Finite Group Databasebeta

← C23×#18246 #19165872

Subgroups — C23×#18246 #19165872

Order 5888.

900 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3563C211yesmaximal, characteristic, complemented
2C23×#3114C211yesmaximal, characteristic
3C23×#3101C211yesmaximal, characteristic, complemented
4C23×#2633C211yesmaximal, characteristic, complemented
5C23×#1611C212yesmaximal, complemented
6C23×#1609C212yesmaximal, complemented
7C23×#1599C212yesmaximal, complemented
8C23×#1597C212yesmaximal
9C23×#1521C211yesmaximal, characteristic, complemented
10C23×#1515C211yesmaximal, characteristic, complemented
11C23×#1470C211yesmaximal, characteristic, complemented
12C23×#564—21no—
13C23×#563—21no—
14C23×#557—21no—
15C23×#554—21no—
16C23×#552—21no—
17C23×#552—22no—
18C23×#552—21no—
19C23×#544—21no—
20C23×#544C2212yes—
21C23×#541—21no—
22C23×#541—22no—
23C23×#539C2212yes—
24C23×#533C2211yescharacteristic
25C23×#529C2211yescharacteristic
26C2×C23×#125C2211yescharacteristic
27C2×C4×D4×C23C2211yescharacteristic
28C23×#483C2212yes—
29C23×#483C2211yescharacteristic
30C23×#480C2211yescharacteristic
31C23×#478—21no—
32C23×#476—21no—
33C23×#463—21no—
34C23×#461—21no—
35C23×#450C2212yes—
36C23×#450C2211yescharacteristic
37C23×#447C2211yescharacteristic
38C23×#442—21no—
39C23×#440—21no—
40C4×SD16×C23—22no—
41C23×#432C2211yescharacteristic
42C23×#431C2211yescharacteristic
43C23×#399—21no—
44C23×#398C2212yes—
45C23×#397—21no—
46C23×#394C2212yes—
47C23×#385—22no—
48C23×#383C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#328C2212yes—
52C23×#327C2212yes—
53C23×#324C2212yes—
54C23×#324C2212yes—
55C22×D4×C23C2311yescharacteristic
56C23×C4×C23—21no—
57C2×SD16×C23—22no—
58C2×SD16×C23—21no—
59C2×D8×C23—21no—
60C23×C4⋊Q8—21no—
61C23×C4⋊Q8—22no—
62C23×C4⋊Q8—22no—
63C23×C4⋊Q8D411yescharacteristic
64C23×C4⋊Q8D411yescharacteristic
65C23×#126—22no—
66C23×#126—22no—
67C23×#126—22no—
68C23×#126—22no—
69C23×#126—21no—
70C23×#125—22no—
71C23×#125—21no—
72C23×#125—22no—
73C23×#125—22no—
74C23×#125—22no—
75C23×#125—22no—
76C23×#125—22no—
77C23×#125—22no—
78C23×#125—22no—
79C23×#125—21no—
80C23×#125—21no—
81C23×#125—21no—
82C23×#125—21no—
83C23×#125—21no—
84C23×#125D411yescharacteristic
85C23×#124D411yescharacteristic
86C23×#123—22no—
87C23×#123—21no—
88C23×#123—22no—
89C23×C22⋊Q8—21no—
90C23×C22⋊Q8—21no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—21no—
95C23×C22⋊Q8—22no—
96C23×C22⋊Q8C2311yescharacteristic
97C23×C22⋊Q8C2312yes—
98C23×#121C2311yescharacteristic
99C4×Q8×C23—21no—
100C4×Q8×C23—22no—
101C4×Q8×C23—21no—
102C4×Q8×C23—21no—
103C4×Q8×C23D411yescharacteristic
104C4×Q8×C23D412yes—
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×C23D411yescharacteristic
112C23×#117—22no—
113C23×#117—21no—
114C23×#117—21no—
115C23×#117—22no—
116C23×#117—21no—
117C2×C4⋊C4×C23—21no—
118C2×C4⋊C4×C23—22no—
119C2×C4⋊C4×C23D412yes—
120C2×C4⋊C4×C23D411yescharacteristic
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×C23C2311yescharacteristic
128C23×#107—21no—
129C23×#106—22no—
130C23×#106—21no—
131C23×C4⋊C8—21no—
132C23×C4⋊C8—21no—
133C23×#103—22no—
