Finite Group Databasebeta

← C23×#18544 #19166170

Subgroups — C23×#18544 #19166170

Order 5888.

954 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3482C211yesmaximal, characteristic
2C23×#3165C211yesmaximal, characteristic, complemented
3C23×#3079C211yesmaximal, characteristic, complemented
4C23×#3039C211yesmaximal, characteristic, complemented
5C23×#1654C212yesmaximal, complemented
6C23×#1653C212yesmaximal, complemented
7C23×#1650C212yesmaximal, complemented
8C23×#1649C212yesmaximal, complemented
9C23×#1561C211yesmaximal, characteristic, complemented
10C23×#1489C211yesmaximal, characteristic, complemented
11C23×#1489C211yesmaximal, characteristic, complemented
12C2×C23×#137—21no—
13C23×#557C2211yescharacteristic
14C23×#556—21no—
15C23×#556C2211yescharacteristic
16C23×#552—22no—
17C23×#552—21no—
18D4×Q8×C23C2211yescharacteristic
19C23×#548—21no—
20C23×#547—21no—
21C23×#547C2211yescharacteristic
22C23×#533—21no—
23C23×#533C2211yescharacteristic
24C2×C23×C4⋊Q8C2211yescharacteristic
25C23×#529—21no—
26C23×#529—22no—
27C4×#41×C23—21no—
28C4×#41×C23C2211yescharacteristic
29C4×#41×C23C2211yescharacteristic
30C2×C4×Q8×C23—21no—
31C23×#501—22no—
32C23×#500C2212yes—
33C23×#498C2212yes—
34C23×#477—21no—
35C23×#476—21no—
36C23×#475C2212yes—
37C23×#474C2212yes—
38C23×#462—21no—
39C23×#462C2212yes—
40C23×#461—21no—
41C23×#461C2212yes—
42C23×#428—22no—
43C23×#428—21no—
44C23×#423C2211yescharacteristic
45C23×#423C2211yescharacteristic
46C23×#419—21no—
47C23×#417—21no—
48C4×M16×C23C2211yescharacteristic
49C23×#332C2212yes—
50C23×#332C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×D4⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C2×#41×C23—21no—
56C2×#41×C23D411yescharacteristic
57C2×#41×C23D411yescharacteristic
58C22×Q8×C23—21no—
59C22×Q8×C23—21no—
60C23×#137—44no—
61C2×Q16×C23—22no—
62C2×SD16×C23—22no—
63C2×M16×C23—21no—
64C2×M16×C23—21no—
65C2×M16×C23—21no—
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×C4⋊Q8C2311yescharacteristic
74C23×#126—22no—
75C23×#126—22no—
76C23×#126—22no—
77C23×#125—22no—
78C23×#125—22no—
79C23×#125—21no—
80C23×#125D412yes—
81C23×#124—22no—
82C23×#124—21no—
83C23×#123—22no—
84C23×#123—22no—
85C23×#123—22no—
86C23×C22⋊Q8—22no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—21no—
90C23×C22⋊Q8—21no—
91C23×C22⋊Q8—21no—
92C23×C22⋊Q8—22no—
93C23×#121—22no—
94C23×#121—21no—
95C4×Q8×C23—22no—
96C4×Q8×C23—22no—
97C4×Q8×C23—21no—
98C4×Q8×C23—21no—
99C4×Q8×C23—21no—
100C4×Q8×C23—21no—
101C4×Q8×C23—21no—
102C4×Q8×C23D411yescharacteristic
103C4×Q8×C23D411yescharacteristic
104C4×Q8×C23C2311yescharacteristic
105C4×Q8×C23C2311yescharacteristic
106C4×D4×C23—22no—
107C4×D4×C23—22no—
108C4×D4×C23—21no—
109C4×D4×C23—21no—
110C4×D4×C23—21no—
111C4×D4×C23—21no—
112C4×D4×C23—21no—
