Finite Group Databasebeta

← C23×#12690 #19160316

Subgroups — C23×#12690 #19160316

Order 5888.

1228 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#534C211yesmaximal, characteristic
2C2×C23×#352C212yesmaximal, complemented
3C2×C23×AΣL(1,𝔽16)C212yesmaximal, complemented
4C23×#1783C211yesmaximal, characteristic, complemented
5C23×#1781C211yesmaximal, characteristic, complemented
6C23×D4×C23C411yescharacteristic, complemented
7C23×D4×C23C2211yescharacteristic
8C23×#534—24no—
9C23×#534—24no—
10C2×C23×#123—21no—
11C2×C23×#123C411yescharacteristic, complemented
12C2×C23×#123C2211yescharacteristic
13C2×C23×#121—21no—
14C2×C23×#121—21no—
15C2×C23×#120—21no—
16C2×C23×#120C411yescharacteristic, complemented
17C2×C23×#120C2211yescharacteristic
18C22×C22⋊C4×C23C411yescharacteristic, complemented
19C2×C23×#100—21no—
20C2×C23×#100C2212yes—
21C2×C23×#99—21no—
22C2×C23×#99C2212yes—
23C23×#407—21no—
24C23×#379—21no—
25C23×#352—24no—
26C23×AΣL(1,𝔽16)—24no—
27C25×C23C2×C411yescharacteristic
28C25×C23C2×C411yescharacteristic
29C25×C23D411yescharacteristic
30C22×D4×C23—21no—
31C22×D4×C23—21no—
32C22×D4×C23—22no—
33C22×D4×C23—21no—
34C22×D4×C23—21no—
35C22×D4×C23—22no—
36C22×D4×C23—22no—
37C22×D4×C23—22no—
38C22×D4×C23—42no—
39C22×D4×C23—42no—
40C22×D4×C23—42no—
41C22×D4×C23—42no—
42C22×D4×C23C2×C411yescharacteristic
43C22×D4×C23D411yescharacteristic
44C22×D4×C23D411yescharacteristic
45C22×D4×C23C2311yescharacteristic
46C23×C4×C23—21no—
47C23×C4×C23C2×C411yescharacteristic
48C23×C4×C23D411yescharacteristic
49C2×M16×C23—22no—
50C23×#123—28no—
51C23×#123—22no—
52C23×#123—22no—
53C23×#123—22no—
54C23×#123D414yes—
55C23×#121—28no—
56C23×#121—28no—
57C23×#120—28no—
58C23×#120—22no—
59C23×#120—22no—
60C23×#120—22no—
61C23×#120D414yes—
62C2×C4⋊C4×C23—21no—
63C2×C4⋊C4×C23—21no—
64C2×C22⋊C4×C23—21no—
65C2×C22⋊C4×C23—21no—
66C2×C22⋊C4×C23—21no—
67C2×C22⋊C4×C23—21no—
68C2×C22⋊C4×C23—21no—
69C2×C22⋊C4×C23—22no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—22no—
72C2×C22⋊C4×C23—42no—
73C2×C22⋊C4×C23C2×C411yescharacteristic
74C2×C22⋊C4×C23C2×C411yescharacteristic
75C23×#100—22no—
76C23×#100—22no—
77C23×#100—42no—
78C23×#99—22no—
79C23×#99—22no—
80C23×#99—42no—
81C23×C22⋊C8—42no—
82C24×C23—22no—
83C24×C23—22no—
84C24×C23—21no—
85C24×C23—21no—
86C24×C23—21no—
87C24×C23—22no—
88C24×C23—21no—
89C24×C23—22no—
90C24×C23—41no—
91C24×C23—41no—
92C24×C23—41no—
93C24×C23—82no—
94C24×C23—82no—
95C24×C23—82no—
96C24×C23C22⋊C411yescharacteristic
97C24×C23C22⋊C411yescharacteristic
98C24×C23C22⋊C411yescharacteristic
99C24×C23C22⋊C412yescomplemented
100C24×C23C22×C411yescharacteristic
101C24×C23C2×D411yescharacteristic
102C2×D4×C23—24no—
103C2×D4×C23—24no—
104C2×D4×C23—22no—
