Finite Group Databasebeta

← C23×#18267 #19165893

Subgroups — C23×#18267 #19165893

Order 5888.

1056 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3536C211yesmaximal, characteristic
2C23×#3092C211yesmaximal, characteristic, complemented
3C2×C23×#484C211yesmaximal, characteristic, complemented
4C23×#2501C211yesmaximal, characteristic, complemented
5C23×#1618C214yesmaximal, complemented
6C23×#1614C214yesmaximal, complemented
7C23×#1508C211yesmaximal, characteristic, complemented
8C23×#1501C211yesmaximal, characteristic, complemented
9C23×#1471C211yesmaximal, characteristic, complemented
10C23×#562—22no—
11C23×#554—21no—
12C23×#548—21no—
13C23×#547—21no—
14C23×#546—22no—
15C23×#546—21no—
16C23×#543C2212yes—
17C23×#542—21no—
18C23×#539—21no—
19C23×#538C2212yes—
20C23×#537—21no—
21C23×#537—21no—
22C23×#525C2211yescharacteristic
23C2×C23×#123—22no—
24C2×C23×C22⋊Q8C2211yescharacteristic
25C2×C23×#121C2211yescharacteristic
26C22×C4⋊C4×C23C2211yescharacteristic
27C23×#484—22no—
28C23×#484C2214yes—
29C23×#484C2212yes—
30C23×#483—21no—
31C23×#481C2212yes—
32C23×#480—21no—
33C23×#428—21no—
34C2×C23×#107—21no—
35C23×#417—21no—
36C2×C23×#103—21no—
37C23×#407C2211yescharacteristic
38C2×C23×C22⋊C8C2211yescharacteristic
39C23×#399C2214yes—
40C23×#390—21no—
41C23×#384—21no—
42C4×C22⋊C4×C23C2211yescharacteristic
43C23×#328C2212yes—
44C23×#328C2214yes—
45C23×#327C2212yes—
46C23×#323C2214yes—
47C2×#41×C23—22no—
48C2×#41×C23—21no—
49C2×#41×C23D411yescharacteristic
50C22×Q8×C23D411yescharacteristic
51C22×D4×C23D411yescharacteristic
52C22×D4×C23D412yes—
53C23×C4×C23—21no—
54C23×C4×C23C2311yescharacteristic
55C2×M16×C23—21no—
56C22×C8×C23—21no—
57C23×C4⋊Q8—21no—
58C23×#126—22no—
59C23×#125—22no—
60C23×#124—21no—
61C23×#124—22no—
62C23×#124—22no—
63C23×#123—22no—
64C23×#123—22no—
65C23×#123—22no—
66C23×#123—21no—
67C23×#123—22no—
68C23×#123—48no—
69C23×C22⋊Q8—21no—
70C23×C22⋊Q8—22no—
71C23×C22⋊Q8—22no—
72C23×C22⋊Q8—22no—
73C23×C22⋊Q8—21no—
74C23×C22⋊Q8—22no—
75C23×C22⋊Q8—21no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—22no—
78C23×C22⋊Q8D412yes—
79C23×C22⋊Q8D411yescharacteristic
80C23×C22⋊Q8C2312yes—
81C23×C22⋊Q8C2311yescharacteristic
82C23×#121—21no—
83C23×#121—22no—
84C23×#121—22no—
85C23×#121—22no—
86C23×#121—22no—
87C23×#121—22no—
88C23×#121—21no—
89C23×#121—21no—
90C23×#121—21no—
91C23×#121—21no—
92C23×#121D411yescharacteristic
93C23×#121C2311yescharacteristic
94C23×#120—21no—
95C23×#120—22no—
96C4×Q8×C23—21no—
97C4×D4×C23—21no—
98C4×D4×C23—22no—
99C4×D4×C23—21no—
100C4×D4×C23—21no—
101C4×D4×C23—22no—
102C23×#117D411yescharacteristic
103C2×C4⋊C4×C23—21no—
104C2×C4⋊C4×C23—21no—
105C2×C4⋊C4×C23—21no—
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23—21no—
108C2×C4⋊C4×C23—21no—
109C2×C4⋊C4×C23—21no—
110C2×C4⋊C4×C23—24no—
111C2×C4⋊C4×C23D411yescharacteristic
112C2×C4⋊C4×C23D411yescharacteristic
113C2×C4⋊C4×C23C2312yes—
114C2×C22⋊C4×C23—21no—
115C2×C22⋊C4×C23—21no—
116C2×C22⋊C4×C23—22no—
117C2×C22⋊C4×C23—24no—
118C2×C22⋊C4×C23C2312yes—
119C2×C42×C23—21no—
120C23×#107—24no—
121C23×#107—22no—
122C23×#106—22no—
