Finite Group Databasebeta

← C23×#18184 #19165810

Subgroups — C23×#18184 #19165810

Order 5888.

1028 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3493C211yesmaximal, characteristic, complemented
2C23×#3114C212yesmaximal, complemented
3C23×#2392C211yesmaximal, characteristic, complemented
4C23×#1612C212yesmaximal, complemented
5C23×#1611C211yesmaximal, characteristic, complemented
6C23×#1610C211yesmaximal, characteristic, complemented
7C23×#1600C212yesmaximal, complemented
8C23×#1599C211yesmaximal, characteristic, complemented
9C23×#1598C211yesmaximal, characteristic, complemented
10C23×#1509C212yesmaximal, complemented
11C23×#1464C211yesmaximal, characteristic, complemented
12C23×#556—22no—
13C23×#555—22no—
14C23×#553—22no—
15C23×#552—22no—
16C23×#544C2211yescharacteristic
17C23×#538—21no—
18C23×#538C2212yes—
19C23×#537C2211yescharacteristic
20C23×#529—21no—
21C23×#529C2212yes—
22C23×#525—21no—
23C23×#520—21no—
24C2×C4×D4×C23—21no—
25C2×C4×D4×C23C2211yescharacteristic
26C2×C23×#117C2211yescharacteristic
27C23×#483C2212yes—
28C23×#480C2212yes—
29C23×#478—22no—
30C23×#463—22no—
31C23×#450C2212yes—
32C23×#447C2212yes—
33C23×#442—22no—
34C23×#440—22no—
35C23×#432C2212yes—
36C23×#398C2211yescharacteristic
37C23×#397C2211yescharacteristic
38C23×#396—21no—
39C23×#394—21no—
40C23×#394C2211yescharacteristic
41C23×#392C2211yescharacteristic
42C4×C4⋊C4×C23C2211yescharacteristic
43C4×C22⋊C4×C23—21no—
44C23×#328C2212yes—
45C23×#328C2212yes—
46C23×#327C2212yes—
47C23×#327C2212yes—
48C23×#324C2211yescharacteristic
49C23×#324C2211yescharacteristic
50C23×#323C2212yes—
51C2×#41×C23—21no—
52C22×D4×C23—21no—
53C22×D4×C23C2311yescharacteristic
54C23×C4×C23—21no—
55C23×C4×C23D411yescharacteristic
56C2×SD16×C23—22no—
57C2×D8×C23—22no—
58C23×C4⋊Q8—22no—
59C23×#126—22no—
60C23×#126—22no—
61C23×#126—22no—
62C23×#126—22no—
63C23×#125—21no—
64C23×#125—22no—
65C23×#125D412yescomplemented
66C23×#124—21no—
67C23×#124—22no—
68C23×#124—22no—
69C23×#124D412yescomplemented
70C23×#123—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×C22⋊Q8—21no—
74C23×C22⋊Q8—21no—
75C23×C22⋊Q8—21no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8C2312yes—
78C23×#121—21no—
79C23×#121—22no—
80C23×#121—21no—
81C23×#121—22no—
82C23×#121—22no—
83C23×#121—21no—
84C23×#121C2312yes—
85C23×#120—22no—
86C4×Q8×C23—22no—
87C4×Q8×C23—21no—
88C4×Q8×C23D412yescomplemented
89C4×D4×C23—22no—
90C4×D4×C23—21no—
91C4×D4×C23—22no—
92C4×D4×C23—22no—
93C4×D4×C23—22no—
94C4×D4×C23—22no—
95C4×D4×C23—22no—
96C4×D4×C23—22no—
97C4×D4×C23—44no—
98C4×D4×C23D412yescomplemented
99C23×#117—22no—
100C23×#117—22no—
101C23×#117—21no—
102C23×#117—22no—
103C23×#117—22no—
104C23×#117—22no—
105C2×C4⋊C4×C23—22no—
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23—21no—
108C2×C4⋊C4×C23D411yescharacteristic
109C2×C4⋊C4×C23C2311yescharacteristic
110C2×C4⋊C4×C23C2311yescharacteristic
111C2×C22⋊C4×C23—21no—
112C2×C22⋊C4×C23—21no—
113C2×C22⋊C4×C23—21no—
114C2×C22⋊C4×C23—21no—
115C2×C22⋊C4×C23—21no—
116C2×C22⋊C4×C23D411yescharacteristic
117C2×C22⋊C4×C23C2311yescharacteristic
118C2×C42×C23—21no—
119C2×C42×C23D411yescharacteristic
120C2×C42×C23C2311yescharacteristic
121C23×#107—22no—
122C23×#106—22no—
123C23×C4⋊C8—22no—