105C2×D4×C23—22no—
106C2×D4×C23—21no—
107C2×D4×C23—21no—
108C2×D4×C23—22no—
109C2×D4×C23—24no—
110C2×D4×C23—24no—
111C2×D4×C23—22no—
112C2×D4×C23—41no—
113C2×D4×C23—41no—
114C2×D4×C23—41no—
115C2×D4×C23—41no—
116C2×D4×C23—41no—
117C2×D4×C23—41no—
118C2×D4×C23—48no—
119C2×D4×C23—48no—
120C2×D4×C23—48no—
121C2×D4×C23—42no—
122C2×D4×C23—42no—
123C2×D4×C23—48no—
124C2×D4×C23—42no—
125C2×D4×C23—42no—
126C2×D4×C23—82no—
127C2×D4×C23—82no—
128C2×D4×C23C2×D412yes—
129C2×D4×C23C2×D412yes—
130C22×C4×C23—22no—
131C22×C4×C23—21no—
132C22×C4×C23—21no—
133C22×C4×C23—21no—
134C22×C4×C23—21no—
135C22×C4×C23—24no—
136C22×C4×C23—22no—
137C22×C4×C23—41no—
138C22×C4×C23—41no—
139C22×C4×C23—41no—
140C22×C4×C23—42no—
141C22×C4×C23—82no—
142C22×C4×C23C22⋊C411yescharacteristic
143C22×C4×C23C22⋊C412yescomplemented
144C22×C4×C23C2×D411yescharacteristic
145M16×C23—42no—
146M16×C23—42no—
147C2×C8×C23—42no—
148C4⋊C4×C23—24no—
149C4⋊C4×C23—24no—
150C22⋊C4×C23—24no—
151C22⋊C4×C23—24no—
152C22⋊C4×C23—24no—
153C22⋊C4×C23—21no—
154C22⋊C4×C23—21no—
155C22⋊C4×C23—24no—
156C22⋊C4×C23—24no—
157C22⋊C4×C23—42no—
158C22⋊C4×C23—42no—
159C22⋊C4×C23—84no—
160C22⋊C4×C23C22⋊C412yescomplemented
161C22⋊C4×C23C22⋊C412yescomplemented
162#12690C2311yesmaximal, characteristic, direct product factor, Sylow
163C23×C23—24no—
164C23×C23—24no—
165C23×C23—21no—
166C23×C23—21no—
167C23×C23—21no—
168C23×C23—21no—
169C23×C23—21no—
170C23×C23—21no—
171C23×C23—21no—
172C23×C23—21no—
173C23×C23—21no—
174C23×C23—22no—
175C23×C23—21no—
176C23×C23—21no—
177C23×C23—42no—
178C23×C23—44no—
179C23×C23—42no—
180C23×C23—42no—
181C23×C23—42no—
182C23×C23—42no—
183C23×C23—44no—
184C23×C23—42no—
185C23×C23—42no—
186C23×C23—42no—
187C23×C23—42no—
188C23×C23—41no—
189C23×C23—41no—
190C23×C23—82no—
191C23×C23—82no—
192C23×C23—82no—
193C23×C23—82no—
194C23×C23—84no—
195C23×C23—82no—
196C23×C23—82no—
197C23×C23—82no—
198C23×C23—84no—
199C23×C23—84no—
200C23×C23—84no—
201C23×C23—84no—
202C23×C23#9912yes—
203C23×C23#9912yes—
204C23×C23C2×C22⋊C411yescharacteristic
205C23×C23C2×C22⋊C411yescharacteristic
206C23×C23C2×C22⋊C411yescharacteristic
207C23×C23#12012yes—
208D4×C23—42no—
209D4×C23—42no—
210D4×C23—44no—
211D4×C23—44no—
212D4×C23—42no—
213D4×C23—42no—
214D4×C23—48no—
215D4×C23—48no—
216D4×C23—84no—
217D4×C23—84no—
218C2×C4×C23—21no—
219C2×C4×C23—21no—
220C2×C4×C23—22no—
221C2×C4×C23—22no—
222C2×C4×C23—21no—
223C2×C4×C23—22no—
224C2×C4×C23—21no—
225C2×C4×C23—21no—
226C2×C4×C23—44no—
227C2×C4×C23—42no—
228C2×C4×C23—42no—
229C2×C4×C23—42no—
230C2×C4×C23—41no—
231C2×C4×C23—41no—
232C2×C4×C23—42no—
233C2×C4×C23—84no—
234C2×C4×C23—82no—
235C2×C4×C23—84no—
236C2×C4×C23#12012yes—
237C8×C23—82no—
238C2×#534C2×C2311yescharacteristic
