Finite Group Databasebeta

← C23×#8403 #19156029

Subgroups — C23×#8403 #19156029

Order 5888.

1364 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#534C211yesmaximal, characteristic
2C23×#1874C212yesmaximal, complemented
3C2×C23×#327C212yesmaximal, complemented
4C2×C23×#323C211yesmaximal, characteristic, complemented
5C23×#1439C211yesmaximal, characteristic, complemented
6C23×D4×C23—21no—
7C23×#534—24no—
8C23×#534—22no—
9C23×#534C414yescomplemented
10C2×C23×#123—22no—
11C2×C23×#121—21no—
12C2×C23×#121C2212yes—
13C2×C23×#120—22no—
14C22×C4⋊C4×C23C2211yescharacteristic, complemented
15C22×C22⋊C4×C23—21no—
16C22×C22⋊C4×C23C2211yescharacteristic
17C2×C23×#102—22no—
18C2×C23×C22⋊C8C2212yescomplemented
19C2×C23×#95—21no—
20C2×C23×#95C2211yescharacteristic, complemented
21C23×#340—22no—
22C23×#327—24no—
23C23×#323—22no—
24C25×C23—21no—
25C25×C23D411yescharacteristic
26C22×D4×C23—22no—
27C22×D4×C23—24no—
28C22×D4×C23—24no—
29C22×D4×C23—22no—
30C22×D4×C23—21no—
31C22×D4×C23—42no—
32C22×D4×C23—42no—
33C22×D4×C23—42no—
34C22×D4×C23—42no—
35C22×D4×C23D412yes—
36C23×C4×C23—21no—
37C23×C4×C23—21no—
38C23×C4×C23—21no—
39C23×C4×C23D411yescharacteristic
40C23×C4×C23C2311yescharacteristic
41C22×C8×C23—22no—
42C23×#123—28no—
43C23×#123—28no—
44C23×#121—24no—
45C23×#121—24no—
46C23×#121—22no—
47C23×#121—22no—
48C23×#121C2×C414yes—
49C23×#121D414yes—
50C23×#120—28no—
51C23×#120—28no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23—22no—
54C2×C4⋊C4×C23—42no—
55C2×C4⋊C4×C23D412yes—
56C2×C22⋊C4×C23—22no—
57C2×C22⋊C4×C23—22no—
58C2×C22⋊C4×C23—22no—
59C2×C22⋊C4×C23—22no—
60C2×C22⋊C4×C23—42no—
61C2×C22⋊C4×C23—42no—
62C2×C22⋊C4×C23—42no—
63C2×C22⋊C4×C23C2×C412yes—
64C2×C22⋊C4×C23D412yes—
65C23×#102—44no—
66C23×C22⋊C8—22no—
67C23×C22⋊C8—22no—
68C23×#95—22no—
69C23×#95—42no—
70C24×C23—21no—
71C24×C23—21no—
72C24×C23—21no—
73C24×C23—21no—
74C24×C23—24no—
75C24×C23—22no—
76C24×C23—41no—
77C24×C23—41no—
78C24×C23—41no—
79C24×C23—41no—
80C24×C23—42no—
81C24×C23—42no—
82C24×C23—81no—
83C24×C23—81no—
84C24×C23—81no—
85C24×C23—81no—
86C24×C23C22⋊C412yes—
87C24×C23D812yes—
88C24×C23SD1612yes—
89C24×C23C2×D411yescharacteristic
90C2×D4×C23—28no—
91C2×D4×C23—24no—
92C2×D4×C23—24no—
93C2×D4×C23—24no—
94C2×D4×C23—22no—
95C2×D4×C23—42no—
96C2×D4×C23—42no—
97C2×D4×C23—48no—
98C2×D4×C23—48no—
99C2×D4×C23—48no—
100C2×D4×C23—44no—
101C2×D4×C23—44no—
102C2×D4×C23—48no—
103C2×D4×C23—42no—
104C2×D4×C23—42no—
105C2×D4×C23—42no—
106C2×D4×C23—42no—
107C2×D4×C23—81no—
108C2×D4×C23—81no—
109C2×D4×C23C22⋊C414yescomplemented
110C22×C4×C23—24no—
111C22×C4×C23—21no—
112C22×C4×C23—22no—
113C22×C4×C23—21no—
114C22×C4×C23—22no—
115C22×C4×C23—22no—
116C22×C4×C23—41no—
117C22×C4×C23—41no—
118C22×C4×C23—42no—
119C22×C4×C23—42no—
120C22×C4×C23—41no—
121C22×C4×C23—41no—
122C22×C4×C23—41no—
123C22×C4×C23—41no—