113C4×D4×C23D411yescharacteristic
114C4×D4×C23D411yescharacteristic
115C4×D4×C23C2311yescharacteristic
116C4×D4×C23C2311yescharacteristic
117C23×#117—22no—
118C23×#117—22no—
119C23×#117—21no—
120C23×#117—21no—
121C23×#117—21no—
122C23×#117D411yescharacteristic
123C23×#117D411yescharacteristic
124C23×#117D412yes—
125C2×C4⋊C4×C23—22no—
126C2×C4⋊C4×C23—21no—
127C2×C4⋊C4×C23—21no—
128C2×C4⋊C4×C23—21no—
129C2×C42×C23—22no—
130C2×C42×C23—21no—
131C2×C42×C23—21no—
132C2×C42×C23C2311yescharacteristic
133C23×#107—22no—
134C23×#107—24no—
135C23×#106—22no—
136C23×#106—24no—
137C23×C4⋊C8—21no—
138C23×C4⋊C8—21no—
139C23×C4⋊C8C2312yes—
140C23×C4⋊C8C2312yes—
141C23×#103—22no—
142C23×#103—22no—
143C23×#103—22no—
144C23×#102—22no—
145C23×#97—21no—
146C4×C8×C23C2312yes—
147#41×C23—22no—
148#41×C23—22no—
149#41×C23—41no—
150#41×C23—41no—
151#41×C23—42no—
152#41×C23—42no—
153C2×Q8×C23—22no—
154C2×Q8×C23—21no—
155C2×Q8×C23—21no—
156C2×Q8×C23—22no—
157C2×Q8×C23—42no—
158C2×Q8×C23—42no—
159C2×Q8×C23—42no—
160C2×Q8×C23—42no—
161C2×Q8×C23—41no—
162C2×Q8×C23C2×D411yescharacteristic
163C2×Q8×C23C2×D411yescharacteristic
164C2×D4×C23—22no—
165C2×D4×C23—21no—
166C2×D4×C23—21no—
167C2×D4×C23C2×D411yescharacteristic
168C2×D4×C23C2×D411yescharacteristic
169C22×C4×C23—22no—
170C22×C4×C23—22no—
171C22×C4×C23—21no—
172C22×C4×C23—21no—
173C22×C4×C23—21no—
174C22×C4×C23—21no—
175C22×C4×C23—21no—
176C22×C4×C23—21no—
177C22×C4×C23—41no—
178C22×C4×C23C2×D411yescharacteristic
179C22×C4×C23C2×D411yescharacteristic
180C22×C4×C23C2×D411yescharacteristic
181Q16×C23—44no—
182SD16×C23—44no—
183M16×C23—24no—
184M16×C23—42no—
185M16×C23—42no—
186C2×C8×C23—22no—
187C2×C8×C23—22no—
188C2×C8×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23—22no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—22no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—21no—
199C4⋊C4×C23—21no—
200C4⋊C4×C23—21no—
201C4⋊C4×C23—21no—
202C4⋊C4×C23—21no—
203C4⋊C4×C23—22no—
204C4⋊C4×C23—22no—
205C4⋊C4×C23—41no—
206C4⋊C4×C23—41no—
207C4⋊C4×C23C2×D411yescharacteristic
208C4⋊C4×C23C2×D411yescharacteristic
209C4⋊C4×C23C2×D411yescharacteristic
210C4⋊C4×C23C2×D412yes—
211C4⋊C4×C23C2×D411yescharacteristic
212C4⋊C4×C23C2×D412yes—
213C22⋊C4×C23—22no—
214C22⋊C4×C23—22no—
215C22⋊C4×C23—22no—
216C22⋊C4×C23—22no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—21no—
219C22⋊C4×C23—22no—
220C22⋊C4×C23—21no—
221C22⋊C4×C23—21no—
222C22⋊C4×C23—21no—
223C22⋊C4×C23—21no—
224C42×C23—22no—
225C42×C23—22no—
226C42×C23—21no—
227C42×C23—21no—
228C42×C23—21no—
229C42×C23—22no—
230C42×C23—21no—
231C42×C23—21no—
232C42×C23—21no—
233C42×C23C2×D411yescharacteristic
234C42×C23C2×D411yescharacteristic
235C42×C23C2×D411yescharacteristic
236C42×C23C2411yescharacteristic
