Finite Group Databasebeta

← C23×#33569 #19181195

Subgroups — C23×#33569 #19181195

Order 5888.

1082 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3544C211yesmaximal, characteristic, complemented
2C23×#3020C212yesmaximal, complemented
3C2×C23×#456C211yesmaximal, characteristic, complemented
4C23×#2205C212yesmaximal, complemented
5C23×#2204C212yesmaximal, complemented
6C23×#2203C214yesmaximal, complemented
7C23×#2133C212yesmaximal, complemented
8C2×C23×#356C211yesmaximal, characteristic, complemented
9C2×C23×2−1+4C2211yescharacteristic
10C23×#575—22no—
11C2×C23×#137—21no—
12C23×#563—21no—
13C23×#563C2212yescomplemented
14Q82×C23C2211yescharacteristic, complemented
15C23×#555C2211yescharacteristic, complemented
16C23×#554—21no—
17C23×#552—22no—
18C23×#547—22no—
19C23×#544—22no—
20C23×#542—22no—
21C23×#536—21no—
22C23×#536C2212yes—
23C2×C23×C4⋊Q8C2211yescharacteristic
24C23×#529—21no—
25C23×#456—21no—
26C23×#456—21no—
27C23×#456—22no—
28C23×#456C2212yes—
29C23×#456C2214yes—
30C23×#456C2212yes—
31C23×#454—22no—
32C23×#454C2214yes—
33C23×#421—22no—
34C2×C23×#104—21no—
35C2×C23×#101C2211yescharacteristic
36C2×C23×#101C2212yes—
37C23×#356C2212yescomplemented
38C23×#356C2214yescomplemented
39C23×#356C2212yescomplemented
40C23×#355C2214yescomplemented
41C23×2−1+4—22no—
42C23×2−1+4—21no—
43C23×2−1+4—42no—
44C23×2−1+4C2312yes—
45C2×#41×C23—21no—
46C2×#41×C23—21no—
47C2×#41×C23—22no—
48C2×#41×C23—41no—
49C2×#41×C23D412yes—
50C22×Q8×C23—41no—
51C22×Q8×C23D411yescharacteristic
52C22×Q8×C23C2311yescharacteristic
53C23×#137—22no—
54C23×#137—22no—
55C23×#137—24no—
56C23×#137—42no—
57C23×#137—42no—
58C23×AGL(1,ℤ/8)—24no—
59C23×AGL(1,ℤ/8)—42no—
60C23×#135—44no—
61C2×Q16×C23—41no—
62C2×SD16×C23—41no—
63C2×M16×C23—21no—
64C2×M16×C23—22no—
65C23×C4⋊Q8—21no—
66C23×C4⋊Q8—22no—
67C23×C4⋊Q8—21no—
68C23×C4⋊Q8—22no—
69C23×C4⋊Q8D411yescharacteristic
70C23×C4⋊Q8D411yescharacteristic
71C23×C4⋊Q8D412yes—
72C23×C4⋊Q8C2311yescharacteristic
73C23×C4⋊Q8C2312yes—
74C23×C4⋊Q8C2311yescharacteristic
75C23×#126—22no—
76C23×#126—22no—
77C23×#126—22no—
78C23×#125—22no—
79C23×#125—22no—
80C23×#125—21no—
81C23×#125—22no—
82C23×#124—21no—
83C23×#124—21no—
84C23×#124—21no—
85C23×#124D412yes—
86C23×#124C2312yes—
87C23×#123—22no—
88C23×#123—22no—
89C23×#123—22no—
90C23×#123—22no—
91C23×#123—22no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—21no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8—22no—
96C23×C22⋊Q8—22no—
97C23×C22⋊Q8—22no—
98C23×C22⋊Q8—22no—
99C23×C22⋊Q8—22no—
100C23×#121—21no—
101C23×#121—22no—
102C4×Q8×C23—22no—
103C4×Q8×C23—21no—
104C4×D4×C23—22no—
105C4×D4×C23—22no—
106C4×D4×C23—21no—
107C23×#117—21no—
108C23×#117—22no—
109C23×#117—22no—
110C23×#117D412yes—
111C2×C4⋊C4×C23—22no—
112C2×C4⋊C4×C23—22no—
113C2×C42×C23D411yescharacteristic
114C23×#104—22no—
115C23×#104—22no—
116C23×#104—24no—
117C23×#104—24no—
118C23×#101—22no—
119C23×#101—21no—
120C23×#101C2312yes—
121C23×#101C2314yes—
