Finite Group Databasebeta

← C23×#18299 #19165925

Subgroups — C23×#18299 #19165925

Order 5888.

974 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3460C211yesmaximal, characteristic
2C23×#3190C212yesmaximal, complemented
3C23×#2638C211yesmaximal, characteristic, complemented
4C23×#1612C212yesmaximal, complemented
5C23×#1611C211yesmaximal, characteristic, complemented
6C23×#1610C211yesmaximal, characteristic, complemented
7C23×#1606C212yesmaximal, complemented
8C23×#1605C211yesmaximal, characteristic, complemented
9C23×#1604C211yesmaximal, characteristic, complemented
10C23×#1516C212yesmaximal, complemented
11C23×#1459C211yesmaximal, characteristic, complemented
12C23×#544C2211yescharacteristic
13C23×#542—22no—
14C23×#540—22no—
15C23×#538—21no—
16C23×#538C2212yes—
17C23×#537C2211yescharacteristic
18C23×#535—21no—
19C23×#532—22no—
20C23×#529—22no—
21C23×#525—21no—
22C23×#525C2212yes—
23C2×C23×C22⋊Q8C2211yescharacteristic
24C4×#41×C23—21no—
25C2×C4×D4×C23C2211yescharacteristic
26C23×#485—22no—
27C23×#483C2212yes—
28C23×#480C2212yes—
29C23×#467—22no—
30C23×#465—22no—
31C23×#452—22no—
32C23×#451C2212yes—
33C23×#448C2212yes—
34C2×C23×C22⋊C8C2212yes—
35C23×#398C2211yescharacteristic
36C23×#397—21no—
37C23×#397C2211yescharacteristic
38C23×#395C2211yescharacteristic
39C23×#394—21no—
40C23×#393C2211yescharacteristic
41C23×#387—21no—
42C23×#385—21no—
43C23×#381C2211yescharacteristic
44C23×#328C2212yes—
45C23×#328C2212yes—
46C23×#327C2212yes—
47C23×#327C2212yes—
48C23×#324C2211yescharacteristic
49C23×#324C2211yescharacteristic
50C23×#323C2212yes—
51C2×#41×C23—21no—
52C2×#41×C23—21no—
53C2×#41×C23D412yes—
54C22×Q8×C23C2311yescharacteristic
55C22×D4×C23D411yescharacteristic
56C23×C4×C23D411yescharacteristic
57C23×C4×C23C2311yescharacteristic
58C2×Q16×C23—22no—
59C2×SD16×C23—22no—
60C22×C8×C23—22no—
61C23×C4⋊Q8—22no—
62C23×#127—22no—
63C23×#126—22no—
64C23×#126—22no—
65C23×#125—22no—
66C23×#124—22no—
67C23×#124—22no—
68C23×#124—22no—
69C23×#123—22no—
70C23×#123—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×C22⋊Q8—21no—
74C23×C22⋊Q8—21no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—22no—
78C23×C22⋊Q8D412yes—
79C23×C22⋊Q8C2312yes—
80C23×#121—21no—
81C23×#121—22no—
82C23×#121—22no—
83C23×#121—22no—
84C23×#121—22no—
85C23×#121—22no—
86C23×#121—21no—
87C23×#121D412yes—
88C23×#121C2312yes—
89C23×#120—22no—
90C23×#120—22no—
91C4×Q8×C23—21no—
92C4×Q8×C23—21no—
93C4×D4×C23—21no—
94C4×D4×C23—21no—
95C4×D4×C23—22no—
96C4×D4×C23—22no—
97C4×D4×C23—22no—
98C4×D4×C23—22no—
99C4×D4×C23—22no—
100C4×D4×C23—22no—
101C23×#117—22no—
102C23×#117—21no—
103C23×#117—21no—
104C23×#117D412yes—
105C2×C4⋊C4×C23—21no—
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23—21no—
108C2×C4⋊C4×C23C2311yescharacteristic
109C2×C4⋊C4×C23C2311yescharacteristic
110C2×C22⋊C4×C23—21no—
111C2×C22⋊C4×C23—21no—
112C2×C22⋊C4×C23—21no—
113C2×C22⋊C4×C23—21no—
114C2×C22⋊C4×C23D411yescharacteristic
115C2×C22⋊C4×C23C2311yescharacteristic
116C2×C42×C23—22no—
