Finite Group Databasebeta

← C23×#9111 #19156737

Subgroups — C23×#9111 #19156737

Order 5888.

984 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×C22⋊C8C211yesmaximal, characteristic
2C23×#2848C211yesmaximal, characteristic
3C2×C23×#328C212yesmaximal, complemented
4C23×C22⋊#95C211yesmaximal, characteristic
5C23×#1294C212yesmaximal, complemented
6C24×C4×C23C2211yescharacteristic
7C23×C8×C23—21no—
8C2×C23×C22⋊Q8C411yescharacteristic, complemented
9C2×C23×C22⋊Q8C411yescharacteristic, complemented
10C2×C23×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C4×C23—21no—
12C2×C23×C22⋊C8—24no—
13C2×C23×C22⋊C8—21no—
14C2×C23×C22⋊C8C2212yes—
15C23×#385—21no—
16C23×#385—21no—
17C23×#385—21no—
18C23×#380C411yescharacteristic, complemented
19C23×#380C411yescharacteristic, complemented
20C23×#380C2211yescharacteristic
21C23×#379—21no—
22C2×C23×#95—21no—
23C2×C23×#95C2212yes—
24C23×#328—24no—
25C25×C23D411yescharacteristic
26C22×Q8×C23C2×C411yescharacteristic
27C23×C4×C23—22no—
28C23×C4×C23—22no—
29C23×C4×C23—22no—
30C23×C4×C23—42no—
31C23×C4×C23—42no—
32C23×C4×C23C2×C411yescharacteristic
33C23×C4×C23C2×C411yescharacteristic
34C23×C4×C23D411yescharacteristic
35C23×C4×C23D411yescharacteristic
36C23×C4×C23D411yescharacteristic
37C23×C4×C23C2311yescharacteristic
38C22×C8×C23—24no—
39C22×C8×C23—22no—
40C22×C8×C23—42no—
41C22×C8×C23—42no—
42C23×C22⋊Q8—22no—
43C23×C22⋊Q8—22no—
44C23×C22⋊Q8—42no—
45C23×C22⋊Q8—42no—
46C2×C4⋊C4×C23—21no—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23—21no—
49C2×C4⋊C4×C23C2×C411yescharacteristic
50C2×C4⋊C4×C23C2×C411yescharacteristic
51C2×C4⋊C4×C23C2×C411yescharacteristic
52C2×C22⋊C4×C23—24no—
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—21no—
55C2×C22⋊C4×C23—21no—
56C2×C22⋊C4×C23—21no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C22⋊C4×C23—42no—
60C2×C22⋊C4×C23—42no—
61C23×C22⋊C8—24no—
62C23×C22⋊C8—22no—
63C23×C22⋊C8D414yes—
64C23×#95—21no—
65C23×#95—21no—
66C23×#95—21no—
67C23×#95—22no—
68C23×#95—22no—
69C23×#95—42no—
70C24×C23—22no—
71C24×C23—22no—
72C24×C23—22no—
73C24×C23—22no—
74C24×C23—41no—
75C24×C23—41no—
76C24×C23C22⋊C411yescharacteristic
77C24×C23C22⋊C411yescharacteristic
78C24×C23SD1612yes—
79C24×C23Q1612yes—
80C24×C23C2×D411yescharacteristic
81C2×Q8×C23—22no—
82C2×Q8×C23—41no—
83C2×Q8×C23—41no—
84C22×C4×C23—22no—
85C22×C4×C23—22no—
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—22no—
89C22×C4×C23—22no—
90C22×C4×C23—22no—
91C22×C4×C23—22no—
92C22×C4×C23—22no—
93C22×C4×C23—22no—
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—24no—
97C22×C4×C23—21no—
98C22×C4×C23—21no—
99C22×C4×C23—21no—
100C22×C4×C23—21no—
101C22×C4×C23—41no—
102C22×C4×C23—41no—
103C22×C4×C23—41no—
104C22×C4×C23—41no—
105C22×C4×C23—41no—
106C22×C4×C23—41no—
107C22×C4×C23—41no—
108C22×C4×C23—48no—
109C22×C4×C23—48no—
110C22×C4×C23—42no—
111C22×C4×C23—42no—
112C22×C4×C23—42no—
113C22×C4×C23—42no—
114C22×C4×C23—42no—
115C22×C4×C23—42no—
116C22×C4×C23C22⋊C411yescharacteristic
117C22×C4×C23C22⋊C411yescharacteristic
118C22×C4×C23C22×C411yescharacteristic
119C22×C4×C23C2×D411yescharacteristic
120C22×C4×C23C2×D412yes—
121C22×C4×C23#4112yes—
122C2×C8×C23—24no—
123C2×C8×C23—48no—
124C2×C8×C23—42no—
125C2×C8×C23—42no—
126C4⋊C4×C23—22no—
127C4⋊C4×C23—41no—
128C4⋊C4×C23—41no—
129C4⋊C4×C23—42no—
130C4⋊C4×C23—42no—
131C4⋊C4×C23—41no—
132C4⋊C4×C23—41no—
133C22⋊C4×C23—42no—
134C22⋊C4×C23—48no—
135C22⋊C4×C23—42no—
136C22⋊C4×C23—42no—
137C22⋊C4×C23—42no—
138C22⋊C4×C23—42no—
139C22⋊C4×C23—42no—
140#9111C2311yesmaximal, characteristic, direct product factor, Sylow
141C23×C23—21no—
