Finite Group Databasebeta

← C23×#18572 #19166198

Subgroups — C23×#18572 #19166198

Order 5888.

954 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3522C211yesmaximal, characteristic
2C23×#3239C211yesmaximal, characteristic, complemented
3C23×#3141C211yesmaximal, characteristic, complemented
4C2×C23×#462C211yesmaximal, characteristic, complemented
5C23×#1645C212yesmaximal, complemented
6C23×#1644C212yesmaximal, complemented
7C23×#1641C212yesmaximal, complemented
8C23×#1640C212yesmaximal, complemented
9C23×#1551C211yesmaximal, characteristic, complemented
10C23×#1491C211yesmaximal, characteristic, complemented
11C23×#1483C211yesmaximal, characteristic, complemented
12C22×Q16×C23—21no—
13C23×#559—21no—
14C23×#559C2211yescharacteristic
15Q82×C23C2211yescharacteristic
16C23×#554—21no—
17C23×#554C2211yescharacteristic
18C23×#550C2211yescharacteristic
19C23×#548—21no—
20C23×#547—22no—
21C23×#547—21no—
22C23×#542—22no—
23C23×#542—21no—
24C23×#532—21no—
25C23×#532C2211yescharacteristic
26C2×C23×C4⋊Q8C2211yescharacteristic
27C23×#529—21no—
28C23×#529C2211yescharacteristic
29C23×#520—21no—
30C2×C4×Q8×C23C2211yescharacteristic
31C23×#501—21no—
32C23×#489—21no—
33C23×#489—22no—
34C23×#487C2212yes—
35C23×#486C2212yes—
36C23×#484—21no—
37C23×#481—21no—
38C23×#475C2212yes—
39C23×#474C2212yes—
40C23×#462—21no—
41C23×#462—21no—
42C23×#462C2212yes—
43C23×#462C2212yes—
44C23×#432C2211yescharacteristic
45C2×C23×C4⋊C8C2211yescharacteristic
46C23×#417—22no—
47C2×C23×#103—21no—
48C4×M16×C23C2211yescharacteristic
49C23×#333C2212yes—
50C23×#331C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×Q8⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C2×#41×C23—22no—
56C2×#41×C23—21no—
57C2×#41×C23D412yes—
58C22×Q8×C23—21no—
59C22×Q8×C23D411yescharacteristic
60C2×Q16×C23—22no—
61C2×Q16×C23—22no—
62C2×Q16×C23—44no—
63C2×M16×C23—21no—
64C22×C8×C23—21no—
65C23×C4⋊Q8—22no—
66C23×C4⋊Q8—21no—
67C23×C4⋊Q8—21no—
68C23×C4⋊Q8—21no—
69C23×C4⋊Q8—21no—
70C23×C4⋊Q8—21no—
71C23×C4⋊Q8C2312yes—
72C23×C4⋊Q8C2311yescharacteristic
73C23×#127—21no—
74C23×#127C2311yescharacteristic
75C23×#126—22no—
76C23×#126—21no—
77C23×#125—21no—
78C23×#125D411yescharacteristic
79C23×#124—22no—
80C23×#124—22no—
81C23×#124—21no—
82C23×#124—21no—
83C23×#124D411yescharacteristic
84C23×#124D412yes—
85C23×#123—22no—
86C23×#123—22no—
87C23×#123—22no—
88C23×#123—22no—
89C23×#123—21no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—21no—
95C23×C22⋊Q8D411yescharacteristic
96C23×#121—22no—
97C23×#121—22no—
98C23×#121—21no—
99C23×#121—21no—
100C23×#121—21no—
101C23×#121D411yescharacteristic
102C4×Q8×C23—21no—
103C4×Q8×C23—21no—
104C4×Q8×C23—21no—
105C4×Q8×C23—21no—
106C4×Q8×C23D411yescharacteristic
107C4×Q8×C23D411yescharacteristic
108C4×Q8×C23C2311yescharacteristic
109C4×Q8×C23C2311yescharacteristic
110C4×Q8×C23C2311yescharacteristic
111C4×D4×C23—22no—
112C4×D4×C23—22no—
113C4×D4×C23—22no—
114C4×D4×C23—21no—
115C4×D4×C23—21no—
116C4×D4×C23—21no—
117C4×D4×C23—21no—
118C4×D4×C23—21no—
119C4×D4×C23—21no—
120C4×D4×C23—21no—
121C4×D4×C23C2311yescharacteristic
122C23×#117—22no—
123C23×#117—21no—
124C2×C4⋊C4×C23—22no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23—21no—
127C2×C4⋊C4×C23—21no—
128C2×C4⋊C4×C23—21no—
129C2×C4⋊C4×C23D411yescharacteristic
130C2×C42×C23—21no—
131C2×C42×C23C2311yescharacteristic
