Finite Group Databasebeta

← C23×#19014 #19166640

Subgroups — C23×#19014 #19166640

Order 5888.

1140 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3558C211yesmaximal, characteristic, complemented
2C23×#3217C211yesmaximal, characteristic, complemented
3C23×#3151C211yesmaximal, characteristic, complemented
4C2×C23×#496C211yesmaximal, characteristic, complemented
5C23×#1656C214yesmaximal, complemented
6C23×#1655C214yesmaximal, complemented
7C23×#1530C211yesmaximal, characteristic, complemented
8C23×#1527C211yesmaximal, characteristic, complemented
9C23×#1524C211yesmaximal, characteristic, complemented
10C2×C23×AGL(1,ℤ/8)—22no—
11C22×SD16×C23—21no—
12C22×D8×C23—21no—
13C23×#562C2212yescomplemented
14C23×H(3,𝔽4)C2212yescomplemented
15C23×#559—21no—
16C23×#548—21no—
17C23×#546—22no—
18C23×#546—21no—
19D42×C23—21no—
20C23×#542—21no—
21C23×#538—22no—
22C23×#538—21no—
23C23×#535—21no—
24C23×#534—21no—
25C23×#532C2211yescharacteristic
26C2×C23×#127C2211yescharacteristic, complemented
27C23×#529C2211yescharacteristic
28C2×C23×#124C2211yescharacteristic
29C23×#496—22no—
30C23×#496C2212yescomplemented
31C23×#496C2214yescomplemented
32C23×#496C2214yescomplemented
33C23×#493—21no—
34C23×#492—21no—
35C23×#490C2212yescomplemented
36C23×#460—21no—
37C23×#459—21no—
38C23×#450—21no—
39C23×#447—21no—
40C23×#432C2211yescharacteristic, complemented
41C4×M16×C23C2211yescharacteristic, complemented
42C2×C23×#97C2211yescharacteristic, complemented
43C23×#330C2212yescomplemented
44C23×#329C2212yescomplemented
45C23×#329C2214yescomplemented
46C23×#329C2214yescomplemented
47C2×#41×C23—21no—
48C2×#41×C23D412yes—
49C22×Q8×C23D411yescharacteristic
50C22×D4×C23—22no—
51C22×D4×C23—21no—
52C22×D4×C23—21no—
53C22×D4×C23—21no—
54C22×D4×C23—21no—
55C22×D4×C23D411yescharacteristic
56C23×AGL(1,ℤ/8)—48no—
57C2×SD16×C23—24no—
58C2×SD16×C23—24no—
59C2×SD16×C23—22no—
60C2×SD16×C23—44no—
61C2×D8×C23—24no—
62C2×D8×C23—24no—
63C2×D8×C23—22no—
64C2×D8×C23—44no—
65C2×M16×C23—21no—
66C22×C8×C23—21no—
67C23×C4⋊Q8D411yescharacteristic
68C23×#127—21no—
69C23×#127—21no—
70C23×#127—21no—
71C23×#127—21no—
72C23×#127D411yescharacteristic
73C23×#127C2312yes—
74C23×#126—22no—
75C23×#126—21no—
76C23×#125C2311yescharacteristic
77C23×#124—21no—
78C23×#124—22no—
79C23×#124—22no—
80C23×#124—22no—
81C23×#124—21no—
82C23×#124—21no—
83C23×#124D412yes—
84C23×#124C2311yescharacteristic
85C23×#124C2312yes—
86C23×#124C2312yes—
87C23×#123—22no—
88C23×#123—22no—
89C23×#123—21no—
90C23×#123—22no—
91C23×#123—22no—
92C23×C22⋊Q8—21no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8D411yescharacteristic
96C23×#121—22no—
97C23×#121—22no—
98C23×#121—22no—
99C23×#121—21no—
100C23×#121—22no—
101C23×#121—22no—
102C23×#121—21no—
103C23×#121—21no—
104C23×#121D411yescharacteristic
