Finite Group Databasebeta

← C23×#18699 #19166325

Subgroups — C23×#18699 #19166325

Order 5888.

878 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3552C211yesmaximal, characteristic
2C23×#3239C211yesmaximal, characteristic, complemented
3C23×#3218C211yesmaximal, characteristic, complemented
4C2×C23×#500C211yesmaximal, characteristic, complemented
5C23×#1650C212yesmaximal, complemented
6C23×#1649C212yesmaximal, complemented
7C23×#1648C212yesmaximal, complemented
8C23×#1647C212yesmaximal, complemented
9C23×#1556C211yesmaximal, characteristic, complemented
10C23×#1494C211yesmaximal, characteristic, complemented
11C23×#1494C211yesmaximal, characteristic, complemented
12C23×#563—22no—
13C23×#557C2211yescharacteristic
14C23×#556C2211yescharacteristic
15C23×#554—21no—
16C23×#553—21no—
17D4×Q8×C23—21no—
18D4×Q8×C23C2211yescharacteristic
19C23×#548—21no—
20C23×#547C2211yescharacteristic
21C23×#544—21no—
22C23×#542—21no—
23C23×#541—21no—
24C23×#539—22no—
25C23×#536—21no—
26C2×C23×C4⋊Q8C2211yescharacteristic
27C2×C23×C4⋊Q8C2211yescharacteristic
28C23×#529C2211yescharacteristic
29C23×#529C2211yescharacteristic
30C23×#501—21no—
31C23×#500—22no—
32C23×#500C2212yes—
33C23×#500C2212yes—
34C23×#497—21no—
35C23×#491—21no—
36C23×#489—21no—
37C23×#484—21no—
38C23×#481—21no—
39C23×#475C2212yes—
40C23×#474C2212yes—
41C23×#462C2212yes—
42C23×#461C2212yes—
43C23×#451—21no—
44C23×#448—21no—
45C23×#432C2211yescharacteristic
46C23×#432C2211yescharacteristic
47C2×C23×#107—22no—
48C2×C4×C8×C23C2211yescharacteristic
49C23×#332C2212yes—
50C23×#332C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×D4⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C2×#41×C23—21no—
56C22×Q8×C23—21no—
57C22×Q8×C23—21no—
58C22×Q8×C23—21no—
59C2×Q16×C23—22no—
60C2×SD16×C23—22no—
61C22×C8×C23—21no—
62C23×C4⋊Q8—22no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8—21no—
66C23×C4⋊Q8—21no—
67C23×C4⋊Q8—21no—
68C23×C4⋊Q8—21no—
69C23×C4⋊Q8—21no—
70C23×C4⋊Q8D412yes—
71C23×C4⋊Q8C2312yes—
72C23×C4⋊Q8C2311yescharacteristic
73C23×C4⋊Q8C2311yescharacteristic
74C23×#126—22no—
75C23×#126—22no—
76C23×#126—22no—
77C23×#126—21no—
78C23×#126—21no—
79C23×#125—22no—
80C23×#125—22no—
81C23×#125—21no—
82C23×#125—22no—
83C23×#125D411yescharacteristic
84C23×#125D411yescharacteristic
85C23×#124—22no—
86C23×#124—21no—
87C23×#124D411yescharacteristic
88C23×#124D411yescharacteristic
89C23×#123—22no—
90C23×#123—22no—
91C23×#123—21no—
92C23×#123—21no—
93C23×#123—22no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8—22no—
96C23×C22⋊Q8—22no—
97C23×C22⋊Q8—22no—
98C23×C22⋊Q8—22no—
99C23×C22⋊Q8—21no—
100C23×C22⋊Q8—21no—
101C23×C22⋊Q8—21no—
102C23×C22⋊Q8—22no—
103C23×C22⋊Q8D411yescharacteristic
104C23×C22⋊Q8D411yescharacteristic
105C23×#121—22no—
106C23×#121—21no—
107C23×#121D411yescharacteristic
108C23×#121D411yescharacteristic
109C4×Q8×C23—21no—
110C4×Q8×C23—21no—
111C4×Q8×C23—21no—
112C4×Q8×C23C2311yescharacteristic
113C4×Q8×C23C2311yescharacteristic
114C4×D4×C23—22no—
115C4×D4×C23—22no—
116C4×D4×C23—21no—
117C4×D4×C23—21no—
118C4×D4×C23—21no—
119C4×D4×C23C2311yescharacteristic
120C4×D4×C23C2311yescharacteristic
121C23×#117—21no—
122C23×#117—21no—
123C23×#117—21no—
124C2×C4⋊C4×C23—21no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23—21no—
