Finite Group Databasebeta

← C23×#18298 #19165924

Subgroups — C23×#18298 #19165924

Order 5888.

1024 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3460C211yesmaximal, characteristic, complemented
2C23×#3188C212yesmaximal, complemented
3C23×#2635C211yesmaximal, characteristic, complemented
4C23×#1618C212yesmaximal, complemented
5C23×#1617C211yesmaximal, characteristic, complemented
6C23×#1616C211yesmaximal, characteristic, complemented
7C23×#1600C212yesmaximal, complemented
8C23×#1599C211yesmaximal, characteristic, complemented
9C23×#1598C211yesmaximal, characteristic, complemented
10C23×#1516C212yesmaximal, complemented
11C23×#1459C211yesmaximal, characteristic, complemented
12C23×#544C2211yescharacteristic
13C23×#542—22no—
14C23×#540—22no—
15C23×#538—21no—
16C23×#538C2212yes—
17C23×#537C2211yescharacteristic
18C23×#535—21no—
19C23×#532—22no—
20C23×#529—22no—
21C23×#525—21no—
22C23×#525C2212yes—
23C2×C23×#121C2211yescharacteristic
24C4×#41×C23—21no—
25C2×C4×D4×C23C2211yescharacteristic
26C23×#484C2212yes—
27C23×#482—22no—
28C23×#481C2212yes—
29C23×#466—22no—
30C23×#465—22no—
31C23×#450C2212yes—
32C23×#449—22no—
33C23×#447C2212yes—
34C2×C23×C22⋊C8C2212yes—
35C23×#400C2211yescharacteristic
36C23×#399C2211yescharacteristic
37C23×#394—21no—
38C23×#394C2211yescharacteristic
39C23×#392—21no—
40C23×#392C2211yescharacteristic
41C23×#388—21no—
42C23×#386—21no—
43C23×#381C2211yescharacteristic
44C23×#328C2212yes—
45C23×#328C2212yes—
46C23×#327C2212yes—
47C23×#327C2212yes—
48C23×#324C2211yescharacteristic
49C23×#324C2211yescharacteristic
50C23×#323C2212yes—
51C2×#41×C23—21no—
52C2×#41×C23—21no—
53C2×#41×C23D412yescomplemented
54C22×D4×C23—21no—
55C22×D4×C23—21no—
56C22×D4×C23D411yescharacteristic
57C22×D4×C23C2311yescharacteristic
58C23×C4×C23D411yescharacteristic
59C23×C4×C23C2311yescharacteristic
60C2×SD16×C23—22no—
61C2×D8×C23—22no—
62C22×C8×C23—22no—
63C23×C4⋊Q8—22no—
64C23×#127—22no—
65C23×#126—22no—
66C23×#126—22no—
67C23×#125—22no—
68C23×#124—22no—
69C23×#124—22no—
70C23×#124—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×#123—22no—
74C23×#123—22no—
75C23×C22⋊Q8—21no—
76C23×C22⋊Q8—21no—
77C23×C22⋊Q8—22no—
78C23×C22⋊Q8D412yescomplemented
79C23×C22⋊Q8C2312yes—
80C23×#121—21no—
81C23×#121—22no—
82C23×#121—22no—
83C23×#121—22no—
84C23×#121—22no—
85C23×#121—22no—
86C23×#121—21no—
87C23×#121—22no—
88C23×#121—22no—
89C23×#121D412yescomplemented
90C23×#121C2312yes—
91C23×#120—22no—
92C23×#120—22no—
93C4×Q8×C23—21no—
94C4×Q8×C23—21no—
95C4×D4×C23—21no—
96C4×D4×C23—21no—
97C4×D4×C23—22no—
98C4×D4×C23—22no—
99C4×D4×C23—22no—
100C4×D4×C23—22no—
101C4×D4×C23—22no—
102C4×D4×C23—22no—
103C23×#117—22no—
104C23×#117—21no—
105C23×#117—21no—
106C23×#117D412yescomplemented
107C2×C4⋊C4×C23—21no—
108C2×C4⋊C4×C23C2311yescharacteristic
109C2×C4⋊C4×C23C2311yescharacteristic
110C2×C22⋊C4×C23—21no—
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—22no—
119C2×C42×C23D411yescharacteristic
120C23×#107—22no—
121C23×#106—22no—
122C23×#103—22no—
123C23×#102—22no—
124C23×#102—22no—
125C23×#102—22no—
126C23×C22⋊C8—22no—
127C23×C22⋊C8C2312yes—
128C23×C22⋊C8C2312yes—
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—42no—
138#41×C23—42no—