122#41×C23—24no—
123#41×C23—22no—
124#41×C23—24no—
125#41×C23—22no—
126#41×C23—42no—
127#41×C23—41no—
128#41×C23—42no—
129#41×C23—42no—
130#41×C23—42no—
131#41×C23—44no—
132#41×C23—42no—
133#41×C23—41no—
134#41×C23—42no—
135#41×C23—81no—
136#41×C23—81no—
137C2×Q8×C23—21no—
138C2×Q8×C23—21no—
139C2×Q8×C23—21no—
140C2×Q8×C23—21no—
141C2×Q8×C23—22no—
142C2×Q8×C23—41no—
143C2×Q8×C23—41no—
144C2×Q8×C23—41no—
145C2×Q8×C23—41no—
146C2×Q8×C23—42no—
147C2×Q8×C23—44no—
148C2×Q8×C23—42no—
149C2×Q8×C23C2×D411yescharacteristic
150C2×Q8×C23C2×D412yes—
151C2×Q8×C23C2×D411yescharacteristic
152C2×Q8×C23C2×D411yescharacteristic
153C2×Q8×C23C2×D412yes—
154C2×Q8×C23C2411yescharacteristic
155C2×D4×C23—21no—
156C2×D4×C23—22no—
157C2×D4×C23—21no—
158C2×D4×C23—22no—
159C2×D4×C23—21no—
160C2×D4×C23—41no—
161C2×D4×C23C2×D412yes—
162C22×C4×C23—22no—
163C22×C4×C23—21no—
164C22×C4×C23—21no—
165C22×C4×C23—22no—
166C22×C4×C23—22no—
167C22×C4×C23—41no—
168C22×C4×C23—42no—
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2×D412yes—
171Q16×C23—42no—
172Q16×C23—42no—
173Q16×C23—44no—
174SD16×C23—42no—
175SD16×C23—42no—
176SD16×C23—44no—
177SD16×C23—44no—
178D8×C23—44no—
179M16×C23—24no—
180M16×C23—22no—
181M16×C23—42no—
182M16×C23—41no—
183C2×C8×C23—41no—
184C2×C8×C23—42no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23—22no—
195C4⋊C4×C23—22no—
196C4⋊C4×C23—22no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—22no—
199C4⋊C4×C23—22no—
200C4⋊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—22no—
210C42×C23—21no—
211C42×C23—22no—
212C42×C23—21no—
213C42×C23—21no—
214C42×C23C2×D411yescharacteristic
215C42×C23C2×D411yescharacteristic
216C42×C23C2×D412yes—
217C42×C23C2×D412yes—
218#33569C2311yesmaximal, characteristic, direct product factor, Sylow
219C23×C23—22no—
220C23×C23—41no—
221C23×C23#12011yescharacteristic
222Q8×C23—24no—
223Q8×C23—22no—
224Q8×C23—42no—
225Q8×C23—41no—
226Q8×C23—41no—
227Q8×C23—41no—
228Q8×C23—41no—
229Q8×C23—42no—
230Q8×C23—42no—
231Q8×C23—42no—
232Q8×C23—41no—
233Q8×C23—81no—
234Q8×C23—81no—
235D4×C23—24no—
236D4×C23—22no—
237D4×C23—42no—
238D4×C23—42no—
239D4×C23—42no—
240D4×C23—41no—
241D4×C23—42no—
242D4×C23—42no—
243D4×C23—42no—
244D4×C23—42no—
245D4×C23—81no—
246C2×C4×C23—21no—
247C2×C4×C23—21no—
248C2×C4×C23—22no—
249C2×C4×C23—22no—
250C2×C4×C23—21no—
251C2×C4×C23—22no—
252C2×C4×C23—21no—
253C2×C4×C23—22no—
254C2×C4×C23—21no—
255C2×C4×C23—22no—
256C2×C4×C23—21no—
257C2×C4×C23—42no—
258C2×C4×C23—44no—
259C2×C4×C23—44no—
260C2×C4×C23—42no—
261C2×C4×C23—44no—
262C2×C4×C23—42no—
263C2×C4×C23—41no—
264C2×C4×C23—41no—
265C2×C4×C23—41no—
266C2×C4×C23—82no—
267C2×C4×C23—82no—
268C2×C4×C23#12011yescharacteristic
269C2×C4×C23#12012yes—
270C2×C4×C23C22×D411yescharacteristic
271C2×C4×C23C22×D412yes—
272C8×C23—44no—
273C8×C23—42no—
274#3544C2×C2311yescharacteristic, complemented
275#3020C2×C2312yescomplemented
276C2×#456C2×C2311yescharacteristic, complemented
277#2205C2×C2312yescomplemented
