Finite Group Databasebeta

← C23×#18323 #19165949

Subgroups — C23×#18323 #19165949

Order 5888.

1008 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3459C211yesmaximal, characteristic
2C23×#3196C212yesmaximal, complemented
3C23×#2564C211yesmaximal, characteristic, complemented
4C23×#1615C212yesmaximal, complemented
5C23×#1614C211yesmaximal, characteristic, complemented
6C23×#1613C211yesmaximal, characteristic, complemented
7C23×#1603C212yesmaximal, complemented
8C23×#1602C211yesmaximal, characteristic, complemented
9C23×#1601C211yesmaximal, characteristic, complemented
10C23×#1504C212yesmaximal, complemented
11C23×#1465C211yesmaximal, characteristic, complemented
12C23×#543C2211yescharacteristic
13C23×#539—22no—
14C23×#539C2212yes—
15C23×#538—22no—
16C23×#538—21no—
17C23×#534—21no—
18C23×#534C2211yescharacteristic
19C23×#529—22no—
20C23×#529—21no—
21C23×#525—22no—
22C23×#525C2212yes—
23C2×C23×C22⋊Q8C2211yescharacteristic, complemented
24C2×C4×D4×C23—21no—
25C2×C4×D4×C23C2211yescharacteristic
26C23×#485—22no—
27C23×#484C2212yescomplemented
28C23×#481C2212yes—
29C23×#470—22no—
30C23×#468—22no—
31C23×#452—22no—
32C23×#451C2212yescomplemented
33C23×#448C2212yes—
34C23×#407C2212yescomplemented
35C23×#400C2211yescharacteristic, complemented
36C23×#399C2211yescharacteristic
37C23×#395C2211yescharacteristic, complemented
38C23×#393C2211yescharacteristic
39C23×#387—21no—
40C23×#386—21no—
41C23×#385—21no—
42C23×#380—21no—
43C2×C23×#95C2211yescharacteristic, complemented
44C23×#328C2212yescomplemented
45C23×#328C2212yescomplemented
46C23×#327C2212yes—
47C23×#327C2212yes—
48C23×#324C2212yescomplemented
49C23×#323C2211yescharacteristic
50C23×#323C2211yescharacteristic
51C2×#41×C23D412yes—
52C22×Q8×C23C2311yescharacteristic
53C22×D4×C23—21no—
54C22×D4×C23D411yescharacteristic
55C23×C4×C23—21no—
56C23×C4×C23—21no—
57C23×C4×C23D411yescharacteristic
58C23×C4×C23C2311yescharacteristic
59C2×Q16×C23—22no—
60C2×SD16×C23—22no—
61C2×M16×C23—22no—
62C23×#126—22no—
63C23×#126—22no—
64C23×#126—22no—
65C23×#125—21no—
66C23×#125—22no—
67C23×#124—22no—
68C23×#124—21no—
69C23×#123—22no—
70C23×#123—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×#123—22no—
74C23×#123—22no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—21no—
78C23×C22⋊Q8—22no—
79C23×C22⋊Q8—22no—
80C23×C22⋊Q8—22no—
81C23×C22⋊Q8D412yes—
82C23×C22⋊Q8C2312yes—
83C23×#121—22no—
84C23×#121—22no—
85C23×#121—22no—
86C23×#121—21no—
87C23×#121D412yes—
88C23×#121C2312yes—
89C23×#120—22no—
90C23×#120—22no—
91C23×#120—22no—
92C4×Q8×C23—21no—
93C4×D4×C23—22no—
94C4×D4×C23—22no—
95C4×D4×C23—21no—
96C4×D4×C23—22no—
97C4×D4×C23—22no—
98C4×D4×C23—22no—
99C4×D4×C23—22no—
100C4×D4×C23—22no—
101C23×#117—22no—
102C23×#117D412yes—
103C2×C4⋊C4×C23—21no—
104C2×C4⋊C4×C23—21no—
105C2×C4⋊C4×C23—21no—
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23C2311yescharacteristic
108C2×C4⋊C4×C23C2311yescharacteristic
109C2×C22⋊C4×C23—21no—
110C2×C22⋊C4×C23—21no—
111C2×C22⋊C4×C23—21no—
112C2×C22⋊C4×C23—21no—
113C2×C22⋊C4×C23—21no—
114C2×C22⋊C4×C23D411yescharacteristic
115C2×C22⋊C4×C23C2311yescharacteristic
116C2×C42×C23—21no—
117C2×C42×C23D411yescharacteristic
118C23×#107—22no—
119C23×#106—22no—
120C23×#103—22no—
121C23×#103—22no—
122C23×#103—22no—
123C23×#102—22no—
