Finite Group Databasebeta

← C23×#18845 #19166471

Subgroups — C23×#18845 #19166471

Order 5888.

876 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3515C211yesmaximal, characteristic
2C23×#3229C211yesmaximal, characteristic, complemented
3C23×#3218C211yesmaximal, characteristic, complemented
4C23×#3079C211yesmaximal, characteristic, complemented
5C23×#1654C212yesmaximal, complemented
6C23×#1653C212yesmaximal, complemented
7C23×#1644C212yesmaximal, complemented
8C23×#1643C212yesmaximal, complemented
9C23×#1559C211yesmaximal, characteristic, complemented
10C23×#1495C211yesmaximal, characteristic, complemented
11C23×#1488C211yesmaximal, characteristic, complemented
12C23×#557C2211yescharacteristic
13C23×#554C2211yescharacteristic
14C23×#554C2211yescharacteristic
15C23×#553—21no—
16C23×#552—21no—
17C23×#552—22no—
18C23×#552—21no—
19C23×#547—21no—
20C23×#547C2211yescharacteristic
21C23×#541—21no—
22C23×#539—22no—
23C23×#533C2211yescharacteristic
24C2×C23×C4⋊Q8C2211yescharacteristic
25C23×#529—21no—
26C23×#529C2211yescharacteristic
27C2×C23×#125—21no—
28C23×#520—21no—
29C4×#41×C23C2211yescharacteristic
30C23×#498C2212yes—
31C23×#497—21no—
32C23×#491—21no—
33C23×#487C2212yes—
34C23×#483—22no—
35C23×#477—21no—
36C23×#476—21no—
37C23×#475—22no—
38C23×#475C2212yes—
39C23×#474C2212yes—
40C23×#462C2212yes—
41C23×#461C2212yes—
42C23×#451—21no—
43C23×#448—21no—
44C23×#432C2211yescharacteristic
45C23×#431C2211yescharacteristic
46C23×#428—21no—
47C23×#423C2211yescharacteristic
48C23×#417—21no—
49C23×#333C2212yes—
50C23×#332C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×D4⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C2×#41×C23—21no—
56C2×#41×C23D411yescharacteristic
57C22×Q8×C23—21no—
58C2×Q16×C23—22no—
59C2×SD16×C23—22no—
60C2×M16×C23—21no—
61C23×C4⋊Q8—21no—
62C23×C4⋊Q8—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8C2312yes—
66C23×C4⋊Q8C2311yescharacteristic
67C23×C4⋊Q8C2311yescharacteristic
68C23×#126—22no—
69C23×#126—22no—
70C23×#126—22no—
71C23×#126—21no—
72C23×#125—22no—
73C23×#125—22no—
74C23×#125—22no—
75C23×#125—21no—
76C23×#125—21no—
77C23×#125—22no—
78C23×#125—21no—
79C23×#125—21no—
80C23×#125D411yescharacteristic
81C23×#124D411yescharacteristic
82C23×#123—22no—
83C23×#123—22no—
84C23×#123—22no—
85C23×#123—21no—
86C23×#123—22no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—21no—
92C23×C22⋊Q8—21no—
93C23×C22⋊Q8D411yescharacteristic
94C23×C22⋊Q8D412yes—
95C23×#121—22no—
96C23×#121—21no—
97C23×#121D411yescharacteristic
98C4×Q8×C23—21no—
99C4×Q8×C23—21no—
100C4×Q8×C23—21no—
101C4×Q8×C23D412yes—
102C4×Q8×C23D411yescharacteristic
103C4×Q8×C23C2311yescharacteristic
104C4×Q8×C23C2311yescharacteristic
105C4×D4×C23—22no—
106C4×D4×C23—22no—
107C4×D4×C23—21no—
108C4×D4×C23—21no—
109C4×D4×C23—21no—
110C4×D4×C23—21no—
111C4×D4×C23—21no—
112C4×D4×C23D411yescharacteristic
113C4×D4×C23C2311yescharacteristic
114C4×D4×C23C2311yescharacteristic
115C23×#117—22no—
116C23×#117—22no—
117C23×#117—21no—
118C23×#117—21no—
119C23×#117—21no—
120C23×#117D411yescharacteristic
121C2×C4⋊C4×C23—22no—
122C2×C4⋊C4×C23—22no—
123C2×C4⋊C4×C23—21no—
