Finite Group Databasebeta

← C23×#19050 #19166676

Subgroups — C23×#19050 #19166676

Order 5888.

1026 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3545C211yesmaximal, characteristic, complemented
2C23×#3217C212yesmaximal, complemented
3C2×C23×#496C211yesmaximal, characteristic, complemented
4C23×#1657C212yesmaximal, complemented
5C23×#1656C214yesmaximal, complemented
6C23×#1655C212yesmaximal, complemented
7C23×#1533C211yesmaximal, characteristic, complemented
8C23×#1532C212yesmaximal, complemented
9C22×SD16×C23—21no—
10C22×D8×C23—21no—
11C23×#563C2211yescharacteristic, complemented
12C23×#562C2212yescomplemented
13C23×H(3,𝔽4)C2211yescharacteristic, complemented
14C23×#555—21no—
15C23×#551—21no—
16C23×#551—22no—
17C23×#546—22no—
18C23×#542—22no—
19C23×#538—22no—
20C23×#536—21no—
21C23×#534—21no—
22C2×C23×#127C2211yescharacteristic, complemented
23C23×#529C2212yes—
24C2×C23×#124C2211yescharacteristic
25C23×#496—21no—
26C23×#496—21no—
27C23×#496C2212yes—
28C23×#496C2214yescomplemented
29C23×#496C2212yescomplemented
30C23×#490C2214yescomplemented
31C23×#460—22no—
32C23×#459—22no—
33C23×#450—22no—
34C23×#447—22no—
35C23×#432C2212yescomplemented
36C2×C23×#97C2211yescharacteristic, complemented
37C23×#330C2214yescomplemented
38C23×#329C2214yescomplemented
39C23×#329C2212yescomplemented
40C23×#329C2212yescomplemented
41C2×#41×C23—21no—
42C22×Q8×C23D411yescharacteristic
43C22×D4×C23—21no—
44C22×D4×C23—21no—
45C22×D4×C23—22no—
46C22×D4×C23D411yescharacteristic
47C2×SD16×C23—22no—
48C2×SD16×C23—22no—
49C2×SD16×C23—24no—
50C2×SD16×C23—44no—
51C2×D8×C23—22no—
52C2×D8×C23—22no—
53C2×D8×C23—24no—
54C2×D8×C23—44no—
55C22×C8×C23—21no—
56C23×C4⋊Q8—21no—
57C23×C4⋊Q8—21no—
58C23×#127—22no—
59C23×#127—21no—
60C23×#127C2312yes—
61C23×#126—22no—
62C23×#126—22no—
63C23×#126—22no—
64C23×#125C2312yes—
65C23×#124—22no—
66C23×#124—21no—
67C23×#124—22no—
68C23×#124—21no—
69C23×#124—22no—
70C23×#124D411yescharacteristic
71C23×#124D411yescharacteristic
72C23×#124C2312yes—
73C23×#124C2311yescharacteristic
74C23×#124C2311yescharacteristic
75C23×#123—22no—
76C23×#123—22no—
77C23×#123—21no—
78C23×#123—21no—
79C23×#123—22no—
80C23×#123—22no—
81C23×#123—22no—
82C23×C22⋊Q8—22no—
83C23×C22⋊Q8—22no—
84C23×C22⋊Q8—22no—
85C23×C22⋊Q8—22no—
86C23×C22⋊Q8D412yes—
87C23×#121—22no—
88C23×#121—22no—
89C23×#121—22no—
90C23×#121—22no—
91C23×#121D412yes—
92C23×#120—22no—
93C23×#120—21no—
94C23×#120—21no—
95C23×#120—22no—
96C4×Q8×C23D412yes—
97C4×D4×C23—22no—
98C4×D4×C23—22no—
99C4×D4×C23—22no—
100C4×D4×C23D412yes—
101C23×#117—21no—
102C23×#117—22no—
103C23×#117—22no—
104C2×C22⋊C4×C23—21no—
105C2×C22⋊C4×C23—22no—
106C2×C22⋊C4×C23—22no—
107C2×C42×C23C2311yescharacteristic
108C23×C4⋊C8—22no—
109C23×#102—24no—
110C23×#102—24no—
111C23×C22⋊C8—22no—
112C23×#97—21no—
113C23×#97C2314yes—
114C23×#97C2312yes—
115C24×C23—21no—
116C24×C23—21no—
117C24×C23—41no—
118#41×C23—82no—
119C2×Q8×C23—21no—