139#41×C23—82no—
140#41×C23—82no—
141C2×Q8×C23—21no—
142C2×Q8×C23—21no—
143C2×Q8×C23C2×D412yes—
144C2×D4×C23—21no—
145C2×D4×C23—22no—
146C2×D4×C23—22no—
147C2×D4×C23—22no—
148C2×D4×C23—22no—
149C2×D4×C23—21no—
150C2×D4×C23—22no—
151C2×D4×C23—22no—
152C2×D4×C23—22no—
153C2×D4×C23—22no—
154C2×D4×C23—41no—
155C2×D4×C23—42no—
156C2×D4×C23—42no—
157C2×D4×C23—81no—
158C2×D4×C23—81no—
159C2×D4×C23—81no—
160C2×D4×C23—81no—
161C2×D4×C23—81no—
162C2×D4×C23C2×D412yes—
163C22×C4×C23—22no—
164C22×C4×C23—22no—
165C22×C4×C23—22no—
166C22×C4×C23—21no—
167C22×C4×C23—21no—
168C22×C4×C23—21no—
169C22×C4×C23—21no—
170C22×C4×C23—21no—
171C22×C4×C23—81no—
172C22×C4×C23—81no—
173C22×C4×C23C2×D411yescharacteristic
174C22×C4×C23C2×D411yescharacteristic
175C22×C4×C23C2×D411yescharacteristic
176C22×C4×C23C2×D411yescharacteristic
177C22×C4×C23C2×D412yes—
178C22×C4×C23C2×D412yes—
179C22×C4×C23C2411yescharacteristic
180SD16×C23—82no—
181D8×C23—82no—
182C2×C8×C23—22no—
183C2×C8×C23—22no—
184C2×C8×C23—82no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—21no—
189C4⋊C4×C23—21no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23—22no—
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—41no—
210C22⋊C4×C23—41no—
211C22⋊C4×C23—42no—
212C22⋊C4×C23—81no—
213C22⋊C4×C23—81no—
214C22⋊C4×C23—81no—
215C22⋊C4×C23—81no—
216C42×C23—22no—
217C42×C23—22no—
218C42×C23—21no—
219C42×C23—22no—
220C42×C23—21no—
221#18298C2311yesmaximal, characteristic, direct product factor, Sylow
222C23×C23—21no—
223C23×C23—21no—
224C23×C23—21no—
225C23×C23—22no—
226C23×C23—21no—
227C23×C23—21no—
228C23×C23—41no—
229C23×C23—41no—
230C23×C23—41no—
231C23×C23—41no—
232C23×C23—41no—
233C23×C23—41no—
234C23×C23—41no—
235C23×C23—41no—
236C23×C23—41no—
237C23×C23—41no—
238C23×C23—41no—
239C23×C23#12012yes—
240C23×C23C22×D411yescharacteristic
241Q8×C23—42no—
242Q8×C23—42no—
243Q8×C23—81no—
244D4×C23—44no—
245D4×C23—42no—
246D4×C23—42no—
247D4×C23—42no—
248D4×C23—42no—
249D4×C23—42no—
250D4×C23—42no—
251D4×C23—82no—
252D4×C23—82no—
253D4×C23—82no—
254D4×C23—81no—
255D4×C23—82no—
256D4×C23—84no—
257D4×C23—84no—
258C2×C4×C23—21no—
259C2×C4×C23—21no—
260C2×C4×C23—21no—
261C2×C4×C23—22no—
262C2×C4×C23—22no—
263C2×C4×C23—22no—
264C2×C4×C23—21no—
265C2×C4×C23—42no—
266C2×C4×C23—41no—
267C2×C4×C23—42no—
268C2×C4×C23—41no—
269C2×C4×C23—42no—
270C2×C4×C23—44no—
271C2×C4×C23—44no—
272C2×C4×C23—41no—
273C2×C4×C23—41no—
274C2×C4×C23—41no—
275C2×C4×C23—41no—
276C2×C4×C23—41no—
277C2×C4×C23—41no—
278C2×C4×C23—41no—
279C2×C4×C23—41no—
280C2×C4×C23—41no—
281C2×C4×C23—41no—
282C2×C4×C23—41no—
283C2×C4×C23—81no—
284C2×C4×C23—82no—
285C2×C4×C23#12011yescharacteristic
286C2×C4×C23#12011yescharacteristic
287C2×C4×C23#13512yes—
288C2×C4×C23C22×D412yes—
289C8×C23—42no—
290C8×C23—42no—
291#3460C2×C2311yescharacteristic, complemented
292#3188C2×C2312yescomplemented
293#2635C2×C2311yescharacteristic, complemented
294#1618C2×C2312yescomplemented
295#1617C2×C2311yescharacteristic, complemented
296#1616C2×C2311yescharacteristic, complemented
297#1600C2×C2312yescomplemented
298#1599C2×C2311yescharacteristic, complemented
299#1598C2×C2311yescharacteristic, complemented
300#1516C2×C2312yescomplemented
301#1459C2×C2311yescharacteristic, complemented