124C2×C4⋊C4×C23—21no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23—21no—
127C2×C4⋊C4×C23—21no—
128C2×C42×C23—21no—
129C2×C42×C23C2311yescharacteristic
130C23×#107—22no—
131C23×#106—22no—
132C23×#106—24no—
133C23×C4⋊C8—21no—
134C23×C4⋊C8—21no—
135C23×C4⋊C8C2312yes—
136C23×C4⋊C8C2312yes—
137C23×#103—22no—
138C23×#103—24no—
139C23×#103—22no—
140C23×#103—22no—
141C23×#102—22no—
142C23×C22⋊C8—21no—
143C23×C22⋊C8—21no—
144C23×#97—21no—
145C4×C8×C23C2312yes—
146#41×C23—22no—
147#41×C23—41no—
148#41×C23—42no—
149#41×C23—42no—
150C2×Q8×C23—21no—
151C2×Q8×C23—22no—
152C2×Q8×C23—41no—
153C2×Q8×C23—82no—
154C2×Q8×C23C2×D411yescharacteristic
155C2×Q8×C23C2×D411yescharacteristic
156C2×Q8×C23C2×D412yes—
157C2×D4×C23—22no—
158C2×D4×C23—21no—
159C2×D4×C23—21no—
160C2×D4×C23C2×D411yescharacteristic
161C2×D4×C23C2×D411yescharacteristic
162C22×C4×C23—22no—
163C22×C4×C23—22no—
164C22×C4×C23—22no—
165C22×C4×C23—21no—
166C22×C4×C23—21no—
167C22×C4×C23—21no—
168C22×C4×C23—21no—
169C22×C4×C23—41no—
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2×D411yescharacteristic
172C22×C4×C23C2×D411yescharacteristic
173Q16×C23—82no—
174SD16×C23—82no—
175M16×C23—42no—
176C2×C8×C23—22no—
177C2×C8×C23—22no—
178C2×C8×C23—22no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—21no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—21no—
198C4⋊C4×C23—22no—
199C4⋊C4×C23—22no—
200C4⋊C4×C23—41no—
201C4⋊C4×C23—41no—
202C4⋊C4×C23C2×D411yescharacteristic
203C4⋊C4×C23C2×D411yescharacteristic
204C4⋊C4×C23C2×D411yescharacteristic
205C4⋊C4×C23C2×D412yes—
206C4⋊C4×C23C2×D411yescharacteristic
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—21no—
213C22⋊C4×C23—21no—
214C22⋊C4×C23—21no—
215C22⋊C4×C23—21no—
216C22⋊C4×C23—21no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—21no—
219C42×C23—22no—
220C42×C23—21no—
221C42×C23—21no—
222C42×C23—22no—
223C42×C23—21no—
224C42×C23—21no—
225C42×C23C2×D411yescharacteristic
226C42×C23C2×D411yescharacteristic
227C42×C23C2×D411yescharacteristic
228C42×C23C2411yescharacteristic
229#18845C2311yesmaximal, characteristic, direct product factor, Sylow
230C23×C23—21no—
231C23×C23—21no—
232C23×C23#12011yescharacteristic
233Q8×C23—21no—
234Q8×C23—21no—
235Q8×C23—22no—
236Q8×C23—42no—
237Q8×C23—41no—
238Q8×C23—41no—
239Q8×C23—42no—
240Q8×C23—42no—
241Q8×C23—82no—
242D4×C23—21no—
243D4×C23—22no—
244D4×C23—42no—
245D4×C23—44no—
246D4×C23—81no—
247D4×C23#13712yes—
248C2×C4×C23—21no—
249C2×C4×C23—21no—
250C2×C4×C23—22no—
251C2×C4×C23—21no—
252C2×C4×C23—21no—
253C2×C4×C23—21no—
254C2×C4×C23—21no—
255C2×C4×C23—22no—
256C2×C4×C23—22no—
257C2×C4×C23—21no—
258C2×C4×C23—21no—
259C2×C4×C23—41no—
260C2×C4×C23—42no—
261C2×C4×C23—44no—
262C2×C4×C23—41no—
263C2×C4×C23—41no—
264C2×C4×C23—42no—
265C2×C4×C23—42no—
266C2×C4×C23—81no—
267C2×C4×C23—82no—
268C2×C4×C23—82no—
269C2×C4×C23—82no—
270C2×C4×C23#12011yescharacteristic
271C2×C4×C23#12011yescharacteristic
272C2×C4×C23#12011yescharacteristic
273C2×C4×C23C22×D411yescharacteristic
274C2×C4×C23C22×D411yescharacteristic
275C2×C4×C23C22×D411yescharacteristic
276C8×C23—42no—
277C8×C23—42no—
278C8×C23—42no—
279#3515C2×C2311yescharacteristic
280#3229C2×C2311yescharacteristic, complemented
