Finite Group Databasebeta

← C23×#18296 #19165922

Subgroups — C23×#18296 #19165922

Order 5888.

1162 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
1C2×C23×#538C211yesmaximal, characteristic, complemented
2C23×#3187C211yesmaximal, characteristic, complemented
3C23×#3186C211yesmaximal, characteristic, complemented
4C23×#2617C211yesmaximal, characteristic, complemented
5C23×#1612C214yesmaximal, complemented
6C23×#1600C214yesmaximal, complemented
7C23×#1520C211yesmaximal, characteristic, complemented
8C23×#1519C211yesmaximal, characteristic, complemented
9C2×C23×#323C211yesmaximal, characteristic, complemented
10C23×#538—22no—
11C23×#538—22no—
12C23×#538—22no—
13C23×#538C2214yes—
14C2×C23×#126—21no—
15C2×C23×#124—21no—
16C2×C23×#123—21no—
17C2×C23×C22⋊Q8C2211yescharacteristic
18C2×C23×#121—21no—
19C2×C23×#121C2211yescharacteristic
20C2×C23×#121C2211yescharacteristic
21C2×C23×#120—21no—
22C2×C23×#120—21no—
23C2×C4×D4×C23—21no—
24C22×C22⋊C4×C23C2211yescharacteristic
25C23×#483C2212yescomplemented
26C23×#481—21no—
27C23×#480—21no—
28C23×#480C2212yescomplemented
29C23×#466—22no—
30C23×#465—22no—
31C23×#450C2212yescomplemented
32C23×#448—21no—
33C23×#447—21no—
34C23×#447C2212yescomplemented
35C2×C23×C22⋊C8C2211yescharacteristic
36C2×C23×C22⋊C8C2211yescharacteristic
37C23×#397C2212yescomplemented
38C23×#394—22no—
39C23×#392C2212yescomplemented
40C23×#386—21no—
41C23×#380C2211yescharacteristic
42C23×#328C2214yescomplemented
43C23×#327C2214yescomplemented
44C23×#323C2214yescomplemented
45C25×C23D411yescharacteristic
46C22×Q8×C23D411yescharacteristic, complemented
47C22×D4×C23—21no—
48C22×D4×C23—21no—
49C22×D4×C23—21no—
50C22×D4×C23—21no—
51C22×D4×C23—22no—
52C22×D4×C23D411yescharacteristic, complemented
53C22×D4×C23C2311yescharacteristic
54C23×C4×C23—21no—
55C23×C4×C23D411yescharacteristic
56C23×C4×C23C2311yescharacteristic
57C2×SD16×C23—22no—
58C2×D8×C23—22no—
59C22×C8×C23—21no—
60C22×C8×C23—21no—
61C23×#126—24no—
62C23×#126—24no—
63C23×#124—24no—
64C23×#124—24no—
65C23×#123—24no—
66C23×#123—24no—
67C23×C22⋊Q8—21no—
68C23×C22⋊Q8—21no—
69C23×C22⋊Q8D412yescomplemented
70C23×C22⋊Q8C2312yes—
71C23×#121—24no—
72C23×#121—24no—
73C23×#121—21no—
74C23×#121—21no—
75C23×#121—21no—
76C23×#121—21no—
77C23×#121—21no—
78C23×#121—21no—
79C23×#121D412yescomplemented
80C23×#121C2312yes—
81C23×#120—24no—
82C23×#120—24no—
83C23×#120—44no—
84C4×D4×C23—24no—
85C4×D4×C23—24no—
86C2×C4⋊C4×C23—21no—
87C2×C4⋊C4×C23—21no—
88C2×C4⋊C4×C23—21no—
89C2×C4⋊C4×C23D411yescharacteristic, complemented
90C2×C4⋊C4×C23D411yescharacteristic, complemented
91C2×C4⋊C4×C23C2311yescharacteristic
92C2×C22⋊C4×C23—21no—
93C2×C22⋊C4×C23—21no—
94C2×C22⋊C4×C23—21no—
95C2×C22⋊C4×C23—21no—
96C2×C22⋊C4×C23—21no—
97C2×C22⋊C4×C23—21no—
98C2×C22⋊C4×C23—21no—
99C2×C22⋊C4×C23—21no—
100C2×C22⋊C4×C23—21no—
101C2×C22⋊C4×C23—21no—
102C2×C22⋊C4×C23—22no—
103C2×C22⋊C4×C23—21no—
104C2×C22⋊C4×C23—22no—
105C2×C22⋊C4×C23D412yes—
106C2×C22⋊C4×C23C2312yes—
107C2×C42×C23—21no—
108C23×#107—22no—
109C23×#106—22no—
110C23×#103—22no—
111C23×#102—22no—
112C23×#102—22no—
113C23×#102—22no—
114C23×C22⋊C8—21no—
115C23×C22⋊C8—21no—
116C23×C22⋊C8C2312yes—
117C23×C22⋊C8C2312yes—
118C23×#95—22no—
119C23×#95C2312yes—
120C24×C23—21no—
121C24×C23—21no—
122C24×C23—21no—
123C24×C23—22no—
124C24×C23—22no—
125C24×C23—21no—
126C24×C23—81no—
127C24×C23—81no—
128C24×C23C2×D412yes—
129C24×C23C2×D411yescharacteristic
130C2×Q8×C23—21no—
