Finite Group Databasebeta

← C23×#24066 #19171692

Subgroups — C23×#24066 #19171692

Order 5888.

1176 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#529C211yesmaximal, characteristic
2C2×C23×#483C211yesmaximal, characteristic, complemented
3C2×C23×#480C211yesmaximal, characteristic
4C2×C23×#389C211yesmaximal, characteristic, complemented
5C23×#2059C212yesmaximal, complemented
6C23×#2042C212yesmaximal, complemented
7C23×#1874C211yesmaximal, characteristic
8C23×#1872C211yesmaximal, characteristic, complemented
9C23×#1864C212yesmaximal, complemented
10C23×#1863C212yesmaximal, complemented
11C23×#1817C211yesmaximal, characteristic, complemented
12C23×#529—24no—
13C23×#529—24no—
14C2×C23×#126—21no—
15C2×C23×#125C2211yescharacteristic
16C2×C23×#124C2211yescharacteristic
17C2×C23×#123—21no—
18C2×C23×C22⋊Q8C2211yescharacteristic
19C2×C23×#121C2211yescharacteristic
20C2×C4×Q8×C23C2211yescharacteristic
21C2×C4×D4×C23—21no—
22C2×C4×D4×C23C2211yescharacteristic
23C2×C23×#117—21no—
24C22×C4⋊C4×C23—21no—
25C22×C4⋊C4×C23C2211yescharacteristic
26C22×C4⋊C4×C23C2211yescharacteristic
27C22×C42×C23C2211yescharacteristic
28C23×#483—24no—
29C23×#480—24no—
30C2×C23×#107C2212yes—
31C2×C23×#106C2212yes—
32C2×C23×#103C2212yes—
33C2×C23×#103C2212yes—
34C2×C23×#102C2212yes—
35C2×C23×#102C2212yes—
36C2×C23×C22⋊C8C2211yescharacteristic
37C2×C23×C22⋊C8C2211yescharacteristic
38C23×#389—21no—
39C23×#389—21no—
40C23×#389C2212yes—
41C23×#389C2212yes—
42C2×C23×#95—21no—
43C23×#340C2212yes—
44C23×#340C2212yes—
45C23×#336C2212yes—
46C23×#336C2212yes—
47C22×Q8×C23C2311yescharacteristic
48C22×D4×C23—21no—
49C22×D4×C23C2311yescharacteristic
50C23×C4×C23—21no—
51C23×C4×C23—21no—
52C23×C4×C23—21no—
53C23×C4×C23D411yescharacteristic
54C23×C4×C23D411yescharacteristic
55C23×C4×C23C2311yescharacteristic
56C22×C8×C23C2312yes—
57C22×C8×C23C2312yes—
58C23×#126—28no—
59C23×#125—22no—
60C23×#125—22no—
61C23×#124—22no—
62C23×#124—22no—
63C23×#123—28no—
64C23×C22⋊Q8—22no—
65C23×C22⋊Q8D414yes—
66C23×#121—22no—
67C23×#121D414yes—
68C4×Q8×C23—22no—
69C4×Q8×C23—22no—
70C4×D4×C23—28no—
71C4×D4×C23—22no—
72C4×D4×C23—22no—
73C23×#117—28no—
74C2×C4⋊C4×C23—21no—
75C2×C4⋊C4×C23—21no—
76C2×C4⋊C4×C23—21no—
77C2×C4⋊C4×C23—21no—
78C2×C4⋊C4×C23—21no—
79C2×C4⋊C4×C23—24no—
80C2×C4⋊C4×C23—22no—
81C2×C4⋊C4×C23—22no—
82C2×C4⋊C4×C23—21no—
83C2×C4⋊C4×C23—22no—
84C2×C4⋊C4×C23—22no—
85C2×C4⋊C4×C23—44no—
86C2×C4⋊C4×C23C2312yes—
87C2×C4⋊C4×C23C2311yescharacteristic
88C2×C4⋊C4×C23C2311yescharacteristic
89C2×C4⋊C4×C23C2312yes—
90C2×C22⋊C4×C23—21no—
91C2×C22⋊C4×C23—21no—
92C2×C22⋊C4×C23—21no—
93C2×C22⋊C4×C23—21no—
94C2×C22⋊C4×C23—21no—
95C2×C42×C23—22no—
96C2×C42×C23—22no—
97C2×C42×C23—21no—
98C2×C42×C23D411yescharacteristic
99C2×C42×C23D411yescharacteristic
100C2×C42×C23C2311yescharacteristic
101C2×C42×C23C2311yescharacteristic
102C23×#107—24no—
103C23×#106—24no—
104C23×#103—24no—
105C23×#103—24no—
106C23×#102—24no—
107C23×#102—24no—
108C23×C22⋊C8—22no—
109C23×C22⋊C8—22no—
110C23×#95—22no—
111C23×#95—22no—
112C24×C23—21no—