120C2×Q8×C23—21no—
121C2×Q8×C23—42no—
122C2×Q8×C23C2×D412yes—
123C2×Q8×C23C2×D411yescharacteristic
124C2×Q8×C23C2×D411yescharacteristic
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23—22no—
128C2×D4×C23—21no—
129C2×D4×C23—22no—
130C2×D4×C23—21no—
131C2×D4×C23—24no—
132C2×D4×C23—22no—
133C2×D4×C23—42no—
134C2×D4×C23—42no—
135C2×D4×C23—42no—
136C2×D4×C23—42no—
137C2×D4×C23—41no—
138C2×D4×C23—84no—
139C2×D4×C23—82no—
140C2×D4×C23C2×D412yes—
141C2×D4×C23C2×D411yescharacteristic
142C2×D4×C23C2×D411yescharacteristic
143C22×C4×C23—21no—
144C22×C4×C23—22no—
145C22×C4×C23—22no—
146C22×C4×C23—21no—
147C22×C4×C23—22no—
148C22×C4×C23C2×D411yescharacteristic
149C22×C4×C23C2×D412yes—
150SD16×C23—44no—
151SD16×C23—44no—
152SD16×C23—84no—
153D8×C23—44no—
154D8×C23—44no—
155D8×C23—84no—
156C2×C8×C23—24no—
157C2×C8×C23—22no—
158C2×C8×C23—24no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—22no—
161C4⋊C4×C23—22no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—22no—
164C4⋊C4×C23—22no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23C2×D412yes—
167C4⋊C4×C23C2×D412yes—
168C22⋊C4×C23—22no—
169C22⋊C4×C23—21no—
170C22⋊C4×C23—21no—
171C22⋊C4×C23—22no—
172C22⋊C4×C23—22no—
173C22⋊C4×C23—22no—
174C22⋊C4×C23—22no—
175C22⋊C4×C23—22no—
176C22⋊C4×C23—22no—
177C22⋊C4×C23—22no—
178C22⋊C4×C23—22no—
179C22⋊C4×C23—22no—
180C22⋊C4×C23—22no—
181C22⋊C4×C23—22no—
182C22⋊C4×C23—22no—
183C42×C23—21no—
184C42×C23—21no—
185C42×C23—22no—
186C42×C23C2×D412yes—
187C42×C23C2×D411yescharacteristic
188C42×C23C2411yescharacteristic
189#19050C2311yesmaximal, characteristic, direct product factor, Sylow
190C23×C23—21no—
191C23×C23—22no—
192C23×C23—21no—
193C23×C23—42no—
194C23×C23—42no—
195C23×C23—82no—
196C23×C23—84no—
197C23×C23—81no—
198C23×C23—82no—
199C23×C23—82no—
200C23×C23#12011yescharacteristic
201Q8×C23—41no—
202Q8×C23—41no—
203Q8×C23—42no—
204Q8×C23—81no—
205Q8×C23—81no—
206D4×C23—41no—
207D4×C23—41no—
208D4×C23—42no—
209D4×C23—42no—
210D4×C23—44no—
211D4×C23—42no—
212D4×C23—44no—
213D4×C23—44no—
214D4×C23—81no—
215D4×C23—82no—
216D4×C23—84no—
217D4×C23—82no—
218D4×C23—82no—
219D4×C23—82no—
220D4×C23—81no—
221C2×C4×C23—22no—
222C2×C4×C23—22no—
223C2×C4×C23—21no—
224C2×C4×C23—22no—
225C2×C4×C23—22no—
226C2×C4×C23—21no—
227C2×C4×C23—42no—
228C2×C4×C23—42no—
229C2×C4×C23—82no—
230C2×C4×C23—82no—
231C2×C4×C23—81no—
232C2×C4×C23—82no—
233C2×C4×C23#12012yes—
234C2×C4×C23#12011yescharacteristic
235C2×C4×C23C22×D412yes—
236C2×C4×C23C22×D411yescharacteristic
237C8×C23—24no—
238C8×C23—44no—
239#3545C2×C2311yescharacteristic, complemented
240#3217C2×C2312yescomplemented
241C2×#496C2×C2311yescharacteristic, complemented
242#1657C2×C2312yescomplemented
243#1656C2×C2314yescomplemented
244#1655C2×C2312yescomplemented
245#1533C2×C2311yescharacteristic, complemented
246#1532C2×C2312yescomplemented
247C22×C23—22no—
248C22×C23—42no—
249C22×C23—41no—
250C22×C23—41no—