124C22×C4×C23—41no—
125C22×C4×C23—41no—
126C22×C4×C23—41no—
127C22×C4×C23—41no—
128C22×C4×C23—41no—
129C22×C4×C23—41no—
130C22×C4×C23—41no—
131C22×C4×C23—41no—
132C22×C4×C23—41no—
133C22×C4×C23—41no—
134C22×C4×C23C22⋊C412yes—
135C22×C4×C23C22×C411yescharacteristic
136C22×C4×C23C2×D412yes—
137C22×C4×C23C2×D412yes—
138C22×C4×C23C2×D411yescharacteristic
139C2×C8×C23—42no—
140C2×C8×C23—42no—
141C2×C8×C23—42no—
142C4⋊C4×C23—24no—
143C4⋊C4×C23—22no—
144C4⋊C4×C23—24no—
145C4⋊C4×C23—42no—
146C4⋊C4×C23C22⋊C414yescomplemented
147C22⋊C4×C23—28no—
148C22⋊C4×C23—24no—
149C22⋊C4×C23—24no—
150C22⋊C4×C23—24no—
151C22⋊C4×C23—24no—
152C22⋊C4×C23—48no—
153#8403C2311yesmaximal, characteristic, direct product factor, Sylow
154C23×C23—21no—
155C23×C23—21no—
156C23×C23—21no—
157C23×C23—21no—
158C23×C23—22no—
159C23×C23—22no—
160C23×C23—22no—
161C23×C23—22no—
162C23×C23—22no—
163C23×C23—22no—
164C23×C23—24no—
165C23×C23—42no—
166C23×C23—42no—
167C23×C23—42no—
168C23×C23—42no—
169C23×C23—42no—
170C23×C23—42no—
171C23×C23—48no—
172C23×C23—41no—
173C23×C23—41no—
174C23×C23—41no—
175C23×C23—41no—
176C23×C23—42no—
177C23×C23—42no—
178C23×C23—42no—
179C23×C23—42no—
180C23×C23—42no—
181C23×C23—42no—
182C23×C23—81no—
183C23×C23—81no—
184C23×C23—81no—
185C23×C23—81no—
186C23×C23—81no—
187C23×C23—81no—
188C23×C23—84no—
189C23×C23—84no—
190C23×C23—84no—
191C23×C23—84no—
192C23×C23—84no—
193C23×C23—84no—
194C23×C23#10214yescomplemented
195C23×C23C2×C22⋊C411yescharacteristic
196C23×C23#12011yescharacteristic
197C23×C23#12011yescharacteristic
198C23×C23C2×D811yescharacteristic
199C23×C23C2×SD1611yescharacteristic
200C23×C23AGL(1,ℤ/8)11yescharacteristic
201C23×C23AGL(1,ℤ/8)11yescharacteristic
202D4×C23—44no—
203D4×C23—416no—
204D4×C23—44no—
205D4×C23—44no—
206D4×C23—44no—
207D4×C23—44no—
208D4×C23—82no—
209D4×C23—84no—
210C2×C4×C23—22no—
211C2×C4×C23—22no—
212C2×C4×C23—22no—
213C2×C4×C23—24no—
214C2×C4×C23—41no—
215C2×C4×C23—41no—
216C2×C4×C23—42no—
217C2×C4×C23—48no—
218C2×C4×C23—42no—
219C2×C4×C23—42no—
220C2×C4×C23—41no—
221C2×C4×C23—41no—
222C2×C4×C23—42no—
223C2×C4×C23—42no—
224C2×C4×C23—42no—
225C2×C4×C23—42no—
226C2×C4×C23—42no—
227C2×C4×C23—42no—
228C2×C4×C23—42no—
229C2×C4×C23—81no—
230C2×C4×C23—81no—
231C2×C4×C23—81no—
232C2×C4×C23—81no—
233C2×C4×C23—81no—
234C2×C4×C23—81no—
235C2×C4×C23—81no—
236C2×C4×C23—81no—
237C2×C4×C23—81no—
238C2×C4×C23C2×C22⋊C412yes—
239C2×C4×C23#12012yes—
240C8×C23—82no—
241C2×#534C2×C2311yescharacteristic
242#1874C2×C2312yescomplemented
243C2×#327C2×C2312yescomplemented
244C2×#323C2×C2311yescharacteristic, complemented
245#1439C2×C2311yescharacteristic, complemented
246C22×C23—21no—
247C22×C23—21no—
248C22×C23—21no—
249C22×C23—21no—
250C22×C23—21no—
251C22×C23—21no—
252C22×C23—21no—
253C22×C23—21no—
254C22×C23—21no—
255C22×C23—21no—
256C22×C23—21no—