124C23×#103—22no—
125C23×#102—22no—
126C23×#102—22no—
127C23×#102—22no—
128C23×C22⋊C8C2312yes—
129C23×C22⋊C8C2312yes—
130C23×#97—22no—
131C23×#95—21no—
132C23×#95—21no—
133C23×#95C2311yescharacteristic
134C23×#95C2311yescharacteristic
135C24×C23—21no—
136C24×C23C2×D411yescharacteristic
137#41×C23—42no—
138#41×C23—42no—
139C2×Q8×C23—21no—
140C2×Q8×C23—21no—
141C2×Q8×C23C2×D412yes—
142C2×D4×C23—21no—
143C2×D4×C23—22no—
144C2×D4×C23—22no—
145C2×D4×C23—22no—
146C2×D4×C23—21no—
147C2×D4×C23—22no—
148C2×D4×C23—42no—
149C2×D4×C23—42no—
150C2×D4×C23—42no—
151C2×D4×C23—42no—
152C2×D4×C23C2×D412yes—
153C22×C4×C23—22no—
154C22×C4×C23—21no—
155C22×C4×C23—22no—
156C22×C4×C23—21no—
157C22×C4×C23—21no—
158C22×C4×C23—22no—
159C22×C4×C23—22no—
160C22×C4×C23—21no—
161C22×C4×C23—21no—
162C22×C4×C23—41no—
163C22×C4×C23—42no—
164C22×C4×C23—42no—
165C22×C4×C23—41no—
166C22×C4×C23—41no—
167C22×C4×C23C2×D411yescharacteristic
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2×D411yescharacteristic
172C22×C4×C23C2411yescharacteristic
173SD16×C23—44no—
174D8×C23—44no—
175C2×C8×C23—22no—
176C2×C8×C23—22no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—21no—
182C4⋊C4×C23—21no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—42no—
191C4⋊C4×C23—42no—
192C4⋊C4×C23C2×D412yes—
193C4⋊C4×C23C2×D412yes—
194C22⋊C4×C23—22no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—22no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—22no—
204C22⋊C4×C23—22no—
205C22⋊C4×C23—42no—
206C22⋊C4×C23—42no—
207C22⋊C4×C23—42no—
208C22⋊C4×C23—42no—
209C22⋊C4×C23—42no—
210C42×C23—22no—
211C42×C23—22no—
212C42×C23—21no—
213C42×C23—21no—
214C42×C23—22no—
215C42×C23—42no—
216C42×C23C2×D412yes—
217C42×C23C2×D412yes—
218#18184C2311yesmaximal, characteristic, direct product factor, Sylow
219C23×C23—22no—
220C23×C23—21no—
221C23×C23—22no—
222C23×C23—21no—
223C23×C23—21no—
224C23×C23—41no—
225C23×C23—41no—
226C23×C23—41no—
227C23×C23—41no—
228C23×C23—41no—
229C23×C23—41no—
230C23×C23—41no—
231C23×C23—41no—
232C23×C23C22×D411yescharacteristic
233Q8×C23—42no—
234Q8×C23—42no—
235Q8×C23—81no—
236D4×C23—24no—
237D4×C23—42no—
238D4×C23—42no—
239D4×C23—42no—
240D4×C23—48no—
241D4×C23—42no—
242D4×C23—42no—
243D4×C23—82no—
244D4×C23—81no—
245D4×C23—82no—
246C2×C4×C23—21no—
247C2×C4×C23—21no—
248C2×C4×C23—21no—
249C2×C4×C23—21no—
250C2×C4×C23—22no—
251C2×C4×C23—21no—
252C2×C4×C23—22no—
253C2×C4×C23—21no—
254C2×C4×C23—22no—
255C2×C4×C23—22no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
258C2×C4×C23—42no—
259C2×C4×C23—42no—
260C2×C4×C23—44no—
261C2×C4×C23—41no—
262C2×C4×C23—41no—
263C2×C4×C23—41no—
264C2×C4×C23—41no—
265C2×C4×C23—41no—
266C2×C4×C23—41no—
267C2×C4×C23—41no—
268C2×C4×C23—41no—
269C2×C4×C23—41no—
270C2×C4×C23—41no—
271C2×C4×C23—82no—
272C2×C4×C23—82no—
273C2×C4×C23—82no—
274C2×C4×C23—82no—
275C2×C4×C23#12012yes—
276C2×C4×C23#12011yescharacteristic
277C2×C4×C23#12011yescharacteristic
278C2×C4×C23AGL(1,ℤ/8)12yes—
279C2×C4×C23C22×D412yes—
280C8×C23—42no—
281C8×C23—42no—
282#3493C2×C2311yescharacteristic, complemented
283#3114C2×C2312yescomplemented
284#2392C2×C2311yescharacteristic, complemented
285#1612C2×C2312yescomplemented
286#1611C2×C2311yescharacteristic, complemented