239C2×#352C2×C2312yescomplemented
240C2×AΣL(1,𝔽16)C2×C2312yescomplemented
241#1783C2×C2311yescharacteristic, complemented
242#1781C2×C2311yescharacteristic, complemented
243C22×C23—22no—
244C22×C23—22no—
245C22×C23—22no—
246C22×C23—22no—
247C22×C23—21no—
248C22×C23—21no—
249C22×C23—22no—
250C22×C23—22no—
251C22×C23—42no—
252C22×C23—41no—
253C22×C23—41no—
254C22×C23—42no—
255C22×C23—42no—
256C22×C23—42no—
257C22×C23—42no—
258C22×C23—42no—
259C22×C23—42no—
260C22×C23—42no—
261C22×C23—42no—
262C22×C23—42no—
263C22×C23—42no—
264C22×C23—42no—
265C22×C23—42no—
266C22×C23—41no—
267C22×C23—42no—
268C22×C23—81no—
269C22×C23—81no—
270C22×C23—82no—
271C22×C23—84no—
272C22×C23—84no—
273C22×C23—88no—
274C22×C23—84no—
275C22×C23—84no—
276C22×C23—82no—
277C22×C23#37911yescharacteristic
278C22×C23C2×#9911yescharacteristic
279C22×C23C2×#9911yescharacteristic
280C4×C23—22no—
281C4×C23—42no—
282C4×C23—42no—
283C4×C23—41no—
284C4×C23—42no—
285C4×C23—162no—
286C23×D4C4×C2311yescharacteristic, complemented
287C23×D4C22×C2311yescharacteristic
288#534—24no—
289#534—24no—
290C2×#123—21no—
291C2×#123C4×C2311yescharacteristic, complemented
292C2×#123C22×C2311yescharacteristic
293C2×#121—21no—
294C2×#121—21no—
295C2×#120—21no—
296C2×#120C4×C2311yescharacteristic, complemented
297C2×#120C22×C2311yescharacteristic
298C22×C22⋊C4C4×C2311yescharacteristic, complemented
299C2×#100—21no—
300C2×#100C22×C2312yes—
301C2×#99—21no—
302C2×#99C22×C2312yes—
303#407—21no—
304#379—21no—
305#352—24no—
306AΣL(1,𝔽16)—24no—
307C2×C23—21no—
308C2×C23—21no—
309C2×C23—42no—
310C2×C23—42no—
311C2×C23—42no—
312C2×C23—41no—
313C2×C23—41no—
314C2×C23—81no—
315C2×C23—82no—
316C2×C23#178111yescharacteristic
317C2×C23#212112yes—
318C25C2×C4×C2311yescharacteristic
319C25C2×C4×C2311yescharacteristic
320C25D4×C2311yescharacteristic
321C22×D4—21no—
322C22×D4—21no—
323C22×D4—22no—
324C22×D4—21no—
325C22×D4—21no—
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4—42no—
330C22×D4—42no—
331C22×D4—42no—
332C22×D4—42no—
333C22×D4C2×C4×C2311yescharacteristic
334C22×D4D4×C2311yescharacteristic
335C22×D4D4×C2311yescharacteristic
336C22×D4C23×C2311yescharacteristic
337C23×C4—21no—
338C23×C4C2×C4×C2311yescharacteristic
339C23×C4D4×C2311yescharacteristic
340C2×M16—22no—
341#123—28no—
342#123—22no—
343#123—22no—
344#123—22no—
345#123D4×C2314yes—
346#121—28no—
347#121—28no—
348#120—28no—
349#120—22no—
350#120—22no—
351#120—22no—
352#120D4×C2314yes—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4C2×C4×C2311yescharacteristic
365C2×C22⋊C4C2×C4×C2311yescharacteristic
366#100—22no—
367#100—22no—
368#100—42no—
369#99—22no—
370#99—22no—
371#99—42no—
372C22⋊C8—42no—
373C23#1269011yescharacteristic, direct product factor, Sylow
