Finite Group Databasebeta

← C23×#9042 #19156668

Subgroups — C23×#9042 #19156668

Order 5888.

966 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#450C211yesmaximal, characteristic
2C23×#2329C211yesmaximal, characteristic, complemented
3C23×#1786C211yesmaximal, characteristic
4C2×C23×#328C211yesmaximal, characteristic, complemented
5C23×#1438C211yesmaximal, characteristic
6C23×#1294C211yesmaximal, characteristic, complemented
7C23×#1280C211yesmaximal, characteristic, complemented
8C23×D4×C23C2211yescharacteristic
9C22×SD16×C23—21no—
10C2×C23×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C4×C23—21no—
12C22×C22⋊C4×C23—21no—
13C23×#450—22no—
14C23×#450C414yescomplemented
15C2×C23×#102—21no—
16C2×C23×#102—21no—
17C2×C23×#102—21no—
18C2×C23×C22⋊C8C2211yescharacteristic
19C2×C23×C22⋊C8C2211yescharacteristic
20C23×#391—21no—
21C23×#389—21no—
22C23×#387—21no—
23C23×#387—21no—
24C23×#385—21no—
25C23×#380C2211yescharacteristic
26C4×C22⋊C4×C23—21no—
27C2×C23×#95C2211yescharacteristic
28C2×C23×#95C2211yescharacteristic
29C23×#328—22no—
30C25×C23—21no—
31C22×Q8×C23D411yescharacteristic
32C22×D4×C23—21no—
33C22×D4×C23—21no—
34C22×D4×C23—21no—
35C22×D4×C23—21no—
36C22×D4×C23—21no—
37C22×D4×C23—42no—
38C22×D4×C23—42no—
39C22×D4×C23C2×C412yes—
40C23×C4×C23—21no—
41C23×C4×C23D411yescharacteristic
42C23×C4×C23D411yescharacteristic
43C23×C4×C23C2311yescharacteristic
44C2×SD16×C23—24no—
45C2×SD16×C23—42no—
46C2×SD16×C23—42no—
47C22×C8×C23—21no—
48C22×C8×C23—21no—
49C23×C22⋊Q8—21no—
50C23×C22⋊Q8—21no—
51C23×C22⋊Q8C2×C412yes—
52C23×C22⋊Q8D412yes—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23—21no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23D411yescharacteristic
57C2×C4⋊C4×C23D411yescharacteristic
58C2×C4⋊C4×C23D411yescharacteristic
59C2×C22⋊C4×C23—21no—
60C2×C22⋊C4×C23—21no—
61C2×C22⋊C4×C23—42no—
62C2×C22⋊C4×C23—42no—
63C2×C22⋊C4×C23—42no—
64C2×C22⋊C4×C23—42no—
65C2×C22⋊C4×C23—42no—
66C2×C22⋊C4×C23—42no—
67C2×C42×C23—21no—
68C2×C42×C23—21no—
69C23×#102—24no—
70C23×#102—42no—
71C23×#102—42no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8—21no—
74C23×C22⋊C8—21no—
75C23×C22⋊C8C2×C412yes—
76C23×#95—21no—
77C23×#95—21no—
78C23×#95—21no—
79C23×#95—21no—
80C23×#95—21no—
81C23×#95—21no—
82C23×#95—21no—
83C23×#95—21no—
84C23×#95—21no—
85C24×C23—22no—
86C24×C23—22no—
87C24×C23—22no—
88C24×C23—41no—
89C24×C23—41no—
90C24×C23—41no—
91C24×C23—41no—
92C24×C23—41no—
93C24×C23—41no—
94C24×C23—41no—
95C24×C23—41no—
96C24×C23—41no—
97C24×C23—41no—
98C24×C23—41no—
99C24×C23—41no—
100C24×C23C2×D411yescharacteristic
101C2×Q8×C23—21no—
102C2×Q8×C23—41no—
103C2×Q8×C23—41no—
104C2×Q8×C23C22⋊C412yescomplemented
105C2×D4×C23—22no—
106C2×D4×C23—22no—
107C2×D4×C23—22no—
108C2×D4×C23—22no—
109C2×D4×C23—22no—
110C2×D4×C23—22no—
111C2×D4×C23—41no—
112C2×D4×C23—41no—
113C2×D4×C23—41no—
114C2×D4×C23—41no—
115C2×D4×C23—44no—
116C2×D4×C23—44no—
117C2×D4×C23—44no—
118C2×D4×C23—44no—
119C2×D4×C23—41no—
120C2×D4×C23—41no—
121C22×C4×C23—21no—
122C22×C4×C23—21no—
123C22×C4×C23—21no—
124C22×C4×C23—21no—
125C22×C4×C23—22no—
126C22×C4×C23—21no—
127C22×C4×C23—21no—
128C22×C4×C23—21no—