287#1610C2×C2311yescharacteristic, complemented
288#1600C2×C2312yescomplemented
289#1599C2×C2311yescharacteristic, complemented
290#1598C2×C2311yescharacteristic, complemented
291#1509C2×C2312yescomplemented
292#1464C2×C2311yescharacteristic, complemented
293C22×C23—21no—
294C22×C23—21no—
295C22×C23—42no—
296C22×C23—42no—
297C22×C23—42no—
298C22×C23—41no—
299C22×C23—42no—
300C22×C23—42no—
301C22×C23—41no—
302C22×C23—42no—
303C22×C23—42no—
304C22×C23—41no—
305C22×C23—81no—
306C22×C23—81no—
307C22×C23—81no—
308C22×C23—81no—
309C22×C23—81no—
310C22×C23C2×#12011yescharacteristic
311C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C23#57511yescharacteristic
313C4×C23—22no—
314C4×C23—21no—
315C4×C23—21no—
316C4×C23—22no—
317C4×C23—22no—
318C4×C23—42no—
319C4×C23—42no—
320C4×C23—41no—
321C4×C23—42no—
322C4×C23—41no—
323C4×C23—42no—
324C4×C23—81no—
325C4×C23—81no—
326#556—22no—
327#555—22no—
328#553—22no—
329#552—22no—
330#544C22×C2311yescharacteristic
331#538—21no—
332#538C22×C2312yes—
333#537C22×C2311yescharacteristic
334#529—21no—
335#529C22×C2312yes—
336#525—21no—
337#520—21no—
338C2×C4×D4—21no—
339C2×C4×D4C22×C2311yescharacteristic
340C2×#117C22×C2311yescharacteristic
341#483C22×C2312yes—
342#480C22×C2312yes—
343#478—22no—
344#463—22no—
345#450C22×C2312yes—
346#447C22×C2312yes—
347#442—22no—
348#440—22no—
349#432C22×C2312yes—
350#398C22×C2311yescharacteristic
351#397C22×C2311yescharacteristic
352#396—21no—
353#394—21no—
354#394C22×C2311yescharacteristic
355#392C22×C2311yescharacteristic
356C4×C4⋊C4C22×C2311yescharacteristic
357C4×C22⋊C4—21no—
358#328C22×C2312yes—
359#328C22×C2312yes—
360#327C22×C2312yes—
361#327C22×C2312yes—
362#324C22×C2311yescharacteristic
363#324C22×C2311yescharacteristic
364#323C22×C2312yes—
365C2×C23—21no—
366C2×C23—21no—
367C2×C23—82no—
368C2×C23—81no—
369C2×C23—82no—
370C2×C23#300611yescharacteristic
371C2×C23#301811yescharacteristic
372C2×C23#302511yescharacteristic
373C2×#41—21no—
374C22×D4—21no—
375C22×D4C23×C2311yescharacteristic
376C23×C4—21no—
377C23×C4D4×C2311yescharacteristic
378C2×SD16—22no—
379C2×D8—22no—
380C4⋊Q8—22no—
381#126—22no—
382#126—22no—
383#126—22no—
384#126—22no—
385#125—21no—
386#125—22no—
387#125D4×C2312yescomplemented
388#124—21no—
389#124—22no—
390#124—22no—
391#124D4×C2312yescomplemented
392#123—22no—
393#123—22no—
394#123—22no—
395C22⋊Q8—21no—
396C22⋊Q8—21no—
397C22⋊Q8—21no—
398C22⋊Q8—22no—
399C22⋊Q8C23×C2312yes—
400#121—21no—
401#121—21no—
402#121—22no—
403#121—22no—
404#121—22no—
405#121—21no—
406#121C23×C2312yes—
407#120—22no—
408C4×Q8—22no—
409C4×Q8—21no—
410C4×Q8D4×C2312yescomplemented
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—21no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—44no—
420C4×D4D4×C2312yescomplemented
421#117—22no—
422#117—22no—
423#117—21no—
424#117—22no—
425#117—22no—
426#117—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4D4×C2311yescharacteristic
431C2×C4⋊C4C23×C2311yescharacteristic
432C2×C4⋊C4C23×C2311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4D4×C2311yescharacteristic
439C2×C22⋊C4C23×C2311yescharacteristic
440C2×C42—21no—
441C2×C42D4×C2311yescharacteristic
442C2×C42C23×C2311yescharacteristic
443#107—22no—
444#106—22no—