142C23×C23—21no—
143C23×C23—21no—
144C23×C23—21no—
145C23×C23—21no—
146C23×C23—21no—
147C23×C23—21no—
148C23×C23—21no—
149C23×C23—24no—
150C23×C23—24no—
151C23×C23—21no—
152C23×C23—21no—
153C23×C23—21no—
154C23×C23—21no—
155C23×C23—21no—
156C23×C23—21no—
157C23×C23—24no—
158C23×C23—24no—
159C23×C23—42no—
160C23×C23—42no—
161C23×C23—44no—
162C23×C23—44no—
163C23×C23—44no—
164C23×C23—44no—
165C23×C23#9912yes—
166C23×C23#10312yes—
167C23×C23#10312yes—
168C23×C23#10412yes—
169C23×C23C2×C22⋊C411yescharacteristic
170C23×C23#12112yes—
171C23×C23C2×SD1611yescharacteristic
172C23×C23C2×Q1611yescharacteristic
173C23×C23#13512yes—
174Q8×C23—82no—
175C2×C4×C23—24no—
176C2×C4×C23—24no—
177C2×C4×C23—21no—
178C2×C4×C23—21no—
179C2×C4×C23—21no—
180C2×C4×C23—21no—
181C2×C4×C23—21no—
182C2×C4×C23—21no—
183C2×C4×C23—21no—
184C2×C4×C23—21no—
185C2×C4×C23—21no—
186C2×C4×C23—42no—
187C2×C4×C23—42no—
188C2×C4×C23—44no—
189C2×C4×C23—44no—
190C2×C4×C23—44no—
191C2×C4×C23—42no—
192C2×C4×C23—42no—
193C2×C4×C23—44no—
194C2×C4×C23—44no—
195C2×C4×C23—44no—
196C2×C4×C23—41no—
197C2×C4×C23—41no—
198C2×C4×C23—41no—
199C2×C4×C23—41no—
200C2×C4×C23—41no—
201C2×C4×C23—41no—
202C2×C4×C23—41no—
203C2×C4×C23—41no—
204C2×C4×C23—41no—
205C2×C4×C23—41no—
206C2×C4×C23—41no—
207C2×C4×C23—41no—
208C2×C4×C23—82no—
209C2×C4×C23—82no—
210C2×C4×C23—82no—
211C2×C4×C23—82no—
212C2×C4×C23#9912yes—
213C2×C4×C23C4×D412yes—
214C2×C4×C23#12011yescharacteristic
215C2×C4×C23#12311yescharacteristic
216C8×C23—44no—
217C22×C22⋊C8C2×C2311yescharacteristic
218#2848C2×C2311yescharacteristic
219C2×#328C2×C2312yescomplemented
220C22⋊#95C2×C2311yescharacteristic
221#1294C2×C2312yescomplemented
222C22×C23—22no—
223C22×C23—22no—
224C22×C23—22no—
225C22×C23—22no—
226C22×C23—22no—
227C22×C23—21no—
228C22×C23—21no—
229C22×C23—22no—
230C22×C23—22no—
231C22×C23—22no—
232C22×C23—22no—
233C22×C23—21no—
234C22×C23—22no—
235C22×C23—21no—
236C22×C23—22no—
237C22×C23—22no—
238C22×C23—22no—
239C22×C23—48no—
240C22×C23—48no—
241C22×C23—41no—
242C22×C23—41no—
243C22×C23—41no—
244C22×C23—41no—
245C22×C23#32814yescomplemented
246C22×C23#38611yescharacteristic
247C22×C23C2×#9911yescharacteristic
248C22×C23C2×#9911yescharacteristic
249C22×C23C2×#10311yescharacteristic
250C22×C23#41611yescharacteristic
251C22×C23C2×#10411yescharacteristic
252C22×C23#42111yescharacteristic
253C22×C23#46512yes—
254C22×C23#46712yes—
255C4×C23—21no—
256C4×C23—22no—
257C4×C23—21no—
258C4×C23—22no—
259C4×C23—41no—
260C4×C23—84no—
261C4×C23—81no—
262C4×C23—81no—
263C4×C23—81no—
264C4×C23—81no—
265C24×C4C22×C2311yescharacteristic
266C23×C8—21no—
267C2×C22⋊Q8C4×C2311yescharacteristic, complemented
268C2×C22⋊Q8C4×C2311yescharacteristic, complemented
269C2×C22⋊Q8C22×C2311yescharacteristic
270C22×C22⋊C4—21no—
271C2×C22⋊C8—21no—
272C2×C22⋊C8—24no—
273C2×C22⋊C8C22×C2312yes—
274#385—21no—
275#385—21no—
276#385—21no—
277#380C4×C2311yescharacteristic, complemented
278#380C4×C2311yescharacteristic, complemented
279#380C22×C2311yescharacteristic
280#379—21no—
281C2×#95—21no—
282C2×#95C22×C2312yes—
283#328—24no—
284C2×C23—22no—
285C2×C23—24no—
286C2×C23—22no—
287C2×C23—42no—
288C2×C23C2×#32811yescharacteristic
289C2×C23#150111yescharacteristic
290C2×C23#151611yescharacteristic
291C2×C23#152011yescharacteristic
292C2×C23#187111yescharacteristic
293C2×C23#189211yescharacteristic
294C2×C23#189811yescharacteristic
295C25D4×C2311yescharacteristic
296C22×Q8C2×C4×C2311yescharacteristic
297C23×C4—22no—
298C23×C4—22no—
299C23×C4—22no—
300C23×C4—42no—
301C23×C4—42no—
302C23×C4C2×C4×C2311yescharacteristic
303C23×C4C2×C4×C2311yescharacteristic