302C22×C23—21no—
303C22×C23—21no—
304C22×C23—22no—
305C22×C23—22no—
306C22×C23—22no—
307C22×C23—42no—
308C22×C23—42no—
309C22×C23—41no—
310C22×C23—42no—
311C22×C23—41no—
312C22×C23—41no—
313C22×C23—42no—
314C22×C23—81no—
315C22×C23—81no—
316C22×C23—81no—
317C22×C23—81no—
318C22×C23—81no—
319C22×C23—81no—
320C22×C23—81no—
321C22×C23—81no—
322C22×C23—81no—
323C22×C23—81no—
324C22×C23C2×#12011yescharacteristic
325C22×C23C2×#13511yescharacteristic
326C22×C23#57511yescharacteristic
327C4×C23—22no—
328C4×C23—22no—
329C4×C23—21no—
330C4×C23—21no—
331C4×C23—42no—
332C4×C23—41no—
333C4×C23—42no—
334C4×C23—41no—
335C4×C23—42no—
336C4×C23—81no—
337C4×C23—81no—
338C4×C23—81no—
339#544C22×C2311yescharacteristic
340#542—22no—
341#540—22no—
342#538—21no—
343#538C22×C2312yes—
344#537C22×C2311yescharacteristic
345#535—21no—
346#532—22no—
347#529—22no—
348#525—21no—
349#525C22×C2312yes—
350C2×#121C22×C2311yescharacteristic
351C4×#41—21no—
352C2×C4×D4C22×C2311yescharacteristic
353#484C22×C2312yes—
354#482—22no—
355#481C22×C2312yes—
356#466—22no—
357#465—22no—
358#450C22×C2312yes—
359#449—22no—
360#447C22×C2312yes—
361C2×C22⋊C8C22×C2312yes—
362#400C22×C2311yescharacteristic
363#399C22×C2311yescharacteristic
364#394—21no—
365#394C22×C2311yescharacteristic
366#392—21no—
367#392C22×C2311yescharacteristic
368#388—21no—
369#386—21no—
370#381C22×C2311yescharacteristic
371#328C22×C2312yes—
372#328C22×C2312yes—
373#327C22×C2312yes—
374#327C22×C2312yes—
375#324C22×C2311yescharacteristic
376#324C22×C2311yescharacteristic
377#323C22×C2312yes—
378C2×C23—21no—
379C2×C23—21no—
380C2×C23—42no—
381C2×C23—82no—
382C2×C23—81no—
383C2×C23—161no—
384C2×C23#300211yescharacteristic
385C2×C23#302111yescharacteristic
386C2×C23#302611yescharacteristic
387C2×#41—21no—
388C2×#41—21no—
389C2×#41D4×C2312yescomplemented
390C22×D4—21no—
391C22×D4—21no—
392C22×D4D4×C2311yescharacteristic
393C22×D4C23×C2311yescharacteristic
394C23×C4D4×C2311yescharacteristic
395C23×C4C23×C2311yescharacteristic
396C2×SD16—22no—
397C2×D8—22no—
398C22×C8—22no—
399C4⋊Q8—22no—
400#127—22no—
401#126—22no—
402#126—22no—
403#125—22no—
404#124—22no—
405#124—22no—
406#124—22no—
407#123—22no—
408#123—22no—
409#123—22no—
410#123—22no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—22no—
414C22⋊Q8D4×C2312yescomplemented
415C22⋊Q8C23×C2312yes—
416#121—21no—
417#121—22no—
418#121—22no—
419#121—22no—
420#121—22no—
421#121—22no—
422#121—21no—
423#121—22no—
424#121—22no—
425#121D4×C2312yescomplemented
426#121C23×C2312yes—
427#120—22no—
428#120—22no—
429C4×Q8—21no—
430C4×Q8—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—22no—
438C4×D4—22no—
439#117—22no—
440#117—21no—
441#117—21no—
442#117D4×C2312yescomplemented
443C2×C4⋊C4—21no—
444C2×C4⋊C4C23×C2311yescharacteristic
445C2×C4⋊C4C23×C2311yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4D4×C2311yescharacteristic
453C2×C22⋊C4C23×C2311yescharacteristic
454C2×C42—22no—
455C2×C42D4×C2311yescharacteristic
456#107—22no—
457#106—22no—
458#103—22no—
459#102—22no—
460#102—22no—
461#102—22no—
462C22⋊C8—22no—
463C22⋊C8C23×C2312yes—
464C22⋊C8C23×C2312yes—
465#95—21no—
466#95—21no—
467#95—21no—
468#95C23×C2311yescharacteristic
469#95C23×C2311yescharacteristic