127C2×C4⋊C4×C23—21no—
128C2×C4⋊C4×C23D412yes—
129C2×C42×C23C2311yescharacteristic
130C23×#107—24no—
131C23×#107—24no—
132C23×#107—22no—
133C23×#106—22no—
134C23×C4⋊C8C2312yes—
135C23×C4⋊C8C2312yes—
136C23×#103—22no—
137C23×#103—22no—
138C23×#103—22no—
139C23×#102—22no—
140C23×C22⋊C8—21no—
141C23×C22⋊C8—21no—
142C23×#97—21no—
143C23×#97—21no—
144C4×C8×C23—21no—
145C4×C8×C23C2312yes—
146#41×C23—82no—
147C2×Q8×C23—22no—
148C2×Q8×C23—22no—
149C2×Q8×C23—21no—
150C2×Q8×C23—21no—
151C2×Q8×C23—22no—
152C2×Q8×C23—41no—
153C2×Q8×C23—82no—
154C2×Q8×C23—82no—
155C2×Q8×C23—82no—
156C2×Q8×C23C2×D411yescharacteristic
157C2×Q8×C23C2×D411yescharacteristic
158C2×D4×C23—21no—
159C2×D4×C23—21no—
160C2×D4×C23—22no—
161C2×D4×C23C2×D411yescharacteristic
162C2×D4×C23C2×D411yescharacteristic
163C22×C4×C23—22no—
164C22×C4×C23—22no—
165C22×C4×C23—22no—
166C22×C4×C23—21no—
167C22×C4×C23—21no—
168C22×C4×C23—41no—
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2×D411yescharacteristic
172Q16×C23—82no—
173SD16×C23—82no—
174C2×C8×C23—24no—
175C2×C8×C23—22no—
176C2×C8×C23—22no—
177C2×C8×C23—22no—
178C4⋊C4×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—21no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—21no—
189C4⋊C4×C23—21no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—22no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—41no—
199C4⋊C4×C23—41no—
200C4⋊C4×C23C2×D411yescharacteristic
201C4⋊C4×C23C2×D411yescharacteristic
202C4⋊C4×C23C2×D411yescharacteristic
203C4⋊C4×C23C2×D411yescharacteristic
204C4⋊C4×C23C2×D412yes—
205C4⋊C4×C23C2×D412yes—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—21no—
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—
219C22⋊C4×C23—21no—
220C42×C23—22no—
221C42×C23—22no—
222C42×C23—21no—
223C42×C23—21no—
224C42×C23C2×D411yescharacteristic
225C42×C23C2×D411yescharacteristic
226C42×C23C2×D411yescharacteristic
227C42×C23C2411yescharacteristic
228#18699C2311yesmaximal, characteristic, direct product factor, Sylow
229C23×C23—21no—
230C23×C23—21no—
231C23×C23#12011yescharacteristic
232Q8×C23—22no—
233Q8×C23—22no—
234Q8×C23—42no—
235Q8×C23—42no—
236Q8×C23—42no—
237Q8×C23—42no—
238Q8×C23—82no—
239Q8×C23—82no—
240Q8×C23—82no—
241D4×C23—22no—
242D4×C23—22no—
243D4×C23—42no—
244D4×C23—42no—
245D4×C23—82no—
246C2×C4×C23—22no—
247C2×C4×C23—22no—
248C2×C4×C23—22no—
249C2×C4×C23—21no—
250C2×C4×C23—21no—
251C2×C4×C23—21no—
252C2×C4×C23—21no—
253C2×C4×C23—21no—
254C2×C4×C23—22no—
255C2×C4×C23—22no—
256C2×C4×C23—21no—
257C2×C4×C23—42no—
258C2×C4×C23—42no—
259C2×C4×C23—42no—
260C2×C4×C23—82no—
261C2×C4×C23—82no—
262C2×C4×C23—82no—
263C2×C4×C23—82no—
264C2×C4×C23#12011yescharacteristic
265C2×C4×C23#12011yescharacteristic
266C2×C4×C23#12011yescharacteristic
267C2×C4×C23C22×D411yescharacteristic
268C2×C4×C23C22×D411yescharacteristic
269C2×C4×C23C22×D411yescharacteristic
270C8×C23—24no—
271C8×C23—42no—
272C8×C23—42no—
273#3552C2×C2311yescharacteristic
274#3239C2×C2311yescharacteristic, complemented
275#3218C2×C2311yescharacteristic, complemented
276C2×#500C2×C2311yescharacteristic, complemented
277#1650C2×C2312yescomplemented
278#1649C2×C2312yescomplemented
279#1648C2×C2312yescomplemented
280#1647C2×C2312yescomplemented