257C22×C23—21no—
258C22×C23—24no—
259C22×C23—44no—
260C22×C23—42no—
261C22×C23—42no—
262C22×C23—42no—
263C22×C23—42no—
264C22×C23—42no—
265C22×C23—42no—
266C22×C23—44no—
267C22×C23—44no—
268C22×C23—44no—
269C22×C23—82no—
270C22×C23—82no—
271C22×C23—82no—
272C22×C23—816no—
273C22×C23—82no—
274C22×C23—82no—
275C22×C23—82no—
276C22×C23—82no—
277C22×C23#37911yescharacteristic
278C22×C23C2×#10211yescharacteristic
279C22×C23#41811yescharacteristic
280C22×C23#44711yescharacteristic
281C22×C23#44711yescharacteristic
282C22×C23#45011yescharacteristic
283C22×C23#45011yescharacteristic
284C22×C23#45311yescharacteristic
285C22×C23#45511yescharacteristic
286C22×C23B(4,𝔽2)11yescharacteristic
287C22×C23#45811yescharacteristic
288C4×C23—22no—
289C4×C23—22no—
290C4×C23—42no—
291C4×C23—44no—
292C4×C23—82no—
293C4×C23—82no—
294C23×D4—21no—
295#534—24no—
296#534—22no—
297#534C4×C2314yescomplemented
298C2×#123—22no—
299C2×#121—21no—
300C2×#121C22×C2312yes—
301C2×#120—22no—
302C22×C4⋊C4C22×C2311yescharacteristic, complemented
303C22×C22⋊C4—21no—
304C22×C22⋊C4C22×C2311yescharacteristic
305C2×#102—22no—
306C2×C22⋊C8C22×C2312yescomplemented
307C2×#95—21no—
308C2×#95C22×C2311yescharacteristic, complemented
309#340—22no—
310#327—24no—
311#323—22no—
312C2×C23—21no—
313C2×C23—21no—
314C2×C23—21no—
315C2×C23—21no—
316C2×C23—44no—
317C2×C23—84no—
318C2×C23#159511yescharacteristic
319C2×C23#160111yescharacteristic
320C2×C23#160711yescharacteristic
321C2×C23#161311yescharacteristic
322C2×C23#178611yescharacteristic
323C2×C23#179011yescharacteristic
324C2×C23#179211yescharacteristic
325C25—21no—
326C25D4×C2311yescharacteristic
327C22×D4—22no—
328C22×D4—24no—
329C22×D4—24no—
330C22×D4—22no—
331C22×D4—21no—
332C22×D4—42no—
333C22×D4—42no—
334C22×D4—42no—
335C22×D4—42no—
336C22×D4D4×C2312yes—
337C23×C4—21no—
338C23×C4—21no—
339C23×C4—21no—
340C23×C4D4×C2311yescharacteristic
341C23×C4C23×C2311yescharacteristic
342C22×C8—22no—
343#123—28no—
344#123—28no—
345#121—24no—
346#121—24no—
347#121—22no—
348#121—22no—
349#121C2×C4×C2314yes—
350#121D4×C2314yes—
351#120—28no—
352#120—28no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4D4×C2312yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—42no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4C2×C4×C2312yes—
365C2×C22⋊C4D4×C2312yes—
366#102—44no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369#95—22no—
370#95—42no—
371C23#840311yescharacteristic, direct product factor, Sylow
372C24—21no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—24no—
377C24—22no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—42no—
383C24—42no—
384C24—81no—
385C24—81no—
386C24—81no—
387C24—81no—
388C24C22⋊C4×C2312yes—
389C24D8×C2312yes—
390C24SD16×C2312yes—
391C24C2×D4×C2311yescharacteristic
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4C22⋊C4×C2314yescomplemented