113C24×C23C2×D411yescharacteristic
114C2×Q8×C23—21no—
115C2×Q8×C23—22no—
116C2×Q8×C23—22no—
117C2×Q8×C23C2×D412yes—
118C2×D4×C23—24no—
119C2×D4×C23—21no—
120C2×D4×C23—22no—
121C2×D4×C23—22no—
122C2×D4×C23—42no—
123C2×D4×C23—42no—
124C2×D4×C23C2×D412yes—
125C22×C4×C23—22no—
126C22×C4×C23—22no—
127C22×C4×C23—24no—
128C22×C4×C23—24no—
129C22×C4×C23—22no—
130C22×C4×C23—22no—
131C22×C4×C23—22no—
132C22×C4×C23—22no—
133C22×C4×C23—21no—
134C22×C4×C23—21no—
135C22×C4×C23—21no—
136C22×C4×C23—22no—
137C22×C4×C23—22no—
138C22×C4×C23—22no—
139C22×C4×C23—41no—
140C22×C4×C23—42no—
141C22×C4×C23—42no—
142C22×C4×C23—42no—
143C22×C4×C23—42no—
144C22×C4×C23—42no—
145C22×C4×C23—42no—
146C22×C4×C23—42no—
147C22×C4×C23C2×D411yescharacteristic
148C22×C4×C23C2×D411yescharacteristic
149C22×C4×C23C2×D411yescharacteristic
150C22×C4×C23C2×D411yescharacteristic
151C22×C4×C23C2×D412yes—
152C22×C4×C23C2×D412yes—
153C22×C4×C23C2×D411yescharacteristic
154C22×C4×C23C2411yescharacteristic
155C2×C8×C23—22no—
156C2×C8×C23—22no—
157C2×C8×C23—22no—
158C2×C8×C23—22no—
159C2×C8×C23—22no—
160C2×C8×C23—22no—
161C4⋊C4×C23—24no—
162C4⋊C4×C23—24no—
163C4⋊C4×C23—24no—
164C4⋊C4×C23—24no—
165C4⋊C4×C23—24no—
166C4⋊C4×C23—21no—
167C4⋊C4×C23—21no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—44no—
172C4⋊C4×C23—44no—
173C4⋊C4×C23—42no—
174C4⋊C4×C23C2×D412yes—
175C4⋊C4×C23C2×D412yes—
176C4⋊C4×C23#4114yes—
177C22⋊C4×C23—24no—
178C22⋊C4×C23—24no—
179C22⋊C4×C23—24no—
180C22⋊C4×C23—24no—
181C22⋊C4×C23—24no—
182C42×C23—24no—
183C42×C23—22no—
184C42×C23—22no—
185C42×C23—21no—
186C42×C23—21no—
187C42×C23—21no—
188C42×C23—21no—
189#24066C2311yesmaximal, characteristic, direct product factor, Sylow
190C23×C23—21no—
191C23×C23—21no—
192C23×C23—21no—
193C23×C23—21no—
194C23×C23—22no—
195C23×C23—41no—
196C23×C23—41no—
197C23×C23—41no—
198C23×C23—41no—
199C23×C23—41no—
200C23×C23—41no—
201C23×C23#12012yes—
202C23×C23#13512yes—
203C23×C23AGL(1,ℤ/8)12yes—
204C23×C23C22×D411yescharacteristic
205Q8×C23—24no—
206Q8×C23—42no—
207D4×C23—24no—
208D4×C23—48no—
209D4×C23—42no—
210C2×C4×C23—21no—
211C2×C4×C23—21no—
212C2×C4×C23—21no—
213C2×C4×C23—21no—
214C2×C4×C23—24no—
215C2×C4×C23—24no—
216C2×C4×C23—21no—
217C2×C4×C23—21no—
218C2×C4×C23—21no—
219C2×C4×C23—22no—
220C2×C4×C23—22no—
221C2×C4×C23—22no—
222C2×C4×C23—24no—
223C2×C4×C23—21no—
224C2×C4×C23—21no—
225C2×C4×C23—21no—
226C2×C4×C23—21no—
227C2×C4×C23—21no—
228C2×C4×C23—21no—
229C2×C4×C23—44no—
230C2×C4×C23—48no—
231C2×C4×C23—48no—
232C2×C4×C23—41no—
233C2×C4×C23—41no—
234C2×C4×C23—41no—
235C2×C4×C23—42no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—42no—
239C2×C4×C23—41no—
240C2×C4×C23—44no—
241C2×C4×C23—42no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—84no—
245C2×C4×C23#12012yes—
246C2×C4×C23#13712yes—
247C2×C4×C23C22×D411yescharacteristic
248C2×C4×C23C22×D411yescharacteristic
249C2×C4×C23C2×#4112yes—
250C2×C4×C232−1+412yes—
251C8×C23—42no—
252C8×C23—42no—
253C2×#529C2×C2311yescharacteristic
254C2×#483C2×C2311yescharacteristic, complemented