131C2×Q8×C23—81no—
132C2×Q8×C23C2×D412yes—
133C2×D4×C23—22no—
134C2×D4×C23—22no—
135C2×D4×C23—24no—
136C2×D4×C23—24no—
137C2×D4×C23—21no—
138C2×D4×C23—21no—
139C2×D4×C23—21no—
140C2×D4×C23—42no—
141C2×D4×C23—42no—
142C2×D4×C23—44no—
143C2×D4×C23—41no—
144C2×D4×C23—41no—
145C2×D4×C23—42no—
146C2×D4×C23—42no—
147C2×D4×C23—81no—
148C2×D4×C23—81no—
149C2×D4×C23—82no—
150C2×D4×C23—82no—
151C2×D4×C23—82no—
152C2×D4×C23—82no—
153C2×D4×C23C2×D412yes—
154C22×C4×C23—21no—
155C22×C4×C23—21no—
156C22×C4×C23—21no—
157C22×C4×C23—21no—
158C22×C4×C23—24no—
159C22×C4×C23—21no—
160C22×C4×C23—22no—
161C22×C4×C23—21no—
162C22×C4×C23—81no—
163C22×C4×C23—81no—
164C22×C4×C23—82no—
165C22×C4×C23C2×D412yes—
166C22×C4×C23C2×D411yescharacteristic
167C22×C4×C23C2×D411yescharacteristic
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2411yescharacteristic
172SD16×C23—82no—
173D8×C23—82no—
174C2×C8×C23—22no—
175C2×C8×C23—22no—
176C2×C8×C23—81no—
177C2×C8×C23—81no—
178C4⋊C4×C23—24no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—21no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—21no—
184C4⋊C4×C23—21no—
185C4⋊C4×C23—81no—
186C4⋊C4×C23C2×D412yes—
187C4⋊C4×C23C2×D412yes—
188C22⋊C4×C23—24no—
189C22⋊C4×C23—24no—
190C22⋊C4×C23—24no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—22no—
194C22⋊C4×C23—22no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—22no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—44no—
200C22⋊C4×C23—42no—
201C22⋊C4×C23—41no—
202C22⋊C4×C23—41no—
203C22⋊C4×C23—82no—
204C42×C23—24no—
205#18296C2311yesmaximal, characteristic, direct product factor, Sylow
206C23×C23—22no—
207C23×C23—22no—
208C23×C23—22no—
209C23×C23—22no—
210C23×C23—41no—
211C23×C23—41no—
212C23×C23—42no—
213C23×C23—42no—
214C23×C23—42no—
215C23×C23—42no—
216C23×C23—42no—
217C23×C23—42no—
218C23×C23—42no—
219C23×C23—42no—
220C23×C23—42no—
221C23×C23—42no—
222C23×C23—42no—
223C23×C23—41no—
224C23×C23—41no—
225C23×C23—41no—
226C23×C23—81no—
227C23×C23—81no—
228C23×C23—84no—
229C23×C23—84no—
230C23×C23#12011yescharacteristic
231C23×C23#12011yescharacteristic
232C23×C23#13512yes—
233C23×C23AGL(1,ℤ/8)12yes—
234C23×C23C22×D411yescharacteristic
235Q8×C23—42no—
236Q8×C23—81no—
237D4×C23—42no—
238D4×C23—42no—
239D4×C23—42no—
240D4×C23—42no—
241D4×C23—42no—
242D4×C23—44no—
243D4×C23—81no—
244D4×C23—84no—
245D4×C23—84no—
246D4×C23—88no—
247D4×C23—82no—
248D4×C23—82no—
249C2×C4×C23—22no—
250C2×C4×C23—22no—
251C2×C4×C23—22no—
252C2×C4×C23—22no—
253C2×C4×C23—41no—
254C2×C4×C23—41no—
255C2×C4×C23—41no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—42no—
260C2×C4×C23—42no—
261C2×C4×C23—42no—
262C2×C4×C23—42no—
263C2×C4×C23—42no—
264C2×C4×C23—42no—
265C2×C4×C23—42no—
266C2×C4×C23—41no—
267C2×C4×C23—41no—
268C2×C4×C23—42no—
269C2×C4×C23—41no—
270C2×C4×C23—81no—
271C2×C4×C23—84no—
272C2×C4×C23#12011yescharacteristic
273C2×C4×C23#12011yescharacteristic
274C2×C4×C23C22×D411yescharacteristic
275C2×C4×C23C22×D411yescharacteristic
276C8×C23—42no—
277C8×C23—42no—
278C2×#538C2×C2311yescharacteristic, complemented
279#3187C2×C2311yescharacteristic, complemented
280#3186C2×C2311yescharacteristic, complemented
281#2617C2×C2311yescharacteristic, complemented
282#1612C2×C2314yescomplemented
283#1600C2×C2314yescomplemented
284#1520C2×C2311yescharacteristic, complemented
285#1519C2×C2311yescharacteristic, complemented
286C2×#323C2×C2311yescharacteristic, complemented
287C22×C23—21no—