237#18544C2311yesmaximal, characteristic, direct product factor, Sylow
238C23×C23—21no—
239C23×C23—21no—
240C23×C23#12011yescharacteristic
241Q8×C23—21no—
242Q8×C23—21no—
243Q8×C23—21no—
244Q8×C23—44no—
245Q8×C23—41no—
246Q8×C23—41no—
247Q8×C23—41no—
248Q8×C23—41no—
249Q8×C23—42no—
250Q8×C23—42no—
251Q8×C23—82no—
252Q8×C23AGL(1,ℤ/8)12yes—
253D4×C23—21no—
254D4×C23—21no—
255D4×C23—21no—
256D4×C23—42no—
257D4×C23—42no—
258D4×C23—44no—
259D4×C23#13712yescomplemented
260C2×C4×C23—21no—
261C2×C4×C23—21no—
262C2×C4×C23—21no—
263C2×C4×C23—21no—
264C2×C4×C23—21no—
265C2×C4×C23—22no—
266C2×C4×C23—21no—
267C2×C4×C23—21no—
268C2×C4×C23—21no—
269C2×C4×C23—22no—
270C2×C4×C23—21no—
271C2×C4×C23—21no—
272C2×C4×C23—22no—
273C2×C4×C23—42no—
274C2×C4×C23—42no—
275C2×C4×C23—44no—
276C2×C4×C23—44no—
277C2×C4×C23—41no—
278C2×C4×C23—41no—
279C2×C4×C23—41no—
280C2×C4×C23—41no—
281C2×C4×C23—42no—
282C2×C4×C23—42no—
283C2×C4×C23—42no—
284C2×C4×C23—82no—
285C2×C4×C23#12011yescharacteristic
286C2×C4×C23#12011yescharacteristic
287C2×C4×C23#12011yescharacteristic
288C2×C4×C23C22×D411yescharacteristic
289C2×C4×C23C22×D411yescharacteristic
290C2×C4×C23C22×D411yescharacteristic
291C8×C23—24no—
292C8×C23—42no—
293C8×C23—42no—
294#3482C2×C2311yescharacteristic
295#3165C2×C2311yescharacteristic, complemented
296#3079C2×C2311yescharacteristic, complemented
297#3039C2×C2311yescharacteristic, complemented
298#1654C2×C2312yescomplemented
299#1653C2×C2312yescomplemented
300#1650C2×C2312yescomplemented
301#1649C2×C2312yescomplemented
302#1561C2×C2311yescharacteristic, complemented
303#1489C2×C2311yescharacteristic, complemented
304#1489C2×C2311yescharacteristic, complemented
305C22×C23—21no—
306C22×C23—21no—
307C22×C23—22no—
308C22×C23—21no—
309C22×C23—41no—
310C22×C23—41no—
311C22×C23—41no—
312C22×C23—41no—
313C22×C23—41no—
314C22×C23C2×#12011yescharacteristic
315C4×C23—21no—
316C4×C23—22no—
317C4×C23—21no—
318C4×C23—21no—
319C4×C23—42no—
320C4×C23—41no—
321C4×C23—41no—
322C4×C23—41no—
323C4×C23—42no—
324C4×C23—41no—
325C4×C23—41no—
326C4×C23—82no—
327C4×C23#45612yes—
328C4×C23C2×AGL(1,ℤ/8)11yescharacteristic
329C4×C23C2×#13711yescharacteristic
330C4×C23#57511yescharacteristic
331C4×C23#57511yescharacteristic
332C2×#137—21no—
333#557C22×C2311yescharacteristic
334#556—21no—
335#556C22×C2311yescharacteristic
336#552—22no—
337#552—21no—
338D4×Q8C22×C2311yescharacteristic
339#548—21no—
340#547—21no—
341#547C22×C2311yescharacteristic
342#533—21no—
343#533C22×C2311yescharacteristic
344C2×C4⋊Q8C22×C2311yescharacteristic
345#529—21no—
346#529—22no—
347C4×#41—21no—
348C4×#41C22×C2311yescharacteristic
349C4×#41C22×C2311yescharacteristic
350C2×C4×Q8—21no—
351#501—22no—
352#500C22×C2312yes—
353#498C22×C2312yes—
354#477—21no—
355#476—21no—
356#475C22×C2312yes—
357#474C22×C2312yes—