134C23×#103—21no—
135C23×#102—21no—
136C23×#102—22no—
137C23×#102—21no—
138C23×#102—21no—
139C23×C22⋊C8C2312yes—
140C23×C22⋊C8C2311yescharacteristic
141C23×C22⋊C8C2311yescharacteristic
142C23×#97—21no—
143C4×C8×C23—21no—
144C23×#95—21no—
145C23×#95—21no—
146C23×#95—22no—
147C23×#95C2312yes—
148C24×C23—21no—
149C2×Q8×C23—22no—
150C2×Q8×C23—21no—
151C2×Q8×C23C2×D412yes—
152C2×Q8×C23C2×D411yescharacteristic
153C2×D4×C23—21no—
154C2×D4×C23—21no—
155C2×D4×C23—21no—
156C2×D4×C23—42no—
157C2×D4×C23C2×D411yescharacteristic
158C22×C4×C23—22no—
159C22×C4×C23—21no—
160C22×C4×C23—22no—
161C22×C4×C23—21no—
162C22×C4×C23—41no—
163C22×C4×C23—41no—
164C22×C4×C23—82no—
165C22×C4×C23C2×D412yes—
166C22×C4×C23C2×D411yescharacteristic
167C22×C4×C23C2×D412yes—
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2411yescharacteristic
170SD16×C23—42no—
171SD16×C23—44no—
172D8×C23—42no—
173C2×C8×C23—21no—
174C2×C8×C23—21no—
175C2×C8×C23—22no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—21no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—21no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—22no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—22no—
199C4⋊C4×C23—21no—
200C4⋊C4×C23—22no—
201C4⋊C4×C23—21no—
202C4⋊C4×C23—21no—
203C4⋊C4×C23—82no—
204C4⋊C4×C23C2×D411yescharacteristic
205C4⋊C4×C23C2×D412yes—
206C4⋊C4×C23C2×D411yescharacteristic
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—21no—
213C22⋊C4×C23—21no—
214C22⋊C4×C23—21no—
215C22⋊C4×C23—21no—
216C22⋊C4×C23—44no—
217C22⋊C4×C23—41no—
218C22⋊C4×C23—41no—
219C22⋊C4×C23—82no—
220C42×C23—22no—
221C42×C23—22no—
222C42×C23—22no—
223C42×C23—22no—
224C42×C23—21no—
225C42×C23C2×D411yescharacteristic
226C42×C23C2×D411yescharacteristic
227C42×C23C2×D411yescharacteristic
228C42×C23C2×D411yescharacteristic
229#18246C2311yesmaximal, characteristic, direct product factor, Sylow
230C23×C23—22no—
231C23×C23—21no—
232C23×C23—41no—
233C23×C23—42no—
234C23×C23—41no—
235C23×C23—42no—
236C23×C23C22×D411yescharacteristic
237Q8×C23—41no—
238Q8×C23—42no—
239Q8×C23—42no—
240Q8×C23—82no—
241D4×C23—22no—
242D4×C23—22no—
243D4×C23—41no—
244D4×C23—41no—
245D4×C23—41no—
246D4×C23—42no—
247C2×C4×C23—22no—
248C2×C4×C23—22no—
249C2×C4×C23—21no—
250C2×C4×C23—21no—
251C2×C4×C23—22no—
252C2×C4×C23—21no—
253C2×C4×C23—22no—
254C2×C4×C23—22no—
255C2×C4×C23—22no—
256C2×C4×C23—22no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—42no—
260C2×C4×C23—42no—
261C2×C4×C23—42no—
262C2×C4×C23—42no—
263C2×C4×C23—41no—
264C2×C4×C23—42no—
265C2×C4×C23—42no—
266C2×C4×C23—82no—
267C2×C4×C23—82no—
268C2×C4×C23—82no—
269C2×C4×C23#12011yescharacteristic
270C2×C4×C23#12011yescharacteristic
271C2×C4×C23#12011yescharacteristic
272C2×C4×C23#12011yescharacteristic
273C2×C4×C23C22×D411yescharacteristic
274C2×C4×C23C22×D411yescharacteristic
275C8×C23—41no—
276C8×C23—41no—
277C8×C23—42no—
278#3563C2×C2311yescharacteristic, complemented
279#3114C2×C2311yescharacteristic
280#3101C2×C2311yescharacteristic, complemented
281#2633C2×C2311yescharacteristic, complemented
282#1611C2×C2312yescomplemented
