Finite Group Databasebeta

← C23×#24085 #19171711

Subgroups — C23×#24085 #19171711

Order 5888.

998 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3495C211yesmaximal, characteristic
2C23×#3092C211yesmaximal, characteristic
3C2×C23×#428C211yesmaximal, characteristic, complemented
4C23×#2503C211yesmaximal, characteristic, complemented
5C23×#2061C212yesmaximal, complemented
6C23×#2038C212yesmaximal, complemented
7C23×#1901C211yesmaximal, characteristic, complemented
8C23×#1871C211yesmaximal, characteristic
9C23×#1860C212yesmaximal, complemented
10C23×#1857C212yesmaximal, complemented
11C23×#1826C211yesmaximal, characteristic, complemented
12C22×M16×C23C2211yescharacteristic
13C23×#556—21no—
14C23×#554—21no—
15C23×#552—21no—
16C23×#552—22no—
17C23×#552—21no—
18C23×#551—22no—
19C23×#544C2211yescharacteristic
20C23×#542—21no—
21C23×#541—21no—
22C23×#536C2211yescharacteristic
23C23×#533—21no—
24C23×#529—21no—
25C23×#525C2211yescharacteristic
26C2×C23×C22⋊Q8C2211yescharacteristic
27C23×#520C2211yescharacteristic
28C23×#519C2211yescharacteristic
29C2×C23×#117C2211yescharacteristic
30C22×C4⋊C4×C23C2211yescharacteristic
31C23×#484—21no—
32C23×#483—21no—
33C23×#481—21no—
34C23×#480—21no—
35C23×#428—21no—
36C23×#428—21no—
37C23×#428C2212yes—
38C23×#428C2212yes—
39C23×#428C2212yes—
40C2×C23×#107—21no—
41C2×C23×#106—21no—
42C2×C23×#104C2212yes—
43C23×#417C2212yes—
44C2×C23×#101C2212yes—
45C23×#407C2211yescharacteristic
46C23×#400—22no—
47C23×#391C2212yes—
48C23×#382C2212yes—
49C23×#381—22no—
50C4×C22⋊C4×C23C2211yescharacteristic
51C23×#343C2212yes—
52C23×#339C2212yes—
53C23×#339C2212yes—
54C23×#339C2212yes—
55C2×#41×C23C2311yescharacteristic
56C22×Q8×C23C2311yescharacteristic
57C23×C4×C23—21no—
58C23×C4×C23C2311yescharacteristic
59C2×M16×C23—21no—
60C2×M16×C23—24no—
61C2×M16×C23C2312yes—
62C2×M16×C23C2312yes—
63C22×C8×C23—21no—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8—21no—
66C23×C4⋊Q8—21no—
67C23×#126—22no—
68C23×#126—22no—
69C23×#126—22no—
70C23×#126—22no—
71C23×#125—22no—
72C23×#125—22no—
73C23×#125—22no—
74C23×#125—21no—
75C23×#125—21no—
76C23×#124—22no—
77C23×#124—21no—
78C23×#123—22no—
79C23×#123—22no—
80C23×#123—22no—
81C23×#123—21no—
82C23×C22⋊Q8—22no—
83C23×C22⋊Q8—22no—
84C23×C22⋊Q8—22no—
85C23×C22⋊Q8—21no—
86C23×C22⋊Q8—21no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—21no—
90C23×C22⋊Q8—21no—
91C23×C22⋊Q8D411yescharacteristic
92C23×C22⋊Q8D411yescharacteristic
93C23×#121D411yescharacteristic
94C23×#121D411yescharacteristic
95C23×#120—21no—
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×D4×C23—22no—
103C4×D4×C23—21no—
104C4×D4×C23—22no—
105C4×D4×C23—21no—
106C4×D4×C23—21no—
107C23×#117—22no—
108C23×#117—22no—
