Finite Group Databasebeta

← C23×#18176 #19165802

Subgroups — C23×#18176 #19165802

Order 5888.

1034 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3491C211yesmaximal, characteristic, complemented
2C23×#3114C212yesmaximal, complemented
3C23×#2392C211yesmaximal, characteristic, complemented
4C23×#1609C212yesmaximal, complemented
5C23×#1608C211yesmaximal, characteristic, complemented
6C23×#1607C211yesmaximal, characteristic, complemented
7C23×#1597C212yesmaximal, complemented
8C23×#1596C211yesmaximal, characteristic, complemented
9C23×#1595C211yesmaximal, characteristic
10C23×#1509C212yesmaximal, complemented
11C23×#1464C211yesmaximal, characteristic, complemented
12C23×#552—22no—
13C23×#552—22no—
14C23×#551—22no—
15C23×#551—22no—
16C23×#543C2211yescharacteristic
17C23×#539—21no—
18C23×#539C2212yes—
19C23×#534C2211yescharacteristic
20C23×#529—21no—
21C23×#529C2212yes—
22C2×C23×#123—21no—
23C23×#518—21no—
24C2×C4×D4×C23—21no—
25C2×C4×D4×C23C2211yescharacteristic, complemented
26C2×C23×#117C2211yescharacteristic
27C23×#483C2212yescomplemented
28C23×#480C2212yes—
29C23×#478—22no—
30C23×#463—22no—
31C23×#450C2212yescomplemented
32C23×#447C2212yes—
33C23×#442—22no—
34C23×#440—22no—
35C23×#432C2212yescomplemented
36C23×#398C2211yescharacteristic, complemented
37C23×#397C2211yescharacteristic
38C23×#396—21no—
39C23×#394—21no—
40C23×#394C2211yescharacteristic, complemented
41C23×#392C2211yescharacteristic
42C4×C4⋊C4×C23C2211yescharacteristic, complemented
43C4×C22⋊C4×C23—21no—
44C23×#328C2212yescomplemented
45C23×#328C2212yescomplemented
46C23×#327C2212yes—
47C23×#327C2212yes—
48C23×#324C2212yescomplemented
49C23×#323C2211yescharacteristic
50C23×#323C2211yescharacteristic
51C22×D4×C23—21no—
52C22×D4×C23—21no—
53C22×D4×C23C2311yescharacteristic
54C23×C4×C23—21no—
55C23×C4×C23D411yescharacteristic
56C2×SD16×C23—22no—
57C2×D8×C23—22no—
58C23×#126—22no—
59C23×#126—22no—
60C23×#126—22no—
61C23×#126—22no—
62C23×#125—21no—
63C23×#125—22no—
64C23×#125—22no—
65C23×#125D412yescomplemented
66C23×#124—21no—
67C23×#124D412yescomplemented
68C23×#123—22no—
69C23×#123—22no—
70C23×#123—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×#123—22no—
74C23×#123—22no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—22no—
78C23×C22⋊Q8—21no—
79C23×C22⋊Q8—22no—
80C23×C22⋊Q8C2312yes—
81C23×#121—21no—
82C23×#121C2312yes—
83C23×#120—22no—
84C4×Q8×C23—21no—
85C4×Q8×C23D412yescomplemented
86C4×D4×C23—21no—
87C4×D4×C23—22no—
88C4×D4×C23—22no—
89C4×D4×C23—22no—
90C4×D4×C23—22no—
91C4×D4×C23—22no—
92C4×D4×C23—22no—
93C4×D4×C23—44no—
94C4×D4×C23D412yescomplemented
95C23×#117—22no—
96C23×#117—22no—
97C23×#117—22no—
98C23×#117—22no—
99C23×#117—22no—
100C2×C4⋊C4×C23—21no—
101C2×C4⋊C4×C23—21no—
102C2×C4⋊C4×C23—21no—
103C2×C4⋊C4×C23—21no—
104C2×C4⋊C4×C23D411yescharacteristic
105C2×C4⋊C4×C23C2311yescharacteristic
106C2×C4⋊C4×C23C2311yescharacteristic
107C2×C22⋊C4×C23—22no—
108C2×C22⋊C4×C23—21no—
109C2×C22⋊C4×C23—21no—
110C2×C22⋊C4×C23—21no—
111C2×C22⋊C4×C23—21no—
112C2×C22⋊C4×C23—21no—
113C2×C22⋊C4×C23—21no—
114C2×C22⋊C4×C23—21no—
115C2×C22⋊C4×C23—21no—
116C2×C22⋊C4×C23D411yescharacteristic
117C2×C22⋊C4×C23C2311yescharacteristic
118C2×C42×C23—21no—
119C2×C42×C23D411yescharacteristic
120C2×C42×C23C2311yescharacteristic
121C23×#107—22no—
122C23×#106—22no—
123C23×C4⋊C8—22no—
124C23×#103—22no—
125C23×#102—22no—