358#462—21no—
359#462C22×C2312yes—
360#461—21no—
361#461C22×C2312yes—
362#428—21no—
363#428—22no—
364#423C22×C2311yescharacteristic
365#423C22×C2311yescharacteristic
366#419—21no—
367#417—21no—
368C4×M16C22×C2311yescharacteristic
369#332C22×C2312yes—
370#332C22×C2312yes—
371Q8⋊C8C22×C2312yes—
372Q8⋊C8C22×C2312yes—
373D4⋊C8C22×C2312yes—
374D4⋊C8C22×C2312yes—
375C2×C23—41no—
376C2×C23—42no—
377C2×C23—81no—
378C2×C23#300611yescharacteristic
379C2×C23#300711yescharacteristic
380C2×C23C2×#45611yescharacteristic
381C2×#41—21no—
382C2×#41D4×C2311yescharacteristic
383C2×#41D4×C2311yescharacteristic
384C22×Q8—21no—
385C22×Q8—21no—
386#137—44no—
387C2×Q16—22no—
388C2×SD16—22no—
389C2×M16—21no—
390C2×M16—21no—
391C2×M16—21no—
392C4⋊Q8—21no—
393C4⋊Q8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8C23×C2312yes—
398C4⋊Q8C23×C2311yescharacteristic
399C4⋊Q8C23×C2311yescharacteristic
400#126—22no—
401#126—22no—
402#126—22no—
403#125—22no—
404#125—22no—
405#125—21no—
406#125D4×C2312yes—
407#124—22no—
408#124—21no—
409#123—22no—
410#123—22no—
411#123—22no—
412C22⋊Q8—22no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8—22no—
418C22⋊Q8—21no—
419#121—22no—
420#121—21no—
421C4×Q8—22no—
422C4×Q8—22no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8D4×C2311yescharacteristic
429C4×Q8D4×C2311yescharacteristic
430C4×Q8C23×C2311yescharacteristic
431C4×Q8C23×C2311yescharacteristic
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4D4×C2311yescharacteristic
440C4×D4D4×C2311yescharacteristic
441C4×D4C23×C2311yescharacteristic
442C4×D4C23×C2311yescharacteristic
443#117—22no—
444#117—22no—
445#117—21no—
446#117—21no—
447#117—21no—
448#117D4×C2311yescharacteristic
449#117D4×C2311yescharacteristic
450#117D4×C2312yes—
451C2×C4⋊C4—22no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—21no—
455C2×C42—22no—
456C2×C42—21no—
457C2×C42—21no—
458C2×C42C23×C2311yescharacteristic
459#107—22no—
460#107—24no—
461#106—22no—
462#106—24no—
463C4⋊C8—21no—
464C4⋊C8—21no—
465C4⋊C8C23×C2312yes—
466C4⋊C8C23×C2312yes—
467#103—22no—
468#103—22no—
469#103—22no—
470#102—22no—
471#97—21no—
472C4×C8C23×C2312yes—
473C23#1854411yescharacteristic, direct product factor, Sylow
474#41—22no—
475#41—22no—
476#41—41no—
477#41—41no—
478#41—42no—
479#41—42no—
480C2×Q8—22no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—22no—
484C2×Q8—42no—
485C2×Q8—42no—
486C2×Q8—42no—
487C2×Q8—42no—
488C2×Q8—41no—
489C2×Q8C2×D4×C2311yescharacteristic
490C2×Q8C2×D4×C2311yescharacteristic
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4C2×D4×C2311yescharacteristic
495C2×D4C2×D4×C2311yescharacteristic
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—41no—
505C22×C4C2×D4×C2311yescharacteristic
506C22×C4C2×D4×C2311yescharacteristic
507C22×C4C2×D4×C2311yescharacteristic