132C23×#107—22no—
133C23×#106—22no—
134C23×C4⋊C8—21no—
135C23×C4⋊C8C2312yes—
136C23×C4⋊C8C2312yes—
137C23×#103—24no—
138C23×#103—22no—
139C23×#103—22no—
140C23×#103—22no—
141C23×#102—24no—
142C23×#102—22no—
143C23×C22⋊C8—21no—
144C23×#97—21no—
145C23×#97—21no—
146C4×C8×C23C2312yes—
147#41×C23—44no—
148#41×C23—42no—
149#41×C23—42no—
150#41×C23—84no—
151C2×Q8×C23—22no—
152C2×Q8×C23—21no—
153C2×Q8×C23—22no—
154C2×Q8×C23—22no—
155C2×Q8×C23—41no—
156C2×Q8×C23—42no—
157C2×Q8×C23—42no—
158C2×Q8×C23—41no—
159C2×Q8×C23C2×D411yescharacteristic
160C2×Q8×C23C2×D411yescharacteristic
161C2×Q8×C23C2×D411yescharacteristic
162C2×Q8×C23C2×D412yes—
163C2×D4×C23—22no—
164C2×D4×C23—22no—
165C2×D4×C23—21no—
166C2×D4×C23—22no—
167C2×D4×C23—21no—
168C2×D4×C23—21no—
169C2×D4×C23C2×D411yescharacteristic
170C2×D4×C23C2×D412yes—
171C22×C4×C23—22no—
172C22×C4×C23—21no—
173C22×C4×C23—21no—
174C22×C4×C23—22no—
175C22×C4×C23—22no—
176C22×C4×C23—21no—
177C22×C4×C23—22no—
178C22×C4×C23—41no—
179C22×C4×C23C2×D411yescharacteristic
180C22×C4×C23C2×D411yescharacteristic
181C22×C4×C23C2×D411yescharacteristic
182Q16×C23—44no—
183Q16×C23—44no—
184M16×C23—42no—
185C2×C8×C23—22no—
186C2×C8×C23—22no—
187C2×C8×C23—22no—
188C2×C8×C23—42no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—21no—
198C4⋊C4×C23—21no—
199C4⋊C4×C23—21no—
200C4⋊C4×C23—21no—
201C4⋊C4×C23—21no—
202C4⋊C4×C23—21no—
203C4⋊C4×C23—21no—
204C4⋊C4×C23—22no—
205C4⋊C4×C23—22no—
206C4⋊C4×C23—41no—
207C4⋊C4×C23—41no—
208C4⋊C4×C23C2×D411yescharacteristic
209C4⋊C4×C23C2×D411yescharacteristic
210C4⋊C4×C23C2×D411yescharacteristic
211C4⋊C4×C23C2×D411yescharacteristic
212C22⋊C4×C23—22no—
213C22⋊C4×C23—22no—
214C22⋊C4×C23—22no—
215C22⋊C4×C23—22no—
216C22⋊C4×C23—22no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—22no—
219C22⋊C4×C23—21no—
220C22⋊C4×C23—21no—
221C22⋊C4×C23—22no—
222C22⋊C4×C23—21no—
223C22⋊C4×C23—21no—
224C42×C23—22no—
225C42×C23—21no—
226C42×C23—21no—
227C42×C23—21no—
228C42×C23—21no—
229C42×C23—21no—
230C42×C23C2×D411yescharacteristic
231C42×C23C2×D411yescharacteristic
232C42×C23C2×D411yescharacteristic
233C42×C23C2411yescharacteristic
234#18572C2311yesmaximal, characteristic, direct product factor, Sylow
235C23×C23—21no—
236C23×C23—22no—
237C23×C23#12011yescharacteristic
238Q8×C23—21no—
239Q8×C23—21no—
240Q8×C23—21no—
241Q8×C23—22no—
242Q8×C23—42no—
243Q8×C23—42no—
244Q8×C23—42no—
245Q8×C23—42no—
246Q8×C23—82no—
247Q8×C23—82no—
248Q8×C23AGL(1,ℤ/8)12yes—
249D4×C23—22no—
250D4×C23—42no—
251D4×C23—44no—
252D4×C23—44no—
253D4×C23—42no—
254D4×C23—41no—
255D4×C23—41no—
256D4×C23—42no—
257D4×C23—82no—
258C2×C4×C23—21no—
259C2×C4×C23—22no—
260C2×C4×C23—21no—
261C2×C4×C23—21no—
262C2×C4×C23—22no—
263C2×C4×C23—21no—
264C2×C4×C23—21no—
265C2×C4×C23—21no—
266C2×C4×C23—21no—
267C2×C4×C23—22no—
268C2×C4×C23—21no—
269C2×C4×C23—21no—
270C2×C4×C23—42no—
271C2×C4×C23—44no—
272C2×C4×C23—42no—
273C2×C4×C23—41no—
274C2×C4×C23—41no—
275C2×C4×C23—42no—
276C2×C4×C23—82no—
277C2×C4×C23—82no—
278C2×C4×C23—82no—
279C2×C4×C23—82no—
280C2×C4×C23#12011yescharacteristic
281C2×C4×C23#12011yescharacteristic
282C2×C4×C23#12011yescharacteristic
283C2×C4×C23C22×D411yescharacteristic
284C2×C4×C23C22×D411yescharacteristic
285C2×C4×C23C22×D411yescharacteristic
286C8×C23—42no—
287C8×C23—42no—
288C8×C23—42no—
289#3522C2×C2311yescharacteristic