445C4⋊C8—22no—
446#103—22no—
447#102—22no—
448#102—22no—
449#102—22no—
450C22⋊C8C23×C2312yes—
451C22⋊C8C23×C2312yes—
452#97—22no—
453#95—21no—
454#95—21no—
455#95C23×C2311yescharacteristic
456#95C23×C2311yescharacteristic
457C23#1818411yescharacteristic, direct product factor, Sylow
458C24—21no—
459C24C2×D4×C2311yescharacteristic
460#41—42no—
461#41—42no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×D4×C2312yes—
465C2×D4—21no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×D4×C2312yes—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—41no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4C2×D4×C2311yescharacteristic
491C22×C4C2×D4×C2311yescharacteristic
492C22×C4C2×D4×C2311yescharacteristic
493C22×C4C2×D4×C2311yescharacteristic
494C22×C4C2×D4×C2311yescharacteristic
495C22×C4C24×C2311yescharacteristic
496SD16—44no—
497D8—44no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4C2×D4×C2312yes—
516C4⋊C4C2×D4×C2312yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—22no—
538C42—42no—
539C42C2×D4×C2312yes—
540C42C2×D4×C2312yes—
541C23—22no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—21no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C22×D4×C2311yescharacteristic
555Q8—42no—
556Q8—42no—
557Q8—81no—
558D4—24no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—48no—
563D4—42no—
564D4—42no—
565D4—82no—
566D4—81no—
567D4—82no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
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—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4C23×#12012yes—
598C2×C4C23×#12011yescharacteristic
599C2×C4C23×#12011yescharacteristic
600C2×C4C23×AGL(1,ℤ/8)12yes—
601C2×C4C22×D4×C2312yes—
602C8—42no—
603C8—42no—
604C22—21no—
605C22—21no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×C23×#12011yescharacteristic
622C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
623C22C23×#57511yescharacteristic
624C4—22no—
625C4—21no—
626C4—21no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—81no—
636C4—81no—
637C2—21no—
638C2—21no—
639C2—81no—
640C2—82no—
641C2—82no—
642C2C23×#300611yescharacteristic
643C2C23×#301811yescharacteristic
644C2C23×#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×#3493C2semidirect
2. C23×#3114C2semidirect
3. C23×#2392C2semidirect
4. C23×#1612C2semidirect
5. C23×#1611C2semidirect
6. C23×#1610C2semidirect
7. C23×#1600C2semidirect
8. C23×#1599C2semidirect
9. C23×#1598C2semidirect
10. C23×#1509C2semidirect
11. C23×#1464C2semidirect
65. C23×#125D4semidirect
69. C23×#124D4semidirect
88. C4×Q8×C23D4semidirect
98. C4×D4×C23D4semidirect
218. #18184C23direct (N × H)
282. #3493C2×C23semidirect
283. #3114C2×C23semidirect
284. #2392C2×C23semidirect
285. #1612C2×C23semidirect
286. #1611C2×C23semidirect
287. #1610C2×C23semidirect
288. #1600C2×C23semidirect
289. #1599C2×C23semidirect
290. #1598C2×C23semidirect
291. #1509C2×C23semidirect
292. #1464C2×C23semidirect
387. #125D4×C23semidirect
391. #124D4×C23semidirect
410. C4×Q8D4×C23semidirect
420. C4×D4D4×C23semidirect
457. C23#18184direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34932
C23×#31142
C23×#23922
C23×#16122
C23×#16112
C23×#16102
C23×#16002
C23×#15992
C23×#15982
C23×#15092
C23×#14642
#1818423