109C23×#117—22no—
110C23×#117—21no—
111C23×#117—21no—
112C23×#117—21no—
113C23×#117D411yescharacteristic
114C23×#117D411yescharacteristic
115C23×#117C2311yescharacteristic
116C23×#117C2311yescharacteristic
117C23×#117C2311yescharacteristic
118C2×C4⋊C4×C23—21no—
119C2×C4⋊C4×C23—21no—
120C2×C4⋊C4×C23—21no—
121C2×C4⋊C4×C23—21no—
122C2×C4⋊C4×C23—21no—
123C2×C4⋊C4×C23—21no—
124C2×C4⋊C4×C23D412yes—
125C2×C4⋊C4×C23D411yescharacteristic
126C2×C4⋊C4×C23D411yescharacteristic
127C2×C4⋊C4×C23C2312yes—
128C2×C4⋊C4×C23C2311yescharacteristic
129C2×C22⋊C4×C23—21no—
130C2×C22⋊C4×C23—21no—
131C2×C22⋊C4×C23D412yes—
132C2×C42×C23—21no—
133C2×C42×C23C2312yes—
134C23×#107—22no—
135C23×#107—22no—
136C23×#107—22no—
137C23×#106—22no—
138C23×#106—22no—
139C23×#106—22no—
140C23×#104—24no—
141C23×#103—22no—
142C23×#102—22no—
143C23×#101—24no—
144C23×C22⋊C8—21no—
145C23×C22⋊C8—21no—
146C23×#95—21no—
147C23×#95—24no—
148C23×#95—24no—
149C24×C23C2×D411yescharacteristic
150#41×C23—22no—
151#41×C23—22no—
152C2×Q8×C23—22no—
153C2×Q8×C23—21no—
154C2×Q8×C23—21no—
155C2×Q8×C23—22no—
156C2×Q8×C23—22no—
157C2×Q8×C23C2×D411yescharacteristic
158C2×D4×C23—21no—
159C2×D4×C23—21no—
160C2×D4×C23—21no—
161C2×D4×C23C2×D411yescharacteristic
162C22×C4×C23—22no—
163C22×C4×C23—22no—
164C22×C4×C23—21no—
165C22×C4×C23—22no—
166C22×C4×C23—21no—
167C22×C4×C23—21no—
168C22×C4×C23—22no—
169C22×C4×C23—22no—
170C22×C4×C23—44no—
171C22×C4×C23—41no—
172C22×C4×C23—41no—
173C22×C4×C23C2×D412yes—
174C22×C4×C23C2×D411yescharacteristic
175C22×C4×C23C2×D411yescharacteristic
176C22×C4×C23C2×D411yescharacteristic
177C22×C4×C23C2×D411yescharacteristic
178C22×C4×C23C2×D411yescharacteristic
179C22×C4×C23C2×D412yes—
180C22×C4×C23C2411yescharacteristic
181M16×C23—24no—
182M16×C23—22no—
183M16×C23—22no—
184M16×C23—22no—
185M16×C23#4114yes—
186C2×C8×C23—24no—
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—22no—
196C4⋊C4×C23—22no—
197C4⋊C4×C23—21no—
198C4⋊C4×C23—21no—
199C4⋊C4×C23—22no—
200C4⋊C4×C23—22no—
201C4⋊C4×C23—21no—
202C4⋊C4×C23—21no—
203C4⋊C4×C23—21no—
204C4⋊C4×C23—21no—
205C4⋊C4×C23—22no—
206C4⋊C4×C23—22no—
207C4⋊C4×C23—41no—
208C4⋊C4×C23—41no—
209C4⋊C4×C23C2×D411yescharacteristic
210C4⋊C4×C23C2×D411yescharacteristic
211C4⋊C4×C23C2×D411yescharacteristic
212C4⋊C4×C23C2×D411yescharacteristic
213C4⋊C4×C23C2×D411yescharacteristic
214C4⋊C4×C23C2×D411yescharacteristic
215C22⋊C4×C23—22no—
216C22⋊C4×C23—22no—
217C22⋊C4×C23—22no—
218C22⋊C4×C23—21no—
219C22⋊C4×C23—21no—
220C22⋊C4×C23—22no—
221C22⋊C4×C23—22no—
222C22⋊C4×C23—21no—
223C22⋊C4×C23—21no—
224C22⋊C4×C23—21no—
225C22⋊C4×C23—21no—
226C22⋊C4×C23—21no—
227C22⋊C4×C23—21no—
228C22⋊C4×C23—21no—
229C22⋊C4×C23—42no—
230C42×C23—22no—
231C42×C23—22no—
232C42×C23—22no—