117C2×C42×C23D411yescharacteristic
118C23×#107—22no—
119C23×#106—22no—
120C23×#103—22no—
121C23×#103—22no—
122C23×#103—22no—
123C23×#102—22no—
124C23×C22⋊C8—22no—
125C23×C22⋊C8C2312yes—
126C23×C22⋊C8C2312yes—
127C23×#95—21no—
128C23×#95—21no—
129C23×#95—21no—
130C23×#95—21no—
131C23×#95—21no—
132C23×#95C2311yescharacteristic
133C23×#95C2311yescharacteristic
134C24×C23C2×D411yescharacteristic
135C24×C23C2×D411yescharacteristic
136#41×C23—42no—
137#41×C23—42no—
138#41×C23—82no—
139#41×C23—82no—
140C2×Q8×C23—21no—
141C2×Q8×C23—21no—
142C2×Q8×C23—22no—
143C2×Q8×C23—81no—
144C2×Q8×C23C2×D412yes—
145C2×D4×C23—21no—
146C2×D4×C23—22no—
147C2×D4×C23—22no—
148C2×D4×C23—22no—
149C2×D4×C23—22no—
150C2×D4×C23—21no—
151C2×D4×C23—22no—
152C2×D4×C23—22no—
153C2×D4×C23—22no—
154C2×D4×C23—41no—
155C2×D4×C23C2×D412yes—
156C22×C4×C23—22no—
157C22×C4×C23—22no—
158C22×C4×C23—22no—
159C22×C4×C23—21no—
160C22×C4×C23—21no—
161C22×C4×C23—21no—
162C22×C4×C23—21no—
163C22×C4×C23—21no—
164C22×C4×C23—21no—
165C22×C4×C23—81no—
166C22×C4×C23—81no—
167C22×C4×C23C2×D411yescharacteristic
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2×D412yes—
172C22×C4×C23C2×D412yes—
173C22×C4×C23C2411yescharacteristic
174Q16×C23—82no—
175SD16×C23—82no—
176C2×C8×C23—22no—
177C2×C8×C23—22no—
178C2×C8×C23—82no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—21no—
183C4⋊C4×C23—21no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—41no—
190C4⋊C4×C23—41no—
191C4⋊C4×C23—42no—
192C4⋊C4×C23—81no—
193C4⋊C4×C23C2×D412yes—
194C4⋊C4×C23C2×D412yes—
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—81no—
209C22⋊C4×C23—81no—
210C42×C23—22no—
211C42×C23—22no—
212C42×C23—21no—
213C42×C23—22no—
214C42×C23—21no—
215#18299C2311yesmaximal, characteristic, direct product factor, Sylow
216C23×C23—21no—
217C23×C23—21no—
218C23×C23—21no—
219C23×C23—22no—
220C23×C23—21no—
221C23×C23—21no—
222C23×C23—41no—
223C23×C23—41no—
224C23×C23—41no—
225C23×C23—41no—
226C23×C23#12012yes—
227C23×C23C22×D411yescharacteristic
228Q8×C23—42no—
229Q8×C23—42no—
230Q8×C23—42no—
231Q8×C23—42no—
232Q8×C23—81no—
233D4×C23—44no—
234D4×C23—42no—
235D4×C23—42no—
236D4×C23—42no—
237D4×C23—42no—
238D4×C23—82no—
239D4×C23—82no—
240D4×C23—82no—
241D4×C23—81no—
242D4×C23—82no—
243C2×C4×C23—21no—
244C2×C4×C23—21no—
245C2×C4×C23—21no—
246C2×C4×C23—22no—
247C2×C4×C23—22no—
248C2×C4×C23—22no—
249C2×C4×C23—21no—
250C2×C4×C23—42no—
251C2×C4×C23—41no—
252C2×C4×C23—42no—
253C2×C4×C23—41no—
254C2×C4×C23—42no—
255C2×C4×C23—44no—
256C2×C4×C23—44no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—41no—
260C2×C4×C23—41no—
261C2×C4×C23—41no—
262C2×C4×C23—41no—
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—81no—
272C2×C4×C23—82no—
273C2×C4×C23#12011yescharacteristic
274C2×C4×C23#12011yescharacteristic
275C2×C4×C23#13512yes—
276C2×C4×C23C22×D412yes—
277C8×C23—42no—
278C8×C23—42no—
279#3460C2×C2311yescharacteristic
280#3190C2×C2312yescomplemented
281#2638C2×C2311yescharacteristic, complemented