281#3218C2×C2311yescharacteristic, complemented
282#3079C2×C2311yescharacteristic, complemented
283#1654C2×C2312yescomplemented
284#1653C2×C2312yescomplemented
285#1644C2×C2312yescomplemented
286#1643C2×C2312yescomplemented
287#1559C2×C2311yescharacteristic, complemented
288#1495C2×C2311yescharacteristic, complemented
289#1488C2×C2311yescharacteristic, complemented
290C22×C23—21no—
291C22×C23—22no—
292C22×C23—21no—
293C22×C23—21no—
294C22×C23—41no—
295C22×C23—41no—
296C22×C23—41no—
297C22×C23—41no—
298C22×C23—41no—
299C22×C23C2×#12011yescharacteristic
300C4×C23—21no—
301C4×C23—21no—
302C4×C23—21no—
303C4×C23—21no—
304C4×C23—41no—
305C4×C23—41no—
306C4×C23—42no—
307C4×C23—41no—
308C4×C23—41no—
309C4×C23—41no—
310C4×C23—41no—
311C4×C23—42no—
312C4×C23—82no—
313C4×C23C2×#13711yescharacteristic
314C4×C23#57511yescharacteristic
315C4×C23#57712yes—
316#557C22×C2311yescharacteristic
317#554C22×C2311yescharacteristic
318#554C22×C2311yescharacteristic
319#553—21no—
320#552—21no—
321#552—22no—
322#552—21no—
323#547—21no—
324#547C22×C2311yescharacteristic
325#541—21no—
326#539—22no—
327#533C22×C2311yescharacteristic
328C2×C4⋊Q8C22×C2311yescharacteristic
329#529—21no—
330#529C22×C2311yescharacteristic
331C2×#125—21no—
332#520—21no—
333C4×#41C22×C2311yescharacteristic
334#498C22×C2312yes—
335#497—21no—
336#491—21no—
337#487C22×C2312yes—
338#483—22no—
339#477—21no—
340#476—21no—
341#475—22no—
342#475C22×C2312yes—
343#474C22×C2312yes—
344#462C22×C2312yes—
345#461C22×C2312yes—
346#451—21no—
347#448—21no—
348#432C22×C2311yescharacteristic
349#431C22×C2311yescharacteristic
350#428—21no—
351#423C22×C2311yescharacteristic
352#417—21no—
353#333C22×C2312yes—
354#332C22×C2312yes—
355Q8⋊C8C22×C2312yes—
356Q8⋊C8C22×C2312yes—
357D4⋊C8C22×C2312yes—
358D4⋊C8C22×C2312yes—
359C2×C23—41no—
360C2×C23—42no—
361C2×C23—81no—
362C2×C23#300711yescharacteristic
363C2×C23#301011yescharacteristic
364C2×C23#302211yescharacteristic
365C2×#41—21no—
366C2×#41D4×C2311yescharacteristic
367C22×Q8—21no—
368C2×Q16—22no—
369C2×SD16—22no—
370C2×M16—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8C23×C2312yes—
376C4⋊Q8C23×C2311yescharacteristic
377C4⋊Q8C23×C2311yescharacteristic
378#126—22no—
379#126—22no—
380#126—22no—
381#126—21no—
382#125—22no—
383#125—22no—
384#125—22no—
385#125—21no—
386#125—21no—
387#125—22no—
388#125—21no—
389#125—21no—
390#125D4×C2311yescharacteristic
391#124D4×C2311yescharacteristic
392#123—22no—
393#123—22no—
394#123—22no—
395#123—21no—
396#123—22no—
397C22⋊Q8—21no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8D4×C2311yescharacteristic
404C22⋊Q8D4×C2312yes—
405#121—22no—
406#121—21no—
407#121D4×C2311yescharacteristic
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8D4×C2311yescharacteristic
412C4×Q8D4×C2312yes—
413C4×Q8C23×C2311yescharacteristic
414C4×Q8C23×C2311yescharacteristic
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4D4×C2311yescharacteristic
423C4×D4C23×C2311yescharacteristic
424C4×D4C23×C2311yescharacteristic
425#117—22no—
426#117—22no—
427#117—21no—
428#117—21no—
429#117—21no—
430#117D4×C2311yescharacteristic
431C2×C4⋊C4—22no—