124C23×C22⋊C8—22no—
125C23×C22⋊C8C2312yes—
126C23×C22⋊C8C2312yes—
127C23×#95—21no—
128C23×#95—21no—
129C23×#95—21no—
130C23×#95—21no—
131C23×#95—21no—
132C23×#95C2311yescharacteristic
133C23×#95C2311yescharacteristic
134C24×C23—21no—
135C24×C23C2×D411yescharacteristic
136C24×C23C2×D411yescharacteristic
137#41×C23—82no—
138C2×Q8×C23—21no—
139C2×Q8×C23—22no—
140C2×Q8×C23—81no—
141C2×Q8×C23C2×D412yes—
142C2×D4×C23—22no—
143C2×D4×C23—22no—
144C2×D4×C23—22no—
145C2×D4×C23—22no—
146C2×D4×C23—21no—
147C2×D4×C23—22no—
148C2×D4×C23—22no—
149C2×D4×C23—41no—
150C2×D4×C23—42no—
151C2×D4×C23—42no—
152C2×D4×C23C2×D412yes—
153C22×C4×C23—21no—
154C22×C4×C23—21no—
155C22×C4×C23—21no—
156C22×C4×C23—21no—
157C22×C4×C23—22no—
158C22×C4×C23—21no—
159C22×C4×C23—22no—
160C22×C4×C23—21no—
161C22×C4×C23—21no—
162C22×C4×C23—21no—
163C22×C4×C23—21no—
164C22×C4×C23—42no—
165C22×C4×C23—42no—
166C22×C4×C23—81no—
167C22×C4×C23—81no—
168C22×C4×C23—82no—
169C22×C4×C23—81no—
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2×D411yescharacteristic
172C22×C4×C23C2×D411yescharacteristic
173C22×C4×C23C2×D411yescharacteristic
174C22×C4×C23C2×D412yes—
175C22×C4×C23C2×D412yes—
176C22×C4×C23C2411yescharacteristic
177Q16×C23—82no—
178SD16×C23—82no—
179M16×C23—82no—
180C2×C8×C23—22no—
181C2×C8×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—42no—
193C4⋊C4×C23—41no—
194C4⋊C4×C23—41no—
195C4⋊C4×C23—81no—
196C4⋊C4×C23C2×D412yes—
197C4⋊C4×C23C2×D412yes—
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—22no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—42no—
211C22⋊C4×C23—42no—
212C22⋊C4×C23—81no—
213C42×C23—22no—
214C42×C23—22no—
215C42×C23—21no—
216C42×C23—21no—
217#18323C2311yesmaximal, characteristic, direct product factor, Sylow
218C23×C23—21no—
219C23×C23—21no—
220C23×C23—21no—
221C23×C23—21no—
222C23×C23—22no—
223C23×C23—41no—
224C23×C23—41no—
225C23×C23—41no—
226C23×C23—41no—
227C23×C23—42no—
228C23×C23—81no—
229C23×C23—81no—
230C23×C23#12012yes—
231C23×C23#13712yes—
232C23×C23C22×D411yescharacteristic
233Q8×C23—42no—
234Q8×C23—42no—
235Q8×C23—42no—
236Q8×C23—81no—
237D4×C23—42no—
238D4×C23—44no—
239D4×C23—44no—
240D4×C23—42no—
241D4×C23—42no—
242D4×C23—82no—
243D4×C23—82no—
244D4×C23—81no—
245C2×C4×C23—22no—
246C2×C4×C23—21no—
247C2×C4×C23—22no—
248C2×C4×C23—21no—
249C2×C4×C23—22no—
250C2×C4×C23—21no—
251C2×C4×C23—22no—
252C2×C4×C23—21no—
253C2×C4×C23—21no—
254C2×C4×C23—42no—
255C2×C4×C23—41no—
256C2×C4×C23—42no—
257C2×C4×C23—41no—
258C2×C4×C23—42no—
259C2×C4×C23—44no—
260C2×C4×C23—41no—
261C2×C4×C23—42no—
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—41no—
272C2×C4×C23—41no—
273C2×C4×C23—41no—
274C2×C4×C23—81no—
275C2×C4×C23—81no—
276C2×C4×C23—82no—
277C2×C4×C23—81no—
278C2×C4×C23—84no—
279C2×C4×C23#12011yescharacteristic
280C2×C4×C23#12011yescharacteristic
281C2×C4×C23C22×D412yes—
282C8×C23—42no—
283C8×C23—42no—
284#3459C2×C2311yescharacteristic
285#3196C2×C2312yescomplemented
286#2564C2×C2311yescharacteristic, complemented
287#1615C2×C2312yescomplemented
288#1614C2×C2311yescharacteristic, complemented
289#1613C2×C2311yescharacteristic, complemented
290#1603C2×C2312yescomplemented
291#1602C2×C2311yescharacteristic, complemented
292#1601C2×C2311yescharacteristic, complemented
293#1504C2×C2312yescomplemented