282#1612C2×C2312yescomplemented
283#1611C2×C2311yescharacteristic, complemented
284#1610C2×C2311yescharacteristic, complemented
285#1606C2×C2312yescomplemented
286#1605C2×C2311yescharacteristic, complemented
287#1604C2×C2311yescharacteristic, complemented
288#1516C2×C2312yescomplemented
289#1459C2×C2311yescharacteristic, complemented
290C22×C23—21no—
291C22×C23—21no—
292C22×C23—22no—
293C22×C23—22no—
294C22×C23—22no—
295C22×C23—42no—
296C22×C23—42no—
297C22×C23—41no—
298C22×C23—42no—
299C22×C23—41no—
300C22×C23—41no—
301C22×C23—42no—
302C22×C23—81no—
303C22×C23—81no—
304C22×C23—81no—
305C22×C23C2×#12011yescharacteristic
306C22×C23C2×#13511yescharacteristic
307C22×C23#57511yescharacteristic
308C4×C23—22no—
309C4×C23—22no—
310C4×C23—21no—
311C4×C23—21no—
312C4×C23—42no—
313C4×C23—41no—
314C4×C23—42no—
315C4×C23—41no—
316C4×C23—42no—
317C4×C23—81no—
318C4×C23—81no—
319C4×C23—81no—
320C4×C23—81no—
321#544C22×C2311yescharacteristic
322#542—22no—
323#540—22no—
324#538—21no—
325#538C22×C2312yes—
326#537C22×C2311yescharacteristic
327#535—21no—
328#532—22no—
329#529—22no—
330#525—21no—
331#525C22×C2312yes—
332C2×C22⋊Q8C22×C2311yescharacteristic
333C4×#41—21no—
334C2×C4×D4C22×C2311yescharacteristic
335#485—22no—
336#483C22×C2312yes—
337#480C22×C2312yes—
338#467—22no—
339#465—22no—
340#452—22no—
341#451C22×C2312yes—
342#448C22×C2312yes—
343C2×C22⋊C8C22×C2312yes—
344#398C22×C2311yescharacteristic
345#397—21no—
346#397C22×C2311yescharacteristic
347#395C22×C2311yescharacteristic
348#394—21no—
349#393C22×C2311yescharacteristic
350#387—21no—
351#385—21no—
352#381C22×C2311yescharacteristic
353#328C22×C2312yes—
354#328C22×C2312yes—
355#327C22×C2312yes—
356#327C22×C2312yes—
357#324C22×C2311yescharacteristic
358#324C22×C2311yescharacteristic
359#323C22×C2312yes—
360C2×C23—21no—
361C2×C23—21no—
362C2×C23—42no—
363C2×C23—82no—
364C2×C23—81no—
365C2×C23#300211yescharacteristic
366C2×C23#302211yescharacteristic
367C2×C23#302611yescharacteristic
368C2×#41—21no—
369C2×#41—21no—
370C2×#41D4×C2312yes—
371C22×Q8C23×C2311yescharacteristic
372C22×D4D4×C2311yescharacteristic
373C23×C4D4×C2311yescharacteristic
374C23×C4C23×C2311yescharacteristic
375C2×Q16—22no—
376C2×SD16—22no—
377C22×C8—22no—
378C4⋊Q8—22no—
379#127—22no—
380#126—22no—
381#126—22no—
382#125—22no—
383#124—22no—
384#124—22no—
385#124—22no—
386#123—22no—
387#123—22no—
388#123—22no—
389#123—22no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8D4×C2312yes—
396C22⋊Q8C23×C2312yes—
397#121—21no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121—22no—
402#121—22no—
403#121—21no—
404#121D4×C2312yes—
405#121C23×C2312yes—
406#120—22no—
407#120—22no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418#117—22no—
419#117—21no—
420#117—21no—
421#117D4×C2312yes—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4C23×C2311yescharacteristic
426C2×C4⋊C4C23×C2311yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4D4×C2311yescharacteristic
432C2×C22⋊C4C23×C2311yescharacteristic
433C2×C42—22no—