105C23×#120—22no—
106C23×#120—22no—
107C23×#120—22no—
108C23×#120—22no—
109C23×#120—22no—
110C4×Q8×C23—21no—
111C4×Q8×C23D411yescharacteristic
112C4×D4×C23—22no—
113C4×D4×C23—22no—
114C4×D4×C23—21no—
115C4×D4×C23—21no—
116C4×D4×C23—21no—
117C4×D4×C23—21no—
118C4×D4×C23D411yescharacteristic
119C23×#117—21no—
120C2×C22⋊C4×C23—22no—
121C2×C22⋊C4×C23—21no—
122C2×C22⋊C4×C23—21no—
123C2×C42×C23C2311yescharacteristic
124C23×C4⋊C8—21no—
125C23×#102—22no—
126C23×#102—22no—
127C23×C22⋊C8—21no—
128C23×#97—21no—
129C23×#97C2312yes—
130C23×#97C2312yes—
131C23×#97C2312yes—
132C4×C8×C23—21no—
133C24×C23—22no—
134C24×C23—21no—
135C24×C23—41no—
136#41×C23—44no—
137#41×C23—82no—
138C2×Q8×C23—21no—
139C2×Q8×C23—42no—
140C2×Q8×C23C2×D411yescharacteristic
141C2×Q8×C23C2×D412yes—
142C2×Q8×C23C2×D412yes—
143C2×D4×C23—22no—
144C2×D4×C23—21no—
145C2×D4×C23—22no—
146C2×D4×C23—22no—
147C2×D4×C23—22no—
148C2×D4×C23—22no—
149C2×D4×C23—22no—
150C2×D4×C23—21no—
151C2×D4×C23—21no—
152C2×D4×C23—21no—
153C2×D4×C23—22no—
154C2×D4×C23—22no—
155C2×D4×C23—22no—
156C2×D4×C23—42no—
157C2×D4×C23—42no—
158C2×D4×C23—42no—
159C2×D4×C23—44no—
160C2×D4×C23—41no—
161C2×D4×C23—41no—
162C2×D4×C23—41no—
163C2×D4×C23—82no—
164C2×D4×C23—82no—
165C2×D4×C23—84no—
166C2×D4×C23C2×D411yescharacteristic
167C2×D4×C23C2×D412yes—
168C2×D4×C23C2×D412yes—
169C22×C4×C23—22no—
170C22×C4×C23—22no—
171C22×C4×C23—21no—
172C22×C4×C23—21no—
173C22×C4×C23—22no—
174C22×C4×C23C2×D411yescharacteristic
175C22×C4×C23C2×D411yescharacteristic
176C22×C4×C23C2×D411yescharacteristic
177SD16×C23—48no—
178SD16×C23—48no—
179SD16×C23—82no—
180D8×C23—48no—
181D8×C23—48no—
182D8×C23—82no—
183M16×C23—24no—
184C2×C8×C23—24no—
185C2×C8×C23—22no—
186C2×C8×C23—22no—
187C2×C8×C23—22no—
188C4⋊C4×C23—22no—
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×C23C2×D411yescharacteristic
197C4⋊C4×C23C2×D411yescharacteristic
198C22⋊C4×C23—22no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—21no—
204C22⋊C4×C23—22no—
205C22⋊C4×C23—21no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—21no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—21no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—22no—
213C22⋊C4×C23—22no—
214C42×C23—22no—
215C42×C23—21no—
216C42×C23C2×D411yescharacteristic
217C42×C23C2×D411yescharacteristic
218C42×C23C2×D411yescharacteristic
219C42×C23C2411yescharacteristic
220#19014C2311yesmaximal, characteristic, direct product factor, Sylow
221C23×C23—21no—
222C23×C23—22no—
223C23×C23—22no—
224C23×C23—42no—
225C23×C23—42no—
226C23×C23—81no—
227C23×C23—81no—
228C23×C23—82no—
229C23×C23—82no—
230C23×C23—82no—
231C23×C23—82no—
232C23×C23—82no—
233C23×C23—82no—
234C23×C23#12011yescharacteristic
235Q8×C23—42no—
236Q8×C23—42no—
237Q8×C23—41no—
238Q8×C23—42no—
239D4×C23—42no—