508Q16—44no—
509SD16—44no—
510M16—24no—
511M16—42no—
512M16—42no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4C2×D4×C2311yescharacteristic
535C4⋊C4C2×D4×C2311yescharacteristic
536C4⋊C4C2×D4×C2311yescharacteristic
537C4⋊C4C2×D4×C2312yes—
538C4⋊C4C2×D4×C2311yescharacteristic
539C4⋊C4C2×D4×C2312yes—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C42—22no—
552C42—22no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—22no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42C2×D4×C2311yescharacteristic
561C42C2×D4×C2311yescharacteristic
562C42C2×D4×C2311yescharacteristic
563C42C24×C2311yescharacteristic
564C23—21no—
565C23—21no—
566C23C23×#12011yescharacteristic
567Q8—21no—
568Q8—21no—
569Q8—21no—
570Q8—44no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—42no—
576Q8—42no—
577Q8—82no—
578Q8C23×AGL(1,ℤ/8)12yes—
579D4—21no—
580D4—21no—
581D4—21no—
582D4—42no—
583D4—42no—
584D4—44no—
585D4C23×#13712yescomplemented
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—82no—
611C2×C4C23×#12011yescharacteristic
612C2×C4C23×#12011yescharacteristic
613C2×C4C23×#12011yescharacteristic
614C2×C4C22×D4×C2311yescharacteristic
615C2×C4C22×D4×C2311yescharacteristic
616C2×C4C22×D4×C2311yescharacteristic
617C8—24no—
618C8—42no—
619C8—42no—
620C22—21no—
621C22—21no—
622C22—22no—
623C22—21no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22C2×C23×#12011yescharacteristic
630C4—21no—
631C4—22no—
632C4—21no—
633C4—21no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—82no—
642C4C23×#45612yes—
643C4C2×C23×AGL(1,ℤ/8)11yescharacteristic
644C4C2×C23×#13711yescharacteristic
645C4C23×#57511yescharacteristic
646C4C23×#57511yescharacteristic
647C2—41no—
648C2—42no—
649C2—81no—
650C2C23×#300611yescharacteristic
651C2C23×#300711yescharacteristic
652C2C2×C23×#45611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3482C2
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×#3165C2semidirect
3. C23×#3079C2semidirect
4. C23×#3039C2semidirect
5. C23×#1654C2semidirect
6. C23×#1653C2semidirect
7. C23×#1650C2semidirect
8. C23×#1649C2semidirect
9. C23×#1561C2semidirect
10. C23×#1489C2semidirect
11. C23×#1489C2semidirect
237. #18544C23direct (N × H)
259. D4×C23#137semidirect
295. #3165C2×C23semidirect
296. #3079C2×C23semidirect
297. #3039C2×C23semidirect
298. #1654C2×C23semidirect
299. #1653C2×C23semidirect
300. #1650C2×C23semidirect
301. #1649C2×C23semidirect
302. #1561C2×C23semidirect
303. #1489C2×C23semidirect
304. #1489C2×C23semidirect
473. C23#18544direct (N × H)
585. D4C23×#137semidirect

Maximal subgroups

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

subgroupindex
C23×#34822
C23×#31652
C23×#30792
C23×#30392
C23×#16542
C23×#16532
C23×#16502
C23×#16492
C23×#15612
C23×#14892
#1854423