255C2×#480C2×C2311yescharacteristic
256C2×#389C2×C2311yescharacteristic, complemented
257#2059C2×C2312yescomplemented
258#2042C2×C2312yescomplemented
259#1874C2×C2311yescharacteristic
260#1872C2×C2311yescharacteristic, complemented
261#1864C2×C2312yescomplemented
262#1863C2×C2312yescomplemented
263#1817C2×C2311yescharacteristic, complemented
264C22×C23—22no—
265C22×C23—22no—
266C22×C23—22no—
267C22×C23—22no—
268C22×C23—22no—
269C22×C23—22no—
270C22×C23—42no—
271C22×C23—42no—
272C22×C23—42no—
273C22×C23—81no—
274C22×C23—81no—
275C22×C23—81no—
276C22×C23—81no—
277C22×C23#44914yes—
278C22×C23C2×#12011yescharacteristic
279C22×C23#52511yescharacteristic
280C22×C23#53611yescharacteristic
281C22×C23C2×#13511yescharacteristic
282C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C23C2×#13711yescharacteristic
284C22×C23#57511yescharacteristic
285C4×C23—22no—
286C4×C23—22no—
287C4×C23—44no—
288C4×C23—42no—
289C4×C23—41no—
290C4×C23—41no—
291C4×C23—42no—
292C4×C23—41no—
293C4×C23—42no—
294C4×C23—82no—
295C4×C23#54712yes—
296C4×C23#54712yes—
297#529—24no—
298#529—24no—
299C2×#126—21no—
300C2×#125C22×C2311yescharacteristic
301C2×#124C22×C2311yescharacteristic
302C2×#123—21no—
303C2×C22⋊Q8C22×C2311yescharacteristic
304C2×#121C22×C2311yescharacteristic
305C2×C4×Q8C22×C2311yescharacteristic
306C2×C4×D4—21no—
307C2×C4×D4C22×C2311yescharacteristic
308C2×#117—21no—
309C22×C4⋊C4—21no—
310C22×C4⋊C4C22×C2311yescharacteristic
311C22×C4⋊C4C22×C2311yescharacteristic
312C22×C42C22×C2311yescharacteristic
313#483—24no—
314#480—24no—
315C2×#107C22×C2312yes—
316C2×#106C22×C2312yes—
317C2×#103C22×C2312yes—
318C2×#103C22×C2312yes—
319C2×#102C22×C2312yes—
320C2×#102C22×C2312yes—
321C2×C22⋊C8C22×C2311yescharacteristic
322C2×C22⋊C8C22×C2311yescharacteristic
323#389—21no—
324#389—21no—
325#389C22×C2312yes—
326#389C22×C2312yes—
327C2×#95—21no—
328#340C22×C2312yes—
329#340C22×C2312yes—
330#336C22×C2312yes—
331#336C22×C2312yes—
332C2×C23—24no—
333C2×C23—82no—
334C2×C23#240611yescharacteristic
335C2×C23C2×#44911yescharacteristic
336C2×C23#300711yescharacteristic
337C2×C23#310811yescharacteristic
338C2×C23#311411yescharacteristic
339C2×C23#323211yescharacteristic
340C2×C23#323811yescharacteristic
341C22×Q8C23×C2311yescharacteristic
342C22×D4—21no—
343C22×D4C23×C2311yescharacteristic
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—21no—
347C23×C4D4×C2311yescharacteristic
348C23×C4D4×C2311yescharacteristic
349C23×C4C23×C2311yescharacteristic
350C22×C8C23×C2312yes—
351C22×C8C23×C2312yes—
352#126—28no—
353#125—22no—
354#125—22no—
355#124—22no—
356#124—22no—
357#123—28no—
358C22⋊Q8—22no—
359C22⋊Q8D4×C2314yes—
360#121—22no—
361#121D4×C2314yes—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×D4—28no—
365C4×D4—22no—
366C4×D4—22no—
367#117—28no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—44no—
380C2×C4⋊C4C23×C2312yes—
381C2×C4⋊C4C23×C2311yescharacteristic
382C2×C4⋊C4C23×C2311yescharacteristic
383C2×C4⋊C4C23×C2312yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C42—21no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42D4×C2311yescharacteristic