233C42×C23—21no—
234C42×C23—21no—
235C42×C23—21no—
236C42×C23—21no—
237C42×C23—21no—
238C42×C23—22no—
239C42×C23—22no—
240#24085C2311yesmaximal, characteristic, direct product factor, Sylow
241C23×C23—22no—
242C23×C23—22no—
243C23×C23—21no—
244C23×C23—41no—
245C23×C23#12011yescharacteristic
246C23×C23#12011yescharacteristic
247C23×C23C22×D411yescharacteristic
248Q8×C23—22no—
249Q8×C23—41no—
250Q8×C23—41no—
251Q8×C23—42no—
252Q8×C23—82no—
253D4×C23—22no—
254D4×C23—42no—
255D4×C23—42no—
256D4×C23—81no—
257C2×C4×C23—22no—
258C2×C4×C23—21no—
259C2×C4×C23—21no—
260C2×C4×C23—21no—
261C2×C4×C23—21no—
262C2×C4×C23—22no—
263C2×C4×C23—22no—
264C2×C4×C23—21no—
265C2×C4×C23—21no—
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—24no—
274C2×C4×C23—42no—
275C2×C4×C23—41no—
276C2×C4×C23—41no—
277C2×C4×C23—42no—
278C2×C4×C23—42no—
279C2×C4×C23—42no—
280C2×C4×C23—44no—
281C2×C4×C23—42no—
282C2×C4×C23—42no—
283C2×C4×C23—81no—
284C2×C4×C23—82no—
285C2×C4×C23—82no—
286C2×C4×C23—82no—
287C2×C4×C23#12011yescharacteristic
288C2×C4×C23#12011yescharacteristic
289C2×C4×C23C22×D411yescharacteristic
290C2×C4×C23C22×D411yescharacteristic
291C2×C4×C23C2×#4112yes—
292C2×C4×C232−1+412yes—
293C8×C23—24no—
294C8×C23—42no—
295#3495C2×C2311yescharacteristic
296#3092C2×C2311yescharacteristic
297C2×#428C2×C2311yescharacteristic, complemented
298#2503C2×C2311yescharacteristic, complemented
299#2061C2×C2312yescomplemented
300#2038C2×C2312yescomplemented
301#1901C2×C2311yescharacteristic, complemented
302#1871C2×C2311yescharacteristic
303#1860C2×C2312yescomplemented
304#1857C2×C2312yescomplemented
305#1826C2×C2311yescharacteristic, complemented
306C22×C23—21no—
307C22×C23—21no—
308C22×C23—21no—
309C22×C23—21no—
310C22×C23—21no—
311C22×C23—21no—
312C22×C23—21no—
313C22×C23—44no—
314C22×C23—41no—
315C22×C23—41no—
316C22×C23—41no—
317C22×C23#45412yes—
318C22×C23C2×#12011yescharacteristic
319C22×C23#54712yes—
320C4×C23—21no—
321C4×C23—21no—
322C4×C23—42no—
323C4×C23—41no—
324C4×C23—41no—
325C4×C23—41no—
326C4×C23—41no—
327C4×C23—42no—
328C4×C23—41no—
329C4×C23—41no—
330C4×C23—41no—
331C4×C23—82no—
332C4×C23—82no—
333C4×C23#52511yescharacteristic
334C4×C23#53611yescharacteristic
335C4×C23#54712yes—
336C22×M16C22×C2311yescharacteristic
337#556—21no—
338#554—21no—
339#552—21no—
340#552—22no—
341#552—21no—
342#551—22no—
343#544C22×C2311yescharacteristic
344#542—21no—
345#541—21no—
346#536C22×C2311yescharacteristic
347#533—21no—
348#529—21no—
349#525C22×C2311yescharacteristic
350C2×C22⋊Q8C22×C2311yescharacteristic
351#520C22×C2311yescharacteristic
352#519C22×C2311yescharacteristic
353C2×#117C22×C2311yescharacteristic
354C22×C4⋊C4C22×C2311yescharacteristic
355#484—21no—
356#483—21no—