251C22×C23—42no—
252C22×C23—42no—
253C22×C23—42no—
254C22×C23—42no—
255C22×C23—82no—
256C22×C23—84no—
257C22×C23—162no—
258C22×C23—164no—
259C22×C23—162no—
260C22×C23#45312yes—
261C22×C23C2×#12011yescharacteristic
262C4×C23—21no—
263C4×C23—21no—
264C4×C23—22no—
265C4×C23—22no—
266C4×C23—41no—
267C4×C23—41no—
268C4×C23—42no—
269C4×C23—42no—
270C4×C23—42no—
271C22×SD16—21no—
272C22×D8—21no—
273#563C22×C2311yescharacteristic, complemented
274#562C22×C2312yescomplemented
275H(3,𝔽4)C22×C2311yescharacteristic, complemented
276#555—21no—
277#551—21no—
278#551—22no—
279#546—22no—
280#542—22no—
281#538—22no—
282#536—21no—
283#534—21no—
284C2×#127C22×C2311yescharacteristic, complemented
285#529C22×C2312yes—
286C2×#124C22×C2311yescharacteristic
287#496—21no—
288#496—21no—
289#496C22×C2312yes—
290#496C22×C2312yescomplemented
291#496C22×C2314yescomplemented
292#490C22×C2314yescomplemented
293#460—22no—
294#459—22no—
295#450—22no—
296#447—22no—
297#432C22×C2312yescomplemented
298C2×#97C22×C2311yescharacteristic, complemented
299#330C22×C2314yescomplemented
300#329C22×C2314yescomplemented
301#329C22×C2312yescomplemented
302#329C22×C2312yescomplemented
303C2×C23—22no—
304C2×C23—82no—
305C2×C23—81no—
306C2×C23—81no—
307C2×C23—162no—
308C2×C23C2×#45311yescharacteristic
309C2×C23#302112yes—
310C2×#41—21no—
311C22×Q8D4×C2311yescharacteristic
312C22×D4—21no—
313C22×D4—21no—
314C22×D4—22no—
315C22×D4D4×C2311yescharacteristic
316C2×SD16—22no—
317C2×SD16—22no—
318C2×SD16—24no—
319C2×SD16—44no—
320C2×D8—22no—
321C2×D8—22no—
322C2×D8—24no—
323C2×D8—44no—
324C22×C8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327#127—22no—
328#127—21no—
329#127C23×C2312yes—
330#126—22no—
331#126—22no—
332#126—22no—
333#125C23×C2312yes—
334#124—21no—
335#124—22no—
336#124—21no—
337#124—22no—
338#124—22no—
339#124D4×C2311yescharacteristic
340#124D4×C2311yescharacteristic
341#124C23×C2311yescharacteristic
342#124C23×C2311yescharacteristic
343#124C23×C2312yes—
344#123—21no—
345#123—22no—
346#123—22no—
347#123—21no—
348#123—22no—
349#123—22no—
350#123—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8D4×C2312yes—
356#121—22no—
357#121—22no—
358#121—22no—
359#121—22no—
360#121D4×C2312yes—
361#120—21no—
362#120—22no—
363#120—21no—
364#120—22no—
365C4×Q8D4×C2312yes—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4D4×C2312yes—
370#117—21no—
371#117—22no—
372#117—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C42C23×C2311yescharacteristic
377C4⋊C8—22no—
378#102—24no—
379#102—24no—
380C22⋊C8—22no—
381#97—21no—
382#97C23×C2314yes—
383#97C23×C2312yes—
384C23#1905011yescharacteristic, direct product factor, Sylow
385C24—21no—
386C24—21no—
387C24—41no—
388#41—82no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—42no—
392C2×Q8C2×D4×C2312yes—
393C2×Q8C2×D4×C2311yescharacteristic
394C2×Q8C2×D4×C2311yescharacteristic
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—84no—