290#3239C2×C2311yescharacteristic, complemented
291#3141C2×C2311yescharacteristic, complemented
292C2×#462C2×C2311yescharacteristic, complemented
293#1645C2×C2312yescomplemented
294#1644C2×C2312yescomplemented
295#1641C2×C2312yescomplemented
296#1640C2×C2312yescomplemented
297#1551C2×C2311yescharacteristic, complemented
298#1491C2×C2311yescharacteristic, complemented
299#1483C2×C2311yescharacteristic, complemented
300C22×C23—21no—
301C22×C23—21no—
302C22×C23—42no—
303C22×C23—41no—
304C22×C23—41no—
305C22×C23—42no—
306C22×C23—42no—
307C22×C23—41no—
308C22×C23#45512yes—
309C22×C23C2×#12011yescharacteristic
310C4×C23—21no—
311C4×C23—22no—
312C4×C23—21no—
313C4×C23—21no—
314C4×C23—21no—
315C4×C23—41no—
316C4×C23—41no—
317C4×C23—41no—
318C4×C23—41no—
319C4×C23—41no—
320C4×C23—42no—
321C4×C23—41no—
322C4×C23—82no—
323C4×C23C2×AGL(1,ℤ/8)11yescharacteristic
324C4×C23#57511yescharacteristic
325C4×C23#57812yes—
326C22×Q16—21no—
327#559—21no—
328#559C22×C2311yescharacteristic
329Q82C22×C2311yescharacteristic
330#554—21no—
331#554C22×C2311yescharacteristic
332#550C22×C2311yescharacteristic
333#548—21no—
334#547—22no—
335#547—21no—
336#542—22no—
337#542—21no—
338#532—21no—
339#532C22×C2311yescharacteristic
340C2×C4⋊Q8C22×C2311yescharacteristic
341#529—21no—
342#529C22×C2311yescharacteristic
343#520—21no—
344C2×C4×Q8C22×C2311yescharacteristic
345#501—21no—
346#489—22no—
347#489—21no—
348#487C22×C2312yes—
349#486C22×C2312yes—
350#484—21no—
351#481—21no—
352#475C22×C2312yes—
353#474C22×C2312yes—
354#462—21no—
355#462—21no—
356#462C22×C2312yes—
357#462C22×C2312yes—
358#432C22×C2311yescharacteristic
359C2×C4⋊C8C22×C2311yescharacteristic
360#417—22no—
361C2×#103—21no—
362C4×M16C22×C2311yescharacteristic
363#333C22×C2312yes—
364#331C22×C2312yes—
365Q8⋊C8C22×C2312yes—
366Q8⋊C8C22×C2312yes—
367Q8⋊C8C22×C2312yes—
368D4⋊C8C22×C2312yes—
369C2×C23—22no—
370C2×C23—81no—
371C2×C23—82no—
372C2×C23#300611yescharacteristic
373C2×C23#301111yescharacteristic
374C2×C23C2×#45511yescharacteristic
375C2×#41—22no—
376C2×#41—21no—
377C2×#41D4×C2312yes—
378C22×Q8—21no—
379C22×Q8D4×C2311yescharacteristic
380C2×Q16—22no—
381C2×Q16—22no—
382C2×Q16—44no—
383C2×M16—21no—
384C22×C8—21no—
385C4⋊Q8—22no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8—21no—
391C4⋊Q8C23×C2312yes—
392C4⋊Q8C23×C2311yescharacteristic
393#127—21no—
394#127C23×C2311yescharacteristic
395#126—22no—
396#126—21no—
397#125—21no—
398#125D4×C2311yescharacteristic
399#124—22no—
400#124—22no—
401#124—21no—
402#124—21no—
403#124D4×C2311yescharacteristic
404#124D4×C2312yes—
405#123—22no—
406#123—22no—
407#123—22no—
408#123—22no—
409#123—21no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—21no—
415C22⋊Q8D4×C2311yescharacteristic
416#121—22no—
417#121—22no—
418#121—21no—
419#121—21no—
420#121—21no—
421#121D4×C2311yescharacteristic
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8D4×C2311yescharacteristic
427C4×Q8D4×C2311yescharacteristic
428C4×Q8C23×C2311yescharacteristic
429C4×Q8C23×C2311yescharacteristic
430C4×Q8C23×C2311yescharacteristic
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4C23×C2311yescharacteristic
442#117—22no—
443#117—21no—
444C2×C4⋊C4—22no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4D4×C2311yescharacteristic
450C2×C42—21no—
451C2×C42C23×C2311yescharacteristic
452#107—22no—
453#106—22no—
454C4⋊C8—21no—