123C23×#103—24no—
124C23×#103—22no—
125C23×#102—22no—
126C23×C22⋊C8—21no—
127C23×C22⋊C8—21no—
128C23×C22⋊C8C2312yes—
129C23×C22⋊C8C2312yes—
130C23×#95C2312yes—
131C24×C23C2×D411yescharacteristic
132C24×C23C2×D412yes—
133#41×C23—42no—
134#41×C23—84no—
135#41×C23—82no—
136C2×Q8×C23—22no—
137C2×Q8×C23—21no—
138C2×Q8×C23—42no—
139C2×Q8×C23C2×D411yescharacteristic
140C2×Q8×C23C2×D412yes—
141C2×D4×C23—21no—
142C2×D4×C23—22no—
143C2×D4×C23—21no—
144C2×D4×C23—22no—
145C2×D4×C23—22no—
146C2×D4×C23—21no—
147C2×D4×C23—22no—
148C2×D4×C23—21no—
149C2×D4×C23—21no—
150C2×D4×C23—42no—
151C2×D4×C23—44no—
152C2×D4×C23C2×D411yescharacteristic
153C22×C4×C23—21no—
154C22×C4×C23—22no—
155C22×C4×C23—22no—
156C22×C4×C23—22no—
157C22×C4×C23—22no—
158C22×C4×C23—21no—
159C22×C4×C23—21no—
160C22×C4×C23—22no—
161C22×C4×C23—42no—
162C22×C4×C23—44no—
163C22×C4×C23—41no—
164C22×C4×C23—41no—
165C22×C4×C23C2×D411yescharacteristic
166C22×C4×C23C2×D411yescharacteristic
167C22×C4×C23C2×D411yescharacteristic
168C22×C4×C23C2×D412yes—
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2411yescharacteristic
172M16×C23—42no—
173C2×C8×C23—22no—
174C2×C8×C23—22no—
175C2×C8×C23—42no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—21no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—21no—
184C4⋊C4×C23—21no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—48no—
190C4⋊C4×C23—41no—
191C4⋊C4×C23—41no—
192C4⋊C4×C23C2×D411yescharacteristic
193C4⋊C4×C23C2×D412yes—
194C4⋊C4×C23C2×D411yescharacteristic
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—24no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—21no—
203C22⋊C4×C23—21no—
204C22⋊C4×C23—22no—
205C22⋊C4×C23—21no—
206C22⋊C4×C23—21no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—21no—
210C22⋊C4×C23—48no—
211C42×C23—22no—
212C42×C23—21no—
213C42×C23—42no—
214#18267C2311yesmaximal, characteristic, direct product factor, Sylow
215C23×C23—22no—
216C23×C23—22no—
217C23×C23—22no—
218C23×C23—21no—
219C23×C23—24no—
220C23×C23—41no—
221C23×C23—42no—
222C23×C23—42no—
223C23×C23#12011yescharacteristic
224C23×C23#12011yescharacteristic
225C23×C23C22×D411yescharacteristic
226Q8×C23—42no—
227Q8×C23—41no—
228Q8×C23—42no—
229Q8×C23—82no—
230D4×C23—41no—
231D4×C23—42no—
232D4×C23—44no—
233D4×C23—42no—
234D4×C23—42no—
235D4×C23—81no—
236D4×C23—82no—
237D4×C23—81no—
238D4×C23—82no—
239D4×C23—82no—
240C2×C4×C23—22no—
241C2×C4×C23—22no—
242C2×C4×C23—22no—
243C2×C4×C23—24no—
244C2×C4×C23—21no—
245C2×C4×C23—22no—
246C2×C4×C23—21no—
247C2×C4×C23—22no—
248C2×C4×C23—21no—
249C2×C4×C23—22no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—42no—
254C2×C4×C23—42no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
257C2×C4×C23—42no—
258C2×C4×C23—44no—
259C2×C4×C23—42no—
260C2×C4×C23—81no—
261C2×C4×C23—82no—
262C2×C4×C23—82no—
263C2×C4×C23—82no—
264C2×C4×C23—82no—
265C2×C4×C23—82no—
266C2×C4×C23#12011yescharacteristic
267C2×C4×C23#12011yescharacteristic
268C2×C4×C23C22×D411yescharacteristic
269C2×C4×C23C22×D411yescharacteristic
270C8×C23—42no—
271C8×C23—42no—
272#3536C2×C2311yescharacteristic