126C23×#102—22no—
127C23×#102—22no—
128C23×C22⋊C8C2312yes—
129C23×C22⋊C8C2312yes—
130C23×#97—22no—
131C23×#95—21no—
132C23×#95—21no—
133C23×#95C2311yescharacteristic
134C23×#95C2311yescharacteristic
135C24×C23—21no—
136C24×C23—21no—
137C24×C23C2×D411yescharacteristic
138C2×Q8×C23—21no—
139C2×Q8×C23C2×D412yes—
140C2×D4×C23—22no—
141C2×D4×C23—22no—
142C2×D4×C23—21no—
143C2×D4×C23—22no—
144C2×D4×C23—42no—
145C2×D4×C23—42no—
146C2×D4×C23—42no—
147C2×D4×C23—42no—
148C2×D4×C23—42no—
149C2×D4×C23—42no—
150C2×D4×C23C2×D412yes—
151C22×C4×C23—21no—
152C22×C4×C23—22no—
153C22×C4×C23—21no—
154C22×C4×C23—21no—
155C22×C4×C23—21no—
156C22×C4×C23—22no—
157C22×C4×C23—21no—
158C22×C4×C23—21no—
159C22×C4×C23—22no—
160C22×C4×C23—21no—
161C22×C4×C23—21no—
162C22×C4×C23—41no—
163C22×C4×C23—42no—
164C22×C4×C23—42no—
165C22×C4×C23—41no—
166C22×C4×C23—41no—
167C22×C4×C23C2×D411yescharacteristic
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2×D411yescharacteristic
172C22×C4×C23C2411yescharacteristic
173SD16×C23—44no—
174D8×C23—44no—
175C2×C8×C23—22no—
176C2×C8×C23—22no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—21no—
182C4⋊C4×C23—21no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—42no—
191C4⋊C4×C23—42no—
192C4⋊C4×C23C2×D412yes—
193C4⋊C4×C23C2×D412yes—
194C22⋊C4×C23—22no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—22no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—22no—
204C22⋊C4×C23—22no—
205C22⋊C4×C23—22no—
206C22⋊C4×C23—42no—
207C22⋊C4×C23—42no—
208C22⋊C4×C23—42no—
209C22⋊C4×C23—42no—
210C22⋊C4×C23—42no—
211C42×C23—22no—
212C42×C23—21no—
213C42×C23—21no—
214C42×C23—22no—
215C42×C23—42no—
216C42×C23C2×D412yes—
217C42×C23C2×D412yes—
218#18176C2311yesmaximal, characteristic, direct product factor, Sylow
219C23×C23—21no—
220C23×C23—21no—
221C23×C23—22no—
222C23×C23—22no—
223C23×C23—41no—
224C23×C23—41no—
225C23×C23—41no—
226C23×C23—41no—
227C23×C23—41no—
228C23×C23—42no—
229C23×C23—41no—
230C23×C23—41no—
231C23×C23—41no—
232C23×C23—81no—
233C23×C23—81no—
234C23×C23AGL(1,ℤ/8)12yes—
235C23×C23C22×D411yescharacteristic
236Q8×C23—42no—
237Q8×C23—81no—
238D4×C23—24no—
239D4×C23—42no—
240D4×C23—42no—
241D4×C23—44no—
242D4×C23—48no—
243D4×C23—42no—
244D4×C23—42no—
245D4×C23—81no—
246C2×C4×C23—21no—
247C2×C4×C23—21no—
248C2×C4×C23—22no—
249C2×C4×C23—21no—
250C2×C4×C23—21no—
251C2×C4×C23—22no—
252C2×C4×C23—22no—
253C2×C4×C23—22no—
254C2×C4×C23—21no—
255C2×C4×C23—21no—
256C2×C4×C23—21no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—42no—
260C2×C4×C23—42no—
261C2×C4×C23—41no—
262C2×C4×C23—42no—
263C2×C4×C23—41no—
264C2×C4×C23—41no—
265C2×C4×C23—41no—
266C2×C4×C23—41no—
267C2×C4×C23—41no—
268C2×C4×C23—41no—
269C2×C4×C23—41no—
270C2×C4×C23—41no—
271C2×C4×C23—41no—
272C2×C4×C23—82no—
273C2×C4×C23—82no—
274C2×C4×C23—82no—
275C2×C4×C23—81no—
276C2×C4×C23—82no—
277C2×C4×C23—81no—
278C2×C4×C23#12012yes—
279C2×C4×C23#12011yescharacteristic
280C2×C4×C23#12011yescharacteristic
281C2×C4×C23C22×D412yes—
282C8×C23—42no—
283C8×C23—42no—
284#3491C2×C2311yescharacteristic, complemented
285#3114C2×C2312yescomplemented
286#2392C2×C2311yescharacteristic, complemented
287#1609C2×C2312yescomplemented
288#1608C2×C2311yescharacteristic, complemented
289#1607C2×C2311yescharacteristic, complemented
290#1597C2×C2312yescomplemented
291#1596C2×C2311yescharacteristic, complemented
