Finite Group Databasebeta

← C23×#33568 #19181194

Subgroups — C23×#33568 #19181194

Order 5888.

1110 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3545C211yesmaximal, characteristic, complemented
2C23×#3020C212yesmaximal, complemented
3C2×C23×#454C211yesmaximal, characteristic, complemented
4C23×#2204C214yesmaximal, complemented
5C23×#2203C212yesmaximal, complemented
6C23×#2202C212yesmaximal, complemented
7C23×#2133C212yesmaximal, complemented
8C2×C23×#355C211yesmaximal, characteristic, complemented
9C2×C23×2−1+4C2211yescharacteristic, complemented
10C23×#575—22no—
11C2×C23×AGL(1,ℤ/8)—21no—
12C23×#563C2211yescharacteristic, complemented
13C23×#562—21no—
14C23×H(3,𝔽4)C2211yescharacteristic, complemented
15C23×#555C2212yescomplemented
16C23×#551—21no—
17C23×#551—22no—
18C23×#546—22no—
19C23×#542—22no—
20C23×#538—22no—
21C23×#536C2212yes—
22C23×#534—21no—
23C23×#529—21no—
24C2×C23×#124C2211yescharacteristic
25C23×#456—22no—
26C23×#456C2214yescomplemented
27C23×#454—21no—
28C23×#454—21no—
29C23×#454—22no—
30C23×#454C2212yes—
31C23×#454C2214yescomplemented
32C23×#454C2212yescomplemented
33C23×#421—22no—
34C2×C23×#104—21no—
35C2×C23×#101C2211yescharacteristic, complemented
36C2×C23×#101C2212yescomplemented
37C23×#356C2214yescomplemented
38C23×#355C2212yescomplemented
39C23×#355C2214yescomplemented
40C23×#355C2212yescomplemented
41C23×2−1+4—21no—
42C23×2−1+4—22no—
43C23×2−1+4—42no—
44C23×2−1+4C2312yes—
45C2×#41×C23—21no—
46C2×#41×C23—22no—
47C2×#41×C23—41no—
48C2×#41×C23D412yes—
49C22×Q8×C23—41no—
50C22×Q8×C23C2311yescharacteristic
51C22×D4×C23—21no—
52C22×D4×C23D411yescharacteristic
53C23×#137—24no—
54C23×#137—42no—
55C23×AGL(1,ℤ/8)—22no—
56C23×AGL(1,ℤ/8)—22no—
57C23×AGL(1,ℤ/8)—24no—
58C23×AGL(1,ℤ/8)—42no—
59C23×AGL(1,ℤ/8)—42no—
60C23×#135—44no—
61C2×SD16×C23—41no—
62C2×D8×C23—41no—
63C2×M16×C23—21no—
64C2×M16×C23—22no—
65C23×C4⋊Q8D412yes—
66C23×C4⋊Q8C2312yes—
67C23×#126—22no—
68C23×#126—22no—
69C23×#126—22no—
70C23×#125—21no—
71C23×#124—22no—
72C23×#124—22no—
73C23×#124—22no—
74C23×#124—21no—
75C23×#124D411yescharacteristic
76C23×#124D411yescharacteristic
77C23×#124D412yes—
78C23×#124C2311yescharacteristic
79C23×#124C2312yes—
80C23×#124C2311yescharacteristic
81C23×#123—22no—
82C23×#123—22no—
83C23×#123—22no—
84C23×#123—21no—
85C23×#123—21no—
86C23×#123—22no—
87C23×#123—22no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—22no—
93C23×#121—22no—
94C23×#121—21no—
95C23×#121—22no—
96C23×#121—22no—
97C23×#121—22no—
98C23×#120—22no—
99C23×#120—21no—
100C23×#120—21no—
101C23×#120—22no—
102C4×Q8×C23—21no—
103C4×D4×C23—22no—
104C4×D4×C23—22no—
105C4×D4×C23—21no—
106C4×D4×C23—22no—
107C23×#117—22no—
108C23×#117—22no—
109C23×#117D412yes—
110C2×C22⋊C4×C23—21no—
111C2×C22⋊C4×C23—22no—
112C2×C22⋊C4×C23—22no—
113C2×C42×C23D411yescharacteristic
114C23×#104—22no—
115C23×#104—22no—
116C23×#104—24no—
117C23×#104—24no—
118C23×#101—21no—
119C23×#101—22no—
120C23×#101C2314yes—
121C23×#101C2312yes—
122C24×C23—21no—
123C24×C23—21no—
124#41×C23—24no—
125#41×C23—22no—
126#41×C23—22no—
127#41×C23—24no—
128#41×C23—42no—
129#41×C23—41no—
130#41×C23—42no—
131#41×C23—42no—
132#41×C23—42no—
133#41×C23—42no—
134#41×C23—44no—
135#41×C23—41no—