470C23#1829811yescharacteristic, direct product factor, Sylow
471C24—21no—
472C24C2×D4×C2311yescharacteristic
473C24C2×D4×C2311yescharacteristic
474#41—42no—
475#41—42no—
476#41—82no—
477#41—82no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8C2×D4×C2312yes—
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—41no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—81no—
495C2×D4—81no—
496C2×D4—81no—
497C2×D4—81no—
498C2×D4—81no—
499C2×D4C2×D4×C2312yes—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4C2×D4×C2311yescharacteristic
511C22×C4C2×D4×C2311yescharacteristic
512C22×C4C2×D4×C2311yescharacteristic
513C22×C4C2×D4×C2311yescharacteristic
514C22×C4C2×D4×C2312yes—
515C22×C4C2×D4×C2312yes—
516C22×C4C24×C2311yescharacteristic
517SD16—82no—
518D8—82no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—82no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—81no—
533C4⋊C4C2×D4×C2312yes—
534C4⋊C4C2×D4×C2312yes—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—81no—
551C22⋊C4—81no—
552C22⋊C4—81no—
553C42—22no—
554C42—22no—
555C42—21no—
556C42—22no—
557C42—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—21no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23C23×#12012yes—
576C23C22×D4×C2311yescharacteristic
577Q8—42no—
578Q8—42no—
579Q8—81no—
580D4—44no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—82no—
588D4—82no—
589D4—81no—
590D4—82no—
591D4—82no—
592D4—84no—
593D4—84no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—81no—
620C2×C4—82no—
621C2×C4C23×#12011yescharacteristic
622C2×C4C23×#12011yescharacteristic
623C2×C4C23×#13512yes—
624C2×C4C22×D4×C2312yes—
625C8—42no—
626C8—42no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22C2×C23×#12011yescharacteristic
650C22C2×C23×#13511yescharacteristic
651C22C23×#57511yescharacteristic
652C4—22no—
653C4—22no—
654C4—21no—
655C4—21no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—81no—
664C2—21no—
665C2—21no—
666C2—42no—
667C2—81no—
668C2—82no—
669C2—161no—
670C2C23×#300211yescharacteristic
671C2C23×#302111yescharacteristic
672C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
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
1. C23×#3460C2semidirect
2. C23×#3188C2semidirect
3. C23×#2635C2semidirect
4. C23×#1618C2semidirect
5. C23×#1617C2semidirect
6. C23×#1616C2semidirect
7. C23×#1600C2semidirect
8. C23×#1599C2semidirect
9. C23×#1598C2semidirect
10. C23×#1516C2semidirect
11. C23×#1459C2semidirect
53. C2×#41×C23D4semidirect
78. C23×C22⋊Q8D4semidirect
89. C23×#121D4semidirect
106. C23×#117D4semidirect
221. #18298C23direct (N × H)
291. #3460C2×C23semidirect
292. #3188C2×C23semidirect
293. #2635C2×C23semidirect
294. #1618C2×C23semidirect
295. #1617C2×C23semidirect
296. #1616C2×C23semidirect
297. #1600C2×C23semidirect
298. #1599C2×C23semidirect
299. #1598C2×C23semidirect
300. #1516C2×C23semidirect
301. #1459C2×C23semidirect
389. C2×#41D4×C23semidirect
414. C22⋊Q8D4×C23semidirect
425. #121D4×C23semidirect
442. #117D4×C23semidirect
470. C23#18298direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34602
C23×#31882
C23×#26352
C23×#16182
C23×#16172
C23×#16162
C23×#16002
C23×#15992
C23×#15982
C23×#15162
C23×#14592
#1829823