273#3092C2×C2311yescharacteristic, complemented
274C2×#484C2×C2311yescharacteristic, complemented
275#2501C2×C2311yescharacteristic, complemented
276#1618C2×C2314yescomplemented
277#1614C2×C2314yescomplemented
278#1508C2×C2311yescharacteristic, complemented
279#1501C2×C2311yescharacteristic, complemented
280#1471C2×C2311yescharacteristic, complemented
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—21no—
284C22×C23—21no—
285C22×C23—21no—
286C22×C23—21no—
287C22×C23—21no—
288C22×C23—44no—
289C22×C23—41no—
290C22×C23—42no—
291C22×C23—41no—
292C22×C23—42no—
293C22×C23—41no—
294C22×C23—42no—
295C22×C23—84no—
296C22×C23#45312yes—
297C22×C23C2×#12011yescharacteristic
298C22×C23#57812yes—
299C4×C23—21no—
300C4×C23—21no—
301C4×C23—21no—
302C4×C23—44no—
303C4×C23—41no—
304C4×C23—42no—
305C4×C23—42no—
306C4×C23—41no—
307C4×C23—41no—
308C4×C23—42no—
309C4×C23—82no—
310C4×C23#45812yes—
311#562—22no—
312#554—21no—
313#548—21no—
314#547—21no—
315#546—22no—
316#546—21no—
317#543C22×C2312yes—
318#542—21no—
319#539—21no—
320#538C22×C2312yes—
321#537—21no—
322#537—21no—
323#525C22×C2311yescharacteristic
324C2×#123—22no—
325C2×C22⋊Q8C22×C2311yescharacteristic
326C2×#121C22×C2311yescharacteristic
327C22×C4⋊C4C22×C2311yescharacteristic
328#484—22no—
329#484C22×C2314yes—
330#484C22×C2312yes—
331#483—21no—
332#481C22×C2312yes—
333#480—21no—
334#428—21no—
335C2×#107—21no—
336#417—21no—
337C2×#103—21no—
338#407C22×C2311yescharacteristic
339C2×C22⋊C8C22×C2311yescharacteristic
340#399C22×C2314yes—
341#390—21no—
342#384—21no—
343C4×C22⋊C4C22×C2311yescharacteristic
344#328C22×C2312yes—
345#328C22×C2314yes—
346#327C22×C2312yes—
347#323C22×C2314yes—
348C2×C23—22no—
349C2×C23—22no—
350C2×C23—41no—
351C2×C23—81no—
352C2×C23—82no—
353C2×C23#301111yescharacteristic
354C2×C23C2×#45311yescharacteristic
355C2×C23C2×#45811yescharacteristic
356C2×#41—22no—
357C2×#41—21no—
358C2×#41D4×C2311yescharacteristic
359C22×Q8D4×C2311yescharacteristic
360C22×D4D4×C2312yes—
361C22×D4D4×C2311yescharacteristic
362C23×C4—21no—
363C23×C4C23×C2311yescharacteristic
364C2×M16—21no—
365C22×C8—21no—
366C4⋊Q8—21no—
367#126—22no—
368#125—22no—
369#124—21no—
370#124—22no—
371#124—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375#123—21no—
376#123—22no—
377#123—48no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8D4×C2312yes—
388C22⋊Q8D4×C2311yescharacteristic
389C22⋊Q8C23×C2312yes—
390C22⋊Q8C23×C2311yescharacteristic
391#121—21no—
392#121—22no—
393#121—22no—
394#121—22no—
395#121—22no—
396#121—22no—
397#121—21no—
398#121—21no—
399#121—21no—
400#121—21no—
401#121D4×C2311yescharacteristic
402#121C23×C2311yescharacteristic
403#120—21no—
404#120—22no—
405C4×Q8—21no—
406C4×D4—22no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—22no—
411#117D4×C2311yescharacteristic
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—24no—
420C2×C4⋊C4D4×C2311yescharacteristic
421C2×C4⋊C4D4×C2311yescharacteristic
422C2×C4⋊C4C23×C2312yes—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—24no—
427C2×C22⋊C4C23×C2312yes—
428C2×C42—21no—
429#107—24no—
430#107—22no—