136#41×C23—42no—
137C2×Q8×C23—21no—
138C2×Q8×C23—21no—
139C2×Q8×C23—22no—
140C2×Q8×C23—41no—
141C2×Q8×C23—41no—
142C2×Q8×C23—42no—
143C2×Q8×C23—42no—
144C2×Q8×C23—44no—
145C2×Q8×C23C2×D412yes—
146C2×Q8×C23C2×D411yescharacteristic
147C2×Q8×C23C2×D412yes—
148C2×Q8×C23C2411yescharacteristic
149C2×D4×C23—21no—
150C2×D4×C23—22no—
151C2×D4×C23—22no—
152C2×D4×C23—22no—
153C2×D4×C23—22no—
154C2×D4×C23—21no—
155C2×D4×C23—22no—
156C2×D4×C23—21no—
157C2×D4×C23—41no—
158C2×D4×C23—41no—
159C2×D4×C23—41no—
160C2×D4×C23—81no—
161C2×D4×C23—81no—
162C2×D4×C23C2×D412yes—
163C2×D4×C23C2×D411yescharacteristic
164C2×D4×C23C2×D411yescharacteristic
165C22×C4×C23—22no—
166C22×C4×C23—22no—
167C22×C4×C23—21no—
168C22×C4×C23—21no—
169C22×C4×C23—22no—
170C22×C4×C23—41no—
171C22×C4×C23—42no—
172C22×C4×C23C2×D411yescharacteristic
173C22×C4×C23C2×D412yes—
174Q16×C23—44no—
175SD16×C23—44no—
176SD16×C23—42no—
177SD16×C23—42no—
178SD16×C23—44no—
179D8×C23—42no—
180D8×C23—42no—
181D8×C23—44no—
182M16×C23—22no—
183M16×C23—24no—
184M16×C23—41no—
185M16×C23—42no—
186C2×C8×C23—41no—
187C2×C8×C23—42no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23—22no—
195C4⋊C4×C23—22no—
196C4⋊C4×C23—22no—
197C22⋊C4×C23—22no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—21no—
200C22⋊C4×C23—21no—
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—22no—
211C22⋊C4×C23—22no—
212C42×C23—22no—
213C42×C23—21no—
214C42×C23C2×D411yescharacteristic
215C42×C23C2×D411yescharacteristic
216C42×C23C2×D412yes—
217C42×C23C2×D412yes—
218#33568C2311yesmaximal, characteristic, direct product factor, Sylow
219C23×C23—21no—
220C23×C23—21no—
221C23×C23—22no—
222C23×C23—41no—
223C23×C23—41no—
224C23×C23—41no—
225C23×C23—81no—
226C23×C23—82no—
227C23×C23—82no—
228C23×C23#12011yescharacteristic
229Q8×C23—22no—
230Q8×C23—24no—
231Q8×C23—41no—
232Q8×C23—42no—
233Q8×C23—41no—
234Q8×C23—42no—
235Q8×C23—42no—
236Q8×C23—41no—
237Q8×C23—81no—
238D4×C23—22no—
239D4×C23—24no—
240D4×C23—41no—
241D4×C23—41no—
242D4×C23—42no—
243D4×C23—42no—
244D4×C23—41no—
245D4×C23—42no—
246D4×C23—42no—
247D4×C23—42no—
248D4×C23—42no—
249D4×C23—81no—
250D4×C23—82no—
251D4×C23—82no—
252D4×C23—81no—
253D4×C23—82no—
254C2×C4×C23—21no—
255C2×C4×C23—22no—
256C2×C4×C23—22no—
257C2×C4×C23—21no—
258C2×C4×C23—21no—
259C2×C4×C23—22no—
260C2×C4×C23—21no—
261C2×C4×C23—22no—
262C2×C4×C23—22no—
263C2×C4×C23—42no—
264C2×C4×C23—44no—
265C2×C4×C23—44no—
266C2×C4×C23—42no—
267C2×C4×C23—41no—
268C2×C4×C23—82no—
269C2×C4×C23—82no—
270C2×C4×C23—81no—
271C2×C4×C23—82no—
272C2×C4×C23—81no—
273C2×C4×C23#12011yescharacteristic
274C2×C4×C23#12012yes—
275C2×C4×C23C22×D412yes—
276C2×C4×C23C22×D411yescharacteristic
277C8×C23—42no—
278C8×C23—44no—
279#3545C2×C2311yescharacteristic, complemented
280#3020C2×C2312yescomplemented
281C2×#454C2×C2311yescharacteristic, complemented
282#2204C2×C2314yescomplemented
283#2203C2×C2312yescomplemented
284#2202C2×C2312yescomplemented
285#2133C2×C2312yescomplemented
286C2×#355C2×C2311yescharacteristic, complemented
287C22×C23—41no—
288C22×C23—42no—
289C22×C23—42no—
290C22×C23—42no—
291C22×C23—42no—
292C22×C23—42no—
293C22×C23—42no—
294C22×C23—41no—
295C22×C23—41no—
296C22×C23—82no—