393C2×C42D4×C2311yescharacteristic
394C2×C42C23×C2311yescharacteristic
395C2×C42C23×C2311yescharacteristic
396#107—24no—
397#106—24no—
398#103—24no—
399#103—24no—
400#102—24no—
401#102—24no—
402C22⋊C8—22no—
403C22⋊C8—22no—
404#95—22no—
405#95—22no—
406C23#2406611yescharacteristic, direct product factor, Sylow
407C24—21no—
408C24C2×D4×C2311yescharacteristic
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8C2×D4×C2312yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×D4×C2312yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4C2×D4×C2311yescharacteristic
443C22×C4C2×D4×C2311yescharacteristic
444C22×C4C2×D4×C2311yescharacteristic
445C22×C4C2×D4×C2311yescharacteristic
446C22×C4C2×D4×C2312yes—
447C22×C4C2×D4×C2312yes—
448C22×C4C2×D4×C2311yescharacteristic
449C22×C4C24×C2311yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4C2×D4×C2312yes—
470C4⋊C4C2×D4×C2312yes—
471C4⋊C4#41×C2314yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23C23×#12012yes—
496C23C23×#13512yes—
497C23C23×AGL(1,ℤ/8)12yes—
498C23C22×D4×C2311yescharacteristic
499Q8—24no—
500Q8—42no—
501D4—24no—
502D4—48no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—84no—
539C2×C4C23×#12012yes—
540C2×C4C23×#13712yes—
541C2×C4C22×D4×C2311yescharacteristic
542C2×C4C22×D4×C2311yescharacteristic
543C2×C4C2×#41×C2312yes—
544C2×C4C23×2−1+412yes—
545C8—42no—
546C8—42no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22C23×#44914yes—
561C22C2×C23×#12011yescharacteristic
562C22C23×#52511yescharacteristic
563C22C23×#53611yescharacteristic
564C22C2×C23×#13511yescharacteristic
565C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
566C22C2×C23×#13711yescharacteristic
567C22C23×#57511yescharacteristic
568C4—22no—
569C4—22no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—44no—
577C4—82no—
578C4C23×#54712yes—
579C4C23×#54712yes—
580C2—24no—
581C2—82no—
582C2C23×#240611yescharacteristic
583C2C2×C23×#44911yescharacteristic
584C2C23×#300711yescharacteristic
585C2C23×#310811yescharacteristic
586C2C23×#311411yescharacteristic
587C2C23×#323211yescharacteristic
588C2C23×#323811yescharacteristic
subgroupG/H
center, socleC23×C23C22×D4
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC2×C23×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C22×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#483C2semidirect
4. C2×C23×#389C2semidirect
5. C23×#2059C2semidirect
6. C23×#2042C2semidirect
8. C23×#1872C2semidirect
9. C23×#1864C2semidirect
10. C23×#1863C2semidirect
11. C23×#1817C2semidirect
189. #24066C23direct (N × H)
254. C2×#483C2×C23semidirect
256. C2×#389C2×C23semidirect
257. #2059C2×C23semidirect
258. #2042C2×C23semidirect
260. #1872C2×C23semidirect
261. #1864C2×C23semidirect
262. #1863C2×C23semidirect
263. #1817C2×C23semidirect
406. C23#24066direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5292
C2×C23×#4832
C2×C23×#4802
C2×C23×#3892
C23×#20592
C23×#20422
C23×#18742
C23×#18722
C23×#18642
C23×#18632
C23×#18172
#2406623