294#1465C2×C2311yescharacteristic, complemented
295C22×C23—21no—
296C22×C23—21no—
297C22×C23—22no—
298C22×C23—22no—
299C22×C23—22no—
300C22×C23—22no—
301C22×C23—42no—
302C22×C23—42no—
303C22×C23—42no—
304C22×C23—41no—
305C22×C23—41no—
306C22×C23—42no—
307C22×C23—42no—
308C22×C23—42no—
309C22×C23—81no—
310C22×C23—84no—
311C22×C23C2×#12011yescharacteristic
312C22×C23C2×#13711yescharacteristic
313C22×C23#57511yescharacteristic
314C4×C23—22no—
315C4×C23—21no—
316C4×C23—21no—
317C4×C23—42no—
318C4×C23—41no—
319C4×C23—42no—
320C4×C23—41no—
321C4×C23—42no—
322C4×C23—41no—
323C4×C23—42no—
324C4×C23—81no—
325C4×C23—81no—
326C4×C23—81no—
327#543C22×C2311yescharacteristic
328#539—22no—
329#539C22×C2312yes—
330#538—22no—
331#538—21no—
332#534—21no—
333#534C22×C2311yescharacteristic
334#529—22no—
335#529—21no—
336#525—22no—
337#525C22×C2312yes—
338C2×C22⋊Q8C22×C2311yescharacteristic, complemented
339C2×C4×D4—21no—
340C2×C4×D4C22×C2311yescharacteristic
341#485—22no—
342#484C22×C2312yescomplemented
343#481C22×C2312yes—
344#470—22no—
345#468—22no—
346#452—22no—
347#451C22×C2312yescomplemented
348#448C22×C2312yes—
349#407C22×C2312yescomplemented
350#400C22×C2311yescharacteristic, complemented
351#399C22×C2311yescharacteristic
352#395C22×C2311yescharacteristic, complemented
353#393C22×C2311yescharacteristic
354#387—21no—
355#386—21no—
356#385—21no—
357#380—21no—
358C2×#95C22×C2311yescharacteristic, complemented
359#328C22×C2312yescomplemented
360#328C22×C2312yescomplemented
361#327C22×C2312yes—
362#327C22×C2312yes—
363#324C22×C2312yescomplemented
364#323C22×C2311yescharacteristic
365#323C22×C2311yescharacteristic
366C2×C23—21no—
367C2×C23—21no—
368C2×C23—42no—
369C2×C23—42no—
370C2×C23—82no—
371C2×C23#300711yescharacteristic
372C2×C23#301811yescharacteristic
373C2×C23#302611yescharacteristic
374C2×#41D4×C2312yes—
375C22×Q8C23×C2311yescharacteristic
376C22×D4—21no—
377C22×D4D4×C2311yescharacteristic
378C23×C4—21no—
379C23×C4—21no—
380C23×C4D4×C2311yescharacteristic
381C23×C4C23×C2311yescharacteristic
382C2×Q16—22no—
383C2×SD16—22no—
384C2×M16—22no—
385#126—22no—
386#126—22no—
387#126—22no—
388#125—21no—
389#125—22no—
390#124—22no—
391#124—21no—
392#123—22no—
393#123—22no—
394#123—22no—
395#123—22no—
396#123—22no—
397#123—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—21no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8D4×C2312yes—
405C22⋊Q8C23×C2312yes—
406#121—22no—
407#121—22no—
408#121—22no—
409#121—21no—
410#121D4×C2312yes—
411#121C23×C2312yes—
412#120—22no—
413#120—22no—
414#120—22no—
415C4×Q8—21no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—21no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424#117—22no—
425#117D4×C2312yes—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4C23×C2311yescharacteristic
431C2×C4⋊C4C23×C2311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4D4×C2311yescharacteristic
438C2×C22⋊C4C23×C2311yescharacteristic
439C2×C42—21no—
440C2×C42D4×C2311yescharacteristic
441#107—22no—
442#106—22no—
443#103—22no—
444#103—22no—
445#103—22no—
446#102—22no—
447C22⋊C8—22no—
448C22⋊C8C23×C2312yes—
449C22⋊C8C23×C2312yes—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95C23×C2311yescharacteristic
456#95C23×C2311yescharacteristic
457C23#1832311yescharacteristic, direct product factor, Sylow