297C22×C23—161no—
298C22×C23—161no—
299C22×C23—162no—
300C22×C23B(4,𝔽2)11yescharacteristic
301C22×C23B(4,𝔽2)11yescharacteristic
302C22×C23C2×#12011yescharacteristic
303C4×C23—21no—
304C4×C23—22no—
305C4×C23—22no—
306C4×C23—21no—
307C4×C23—41no—
308C4×C23—42no—
309C4×C23—42no—
310C4×C23—41no—
311C4×C23—42no—
312C4×C23—42no—
313C2×2−1+4C22×C2311yescharacteristic, complemented
314#575—22no—
315C2×AGL(1,ℤ/8)—21no—
316#563C22×C2311yescharacteristic, complemented
317#562—21no—
318H(3,𝔽4)C22×C2311yescharacteristic, complemented
319#555C22×C2312yescomplemented
320#551—21no—
321#551—22no—
322#546—22no—
323#542—22no—
324#538—22no—
325#536C22×C2312yes—
326#534—21no—
327#529—21no—
328C2×#124C22×C2311yescharacteristic
329#456—22no—
330#456C22×C2314yescomplemented
331#454—21no—
332#454—21no—
333#454—22no—
334#454C22×C2312yes—
335#454C22×C2314yescomplemented
336#454C22×C2312yescomplemented
337#421—22no—
338C2×#104—21no—
339C2×#101C22×C2312yescomplemented
340C2×#101C22×C2311yescharacteristic, complemented
341#356C22×C2314yescomplemented
342#355C22×C2312yescomplemented
343#355C22×C2312yescomplemented
344#355C22×C2314yescomplemented
345C2×C23—21no—
346C2×C23—21no—
347C2×C23—21no—
348C2×C23—81no—
349C2×C23—82no—
350C2×C23—81no—
351C2×C23—82no—
352C2×C23C2×B(4,𝔽2)11yescharacteristic
3532−1+4—21no—
3542−1+4—22no—
3552−1+4—42no—
3562−1+4C23×C2312yes—
357C2×#41—21no—
358C2×#41—22no—
359C2×#41—41no—
360C2×#41D4×C2312yes—
361C22×Q8—41no—
362C22×Q8C23×C2311yescharacteristic
363C22×D4—21no—
364C22×D4D4×C2311yescharacteristic
365#137—24no—
366#137—42no—
367AGL(1,ℤ/8)—22no—
368AGL(1,ℤ/8)—22no—
369AGL(1,ℤ/8)—24no—
370AGL(1,ℤ/8)—42no—
371AGL(1,ℤ/8)—42no—
372#135—44no—
373C2×SD16—41no—
374C2×D8—41no—
375C2×M16—21no—
376C2×M16—22no—
377C4⋊Q8D4×C2312yes—
378C4⋊Q8C23×C2312yes—
379#126—22no—
380#126—22no—
381#126—22no—
382#125—21no—
383#124—22no—
384#124—22no—
385#124—21no—
386#124—22no—
387#124D4×C2311yescharacteristic
388#124D4×C2311yescharacteristic
389#124D4×C2312yes—
390#124C23×C2311yescharacteristic
391#124C23×C2311yescharacteristic
392#124C23×C2312yes—
393#123—21no—
394#123—22no—
395#123—22no—
396#123—22no—
397#123—21no—
398#123—22no—
399#123—22no—
400C22⋊Q8—21no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405#121—21no—
406#121—22no—
407#121—22no—
408#121—22no—
409#121—22no—
410#120—21no—
411#120—22no—
412#120—21no—
413#120—22no—
414C4×Q8—21no—
415C4×D4—21no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419#117—22no—
420#117—22no—
421#117D4×C2312yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C42D4×C2311yescharacteristic
426#104—22no—
427#104—22no—
428#104—24no—
429#104—24no—
430#101—21no—
431#101—22no—
432#101C23×C2314yes—
433#101C23×C2312yes—
434C23#3356811yescharacteristic, direct product factor, Sylow
435C24—21no—
436C24—21no—
437#41—24no—
438#41—22no—
439#41—22no—
440#41—24no—
441#41—42no—
442#41—41no—
443#41—42no—
444#41—42no—
445#41—42no—
446#41—42no—
447#41—44no—
448#41—41no—
449#41—42no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—44no—
458C2×Q8C2×D4×C2312yes—
459C2×Q8C2×D4×C2311yescharacteristic
460C2×Q8C2×D4×C2312yes—
461C2×Q8C24×C2311yescharacteristic
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—81no—