434C2×C42D4×C2311yescharacteristic
435#107—22no—
436#106—22no—
437#103—22no—
438#103—22no—
439#103—22no—
440#102—22no—
441C22⋊C8—22no—
442C22⋊C8C23×C2312yes—
443C22⋊C8C23×C2312yes—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95C23×C2311yescharacteristic
450#95C23×C2311yescharacteristic
451C23#1829911yescharacteristic, direct product factor, Sylow
452C24C2×D4×C2311yescharacteristic
453C24C2×D4×C2311yescharacteristic
454#41—42no—
455#41—42no—
456#41—82no—
457#41—82no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—81no—
462C2×Q8C2×D4×C2312yes—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—41no—
473C2×D4C2×D4×C2312yes—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—81no—
484C22×C4—81no—
485C22×C4C2×D4×C2311yescharacteristic
486C22×C4C2×D4×C2311yescharacteristic
487C22×C4C2×D4×C2311yescharacteristic
488C22×C4C2×D4×C2311yescharacteristic
489C22×C4C2×D4×C2312yes—
490C22×C4C2×D4×C2312yes—
491C22×C4C24×C2311yescharacteristic
492Q16—82no—
493SD16—82no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—82no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4C2×D4×C2312yes—
512C4⋊C4C2×D4×C2312yes—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—22no—
532C42—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—21no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23C23×#12012yes—
544C23C22×D4×C2311yescharacteristic
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—81no—
550D4—44no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—82no—
556D4—82no—
557D4—81no—
558D4—82no—
559D4—82no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—81no—
589C2×C4—82no—
590C2×C4C23×#12011yescharacteristic
591C2×C4C23×#12011yescharacteristic
592C2×C4C23×#13512yes—
593C2×C4C22×D4×C2312yes—
594C8—42no—
595C8—42no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×C23×#12011yescharacteristic
612C22C2×C23×#13511yescharacteristic
613C22C23×#57511yescharacteristic
614C4—22no—
615C4—22no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C2—21no—
628C2—21no—
629C2—42no—
630C2—81no—
631C2—82no—
632C2C23×#300211yescharacteristic
633C2C23×#302211yescharacteristic
634C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3460C2
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
2. C23×#3190C2semidirect
3. C23×#2638C2semidirect
4. C23×#1612C2semidirect
5. C23×#1611C2semidirect
6. C23×#1610C2semidirect
7. C23×#1606C2semidirect
8. C23×#1605C2semidirect
9. C23×#1604C2semidirect
10. C23×#1516C2semidirect
11. C23×#1459C2semidirect
215. #18299C23direct (N × H)
280. #3190C2×C23semidirect
281. #2638C2×C23semidirect
282. #1612C2×C23semidirect
283. #1611C2×C23semidirect
284. #1610C2×C23semidirect
285. #1606C2×C23semidirect
286. #1605C2×C23semidirect
287. #1604C2×C23semidirect
288. #1516C2×C23semidirect
289. #1459C2×C23semidirect
451. C23#18299direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34602
C23×#31902
C23×#26382
C23×#16122
C23×#16112
C23×#16102
C23×#16062
C23×#16052
C23×#16042
C23×#15162
C23×#14592
#1829923