292#1595C2×C2311yescharacteristic
293#1509C2×C2312yescomplemented
294#1464C2×C2311yescharacteristic, complemented
295C22×C23—21no—
296C22×C23—21no—
297C22×C23—22no—
298C22×C23—42no—
299C22×C23—42no—
300C22×C23—42no—
301C22×C23—41no—
302C22×C23—41no—
303C22×C23—42no—
304C22×C23—42no—
305C22×C23—42no—
306C22×C23—42no—
307C22×C23—81no—
308C22×C23—81no—
309C22×C23—81no—
310C22×C23—81no—
311C22×C23—81no—
312C22×C23—162no—
313C22×C23C2×#12011yescharacteristic
314C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C23#57511yescharacteristic
316C4×C23—21no—
317C4×C23—21no—
318C4×C23—22no—
319C4×C23—22no—
320C4×C23—42no—
321C4×C23—41no—
322C4×C23—42no—
323C4×C23—41no—
324C4×C23—42no—
325C4×C23—41no—
326C4×C23—42no—
327C4×C23—81no—
328C4×C23—81no—
329#552—22no—
330#552—22no—
331#551—22no—
332#551—22no—
333#543C22×C2311yescharacteristic
334#539—21no—
335#539C22×C2312yes—
336#534C22×C2311yescharacteristic
337#529—21no—
338#529C22×C2312yes—
339C2×#123—21no—
340#518—21no—
341C2×C4×D4—21no—
342C2×C4×D4C22×C2311yescharacteristic, complemented
343C2×#117C22×C2311yescharacteristic
344#483C22×C2312yescomplemented
345#480C22×C2312yes—
346#478—22no—
347#463—22no—
348#450C22×C2312yescomplemented
349#447C22×C2312yes—
350#442—22no—
351#440—22no—
352#432C22×C2312yescomplemented
353#398C22×C2311yescharacteristic, complemented
354#397C22×C2311yescharacteristic
355#396—21no—
356#394—21no—
357#394C22×C2311yescharacteristic, complemented
358#392C22×C2311yescharacteristic
359C4×C4⋊C4C22×C2311yescharacteristic, complemented
360C4×C22⋊C4—21no—
361#328C22×C2312yescomplemented
362#328C22×C2312yescomplemented
363#327C22×C2312yes—
364#327C22×C2312yes—
365#324C22×C2312yescomplemented
366#323C22×C2311yescharacteristic
367#323C22×C2311yescharacteristic
368C2×C23—21no—
369C2×C23—21no—
370C2×C23—42no—
371C2×C23—82no—
372C2×C23—82no—
373C2×C23#300611yescharacteristic
374C2×C23#301811yescharacteristic
375C2×C23#302511yescharacteristic
376C22×D4—21no—
377C22×D4—21no—
378C22×D4C23×C2311yescharacteristic
379C23×C4—21no—
380C23×C4D4×C2311yescharacteristic
381C2×SD16—22no—
382C2×D8—22no—
383#126—22no—
384#126—22no—
385#126—22no—
386#126—22no—
387#125—21no—
388#125—22no—
389#125—22no—
390#125D4×C2312yescomplemented
391#124—21no—
392#124D4×C2312yescomplemented
393#123—22no—
394#123—22no—
395#123—22no—
396#123—22no—
397#123—22no—
398#123—22no—
399#123—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—21no—
404C22⋊Q8—22no—
405C22⋊Q8C23×C2312yes—
406#121—21no—
407#121C23×C2312yes—
408#120—22no—
409C4×Q8—21no—
410C4×Q8D4×C2312yescomplemented
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—21no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—44no—
419C4×D4D4×C2312yescomplemented
420#117—22no—
421#117—22no—
422#117—22no—
423#117—22no—
424#117—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4D4×C2311yescharacteristic
430C2×C4⋊C4C23×C2311yescharacteristic
431C2×C4⋊C4C23×C2311yescharacteristic
432C2×C22⋊C4—22no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4D4×C2311yescharacteristic
442C2×C22⋊C4C23×C2311yescharacteristic
443C2×C42—21no—
444C2×C42D4×C2311yescharacteristic
445C2×C42C23×C2311yescharacteristic
446#107—22no—
447#106—22no—
448C4⋊C8—22no—
449#103—22no—
450#102—22no—
451#102—22no—
452#102—22no—
453C22⋊C8C23×C2312yes—
454C22⋊C8C23×C2312yes—
455#97—22no—
456#95—21no—
457#95—21no—
458#95C23×C2311yescharacteristic