357#481—21no—
358#480—21no—
359#428—21no—
360#428—21no—
361#428C22×C2312yes—
362#428C22×C2312yes—
363#428C22×C2312yes—
364C2×#107—21no—
365C2×#106—21no—
366C2×#104C22×C2312yes—
367#417C22×C2312yes—
368C2×#101C22×C2312yes—
369#407C22×C2311yescharacteristic
370#400—22no—
371#391C22×C2312yes—
372#382C22×C2312yes—
373#381—22no—
374C4×C22⋊C4C22×C2311yescharacteristic
375#343C22×C2312yes—
376#339C22×C2312yes—
377#339C22×C2312yes—
378#339C22×C2312yes—
379C2×C23—22no—
380C2×C23—22no—
381C2×C23—41no—
382C2×C23—81no—
383C2×C23#240611yescharacteristic
384C2×C23C2×#45411yescharacteristic
385C2×C23#302211yescharacteristic
386C2×#41C23×C2311yescharacteristic
387C22×Q8C23×C2311yescharacteristic
388C23×C4—21no—
389C23×C4C23×C2311yescharacteristic
390C2×M16—21no—
391C2×M16—24no—
392C2×M16C23×C2312yes—
393C2×M16C23×C2312yes—
394C22×C8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8—21no—
398#126—22no—
399#126—22no—
400#126—22no—
401#126—22no—
402#125—22no—
403#125—22no—
404#125—22no—
405#125—21no—
406#125—21no—
407#124—22no—
408#124—21no—
409#123—22no—
410#123—22no—
411#123—21no—
412#123—22no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418C22⋊Q8—21no—
419C22⋊Q8—22no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8D4×C2311yescharacteristic
423C22⋊Q8D4×C2311yescharacteristic
424#121D4×C2311yescharacteristic
425#121D4×C2311yescharacteristic
426#120—21no—
427C4×Q8—22no—
428C4×Q8—21no—
429C4×Q8—21no—
430C4×Q8—21no—
431C4×Q8—21no—
432C4×Q8—21no—
433C4×D4—22no—
434C4×D4—21no—
435C4×D4—22no—
436C4×D4—21no—
437C4×D4—21no—
438#117—22no—
439#117—22no—
440#117—21no—
441#117—21no—
442#117—22no—
443#117—21no—
444#117D4×C2311yescharacteristic
445#117D4×C2311yescharacteristic
446#117C23×C2311yescharacteristic
447#117C23×C2311yescharacteristic
448#117C23×C2311yescharacteristic
449C2×C4⋊C4—21no—
450C2×C4⋊C4—21no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4D4×C2312yes—
456C2×C4⋊C4D4×C2311yescharacteristic
457C2×C4⋊C4D4×C2311yescharacteristic
458C2×C4⋊C4C23×C2312yes—
459C2×C4⋊C4C23×C2311yescharacteristic
460C2×C22⋊C4—21no—
461C2×C22⋊C4—21no—
462C2×C22⋊C4D4×C2312yes—
463C2×C42—21no—
464C2×C42C23×C2312yes—
465#107—22no—
466#107—22no—
467#107—22no—
468#106—22no—
469#106—22no—
470#106—22no—
471#104—24no—
472#103—22no—
473#102—22no—
474#101—24no—
475C22⋊C8—21no—
476C22⋊C8—21no—
477#95—21no—
478#95—24no—
479#95—24no—
480C23#2408511yescharacteristic, direct product factor, Sylow
481C24C2×D4×C2311yescharacteristic
482#41—22no—
483#41—22no—
484C2×Q8—22no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8C2×D4×C2311yescharacteristic
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4C2×D4×C2311yescharacteristic