240D4×C23—42no—
241D4×C23—41no—
242D4×C23—42no—
243D4×C23—44no—
244D4×C23—42no—
245D4×C23—42no—
246D4×C23—42no—
247D4×C23—42no—
248D4×C23—44no—
249D4×C23—42no—
250D4×C23—42no—
251D4×C23—81no—
252D4×C23—82no—
253D4×C23—82no—
254D4×C23—82no—
255D4×C23—82no—
256D4×C23—82no—
257D4×C23—82no—
258D4×C23—82no—
259D4×C23—82no—
260C2×C4×C23—22no—
261C2×C4×C23—22no—
262C2×C4×C23—21no—
263C2×C4×C23—22no—
264C2×C4×C23—22no—
265C2×C4×C23—21no—
266C2×C4×C23—21no—
267C2×C4×C23—41no—
268C2×C4×C23—42no—
269C2×C4×C23—82no—
270C2×C4×C23—81no—
271C2×C4×C23—82no—
272C2×C4×C23—82no—
273C2×C4×C23#12011yescharacteristic
274C2×C4×C23#12011yescharacteristic
275C2×C4×C23#12011yescharacteristic
276C2×C4×C23C22×D411yescharacteristic
277C2×C4×C23C22×D411yescharacteristic
278C2×C4×C23C22×D411yescharacteristic
279C8×C23—24no—
280C8×C23—24no—
281C8×C23—42no—
282#3558C2×C2311yescharacteristic, complemented
283#3217C2×C2311yescharacteristic, complemented
284#3151C2×C2311yescharacteristic, complemented
285C2×#496C2×C2311yescharacteristic, complemented
286#1656C2×C2314yescomplemented
287#1655C2×C2314yescomplemented
288#1530C2×C2311yescharacteristic, complemented
289#1527C2×C2311yescharacteristic, complemented
290#1524C2×C2311yescharacteristic, complemented
291C22×C23—21no—
292C22×C23—21no—
293C22×C23—41no—
294C22×C23—42no—
295C22×C23—42no—
296C22×C23—42no—
297C22×C23—42no—
298C22×C23—41no—
299C22×C23—41no—
300C22×C23—42no—
301C22×C23—42no—
302C22×C23—82no—
303C22×C23—82no—
304C22×C23—82no—
305C22×C23—162no—
306C22×C23—164no—
307C22×C23—164no—
308C22×C23#45312yes—
309C22×C23C2×#12011yescharacteristic
310C4×C23—21no—
311C4×C23—21no—
312C4×C23—21no—
313C4×C23—21no—
314C4×C23—21no—
315C4×C23—41no—
316C4×C23—41no—
317C4×C23—42no—
318C4×C23—42no—
319C4×C23—41no—
320C4×C23#45312yes—
321C2×AGL(1,ℤ/8)—22no—
322C22×SD16—21no—
323C22×D8—21no—
324#562C22×C2312yescomplemented
325H(3,𝔽4)C22×C2312yescomplemented
326#559—21no—
327#548—21no—
328#546—22no—
329#546—21no—
330D42—21no—
331#542—21no—
332#538—22no—
333#538—21no—
334#535—21no—
335#534—21no—
336#532C22×C2311yescharacteristic
337C2×#127C22×C2311yescharacteristic, complemented
338#529C22×C2311yescharacteristic
339C2×#124C22×C2311yescharacteristic
340#496—22no—
341#496C22×C2314yescomplemented
342#496C22×C2314yescomplemented
343#496C22×C2312yescomplemented
344#493—21no—
345#492—21no—
346#490C22×C2312yescomplemented
347#460—21no—
348#459—21no—
349#450—21no—
350#447—21no—
351#432C22×C2311yescharacteristic, complemented
352C4×M16C22×C2311yescharacteristic, complemented
353C2×#97C22×C2311yescharacteristic, complemented
354#330C22×C2312yescomplemented
355#329C22×C2312yescomplemented
356#329C22×C2314yescomplemented
357#329C22×C2314yescomplemented
358C2×C23—22no—
359C2×C23—82no—
360C2×C23—81no—
361C2×C23—82no—