288C22×C23—21no—
289C22×C23—21no—
290C22×C23—21no—
291C22×C23—21no—
292C22×C23—21no—
293C22×C23—42no—
294C22×C23—42no—
295C22×C23—44no—
296C22×C23—42no—
297C22×C23—42no—
298C22×C23—42no—
299C22×C23—42no—
300C22×C23—42no—
301C22×C23—42no—
302C22×C23—82no—
303C22×C23—84no—
304C22×C23—84no—
305C22×C23—84no—
306C22×C23—82no—
307C22×C23—82no—
308C22×C23—81no—
309C22×C23—81no—
310C22×C23—81no—
311C22×C23#44912yes—
312C22×C23#44912yes—
313C22×C23C2×#12011yescharacteristic
314C22×C23C2×#13511yescharacteristic
315C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
316C4×C23—21no—
317C4×C23—21no—
318C4×C23—21no—
319C4×C23—21no—
320C4×C23—42no—
321C4×C23—42no—
322C4×C23—42no—
323C4×C23—42no—
324C4×C23—82no—
325C4×C23—81no—
326#538—22no—
327#538—22no—
328#538—22no—
329#538C22×C2314yes—
330C2×#126—21no—
331C2×#124—21no—
332C2×#123—21no—
333C2×C22⋊Q8C22×C2311yescharacteristic
334C2×#121—21no—
335C2×#121C22×C2311yescharacteristic
336C2×#121C22×C2311yescharacteristic
337C2×#120—21no—
338C2×#120—21no—
339C2×C4×D4—21no—
340C22×C22⋊C4C22×C2311yescharacteristic
341#483C22×C2312yescomplemented
342#481—21no—
343#480—21no—
344#480C22×C2312yescomplemented
345#466—22no—
346#465—22no—
347#450C22×C2312yescomplemented
348#448—21no—
349#447—21no—
350#447C22×C2312yescomplemented
351C2×C22⋊C8C22×C2311yescharacteristic
352C2×C22⋊C8C22×C2311yescharacteristic
353#397C22×C2312yescomplemented
354#394—22no—
355#392C22×C2312yescomplemented
356#386—21no—
357#380C22×C2311yescharacteristic
358#328C22×C2314yescomplemented
359#327C22×C2314yescomplemented
360#323C22×C2314yescomplemented
361C2×C23—21no—
362C2×C23—21no—
363C2×C23—22no—
364C2×C23—22no—
365C2×C23—82no—
366C2×C23—82no—
367C2×C23—161no—
368C2×C23C2×#44911yescharacteristic
369C2×C23#301911yescharacteristic
370C2×C23#302611yescharacteristic
371C25D4×C2311yescharacteristic
372C22×Q8D4×C2311yescharacteristic, complemented
373C22×D4—21no—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4—21no—
377C22×D4—22no—
378C22×D4D4×C2311yescharacteristic, complemented
379C22×D4C23×C2311yescharacteristic
380C23×C4—21no—
381C23×C4D4×C2311yescharacteristic
382C23×C4C23×C2311yescharacteristic
383C2×SD16—22no—
384C2×D8—22no—
385C22×C8—21no—
386C22×C8—21no—
387#126—24no—
388#126—24no—
389#124—24no—
390#124—24no—
391#123—24no—
392#123—24no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8D4×C2312yescomplemented
396C22⋊Q8C23×C2312yes—
397#121—24no—
398#121—24no—
399#121—21no—
400#121—21no—
401#121—21no—
402#121—21no—
403#121—21no—
404#121—21no—
405#121D4×C2312yescomplemented
406#121C23×C2312yes—
407#120—24no—
408#120—24no—
409#120—44no—
410C4×D4—24no—
411C4×D4—24no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4D4×C2311yescharacteristic, complemented
416C2×C4⋊C4D4×C2311yescharacteristic, complemented
417C2×C4⋊C4C23×C2311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4D4×C2312yes—
432C2×C22⋊C4C23×C2312yes—
433C2×C42—21no—
434#107—22no—
435#106—22no—
436#103—22no—
437#102—22no—
438#102—22no—
439#102—22no—
440C22⋊C8—21no—
441C22⋊C8—21no—
442C22⋊C8C23×C2312yes—
443C22⋊C8C23×C2312yes—
444#95—22no—
445#95C23×C2312yes—
446C23#1829611yescharacteristic, direct product factor, Sylow
447C24—21no—
448C24—21no—
449C24—21no—
450C24—22no—
451C24—22no—
452C24—21no—
453C24—81no—
454C24—81no—
455C24C2×D4×C2312yes—
456C24C2×D4×C2311yescharacteristic
457C2×Q8—21no—
458C2×Q8—81no—