431#106—22no—
432#103—24no—
433#103—22no—
434#102—22no—
435C22⋊C8—21no—
436C22⋊C8—21no—
437C22⋊C8C23×C2312yes—
438C22⋊C8C23×C2312yes—
439#95C23×C2312yes—
440C23#1826711yescharacteristic, direct product factor, Sylow
441C24C2×D4×C2311yescharacteristic
442C24C2×D4×C2312yes—
443#41—42no—
444#41—84no—
445#41—82no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—42no—
449C2×Q8C2×D4×C2311yescharacteristic
450C2×Q8C2×D4×C2312yes—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—42no—
461C2×D4—44no—
462C2×D4C2×D4×C2311yescharacteristic
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4C2×D4×C2311yescharacteristic
476C22×C4C2×D4×C2311yescharacteristic
477C22×C4C2×D4×C2311yescharacteristic
478C22×C4C2×D4×C2312yes—
479C22×C4C2×D4×C2311yescharacteristic
480C22×C4C2×D4×C2311yescharacteristic
481C22×C4C24×C2311yescharacteristic
482M16—42no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—42no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—48no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4C2×D4×C2311yescharacteristic
503C4⋊C4C2×D4×C2312yes—
504C4⋊C4C2×D4×C2311yescharacteristic
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—48no—
521C42—22no—
522C42—21no—
523C42—42no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—21no—
528C23—24no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23C23×#12011yescharacteristic
533C23C23×#12011yescharacteristic
534C23C22×D4×C2311yescharacteristic
535Q8—42no—
536Q8—41no—
537Q8—42no—
538Q8—82no—
539D4—41no—
540D4—42no—
541D4—44no—
542D4—42no—
543D4—42no—
544D4—81no—
545D4—82no—
546D4—81no—
547D4—82no—
548D4—82no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4C23×#12011yescharacteristic
576C2×C4C23×#12011yescharacteristic
577C2×C4C22×D4×C2311yescharacteristic
578C2×C4C22×D4×C2311yescharacteristic
579C8—42no—
580C8—42no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—44no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—42no—
595C22—84no—
596C22C23×#45312yes—
597C22C2×C23×#12011yescharacteristic
598C22C23×#57812yes—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—44no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—42no—
609C4—82no—
610C4C23×#45812yes—
611C2—22no—
612C2—22no—
613C2—41no—
614C2—82no—
615C2—81no—
616C2C23×#301111yescharacteristic
617C2C2×C23×#45311yescharacteristic
618C2C2×C23×#45811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3536C2
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×#3092C2semidirect
3. C2×C23×#484C2semidirect
4. C23×#2501C2semidirect
5. C23×#1618C2semidirect
6. C23×#1614C2semidirect
7. C23×#1508C2semidirect
8. C23×#1501C2semidirect
9. C23×#1471C2semidirect
214. #18267C23direct (N × H)
273. #3092C2×C23semidirect
274. C2×#484C2×C23semidirect
275. #2501C2×C23semidirect
276. #1618C2×C23semidirect
277. #1614C2×C23semidirect
278. #1508C2×C23semidirect
279. #1501C2×C23semidirect
280. #1471C2×C23semidirect
440. C23#18267direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35362
C23×#30922
C2×C23×#4842
C23×#25012
C23×#16182
C23×#16142
C23×#15082
C23×#15012
C23×#14712
#1826723