459#95C23×C2311yescharacteristic
460C23#1817611yescharacteristic, direct product factor, Sylow
461C24—21no—
462C24—21no—
463C24C2×D4×C2311yescharacteristic
464C2×Q8—21no—
465C2×Q8C2×D4×C2312yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4C2×D4×C2312yes—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—41no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4C2×D4×C2311yescharacteristic
494C22×C4C2×D4×C2311yescharacteristic
495C22×C4C2×D4×C2311yescharacteristic
496C22×C4C2×D4×C2311yescharacteristic
497C22×C4C2×D4×C2311yescharacteristic
498C22×C4C24×C2311yescharacteristic
499SD16—44no—
500D8—44no—
501C2×C8—22no—
502C2×C8—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4C2×D4×C2312yes—
519C4⋊C4C2×D4×C2312yes—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—22no—
541C42—42no—
542C42C2×D4×C2312yes—
543C42C2×D4×C2312yes—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—81no—
558C23—81no—
559C23C23×AGL(1,ℤ/8)12yes—
560C23C22×D4×C2311yescharacteristic
561Q8—42no—
562Q8—81no—
563D4—24no—
564D4—42no—
565D4—42no—
566D4—44no—
567D4—48no—
568D4—42no—
569D4—42no—
570D4—81no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4C23×#12012yes—
604C2×C4C23×#12011yescharacteristic
605C2×C4C23×#12011yescharacteristic
606C2×C4C22×D4×C2312yes—
607C8—42no—
608C8—42no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—162no—
627C22C2×C23×#12011yescharacteristic
628C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
629C22C23×#57511yescharacteristic
630C4—21no—
631C4—21no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—42no—
641C4—81no—
642C4—81no—
643C2—21no—
644C2—21no—
645C2—42no—
646C2—82no—
647C2—82no—
648C2C23×#300611yescharacteristic
649C2C23×#301811yescharacteristic
650C2C23×#302511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#1595C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3491C2semidirect
2. C23×#3114C2semidirect
3. C23×#2392C2semidirect
4. C23×#1609C2semidirect
5. C23×#1608C2semidirect
6. C23×#1607C2semidirect
7. C23×#1597C2semidirect
8. C23×#1596C2semidirect
10. C23×#1509C2semidirect
11. C23×#1464C2semidirect
25. C2×C4×D4×C23C22semidirect
27. C23×#483C22semidirect
31. C23×#450C22semidirect
35. C23×#432C22semidirect
36. C23×#398C22semidirect
40. C23×#394C22semidirect
42. C4×C4⋊C4×C23C22semidirect
44. C23×#328C22semidirect
45. C23×#328C22semidirect
48. C23×#324C22semidirect
65. C23×#125D4semidirect
67. C23×#124D4semidirect
85. C4×Q8×C23D4semidirect
94. C4×D4×C23D4semidirect
218. #18176C23direct (N × H)
284. #3491C2×C23semidirect
285. #3114C2×C23semidirect
286. #2392C2×C23semidirect
287. #1609C2×C23semidirect
288. #1608C2×C23semidirect
289. #1607C2×C23semidirect
290. #1597C2×C23semidirect
291. #1596C2×C23semidirect
293. #1509C2×C23semidirect
294. #1464C2×C23semidirect
342. C2×C4×D4C22×C23semidirect
344. #483C22×C23semidirect
348. #450C22×C23semidirect
352. #432C22×C23semidirect
353. #398C22×C23semidirect
357. #394C22×C23semidirect
359. C4×C4⋊C4C22×C23semidirect
361. #328C22×C23semidirect
362. #328C22×C23semidirect
365. #324C22×C23semidirect
390. #125D4×C23semidirect
392. #124D4×C23semidirect
410. C4×Q8D4×C23semidirect
419. C4×D4D4×C23semidirect
460. C23#18176direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34912
C23×#31142
C23×#23922
C23×#16092
C23×#16082
C23×#16072
C23×#15972
C23×#15962
C23×#15952
C23×#15092
C23×#14642
#1817623