458C24—21no—
459C24C2×D4×C2311yescharacteristic
460C24C2×D4×C2311yescharacteristic
461#41—82no—
462C2×Q8—21no—
463C2×Q8—22no—
464C2×Q8—81no—
465C2×Q8C2×D4×C2312yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—41no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4C2×D4×C2312yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—82no—
493C22×C4—81no—
494C22×C4C2×D4×C2311yescharacteristic
495C22×C4C2×D4×C2311yescharacteristic
496C22×C4C2×D4×C2311yescharacteristic
497C22×C4C2×D4×C2311yescharacteristic
498C22×C4C2×D4×C2312yes—
499C22×C4C2×D4×C2312yes—
500C22×C4C24×C2311yescharacteristic
501Q16—82no—
502SD16—82no—
503M16—82no—
504C2×C8—22no—
505C2×C8—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—81no—
520C4⋊C4C2×D4×C2312yes—
521C4⋊C4C2×D4×C2312yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—81no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—81no—
552C23—81no—
553C23C23×#12012yes—
554C23C23×#13712yes—
555C23C22×D4×C2311yescharacteristic
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—81no—
560D4—42no—
561D4—44no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4—82no—
566D4—82no—
567D4—81no—
568C2×C4—22no—
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—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
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—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—84no—
602C2×C4C23×#12011yescharacteristic
603C2×C4C23×#12011yescharacteristic
604C2×C4C22×D4×C2312yes—
605C8—42no—
606C8—42no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—84no—
623C22C2×C23×#12011yescharacteristic
624C22C2×C23×#13711yescharacteristic
625C22C23×#57511yescharacteristic
626C4—22no—
627C4—21no—
628C4—21no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—81no—
639C2—21no—
640C2—21no—
641C2—42no—
642C2—42no—
643C2—82no—
644C2C23×#300711yescharacteristic
645C2C23×#301811yescharacteristic
646C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3196C2semidirect
3. C23×#2564C2semidirect
4. C23×#1615C2semidirect
5. C23×#1614C2semidirect
6. C23×#1613C2semidirect
7. C23×#1603C2semidirect
8. C23×#1602C2semidirect
9. C23×#1601C2semidirect
10. C23×#1504C2semidirect
11. C23×#1465C2semidirect
23. C2×C23×C22⋊Q8C22semidirect
27. C23×#484C22semidirect
32. C23×#451C22semidirect
34. C23×#407C22semidirect
35. C23×#400C22semidirect
37. C23×#395C22semidirect
43. C2×C23×#95C22semidirect
44. C23×#328C22semidirect
45. C23×#328C22semidirect
48. C23×#324C22semidirect
217. #18323C23direct (N × H)
285. #3196C2×C23semidirect
286. #2564C2×C23semidirect
287. #1615C2×C23semidirect
288. #1614C2×C23semidirect
289. #1613C2×C23semidirect
290. #1603C2×C23semidirect
291. #1602C2×C23semidirect
292. #1601C2×C23semidirect
293. #1504C2×C23semidirect
294. #1465C2×C23semidirect
338. C2×C22⋊Q8C22×C23semidirect
342. #484C22×C23semidirect
347. #451C22×C23semidirect
349. #407C22×C23semidirect
350. #400C22×C23semidirect
352. #395C22×C23semidirect
358. C2×#95C22×C23semidirect
359. #328C22×C23semidirect
360. #328C22×C23semidirect
363. #324C22×C23semidirect
457. C23#18323direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34592
C23×#31962
C23×#25642
C23×#16152
C23×#16142
C23×#16132
C23×#16032
C23×#16022
C23×#16012
C23×#15042
C23×#14652
#1832323