129C22×C4×C23—21no—
130C22×C4×C23—21no—
131C22×C4×C23—41no—
132C22×C4×C23—41no—
133C22×C4×C23—41no—
134C22×C4×C23—41no—
135C22×C4×C23—41no—
136C22×C4×C23—41no—
137C22×C4×C23—41no—
138C22×C4×C23—41no—
139C22×C4×C23—41no—
140C22×C4×C23—41no—
141C22×C4×C23—41no—
142C22×C4×C23—41no—
143C22×C4×C23SD1612yescomplemented
144C22×C4×C23C22×C411yescharacteristic
145C22×C4×C23C2×D411yescharacteristic
146C22×C4×C23C2×D411yescharacteristic
147C22×C4×C23C2×D411yescharacteristic
148C22×C4×C23#4111yescharacteristic
149C22×C4×C23#4111yescharacteristic
150SD16×C23—48no—
151C2×C8×C23—22no—
152C2×C8×C23—41no—
153C2×C8×C23—41no—
154C2×C8×C23—41no—
155C2×C8×C23—41no—
156C2×C8×C23—41no—
157C4⋊C4×C23—22no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—21no—
160C4⋊C4×C23—41no—
161C4⋊C4×C23—41no—
162C4⋊C4×C23—81no—
163C4⋊C4×C23—81no—
164C4⋊C4×C23C22⋊C412yescomplemented
165C22⋊C4×C23—22no—
166C22⋊C4×C23—22no—
167C22⋊C4×C23—48no—
168C22⋊C4×C23—48no—
169C22⋊C4×C23—41no—
170C22⋊C4×C23—41no—
171C42×C23—81no—
172C42×C23—81no—
173#9042C2311yesmaximal, characteristic, direct product factor, Sylow
174C23×C23—21no—
175C23×C23—21no—
176C23×C23—21no—
177C23×C23—21no—
178C23×C23—24no—
179C23×C23—42no—
180C23×C23—42no—
181C23×C23—42no—
182C23×C23—42no—
183C23×C23—42no—
184C23×C23—42no—
185C23×C23—42no—
186C23×C23—42no—
187C23×C23—42no—
188C23×C23—42no—
189C23×C23—42no—
190C23×C23—42no—
191C23×C23—42no—
192C23×C23—42no—
193C23×C23—42no—
194C23×C23—42no—
195C23×C23—42no—
196C23×C23—42no—
197C23×C23—81no—
198C23×C23—81no—
199C23×C23—81no—
200C23×C23—81no—
201C23×C23—81no—
202C23×C23—81no—
203C23×C23—81no—
204C23×C23—81no—
205C23×C23—81no—
206C23×C23—81no—
207C23×C23—81no—
208C23×C23—81no—
209C23×C23—81no—
210C23×C23—81no—
211C23×C23—81no—
212C23×C23—81no—
213C23×C23C4×D411yescharacteristic
214C23×C23#12011yescharacteristic
215C23×C23#12111yescharacteristic
216C23×C23C2×SD1611yescharacteristic
217C23×C23#13511yescharacteristic
218C23×C23AGL(1,ℤ/8)11yescharacteristic
219C23×C23#13711yescharacteristic
220Q8×C23—42no—
221Q8×C23—81no—
222D4×C23—24no—
223D4×C23—24no—
224D4×C23—42no—
225D4×C23—42no—
226D4×C23—42no—
227D4×C23—42no—
228D4×C23—42no—
229D4×C23—42no—
230C2×C4×C23—21no—
231C2×C4×C23—21no—
232C2×C4×C23—21no—
233C2×C4×C23—21no—
234C2×C4×C23—22no—
235C2×C4×C23—22no—
236C2×C4×C23—22no—
237C2×C4×C23—41no—
238C2×C4×C23—41no—
239C2×C4×C23—41no—
240C2×C4×C23—41no—
241C2×C4×C23—41no—
242C2×C4×C23—41no—
243C2×C4×C23—41no—
244C2×C4×C23—41no—
245C2×C4×C23—41no—
246C2×C4×C23—41no—
247C2×C4×C23—42no—
248C2×C4×C23—42no—
249C2×C4×C23—42no—
250C2×C4×C23—42no—
251C2×C4×C23—42no—
252C2×C4×C23—41no—
253C2×C4×C23—41no—
254C2×C4×C23—41no—
255C2×C4×C23—41no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—41no—
260C2×C4×C23—41no—
261C2×C4×C23—81no—
262C2×C4×C23—81no—
263C2×C4×C23—81no—
264C2×C4×C23—81no—
265C2×C4×C23—81no—
266C2×C4×C23—81no—
267C2×C4×C23C2×C22⋊C411yescharacteristic
268C2×C4×C23C4×D411yescharacteristic
269C2×C4×C23#12111yescharacteristic
270C2×C4×C23#12311yescharacteristic
271C8×C23—42no—
272C8×C23—81no—
273C2×#450C2×C2311yescharacteristic
274#2329C2×C2311yescharacteristic, complemented
