
249D4×C23—22no—
250D4×C23—22no—
251D4×C23—42no—
252D4×C23—41no—
253D4×C23—41no—
254D4×C23—41no—
255D4×C23—42no—
256D4×C23—41no—
257D4×C23—42no—
258D4×C23—41no—
259D4×C23—42no—
260D4×C23—81no—
261D4×C23—82no—
262D4×C23—82no—
263C2×C4×C23—21no—
264C2×C4×C23—21no—
265C2×C4×C23—22no—
266C2×C4×C23—21no—
267C2×C4×C23—21no—
268C2×C4×C23—22no—
269C2×C4×C23—22no—
270C2×C4×C23—21no—
271C2×C4×C23—22no—
272C2×C4×C23—22no—
273C2×C4×C23—22no—
274C2×C4×C23—41no—
275C2×C4×C23—42no—
276C2×C4×C23—42no—
277C2×C4×C23—42no—
278C2×C4×C23—42no—
279C2×C4×C23—41no—
280C2×C4×C23—41no—
281C2×C4×C23—41no—
282C2×C4×C23—42no—
283C2×C4×C23—82no—
284C2×C4×C23—82no—
285C2×C4×C23—82no—
286C2×C4×C23—82no—
287C2×C4×C23—82no—
288C2×C4×C23—82no—
289C2×C4×C23#12011yescharacteristic
290C2×C4×C23#12011yescharacteristic
291C2×C4×C23#12011yescharacteristic
292C2×C4×C23#12011yescharacteristic
293C2×C4×C23C22×D411yescharacteristic
294C2×C4×C23C22×D411yescharacteristic
295C8×C23—42no—
296C8×C23—41no—
297C8×C23—41no—
298#3559C2×C2311yescharacteristic, complemented
299#3109C2×C2311yescharacteristic, complemented
300#3105C2×C2311yescharacteristic, complemented
301#2562C2×C2311yescharacteristic, complemented
302#1616C2×C2312yescomplemented
303#1615C2×C2312yescomplemented
304#1598C2×C2312yescomplemented
305#1597C2×C2312yescomplemented
306#1509C2×C2311yescharacteristic, complemented
307#1502C2×C2311yescharacteristic, complemented
308#1470C2×C2311yescharacteristic, complemented
309C22×C23—21no—
310C22×C23—21no—
311C22×C23—42no—
312C22×C23—42no—
313C22×C23—42no—
314C22×C23—41no—
315C22×C23—41no—
316C22×C23—42no—
317C22×C23—42no—
318C22×C23—42no—
319C22×C23—41no—
320C22×C23—81no—
321C22×C23—81no—
322C22×C23—81no—
323C22×C23—81no—
324C22×C23C2×#12011yescharacteristic
325C22×C23#57611yescharacteristic
326C22×C23#57711yescharacteristic
327C4×C23—21no—
328C4×C23—21no—
329C4×C23—21no—
330C4×C23—21no—
331C4×C23—21no—
332C4×C23—42no—
333C4×C23—42no—
334C4×C23—41no—
335C4×C23—41no—
336C4×C23—42no—
337C4×C23—41no—
338C4×C23—82no—
339C4×C23—81no—
340C4×C23#45412yes—
341#563—21no—
342#562—21no—
343#554—21no—
344#553—21no—
345#553—21no—
346#547—22no—
347#547—21no—
348#544—21no—
349#540—21no—
350#539—22no—
351#539C22×C2312yes—
352#537C22×C2312yes—
353#532C22×C2311yescharacteristic
354#529C22×C2311yescharacteristic
355C2×#125C22×C2311yescharacteristic
356C2×C4×D4C22×C2311yescharacteristic
357#484C22×C2311yescharacteristic
358#481C22×C2311yescharacteristic
359#481C22×C2312yes—
360#478—21no—
361#476—21no—
362#463—21no—
363#459—21no—
364#450C22×C2311yescharacteristic
365#447C22×C2311yescharacteristic
366#447C22×C2312yes—
367#442—22no—
368C4×SD16—21no—
369C4×D8—21no—
370#432C22×C2311yescharacteristic
371#431C22×C2311yescharacteristic
372#400C22×C2312yes—
373#394C22×C2312yes—
374#388—22no—
375#380—21no—
376#380—21no—
377C4×C4⋊C4C22×C2311yescharacteristic
378#328C22×C2312yes—
379#327C22×C2312yes—
380#327C22×C2312yes—
381#327C22×C2312yes—
382#324C22×C2312yes—
383#324C22×C2312yes—
384C2×C23—21no—
385C2×C23—21no—
386C2×C23—81no—
387C2×C23—82no—
388C2×C23—82no—
389C2×C23#301211yescharacteristic
390C2×C23C2×#45411yescharacteristic
391C2×C23#302511yescharacteristic
392C2×#41—21no—
393C2×#41—22no—
394C22×D4C23×C2311yescharacteristic
395C23×C4—21no—
396C2×SD16—21no—
397C2×D8—21no—
398C2×D8—22no—
399C4⋊Q8—21no—
400C4⋊Q8—22no—
401C4⋊Q8D4×C2311yescharacteristic
402#127D4×C2311yescharacteristic
403#126—22no—
404#126—22no—
405#126—21no—
406#125—21no—
407#125—22no—
408#125—21no—
409#125—21no—
410#125—21no—
411#125—21no—
412#125D4×C2311yescharacteristic
413#124—21no—
414#124D4×C2311yescharacteristic
415#123—22no—
416#123—22no—
417#123—22no—
418#123—22no—
419#123—22no—
420#123—21no—
421#123—22no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8—22no—
426C22⋊Q8—22no—
427C22⋊Q8—22no—
428C22⋊Q8C23×C2311yescharacteristic
429#121—21no—
430#121—22no—
431#121—21no—
432#121—22no—
433#121—22no—
434#121—21no—
435#121—22no—
436#121C23×C2311yescharacteristic
437#121C23×C2312yes—
438C4×Q8D4×C2311yescharacteristic
439C4×D4—22no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—22no—