304C23×C4D4×C2311yescharacteristic
305C23×C4D4×C2311yescharacteristic
306C23×C4D4×C2311yescharacteristic
307C23×C4C23×C2311yescharacteristic
308C22×C8—24no—
309C22×C8—22no—
310C22×C8—42no—
311C22×C8—42no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—42no—
315C22⋊Q8—42no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C2×C4×C2311yescharacteristic
320C2×C4⋊C4C2×C4×C2311yescharacteristic
321C2×C4⋊C4C2×C4×C2311yescharacteristic
322C2×C22⋊C4—24no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—42no—
331C22⋊C8—24no—
332C22⋊C8—22no—
333C22⋊C8D4×C2314yes—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—22no—
338#95—22no—
339#95—42no—
340C23#911111yescharacteristic, direct product factor, Sylow
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—41no—
346C24—41no—
347C24C22⋊C4×C2311yescharacteristic
348C24C22⋊C4×C2311yescharacteristic
349C24SD16×C2312yes—
350C24Q16×C2312yes—
351C24C2×D4×C2311yescharacteristic
352C2×Q8—22no—
353C2×Q8—41no—
354C2×Q8—41no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4C22⋊C4×C2311yescharacteristic
388C22×C4C22⋊C4×C2311yescharacteristic
389C22×C4C22×C4×C2311yescharacteristic
390C22×C4C2×D4×C2311yescharacteristic
391C22×C4C2×D4×C2312yes—
392C22×C4#41×C2312yes—
393C2×C8—24no—
394C2×C8—48no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C22⋊C4—48no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—24no—
420C23—24no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—24no—
428C23—24no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23C23×#9912yes—
436C23C23×#10312yes—
437C23C23×#10312yes—
438C23C23×#10412yes—
439C23C2×C22⋊C4×C2311yescharacteristic
440C23C23×#12112yes—
441C23C2×SD16×C2311yescharacteristic
442C23C2×Q16×C2311yescharacteristic
443C23C23×#13512yes—
444Q8—82no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4C23×#9912yes—
483C2×C4C4×D4×C2312yes—
484C2×C4C23×#12011yescharacteristic
485C2×C4C23×#12311yescharacteristic
486C8—44no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—21no—
497C22—22no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—48no—
505C22—48no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22C23×#32814yescomplemented
511C22C23×#38611yescharacteristic
512C22C2×C23×#9911yescharacteristic
513C22C2×C23×#9911yescharacteristic
514C22C2×C23×#10311yescharacteristic
515C22C23×#41611yescharacteristic
516C22C2×C23×#10411yescharacteristic
517C22C23×#42111yescharacteristic
518C22C23×#46512yes—
519C22C23×#46712yes—
520C4—21no—
521C4—22no—
522C4—21no—
523C4—22no—
524C4—41no—
525C4—84no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C2—22no—
531C2—24no—
532C2—22no—
533C2—42no—
534C2C2×C23×#32811yescharacteristic
535C2C23×#150111yescharacteristic
536C2C23×#151611yescharacteristic
537C2C23×#152011yescharacteristic
538C2C23×#187111yescharacteristic
539C2C23×#189211yescharacteristic
540C2C23×#189811yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC25×C23D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#328C2semidirect
5. C23×#1294C2semidirect
8. C2×C23×C22⋊Q8C4semidirect
9. C2×C23×C22⋊Q8C4semidirect
18. C23×#380C4semidirect
19. C23×#380C4semidirect
140. #9111C23direct (N × H)
219. C2×#328C2×C23semidirect
221. #1294C2×C23semidirect
245. C22×C23#328semidirect
267. C2×C22⋊Q8C4×C23semidirect
268. C2×C22⋊Q8C4×C23semidirect
277. #380C4×C23semidirect
278. #380C4×C23semidirect
340. C23#9111direct (N × H)
510. C22C23×#328semidirect

Maximal subgroups

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

subgroupindex
C22×C23×C22⋊C82
C23×#28482
C2×C23×#3282
C23×C22⋊#952
C23×#12942
#911123