455C4⋊C8C23×C2312yes—
456C4⋊C8C23×C2312yes—
457#103—24no—
458#103—22no—
459#103—22no—
460#103—22no—
461#102—24no—
462#102—22no—
463C22⋊C8—21no—
464#97—21no—
465#97—21no—
466C4×C8C23×C2312yes—
467C23#1857211yescharacteristic, direct product factor, Sylow
468#41—44no—
469#41—42no—
470#41—42no—
471#41—84no—
472C2×Q8—22no—
473C2×Q8—21no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—41no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—41no—
480C2×Q8C2×D4×C2311yescharacteristic
481C2×Q8C2×D4×C2311yescharacteristic
482C2×Q8C2×D4×C2311yescharacteristic
483C2×Q8C2×D4×C2312yes—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4C2×D4×C2311yescharacteristic
491C2×D4C2×D4×C2312yes—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—41no—
500C22×C4C2×D4×C2311yescharacteristic
501C22×C4C2×D4×C2311yescharacteristic
502C22×C4C2×D4×C2311yescharacteristic
503Q16—44no—
504Q16—44no—
505M16—42no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—42no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4C2×D4×C2311yescharacteristic
530C4⋊C4C2×D4×C2311yescharacteristic
531C4⋊C4C2×D4×C2311yescharacteristic
532C4⋊C4C2×D4×C2311yescharacteristic
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C42—22no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42C2×D4×C2311yescharacteristic
552C42C2×D4×C2311yescharacteristic
553C42C2×D4×C2311yescharacteristic
554C42C24×C2311yescharacteristic
555C23—21no—
556C23—22no—
557C23C23×#12011yescharacteristic
558Q8—21no—
559Q8—21no—
560Q8—21no—
561Q8—22no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—82no—
567Q8—82no—
568Q8C23×AGL(1,ℤ/8)12yes—
569D4—22no—
570D4—42no—
571D4—44no—
572D4—44no—
573D4—42no—
574D4—41no—
575D4—41no—
576D4—42no—
577D4—82no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4C23×#12011yescharacteristic
601C2×C4C23×#12011yescharacteristic
602C2×C4C23×#12011yescharacteristic
603C2×C4C22×D4×C2311yescharacteristic
604C2×C4C22×D4×C2311yescharacteristic
605C2×C4C22×D4×C2311yescharacteristic
606C8—42no—
607C8—42no—
608C8—42no—
609C22—21no—
610C22—21no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22C23×#45512yes—
618C22C2×C23×#12011yescharacteristic
619C4—21no—
620C4—22no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—82no—
632C4C2×C23×AGL(1,ℤ/8)11yescharacteristic
633C4C23×#57511yescharacteristic
634C4C23×#57812yes—
635C2—22no—
636C2—81no—
637C2—82no—
638C2C23×#300611yescharacteristic
639C2C23×#301111yescharacteristic
640C2C2×C23×#45511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3522C2
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
2. C23×#3239C2semidirect
3. C23×#3141C2semidirect
4. C2×C23×#462C2semidirect
5. C23×#1645C2semidirect
6. C23×#1644C2semidirect
7. C23×#1641C2semidirect
8. C23×#1640C2semidirect
9. C23×#1551C2semidirect
10. C23×#1491C2semidirect
11. C23×#1483C2semidirect
234. #18572C23direct (N × H)
290. #3239C2×C23semidirect
291. #3141C2×C23semidirect
292. C2×#462C2×C23semidirect
293. #1645C2×C23semidirect
294. #1644C2×C23semidirect
295. #1641C2×C23semidirect
296. #1640C2×C23semidirect
297. #1551C2×C23semidirect
298. #1491C2×C23semidirect
299. #1483C2×C23semidirect
467. C23#18572direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35222
C23×#32392
C23×#31412
C2×C23×#4622
C23×#16452
C23×#16442
C23×#16412
C23×#16402
C23×#15512
C23×#14912
C23×#14832
#1857223