374C24—22no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—82no—
386C24—82no—
387C24—82no—
388C24C22⋊C4×C2311yescharacteristic
389C24C22⋊C4×C2311yescharacteristic
390C24C22⋊C4×C2311yescharacteristic
391C24C22⋊C4×C2312yescomplemented
392C24C22×C4×C2311yescharacteristic
393C24C2×D4×C2311yescharacteristic
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4C2×D4×C2312yes—
421C2×D4C2×D4×C2312yes—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—82no—
434C22×C4C22⋊C4×C2311yescharacteristic
435C22×C4C22⋊C4×C2312yescomplemented
436C22×C4C2×D4×C2311yescharacteristic
437M16—42no—
438M16—42no—
439C2×C8—42no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—84no—
452C22⋊C4C22⋊C4×C2312yescomplemented
453C22⋊C4C22⋊C4×C2312yescomplemented
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—42no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23C23×#9912yes—
494C23C23×#9912yes—
495C23C2×C22⋊C4×C2311yescharacteristic
496C23C2×C22⋊C4×C2311yescharacteristic
497C23C2×C22⋊C4×C2311yescharacteristic
498C23C23×#12012yes—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—48no—
506D4—48no—
507D4—84no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4C23×#12012yes—
528C8—82no—
529C22—22no—
530C22—22no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—88no—
560C22—84no—
561C22—84no—
562C22—82no—
563C22C23×#37911yescharacteristic
564C22C2×C23×#9911yescharacteristic
565C22C2×C23×#9911yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—162no—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—41no—
577C2—41no—
578C2—42no—
579C2—81no—
580C2—82no—
581C2C23×#178111yescharacteristic
582C2C23×#212112yes—
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×#534C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C23×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#352C2semidirect
3. C2×C23×AΣL(1,𝔽16)C2semidirect
4. C23×#1783C2semidirect
5. C23×#1781C2semidirect
6. C23×D4×C23C4semidirect
11. C2×C23×#123C4semidirect
16. C2×C23×#120C4semidirect
18. C22×C22⋊C4×C23C4semidirect
99. C24×C23C22⋊C4semidirect
143. C22×C4×C23C22⋊C4semidirect
160. C22⋊C4×C23C22⋊C4semidirect
161. C22⋊C4×C23C22⋊C4semidirect
162. #12690C23direct (N × H)
239. C2×#352C2×C23semidirect
240. C2×AΣL(1,𝔽16)C2×C23semidirect
241. #1783C2×C23semidirect
242. #1781C2×C23semidirect
286. C23×D4C4×C23semidirect
291. C2×#123C4×C23semidirect
296. C2×#120C4×C23semidirect
298. C22×C22⋊C4C4×C23semidirect
373. C23#12690direct (N × H)
391. C24C22⋊C4×C23semidirect
435. C22×C4C22⋊C4×C23semidirect
452. C22⋊C4C22⋊C4×C23semidirect
453. C22⋊C4C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#5342
C2×C23×#3522
C2×C23×AΣL(1,𝔽16)2
C23×#17832
C23×#17812
#1269023