474C2×D4—81no—
475C2×D4C2×D4×C2312yes—
476C2×D4C2×D4×C2311yescharacteristic
477C2×D4C2×D4×C2311yescharacteristic
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4C2×D4×C2311yescharacteristic
486C22×C4C2×D4×C2312yes—
487Q16—44no—
488SD16—44no—
489SD16—42no—
490SD16—42no—
491SD16—44no—
492D8—42no—
493D8—42no—
494D8—44no—
495M16—22no—
496M16—24no—
497M16—41no—
498M16—42no—
499C2×C8—41no—
500C2×C8—42no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C42—22no—
526C42—21no—
527C42C2×D4×C2311yescharacteristic
528C42C2×D4×C2311yescharacteristic
529C42C2×D4×C2312yes—
530C42C2×D4×C2312yes—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—81no—
538C23—82no—
539C23—82no—
540C23C23×#12011yescharacteristic
541Q8—22no—
542Q8—24no—
543Q8—41no—
544Q8—42no—
545Q8—41no—
546Q8—42no—
547Q8—42no—
548Q8—41no—
549Q8—81no—
550D4—22no—
551D4—24no—
552D4—41no—
553D4—41no—
554D4—42no—
555D4—42no—
556D4—41no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—81no—
562D4—82no—
563D4—82no—
564D4—82no—
565D4—81no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4C23×#12011yescharacteristic
586C2×C4C23×#12012yes—
587C2×C4C22×D4×C2312yes—
588C2×C4C22×D4×C2311yescharacteristic
589C8—42no—
590C8—44no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—41no—
599C22—41no—
600C22—82no—
601C22—161no—
602C22—161no—
603C22—162no—
604C22C23×B(4,𝔽2)11yescharacteristic
605C22C23×B(4,𝔽2)11yescharacteristic
606C22C2×C23×#12011yescharacteristic
607C4—21no—
608C4—22no—
609C4—22no—
610C4—21no—
611C4—41no—
612C4—42no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—81no—
621C2—81no—
622C2—82no—
623C2—82no—
624C2C2×C23×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C23C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C23
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C22×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3545C2semidirect
2. C23×#3020C2semidirect
3. C2×C23×#454C2semidirect
4. C23×#2204C2semidirect
5. C23×#2203C2semidirect
6. C23×#2202C2semidirect
7. C23×#2133C2semidirect
8. C2×C23×#355C2semidirect
9. C2×C23×2−1+4C22semidirect
12. C23×#563C22semidirect
14. C23×H(3,𝔽4)C22semidirect
15. C23×#555C22semidirect
26. C23×#456C22semidirect
31. C23×#454C22semidirect
32. C23×#454C22semidirect
35. C2×C23×#101C22semidirect
36. C2×C23×#101C22semidirect
37. C23×#356C22semidirect
38. C23×#355C22semidirect
39. C23×#355C22semidirect
40. C23×#355C22semidirect
218. #33568C23direct (N × H)
279. #3545C2×C23semidirect
280. #3020C2×C23semidirect
281. C2×#454C2×C23semidirect
282. #2204C2×C23semidirect
283. #2203C2×C23semidirect
284. #2202C2×C23semidirect
285. #2133C2×C23semidirect
286. C2×#355C2×C23semidirect
313. C2×2−1+4C22×C23semidirect
316. #563C22×C23semidirect
318. H(3,𝔽4)C22×C23semidirect
319. #555C22×C23semidirect
330. #456C22×C23semidirect
335. #454C22×C23semidirect
336. #454C22×C23semidirect
339. C2×#101C22×C23semidirect
340. C2×#101C22×C23semidirect
341. #356C22×C23semidirect
342. #355C22×C23semidirect
343. #355C22×C23semidirect
344. #355C22×C23semidirect
434. C23#33568direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35452
C23×#30202
C2×C23×#4542
C23×#22042
C23×#22032
C23×#22022
C23×#21332
C2×C23×#3552
#3356823