283#1609C2×C2312yescomplemented
284#1599C2×C2312yescomplemented
285#1597C2×C2312yes—
286#1521C2×C2311yescharacteristic, complemented
287#1515C2×C2311yescharacteristic, complemented
288#1470C2×C2311yescharacteristic, complemented
289C22×C23—21no—
290C22×C23—21no—
291C22×C23—44no—
292C22×C23—41no—
293C22×C23—41no—
294C22×C23—41no—
295C22×C23—42no—
296C22×C23—82no—
297C22×C23—81no—
298C22×C23—81no—
299C22×C23C2×#12011yescharacteristic
300C22×C23#57712yes—
301C4×C23—21no—
302C4×C23—21no—
303C4×C23—21no—
304C4×C23—21no—
305C4×C23—21no—
306C4×C23—21no—
307C4×C23—42no—
308C4×C23—41no—
309C4×C23—41no—
310C4×C23—42no—
311C4×C23—42no—
312C4×C23—42no—
313C4×C23—41no—
314C4×C23—82no—
315C4×C23—81no—
316#564—21no—
317#563—21no—
318#557—21no—
319#554—21no—
320#552—21no—
321#552—22no—
322#552—21no—
323#544—21no—
324#544C22×C2312yes—
325#541—21no—
326#541—22no—
327#539C22×C2312yes—
328#533C22×C2311yescharacteristic
329#529C22×C2311yescharacteristic
330C2×#125C22×C2311yescharacteristic
331C2×C4×D4C22×C2311yescharacteristic
332#483C22×C2312yes—
333#483C22×C2311yescharacteristic
334#480C22×C2311yescharacteristic
335#478—21no—
336#476—21no—
337#463—21no—
338#461—21no—
339#450C22×C2312yes—
340#450C22×C2311yescharacteristic
341#447C22×C2311yescharacteristic
342#442—21no—
343#440—21no—
344C4×SD16—22no—
345#432C22×C2311yescharacteristic
346#431C22×C2311yescharacteristic
347#399—21no—
348#398C22×C2312yes—
349#397—21no—
350#394C22×C2312yes—
351#385—22no—
352#383C22×C2311yescharacteristic
353#328C22×C2312yes—
354#328C22×C2312yes—
355#328C22×C2312yes—
356#327C22×C2312yes—
357#324C22×C2312yes—
358#324C22×C2312yes—
359C2×C23—22no—
360C2×C23—81no—
361C2×C23—82no—
362C2×C23#301011yescharacteristic
363C2×C23#302011yescharacteristic
364C2×C23#302511yescharacteristic
365C22×D4C23×C2311yescharacteristic
366C23×C4—21no—
367C2×SD16—22no—
368C2×SD16—21no—
369C2×D8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—22no—
372C4⋊Q8—22no—
373C4⋊Q8D4×C2311yescharacteristic
374C4⋊Q8D4×C2311yescharacteristic
375#126—22no—
376#126—22no—
377#126—22no—
378#126—22no—
379#126—21no—
380#125—22no—
381#125—22no—
382#125—22no—
383#125—22no—
384#125—21no—
385#125—22no—
386#125—22no—
387#125—22no—
388#125—22no—
389#125—21no—
390#125—21no—
391#125—21no—
392#125—21no—
393#125—21no—
394#125D4×C2311yescharacteristic
395#124D4×C2311yescharacteristic
396#123—22no—
397#123—21no—
398#123—22no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8C23×C2311yescharacteristic
407C22⋊Q8C23×C2312yes—
408#121C23×C2311yescharacteristic
409C4×Q8—21no—
410C4×Q8—22no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8D4×C2311yescharacteristic
414C4×Q8D4×C2312yes—
415C4×D4—21no—
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4D4×C2311yescharacteristic
422#117—22no—
423#117—21no—
424#117—21no—
425#117—22no—
426#117—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4D4×C2312yes—
430C2×C4⋊C4D4×C2311yescharacteristic
431C2×C4⋊C4D4×C2311yescharacteristic
432C2×C4⋊C4C23×C2311yescharacteristic
433C2×C4⋊C4C23×C2312yes—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—22no—