409C2×D4—82no—
410C2×D4C2×D4×C2312yes—
411C2×D4C2×D4×C2311yescharacteristic
412C2×D4C2×D4×C2311yescharacteristic
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4C2×D4×C2311yescharacteristic
419C22×C4C2×D4×C2312yes—
420SD16—44no—
421SD16—44no—
422SD16—84no—
423D8—44no—
424D8—44no—
425D8—84no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4C2×D4×C2312yes—
437C4⋊C4C2×D4×C2312yes—
438C22⋊C4—22no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42C2×D4×C2311yescharacteristic
457C42C2×D4×C2312yes—
458C42C24×C2311yescharacteristic
459C23—21no—
460C23—22no—
461C23—21no—
462C23—42no—
463C23—42no—
464C23—82no—
465C23—84no—
466C23—81no—
467C23—82no—
468C23—82no—
469C23C23×#12011yescharacteristic
470Q8—41no—
471Q8—41no—
472Q8—42no—
473Q8—81no—
474Q8—81no—
475D4—41no—
476D4—41no—
477D4—42no—
478D4—42no—
479D4—44no—
480D4—42no—
481D4—44no—
482D4—44no—
483D4—81no—
484D4—82no—
485D4—84no—
486D4—82no—
487D4—81no—
488D4—82no—
489D4—82no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4C23×#12012yes—
503C2×C4C23×#12011yescharacteristic
504C2×C4C22×D4×C2312yes—
505C2×C4C22×D4×C2311yescharacteristic
506C8—24no—
507C8—44no—
508C22—22no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—84no—
518C22—162no—
519C22—164no—
520C22—162no—
521C22C23×#45312yes—
522C22C2×C23×#12011yescharacteristic
523C4—21no—
524C4—21no—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—42no—
531C4—42no—
532C2—22no—
533C2—81no—
534C2—82no—
535C2—81no—
536C2—162no—
537C2C2×C23×#45311yescharacteristic
538C2C23×#302112yes—
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×#3545C2semidirect
2. C23×#3217C2semidirect
3. C2×C23×#496C2semidirect
4. C23×#1657C2semidirect
5. C23×#1656C2semidirect
6. C23×#1655C2semidirect
7. C23×#1533C2semidirect
8. C23×#1532C2semidirect
11. C23×#563C22semidirect
12. C23×#562C22semidirect
13. C23×H(3,𝔽4)C22semidirect
22. C2×C23×#127C22semidirect
28. C23×#496C22semidirect
29. C23×#496C22semidirect
30. C23×#490C22semidirect
35. C23×#432C22semidirect
36. C2×C23×#97C22semidirect
37. C23×#330C22semidirect
38. C23×#329C22semidirect
39. C23×#329C22semidirect
40. C23×#329C22semidirect
189. #19050C23direct (N × H)
239. #3545C2×C23semidirect
240. #3217C2×C23semidirect
241. C2×#496C2×C23semidirect
242. #1657C2×C23semidirect
243. #1656C2×C23semidirect
244. #1655C2×C23semidirect
245. #1533C2×C23semidirect
246. #1532C2×C23semidirect
273. #563C22×C23semidirect
274. #562C22×C23semidirect
275. H(3,𝔽4)C22×C23semidirect
284. C2×#127C22×C23semidirect
290. #496C22×C23semidirect
291. #496C22×C23semidirect
292. #490C22×C23semidirect
297. #432C22×C23semidirect
298. C2×#97C22×C23semidirect
299. #330C22×C23semidirect
300. #329C22×C23semidirect
301. #329C22×C23semidirect
302. #329C22×C23semidirect
384. C23#19050direct (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×#32172
C2×C23×#4962
C23×#16572
C23×#16562
C23×#16552
C23×#15332
C23×#15322
#1905023