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4C22⋊C4×C2312yes—
437C22×C4C22×C4×C2311yescharacteristic
438C22×C4C2×D4×C2312yes—
439C22×C4C2×D4×C2312yes—
440C22×C4C2×D4×C2311yescharacteristic
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—42no—
448C4⋊C4C22⋊C4×C2314yescomplemented
449C22⋊C4—28no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—48no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—48no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23C23×#10214yescomplemented
496C23C2×C22⋊C4×C2311yescharacteristic
497C23C23×#12011yescharacteristic
498C23C23×#12011yescharacteristic
499C23C2×D8×C2311yescharacteristic
500C23C2×SD16×C2311yescharacteristic
501C23C23×AGL(1,ℤ/8)11yescharacteristic
502C23C23×AGL(1,ℤ/8)11yescharacteristic
503D4—44no—
504D4—416no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—82no—
510D4—84no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4C2×C22⋊C4×C2312yes—
540C2×C4C23×#12012yes—
541C8—82no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—24no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—816no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22C23×#37911yescharacteristic
574C22C2×C23×#10211yescharacteristic
575C22C23×#41811yescharacteristic
576C22C23×#44711yescharacteristic
577C22C23×#44711yescharacteristic
578C22C23×#45011yescharacteristic
579C22C23×#45011yescharacteristic
580C22C23×#45311yescharacteristic
581C22C23×#45511yescharacteristic
582C22C23×B(4,𝔽2)11yescharacteristic
583C22C23×#45811yescharacteristic
584C4—22no—
585C4—22no—
586C4—44no—
587C4—42no—
588C4—82no—
589C4—82no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—44no—
595C2—84no—
596C2C23×#159511yescharacteristic
597C2C23×#160111yescharacteristic
598C2C23×#160711yescharacteristic
599C2C23×#161311yescharacteristic
600C2C23×#178611yescharacteristic
601C2C23×#179011yescharacteristic
602C2C23×#179211yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#534C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1874C2semidirect
3. C2×C23×#327C2semidirect
4. C2×C23×#323C2semidirect
5. C23×#1439C2semidirect
9. C23×#534C4semidirect
14. C22×C4⋊C4×C23C22semidirect
18. C2×C23×C22⋊C8C22semidirect
20. C2×C23×#95C22semidirect
109. C2×D4×C23C22⋊C4semidirect
146. C4⋊C4×C23C22⋊C4semidirect
153. #8403C23direct (N × H)
194. C23×C23#102semidirect
242. #1874C2×C23semidirect
243. C2×#327C2×C23semidirect
244. C2×#323C2×C23semidirect
245. #1439C2×C23semidirect
297. #534C4×C23semidirect
302. C22×C4⋊C4C22×C23semidirect
306. C2×C22⋊C8C22×C23semidirect
308. C2×#95C22×C23semidirect
371. C23#8403direct (N × H)
411. C2×D4C22⋊C4×C23semidirect
448. C4⋊C4C22⋊C4×C23semidirect
495. C23C23×#102semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#5342
C23×#18742
C2×C23×#3272
C2×C23×#3232
C23×#14392
#840323