459C2×Q8C2×D4×C2312yes—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—44no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—81no—
475C2×D4—81no—
476C2×D4—82no—
477C2×D4—82no—
478C2×D4—82no—
479C2×D4—82no—
480C2×D4C2×D4×C2312yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—82no—
492C22×C4C2×D4×C2312yes—
493C22×C4C2×D4×C2311yescharacteristic
494C22×C4C2×D4×C2311yescharacteristic
495C22×C4C2×D4×C2311yescharacteristic
496C22×C4C2×D4×C2311yescharacteristic
497C22×C4C2×D4×C2311yescharacteristic
498C22×C4C24×C2311yescharacteristic
499SD16—82no—
500D8—82no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—81no—
504C2×C8—81no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—81no—
513C4⋊C4C2×D4×C2312yes—
514C4⋊C4C2×D4×C2312yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—82no—
531C42—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—81no—
553C23—81no—
554C23—84no—
555C23—84no—
556C23C23×#12011yescharacteristic
557C23C23×#12011yescharacteristic
558C23C23×#13512yes—
559C23C23×AGL(1,ℤ/8)12yes—
560C23C22×D4×C2311yescharacteristic
561Q8—42no—
562Q8—81no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—44no—
569D4—81no—
570D4—84no—
571D4—84no—
572D4—88no—
573D4—82no—
574D4—82no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—81no—
597C2×C4—84no—
598C2×C4C23×#12011yescharacteristic
599C2×C4C23×#12011yescharacteristic
600C2×C4C22×D4×C2311yescharacteristic
601C2×C4C22×D4×C2311yescharacteristic
602C8—42no—
603C8—42no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—82no—
624C22—82no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22C23×#44912yes—
629C22C23×#44912yes—
630C22C2×C23×#12011yescharacteristic
631C22C2×C23×#13511yescharacteristic
632C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—81no—
643C2—21no—
644C2—21no—
645C2—22no—
646C2—22no—
647C2—82no—
648C2—82no—
649C2—161no—
650C2C2×C23×#44911yescharacteristic
651C2C23×#301911yescharacteristic
652C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C23×#538C2semidirect
2. C23×#3187C2semidirect
3. C23×#3186C2semidirect
4. C23×#2617C2semidirect
5. C23×#1612C2semidirect
6. C23×#1600C2semidirect
7. C23×#1520C2semidirect
8. C23×#1519C2semidirect
9. C2×C23×#323C2semidirect
25. C23×#483C22semidirect
28. C23×#480C22semidirect
31. C23×#450C22semidirect
34. C23×#447C22semidirect
37. C23×#397C22semidirect
39. C23×#392C22semidirect
42. C23×#328C22semidirect
43. C23×#327C22semidirect
44. C23×#323C22semidirect
46. C22×Q8×C23D4semidirect
52. C22×D4×C23D4semidirect
69. C23×C22⋊Q8D4semidirect
79. C23×#121D4semidirect
89. C2×C4⋊C4×C23D4semidirect
90. C2×C4⋊C4×C23D4semidirect
205. #18296C23direct (N × H)
278. C2×#538C2×C23semidirect
279. #3187C2×C23semidirect
280. #3186C2×C23semidirect
281. #2617C2×C23semidirect
282. #1612C2×C23semidirect
283. #1600C2×C23semidirect
284. #1520C2×C23semidirect
285. #1519C2×C23semidirect
286. C2×#323C2×C23semidirect
341. #483C22×C23semidirect
344. #480C22×C23semidirect
347. #450C22×C23semidirect
350. #447C22×C23semidirect
353. #397C22×C23semidirect
355. #392C22×C23semidirect
358. #328C22×C23semidirect
359. #327C22×C23semidirect
360. #323C22×C23semidirect
372. C22×Q8D4×C23semidirect
378. C22×D4D4×C23semidirect
395. C22⋊Q8D4×C23semidirect
405. #121D4×C23semidirect
415. C2×C4⋊C4D4×C23semidirect
416. C2×C4⋊C4D4×C23semidirect
446. C23#18296direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5382
C23×#31872
C23×#31862
C23×#26172
C23×#16122
C23×#16002
C23×#15202
C23×#15192
C2×C23×#3232
#1829623