444C4×D4—21no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—21no—
449C4×D4D4×C2311yescharacteristic
450C4×D4D4×C2312yes—
451#117—21no—
452#117—21no—
453C2×C4⋊C4—22no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4—21no—
456C2×C4⋊C4D4×C2311yescharacteristic
457C2×C4⋊C4D4×C2312yes—
458C2×C4⋊C4D4×C2311yescharacteristic
459C2×C4⋊C4C23×C2311yescharacteristic
460C2×C4⋊C4C23×C2312yes—
461C2×C22⋊C4—21no—
462C2×C22⋊C4—22no—
463C2×C22⋊C4—22no—
464C2×C42—21no—
465C2×C42C23×C2311yescharacteristic
466#107—21no—
467#106—21no—
468#106—22no—
469C4⋊C8—21no—
470C4⋊C8—21no—
471#103—21no—
472#102—21no—
473#102—22no—
474#102—21no—
475#102—21no—
476#102—22no—
477C22⋊C8C23×C2311yescharacteristic
478C22⋊C8C23×C2311yescharacteristic
479C22⋊C8C23×C2312yes—
480#97—21no—
481C4×C8—21no—
482#95—22no—
483#95—22no—
484#95C23×C2312yes—
485C23#1827611yescharacteristic, direct product factor, Sylow
486C24—21no—
487#41—81no—
488#41—81no—
489#41—82no—
490#41—82no—
491C2×Q8—22no—
492C2×Q8—21no—
493C2×Q8C2×D4×C2311yescharacteristic
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—42no—
503C2×D4C2×D4×C2311yescharacteristic
504C2×D4C2×D4×C2312yes—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—41no—
515C22×C4—41no—
516C22×C4C2×D4×C2312yes—
517C22×C4C2×D4×C2311yescharacteristic
518C22×C4C2×D4×C2311yescharacteristic
519C22×C4C2×D4×C2312yes—
520C22×C4C24×C2311yescharacteristic
521SD16—42no—
522D8—44no—
523D8—42no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—42no—
547C4⋊C4C2×D4×C2312yes—
548C4⋊C4C2×D4×C2311yescharacteristic
549C4⋊C4C2×D4×C2311yescharacteristic
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—21no—
556C22⋊C4—21no—
557C22⋊C4—21no—
558C22⋊C4—21no—
559C22⋊C4—22no—
560C22⋊C4—21no—
561C22⋊C4—22no—
562C22⋊C4—44no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—82no—
566C42—22no—
567C42—21no—
568C42—81no—
569C42C2×D4×C2311yescharacteristic
570C42C2×D4×C2311yescharacteristic
571C42C2×D4×C2311yescharacteristic
572C42C2×D4×C2311yescharacteristic
573C23—22no—
574C23—21no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C22×D4×C2311yescharacteristic
583Q8—41no—
584Q8—81no—
585Q8—82no—
586D4—22no—
587D4—22no—
588D4—42no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—42no—
593D4—41no—
594D4—42no—
595D4—41no—
596D4—42no—
597D4—81no—
598D4—82no—
599D4—82no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4C23×#12011yescharacteristic
627C2×C4C23×#12011yescharacteristic
628C2×C4C23×#12011yescharacteristic
629C2×C4C23×#12011yescharacteristic
630C2×C4C22×D4×C2311yescharacteristic
631C2×C4C22×D4×C2311yescharacteristic
632C8—42no—
633C8—41no—
634C8—41no—
635C22—21no—
636C22—21no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—41no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22C2×C23×#12011yescharacteristic
651C22C23×#57611yescharacteristic
652C22C23×#57711yescharacteristic
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—42no—
659C4—42no—
660C4—41no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—82no—
665C4—81no—
666C4C23×#45412yes—
667C2—21no—
668C2—21no—
669C2—82no—
670C2—81no—
671C2—82no—
672C2C23×#301211yescharacteristic
673C2C2×C23×#45411yescharacteristic
674C2C23×#302511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
LCSG ▷ C22×C4 ▷ 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×#3559C2semidirect
2. C23×#3109C2semidirect
3. C23×#3105C2semidirect
4. C23×#2562C2semidirect
5. C23×#1616C2semidirect
6. C23×#1615C2semidirect
7. C23×#1598C2semidirect
8. C23×#1597C2semidirect
9. C23×#1509C2semidirect
10. C23×#1502C2semidirect
11. C23×#1470C2semidirect
235. #18276C23direct (N × H)
298. #3559C2×C23semidirect
299. #3109C2×C23semidirect
300. #3105C2×C23semidirect
301. #2562C2×C23semidirect
302. #1616C2×C23semidirect
303. #1615C2×C23semidirect
304. #1598C2×C23semidirect
305. #1597C2×C23semidirect
306. #1509C2×C23semidirect
307. #1502C2×C23semidirect
308. #1470C2×C23semidirect
485. C23#18276direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35592
C23×#31092
C23×#31052
C23×#25622
C23×#16162
C23×#16152
C23×#15982
C23×#15972
C23×#15092
C23×#15022
C23×#14702
#1827623