432C2×C4⋊C4—22no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C42—21no—
439C2×C42C23×C2311yescharacteristic
440#107—22no—
441#106—22no—
442#106—24no—
443C4⋊C8—21no—
444C4⋊C8—21no—
445C4⋊C8C23×C2312yes—
446C4⋊C8C23×C2312yes—
447#103—22no—
448#103—24no—
449#103—22no—
450#103—22no—
451#102—22no—
452C22⋊C8—21no—
453C22⋊C8—21no—
454#97—21no—
455C4×C8C23×C2312yes—
456C23#1884511yescharacteristic, direct product factor, Sylow
457#41—22no—
458#41—41no—
459#41—42no—
460#41—42no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—41no—
464C2×Q8—82no—
465C2×Q8C2×D4×C2311yescharacteristic
466C2×Q8C2×D4×C2311yescharacteristic
467C2×Q8C2×D4×C2312yes—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4C2×D4×C2311yescharacteristic
472C2×D4C2×D4×C2311yescharacteristic
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—41no—
481C22×C4C2×D4×C2311yescharacteristic
482C22×C4C2×D4×C2311yescharacteristic
483C22×C4C2×D4×C2311yescharacteristic
484Q16—82no—
485SD16—82no—
486M16—42no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4C2×D4×C2311yescharacteristic
514C4⋊C4C2×D4×C2311yescharacteristic
515C4⋊C4C2×D4×C2311yescharacteristic
516C4⋊C4C2×D4×C2312yes—
517C4⋊C4C2×D4×C2311yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42C2×D4×C2311yescharacteristic
537C42C2×D4×C2311yescharacteristic
538C42C2×D4×C2311yescharacteristic
539C42C24×C2311yescharacteristic
540C23—21no—
541C23—21no—
542C23C23×#12011yescharacteristic
543Q8—21no—
544Q8—21no—
545Q8—22no—
546Q8—42no—
547Q8—41no—
548Q8—41no—
549Q8—42no—
550Q8—42no—
551Q8—82no—
552D4—21no—
553D4—22no—
554D4—42no—
555D4—44no—
556D4—81no—
557D4C23×#13712yes—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4C23×#12011yescharacteristic
581C2×C4C23×#12011yescharacteristic
582C2×C4C23×#12011yescharacteristic
583C2×C4C22×D4×C2311yescharacteristic
584C2×C4C22×D4×C2311yescharacteristic
585C2×C4C22×D4×C2311yescharacteristic
586C8—42no—
587C8—42no—
588C8—42no—
589C22—21no—
590C22—22no—
591C22—21no—
592C22—21no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22C2×C23×#12011yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—82no—
612C4C2×C23×#13711yescharacteristic
613C4C23×#57511yescharacteristic
614C4C23×#57712yes—
615C2—41no—
616C2—42no—
617C2—81no—
618C2C23×#300711yescharacteristic
619C2C23×#301011yescharacteristic
620C2C23×#302211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3515C2
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×#3229C2semidirect
3. C23×#3218C2semidirect
4. C23×#3079C2semidirect
5. C23×#1654C2semidirect
6. C23×#1653C2semidirect
7. C23×#1644C2semidirect
8. C23×#1643C2semidirect
9. C23×#1559C2semidirect
10. C23×#1495C2semidirect
11. C23×#1488C2semidirect
229. #18845C23direct (N × H)
280. #3229C2×C23semidirect
281. #3218C2×C23semidirect
282. #3079C2×C23semidirect
283. #1654C2×C23semidirect
284. #1653C2×C23semidirect
285. #1644C2×C23semidirect
286. #1643C2×C23semidirect
287. #1559C2×C23semidirect
288. #1495C2×C23semidirect
289. #1488C2×C23semidirect
456. C23#18845direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35152
C23×#32292
C23×#32182
C23×#30792
C23×#16542
C23×#16532
C23×#16442
C23×#16432
C23×#15592
C23×#14952
C23×#14882
#1884523