278#2204C2×C2312yescomplemented
279#2203C2×C2314yescomplemented
280#2133C2×C2312yescomplemented
281C2×#356C2×C2311yescharacteristic, complemented
282C22×C23—41no—
283C22×C23—42no—
284C22×C23—42no—
285C22×C23—42no—
286C22×C23—42no—
287C22×C23—82no—
288C22×C23B(4,𝔽2)11yescharacteristic
289C22×C23B(4,𝔽2)11yescharacteristic
290C22×C23C2×#12011yescharacteristic
291C4×C23—22no—
292C4×C23—22no—
293C4×C23—21no—
294C4×C23—22no—
295C4×C23—21no—
296C4×C23—42no—
297C4×C23—41no—
298C4×C23—42no—
299C4×C23—41no—
300C4×C23—41no—
301C4×C23—42no—
302C4×C23—42no—
303C2×2−1+4C22×C2311yescharacteristic
304#575—22no—
305C2×#137—21no—
306#563—21no—
307#563C22×C2312yescomplemented
308Q82C22×C2311yescharacteristic, complemented
309#555C22×C2311yescharacteristic, complemented
310#554—21no—
311#552—22no—
312#547—22no—
313#544—22no—
314#542—22no—
315#536—21no—
316#536C22×C2312yes—
317C2×C4⋊Q8C22×C2311yescharacteristic
318#529—21no—
319#456—21no—
320#456—21no—
321#456—22no—
322#456C22×C2312yes—
323#456C22×C2314yes—
324#456C22×C2312yes—
325#454—22no—
326#454C22×C2314yes—
327#421—22no—
328C2×#104—21no—
329C2×#101C22×C2312yes—
330C2×#101C22×C2311yescharacteristic
331#356C22×C2312yescomplemented
332#356C22×C2312yescomplemented
333#356C22×C2314yescomplemented
334#355C22×C2314yescomplemented
335C2×C23—21no—
336C2×C23—21no—
337C2×C23—21no—
338C2×C23—82no—
339C2×C23—82no—
340C2×C23C2×B(4,𝔽2)11yescharacteristic
3412−1+4—22no—
3422−1+4—21no—
3432−1+4—42no—
3442−1+4C23×C2312yes—
345C2×#41—21no—
346C2×#41—21no—
347C2×#41—22no—
348C2×#41—41no—
349C2×#41D4×C2312yes—
350C22×Q8—41no—
351C22×Q8D4×C2311yescharacteristic
352C22×Q8C23×C2311yescharacteristic
353#137—22no—
354#137—22no—
355#137—24no—
356#137—42no—
357#137—42no—
358AGL(1,ℤ/8)—24no—
359AGL(1,ℤ/8)—42no—
360#135—44no—
361C2×Q16—41no—
362C2×SD16—41no—
363C2×M16—21no—
364C2×M16—22no—
365C4⋊Q8—21no—
366C4⋊Q8—22no—
367C4⋊Q8—21no—
368C4⋊Q8—22no—
369C4⋊Q8D4×C2311yescharacteristic
370C4⋊Q8D4×C2311yescharacteristic
371C4⋊Q8D4×C2312yes—
372C4⋊Q8C23×C2311yescharacteristic
373C4⋊Q8C23×C2311yescharacteristic
374C4⋊Q8C23×C2312yes—
375#126—22no—
376#126—22no—
377#126—22no—
378#125—21no—
379#125—22no—
380#125—22no—
381#125—22no—
382#124—21no—
383#124—21no—
384#124—21no—
385#124D4×C2312yes—
386#124C23×C2312yes—
387#123—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392C22⋊Q8—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400#121—21no—
401#121—22no—
402C4×Q8—21no—
403C4×Q8—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—21no—
407#117—22no—
408#117—21no—
409#117—22no—
410#117D4×C2312yes—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C42D4×C2311yescharacteristic
414#104—22no—
415#104—22no—
416#104—24no—
417#104—24no—
418#101—22no—
419#101—21no—
420#101C23×C2312yes—
421#101C23×C2314yes—
422C23#3356911yescharacteristic, direct product factor, Sylow
423#41—24no—
424#41—22no—
425#41—24no—
426#41—22no—
427#41—42no—
428#41—41no—
429#41—42no—
430#41—42no—