436C2×C22⋊C4—22no—
437C2×C42C23×C2311yescharacteristic
438#107—21no—
439#106—22no—
440#106—21no—
441C4⋊C8—21no—
442C4⋊C8—21no—
443#103—22no—
444#103—21no—
445#102—21no—
446#102—22no—
447#102—21no—
448#102—21no—
449C22⋊C8C23×C2312yes—
450C22⋊C8C23×C2311yescharacteristic
451C22⋊C8C23×C2311yescharacteristic
452#97—21no—
453C4×C8—21no—
454#95—21no—
455#95—21no—
456#95—22no—
457#95C23×C2312yes—
458C23#1824611yescharacteristic, direct product factor, Sylow
459C24—21no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8C2×D4×C2312yes—
463C2×Q8C2×D4×C2311yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—42no—
468C2×D4C2×D4×C2311yescharacteristic
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—82no—
476C22×C4C2×D4×C2312yes—
477C22×C4C2×D4×C2311yescharacteristic
478C22×C4C2×D4×C2312yes—
479C22×C4C2×D4×C2311yescharacteristic
480C22×C4C24×C2311yescharacteristic
481SD16—42no—
482SD16—44no—
483D8—42no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—82no—
515C4⋊C4C2×D4×C2311yescharacteristic
516C4⋊C4C2×D4×C2312yes—
517C4⋊C4C2×D4×C2311yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—44no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—82no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42C2×D4×C2311yescharacteristic
537C42C2×D4×C2311yescharacteristic
538C42C2×D4×C2311yescharacteristic
539C42C2×D4×C2311yescharacteristic
540C23—22no—
541C23—21no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23C22×D4×C2311yescharacteristic
547Q8—41no—
548Q8—42no—
549Q8—42no—
550Q8—82no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—42no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4C23×#12011yescharacteristic
580C2×C4C23×#12011yescharacteristic
581C2×C4C23×#12011yescharacteristic
582C2×C4C23×#12011yescharacteristic
583C2×C4C22×D4×C2311yescharacteristic
584C2×C4C22×D4×C2311yescharacteristic
585C8—41no—
586C8—41no—
587C8—42no—
588C22—21no—
589C22—21no—
590C22—44no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22C2×C23×#12011yescharacteristic
599C22C23×#57712yes—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—42no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—82no—
614C4—81no—
615C2—22no—
616C2—82no—
617C2—81no—
618C2C23×#301011yescharacteristic
619C2C23×#302011yescharacteristic
620C2C23×#302511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#447C22
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×#3563C2semidirect
3. C23×#3101C2semidirect
4. C23×#2633C2semidirect
5. C23×#1611C2semidirect
6. C23×#1609C2semidirect
7. C23×#1599C2semidirect
9. C23×#1521C2semidirect
10. C23×#1515C2semidirect
11. C23×#1470C2semidirect
229. #18246C23direct (N × H)
278. #3563C2×C23semidirect
280. #3101C2×C23semidirect
281. #2633C2×C23semidirect
282. #1611C2×C23semidirect
283. #1609C2×C23semidirect
284. #1599C2×C23semidirect
286. #1521C2×C23semidirect
287. #1515C2×C23semidirect
288. #1470C2×C23semidirect
458. C23#18246direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35632
C23×#31142
C23×#31012
C23×#26332
C23×#16112
C23×#16092
C23×#15992
C23×#15972
C23×#15212
C23×#15152
C23×#14702
#1824623