362C2×C23—162no—
363C2×C23C2×#45311yescharacteristic
364C2×C23C2×#45311yescharacteristic
365C2×C23#302111yescharacteristic
366C2×#41—21no—
367C2×#41D4×C2312yes—
368C22×Q8D4×C2311yescharacteristic
369C22×D4—22no—
370C22×D4—21no—
371C22×D4—21no—
372C22×D4—21no—
373C22×D4—21no—
374C22×D4D4×C2311yescharacteristic
375AGL(1,ℤ/8)—48no—
376C2×SD16—24no—
377C2×SD16—24no—
378C2×SD16—22no—
379C2×SD16—44no—
380C2×D8—24no—
381C2×D8—24no—
382C2×D8—22no—
383C2×D8—44no—
384C2×M16—21no—
385C22×C8—21no—
386C4⋊Q8D4×C2311yescharacteristic
387#127—21no—
388#127—21no—
389#127—21no—
390#127—21no—
391#127D4×C2311yescharacteristic
392#127C23×C2312yes—
393#126—22no—
394#126—21no—
395#125C23×C2311yescharacteristic
396#124—21no—
397#124—22no—
398#124—22no—
399#124—22no—
400#124—21no—
401#124—21no—
402#124D4×C2312yes—
403#124C23×C2312yes—
404#124C23×C2311yescharacteristic
405#124C23×C2312yes—
406#123—22no—
407#123—22no—
408#123—21no—
409#123—22no—
410#123—22no—
411C22⋊Q8—21no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8D4×C2311yescharacteristic
415#121—22no—
416#121—22no—
417#121—22no—
418#121—21no—
419#121—22no—
420#121—22no—
421#121—21no—
422#121—21no—
423#121D4×C2311yescharacteristic
424#120—22no—
425#120—22no—
426#120—22no—
427#120—22no—
428#120—22no—
429C4×Q8—21no—
430C4×Q8D4×C2311yescharacteristic
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4D4×C2311yescharacteristic
438#117—21no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C42C23×C2311yescharacteristic
443C4⋊C8—21no—
444#102—22no—
445#102—22no—
446C22⋊C8—21no—
447#97—21no—
448#97C23×C2312yes—
449#97C23×C2312yes—
450#97C23×C2312yes—
451C4×C8—21no—
452C23#1901411yescharacteristic, direct product factor, Sylow
453C24—22no—
454C24—21no—
455C24—41no—
456#41—44no—
457#41—82no—
458C2×Q8—21no—
459C2×Q8—42no—
460C2×Q8C2×D4×C2311yescharacteristic
461C2×Q8C2×D4×C2312yes—
462C2×Q8C2×D4×C2312yes—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—44no—
480C2×D4—41no—
481C2×D4—41no—
482C2×D4—41no—
483C2×D4—82no—
484C2×D4—82no—
485C2×D4—84no—
486C2×D4C2×D4×C2311yescharacteristic
487C2×D4C2×D4×C2312yes—
488C2×D4C2×D4×C2312yes—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4C2×D4×C2311yescharacteristic
495C22×C4C2×D4×C2311yescharacteristic
496C22×C4C2×D4×C2311yescharacteristic
497SD16—48no—
498SD16—48no—
499SD16—82no—
500D8—48no—
501D8—48no—
502D8—82no—
503M16—24no—
504C2×C8—24no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4C2×D4×C2311yescharacteristic
517C4⋊C4C2×D4×C2311yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—22no—
525C22⋊C4—21no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C42—22no—
535C42—21no—
536C42C2×D4×C2311yescharacteristic
537C42C2×D4×C2311yescharacteristic
538C42C2×D4×C2311yescharacteristic
539C42C24×C2311yescharacteristic