275#1786C2×C2311yescharacteristic
276C2×#328C2×C2311yescharacteristic, complemented
277#1438C2×C2311yescharacteristic
278#1294C2×C2311yescharacteristic, complemented
279#1280C2×C2311yescharacteristic, complemented
280C22×C23—21no—
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—21no—
284C22×C23—21no—
285C22×C23—21no—
286C22×C23—21no—
287C22×C23—21no—
288C22×C23—21no—
289C22×C23—21no—
290C22×C23—21no—
291C22×C23—21no—
292C22×C23—44no—
293C22×C23—44no—
294C22×C23—44no—
295C22×C23—82no—
296C22×C23—82no—
297C22×C23—82no—
298C22×C23—82no—
299C22×C23—82no—
300C22×C23—82no—
301C22×C23—82no—
302C22×C23—82no—
303C22×C23—82no—
304C22×C23—82no—
305C22×C23—82no—
306C22×C23—82no—
307C22×C23#38611yescharacteristic
308C22×C23C4×SD1611yescharacteristic
309C22×C23#44011yescharacteristic
310C22×C23#44811yescharacteristic
311C22×C23#44911yescharacteristic
312C22×C23#45311yescharacteristic
313C22×C23#45411yescharacteristic
314C22×C23B(4,𝔽2)11yescharacteristic
315C22×C23B(4,𝔽2)11yescharacteristic
316C22×C23#46011yescharacteristic
317C22×C23#46411yescharacteristic
318C4×C23—21no—
319C4×C23—21no—
320C4×C23—21no—
321C4×C23—21no—
322C4×C23—42no—
323C4×C23—42no—
324C4×C23—42no—
325C4×C23—82no—
326C4×C23—81no—
327C4×C23—81no—
328C4×C23—81no—
329C23×D4C22×C2311yescharacteristic
330C22×SD16—21no—
331C2×C22⋊Q8C22×C2311yescharacteristic
332C22×C22⋊C4—21no—
333C22×C22⋊C4—21no—
334#450—22no—
335#450C4×C2314yescomplemented
336C2×#102—21no—
337C2×#102—21no—
338C2×#102—21no—
339C2×C22⋊C8C22×C2311yescharacteristic
340C2×C22⋊C8C22×C2311yescharacteristic
341#391—21no—
342#389—21no—
343#387—21no—
344#387—21no—
345#385—21no—
346#380C22×C2311yescharacteristic
347C4×C22⋊C4—21no—
348C2×#95C22×C2311yescharacteristic
349C2×#95C22×C2311yescharacteristic
350#328—22no—
351C2×C23—21no—
352C2×C23—21no—
353C2×C23—21no—
354C2×C23—21no—
355C2×C23—84no—
356C2×C23#159611yescharacteristic
357C2×C23#159711yescharacteristic
358C2×C23#159911yescharacteristic
359C2×C23#160011yescharacteristic
360C2×C23#188011yescharacteristic
361C2×C23#188311yescharacteristic
362C2×C23#188911yescharacteristic
363C25—21no—
364C22×Q8D4×C2311yescharacteristic
365C22×D4—21no—
366C22×D4—21no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—42no—
371C22×D4—42no—
372C22×D4C2×C4×C2312yes—
373C23×C4—21no—
374C23×C4D4×C2311yescharacteristic
375C23×C4D4×C2311yescharacteristic
376C23×C4C23×C2311yescharacteristic
377C2×SD16—24no—
378C2×SD16—42no—
379C2×SD16—42no—
380C22×C8—21no—
381C22×C8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8C2×C4×C2312yes—
385C22⋊Q8D4×C2312yes—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4D4×C2311yescharacteristic
390C2×C4⋊C4D4×C2311yescharacteristic
391C2×C4⋊C4D4×C2311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—42no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C42—21no—
401C2×C42—21no—
402#102—24no—
403#102—42no—
404#102—42no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—21no—
408C22⋊C8C2×C4×C2312yes—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418C23#904211yescharacteristic, direct product factor, Sylow
419C24—22no—
420C24—22no—
421C24—22no—
422C24—41no—
423C24—41no—