431#41—42no—
432#41—44no—
433#41—42no—
434#41—41no—
435#41—42no—
436#41—81no—
437#41—81no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—42no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8C2×D4×C2311yescharacteristic
451C2×Q8C2×D4×C2312yes—
452C2×Q8C2×D4×C2311yescharacteristic
453C2×Q8C2×D4×C2311yescharacteristic
454C2×Q8C2×D4×C2312yes—
455C2×Q8C24×C2311yescharacteristic
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—41no—
462C2×D4C2×D4×C2312yes—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4C2×D4×C2311yescharacteristic
471C22×C4C2×D4×C2312yes—
472Q16—42no—
473Q16—42no—
474Q16—44no—
475SD16—42no—
476SD16—42no—
477SD16—44no—
478SD16—44no—
479D8—44no—
480M16—24no—
481M16—22no—
482M16—42no—
483M16—41no—
484C2×C8—41no—
485C2×C8—42no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C42—21no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42C2×D4×C2311yescharacteristic
516C42C2×D4×C2311yescharacteristic
517C42C2×D4×C2312yes—
518C42C2×D4×C2312yes—
519C23—22no—
520C23—41no—
521C23C23×#12011yescharacteristic
522Q8—24no—
523Q8—22no—
524Q8—42no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—42no—
530Q8—42no—
531Q8—41no—
532Q8—42no—
533Q8—81no—
534Q8—81no—
535D4—24no—
536D4—22no—
537D4—42no—
538D4—42no—
539D4—41no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—81no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4C23×#12011yescharacteristic
569C2×C4C23×#12012yes—
570C2×C4C22×D4×C2311yescharacteristic
571C2×C4C22×D4×C2312yes—
572C8—44no—
573C8—42no—
574C22—41no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—82no—
580C22C23×B(4,𝔽2)11yescharacteristic
581C22C23×B(4,𝔽2)11yescharacteristic
582C22C2×C23×#12011yescharacteristic
583C4—22no—
584C4—22no—
585C4—21no—
586C4—22no—
587C4—21no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—42no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—82no—
599C2—82no—
600C2C2×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×#3544C2semidirect
2. C23×#3020C2semidirect
3. C2×C23×#456C2semidirect
4. C23×#2205C2semidirect
5. C23×#2204C2semidirect
6. C23×#2203C2semidirect
7. C23×#2133C2semidirect
8. C2×C23×#356C2semidirect
13. C23×#563C22semidirect
14. Q82×C23C22semidirect
15. C23×#555C22semidirect
37. C23×#356C22semidirect
38. C23×#356C22semidirect
39. C23×#356C22semidirect
40. C23×#355C22semidirect
218. #33569C23direct (N × H)
274. #3544C2×C23semidirect
275. #3020C2×C23semidirect
276. C2×#456C2×C23semidirect
277. #2205C2×C23semidirect
278. #2204C2×C23semidirect
279. #2203C2×C23semidirect
280. #2133C2×C23semidirect
281. C2×#356C2×C23semidirect
307. #563C22×C23semidirect
308. Q82C22×C23semidirect
309. #555C22×C23semidirect
331. #356C22×C23semidirect
332. #356C22×C23semidirect
333. #356C22×C23semidirect
334. #355C22×C23semidirect
422. C23#33569direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35442
C23×#30202
C2×C23×#4562
C23×#22052
C23×#22042
C23×#22032
C23×#21332
C2×C23×#3562
#3356923