540C23—21no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23—81no—
546C23—81no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23—82no—
553C23C23×#12011yescharacteristic
554Q8—42no—
555Q8—42no—
556Q8—41no—
557Q8—42no—
558D4—42no—
559D4—42no—
560D4—41no—
561D4—42no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—44no—
568D4—42no—
569D4—42no—
570D4—81no—
571D4—82no—
572D4—82no—
573D4—82no—
574D4—82no—
575D4—82no—
576D4—82no—
577D4—82no—
578D4—82no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—82no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4C23×#12011yescharacteristic
593C2×C4C23×#12011yescharacteristic
594C2×C4C23×#12011yescharacteristic
595C2×C4C22×D4×C2311yescharacteristic
596C2×C4C22×D4×C2311yescharacteristic
597C2×C4C22×D4×C2311yescharacteristic
598C8—24no—
599C8—24no—
600C8—42no—
601C22—21no—
602C22—21no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—162no—
616C22—164no—
617C22—164no—
618C22C23×#45312yes—
619C22C2×C23×#12011yescharacteristic
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4C23×#45312yes—
631C2—22no—
632C2—81no—
633C2—82no—
634C2—82no—
635C2—162no—
636C2C2×C23×#45311yescharacteristic
637C2C2×C23×#45311yescharacteristic
638C2C23×#302111yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
LCSG ▷ C42 ▷ 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×#3558C2semidirect
2. C23×#3217C2semidirect
3. C23×#3151C2semidirect
4. C2×C23×#496C2semidirect
5. C23×#1656C2semidirect
6. C23×#1655C2semidirect
7. C23×#1530C2semidirect
8. C23×#1527C2semidirect
9. C23×#1524C2semidirect
13. C23×#562C22semidirect
14. C23×H(3,𝔽4)C22semidirect
26. C2×C23×#127C22semidirect
30. C23×#496C22semidirect
31. C23×#496C22semidirect
32. C23×#496C22semidirect
35. C23×#490C22semidirect
40. C23×#432C22semidirect
41. C4×M16×C23C22semidirect
42. C2×C23×#97C22semidirect
43. C23×#330C22semidirect
44. C23×#329C22semidirect
45. C23×#329C22semidirect
46. C23×#329C22semidirect
220. #19014C23direct (N × H)
282. #3558C2×C23semidirect
283. #3217C2×C23semidirect
284. #3151C2×C23semidirect
285. C2×#496C2×C23semidirect
286. #1656C2×C23semidirect
287. #1655C2×C23semidirect
288. #1530C2×C23semidirect
289. #1527C2×C23semidirect
290. #1524C2×C23semidirect
324. #562C22×C23semidirect
325. H(3,𝔽4)C22×C23semidirect
337. C2×#127C22×C23semidirect
341. #496C22×C23semidirect
342. #496C22×C23semidirect
343. #496C22×C23semidirect
346. #490C22×C23semidirect
351. #432C22×C23semidirect
352. C4×M16C22×C23semidirect
353. C2×#97C22×C23semidirect
354. #330C22×C23semidirect
355. #329C22×C23semidirect
356. #329C22×C23semidirect
357. #329C22×C23semidirect
452. C23#19014direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35582
C23×#32172
C23×#31512
C2×C23×#4962
C23×#16562
C23×#16552
C23×#15302
C23×#15272
C23×#15242
#1901423