424C24—41no—
425C24—41no—
426C24—41no—
427C24—41no—
428C24—41no—
429C24—41no—
430C24—41no—
431C24—41no—
432C24—41no—
433C24—41no—
434C24C2×D4×C2311yescharacteristic
435C2×Q8—21no—
436C2×Q8—41no—
437C2×Q8—41no—
438C2×Q8C22⋊C4×C2312yescomplemented
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—41no—
454C2×D4—41no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4SD16×C2312yescomplemented
478C22×C4C22×C4×C2311yescharacteristic
479C22×C4C2×D4×C2311yescharacteristic
480C22×C4C2×D4×C2311yescharacteristic
481C22×C4C2×D4×C2311yescharacteristic
482C22×C4#41×C2311yescharacteristic
483C22×C4#41×C2311yescharacteristic
484SD16—48no—
485C2×C8—22no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4C22⋊C4×C2312yescomplemented
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C42—81no—
506C42—81no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—24no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—81no—
531C23—81no—
532C23—81no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—81no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—81no—
544C23—81no—
545C23—81no—
546C23C4×D4×C2311yescharacteristic
547C23C23×#12011yescharacteristic
548C23C23×#12111yescharacteristic
549C23C2×SD16×C2311yescharacteristic
550C23C23×#13511yescharacteristic
551C23C23×AGL(1,ℤ/8)11yescharacteristic
552C23C23×#13711yescharacteristic
553Q8—42no—
554Q8—81no—
555D4—24no—
556D4—24no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4C2×C22⋊C4×C2311yescharacteristic
601C2×C4C4×D4×C2311yescharacteristic
602C2×C4C23×#12111yescharacteristic
603C2×C4C23×#12311yescharacteristic
604C8—42no—
605C8—81no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C23×#38611yescharacteristic
634C22C4×SD16×C2311yescharacteristic
635C22C23×#44011yescharacteristic
636C22C23×#44811yescharacteristic
637C22C23×#44911yescharacteristic
638C22C23×#45311yescharacteristic
639C22C23×#45411yescharacteristic
640C22C23×B(4,𝔽2)11yescharacteristic
641C22C23×B(4,𝔽2)11yescharacteristic
642C22C23×#46011yescharacteristic
643C22C23×#46411yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—81no—
653C4—81no—
654C4—81no—
655C2—21no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—84no—
660C2C23×#159611yescharacteristic
661C2C23×#159711yescharacteristic
662C2C23×#159911yescharacteristic
663C2C23×#160011yescharacteristic
664C2C23×#188011yescharacteristic
665C2C23×#188311yescharacteristic
666C2C23×#188911yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×D4×C23C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2329C2semidirect
4. C2×C23×#328C2semidirect
6. C23×#1294C2semidirect
7. C23×#1280C2semidirect
14. C23×#450C4semidirect
104. C2×Q8×C23C22⋊C4semidirect
143. C22×C4×C23SD16semidirect
164. C4⋊C4×C23C22⋊C4semidirect
173. #9042C23direct (N × H)
274. #2329C2×C23semidirect
276. C2×#328C2×C23semidirect
278. #1294C2×C23semidirect
279. #1280C2×C23semidirect
335. #450C4×C23semidirect
418. C23#9042direct (N × H)
438. C2×Q8C22⋊C4×C23semidirect
477. C22×C4SD16×C23semidirect
498. C4⋊C4C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#4502
C23×#23292
C23×#17862
C2×C23×#3282
C23×#14382
C23×#12942
C23×#12802
#904223