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—44no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4C2×D4×C2312yes—
506C22×C4C2×D4×C2311yescharacteristic
507C22×C4C2×D4×C2311yescharacteristic
508C22×C4C2×D4×C2311yescharacteristic
509C22×C4C2×D4×C2311yescharacteristic
510C22×C4C2×D4×C2311yescharacteristic
511C22×C4C2×D4×C2312yes—
512C22×C4C24×C2311yescharacteristic
513M16—24no—
514M16—22no—
515M16—22no—
516M16—22no—
517M16#41×C2314yes—
518C2×C8—24no—
519C2×C8—22no—
520C2×C8—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4C2×D4×C2311yescharacteristic
542C4⋊C4C2×D4×C2311yescharacteristic
543C4⋊C4C2×D4×C2311yescharacteristic
544C4⋊C4C2×D4×C2311yescharacteristic
545C4⋊C4C2×D4×C2311yescharacteristic
546C4⋊C4C2×D4×C2311yescharacteristic
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C22⋊C4—21no—
557C22⋊C4—21no—
558C22⋊C4—21no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—42no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—22no—
571C42—22no—
572C23—22no—
573C23—22no—
574C23—21no—
575C23—41no—
576C23C23×#12011yescharacteristic
577C23C23×#12011yescharacteristic
578C23C22×D4×C2311yescharacteristic
579Q8—22no—
580Q8—42no—
581Q8—41no—
582Q8—41no—
583Q8—82no—
584D4—22no—
585D4—42no—
586D4—42no—
587D4—81no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—81no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4C23×#12011yescharacteristic
619C2×C4C23×#12011yescharacteristic
620C2×C4C22×D4×C2311yescharacteristic
621C2×C4C22×D4×C2311yescharacteristic
622C2×C4C2×#41×C2312yes—
623C2×C4C23×2−1+412yes—
624C8—24no—
625C8—42no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—44no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C23×#45412yes—
638C22C2×C23×#12011yescharacteristic
639C22C23×#54712yes—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—82no—
652C4—82no—
653C4C23×#52511yescharacteristic
654C4C23×#53611yescharacteristic
655C4C23×#54712yes—
656C2—22no—
657C2—22no—
658C2—41no—
659C2—81no—
660C2C23×#240611yescharacteristic
661C2C2×C23×#45411yescharacteristic
662C2C23×#302211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#428C2semidirect
4. C23×#2503C2semidirect
5. C23×#2061C2semidirect
6. C23×#2038C2semidirect
7. C23×#1901C2semidirect
9. C23×#1860C2semidirect
10. C23×#1857C2semidirect
11. C23×#1826C2semidirect
240. #24085C23direct (N × H)
297. C2×#428C2×C23semidirect
298. #2503C2×C23semidirect
299. #2061C2×C23semidirect
300. #2038C2×C23semidirect
301. #1901C2×C23semidirect
303. #1860C2×C23semidirect
304. #1857C2×C23semidirect
305. #1826C2×C23semidirect
480. C23#24085direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34952
C23×#30922
C2×C23×#4282
C23×#25032
C23×#20612
C23×#20382
C23×#19012
C23×#18712
C23×#18602
C23×#18572
C23×#18262
#2408523