281#1556C2×C2311yescharacteristic, complemented
282#1494C2×C2311yescharacteristic, complemented
283#1494C2×C2311yescharacteristic, complemented
284C22×C23—21no—
285C22×C23—21no—
286C22×C23—41no—
287C22×C23—41no—
288C22×C23—41no—
289C22×C23—41no—
290C22×C23—41no—
291C22×C23—41no—
292C22×C23#45612yes—
293C22×C23C2×#12011yescharacteristic
294C4×C23—21no—
295C4×C23—21no—
296C4×C23—21no—
297C4×C23—21no—
298C4×C23—41no—
299C4×C23—42no—
300C4×C23—41no—
301C4×C23—42no—
302C4×C23—41no—
303C4×C23—41no—
304C4×C23—41no—
305C4×C23—41no—
306C4×C23—82no—
307C4×C23#57612yes—
308C4×C23#57812yes—
309#563—22no—
310#557C22×C2311yescharacteristic
311#556C22×C2311yescharacteristic
312#554—21no—
313#553—21no—
314D4×Q8—21no—
315D4×Q8C22×C2311yescharacteristic
316#548—21no—
317#547C22×C2311yescharacteristic
318#544—21no—
319#542—21no—
320#541—21no—
321#539—22no—
322#536—21no—
323C2×C4⋊Q8C22×C2311yescharacteristic
324C2×C4⋊Q8C22×C2311yescharacteristic
325#529C22×C2311yescharacteristic
326#529C22×C2311yescharacteristic
327#501—21no—
328#500—22no—
329#500C22×C2312yes—
330#500C22×C2312yes—
331#497—21no—
332#491—21no—
333#489—21no—
334#484—21no—
335#481—21no—
336#475C22×C2312yes—
337#474C22×C2312yes—
338#462C22×C2312yes—
339#461C22×C2312yes—
340#451—21no—
341#448—21no—
342#432C22×C2311yescharacteristic
343#432C22×C2311yescharacteristic
344C2×#107—22no—
345C2×C4×C8C22×C2311yescharacteristic
346#332C22×C2312yes—
347#332C22×C2312yes—
348Q8⋊C8C22×C2312yes—
349Q8⋊C8C22×C2312yes—
350D4⋊C8C22×C2312yes—
351D4⋊C8C22×C2312yes—
352C2×C23—22no—
353C2×C23—81no—
354C2×C23—81no—
355C2×C23#300811yescharacteristic
356C2×C23#301111yescharacteristic
357C2×C23C2×#45611yescharacteristic
358C2×#41—21no—
359C22×Q8—21no—
360C22×Q8—21no—
361C22×Q8—21no—
362C2×Q16—22no—
363C2×SD16—22no—
364C22×C8—21no—
365C4⋊Q8—22no—
366C4⋊Q8—21no—
367C4⋊Q8—21no—
368C4⋊Q8—21no—
369C4⋊Q8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8D4×C2312yes—
374C4⋊Q8C23×C2312yes—
375C4⋊Q8C23×C2311yescharacteristic
376C4⋊Q8C23×C2311yescharacteristic
377#126—22no—
378#126—22no—
379#126—22no—
380#126—21no—
381#126—21no—
382#125—22no—
383#125—22no—
384#125—22no—
385#125—21no—
386#125D4×C2311yescharacteristic
387#125D4×C2311yescharacteristic
388#124—22no—
389#124—21no—
390#124D4×C2311yescharacteristic
391#124D4×C2311yescharacteristic
392#123—22no—
393#123—22no—
394#123—21no—
395#123—21no—
396#123—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—21no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8D4×C2311yescharacteristic
407C22⋊Q8D4×C2311yescharacteristic
408#121—22no—
409#121—21no—
410#121D4×C2311yescharacteristic
411#121D4×C2311yescharacteristic
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8C23×C2311yescharacteristic
416C4×Q8C23×C2311yescharacteristic
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4C23×C2311yescharacteristic
423C4×D4C23×C2311yescharacteristic
424#117—21no—
425#117—21no—
426#117—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4D4×C2312yes—
432C2×C42C23×C2311yescharacteristic
433#107—24no—
434#107—24no—
435#107—22no—
436#106—22no—
437C4⋊C8C23×C2312yes—
438C4⋊C8C23×C2312yes—
439#103—22no—
440#103—22no—
441#103—22no—
442#102—22no—
443C22⋊C8—21no—
444C22⋊C8—21no—
445#97—21no—
446#97—21no—
447C4×C8—21no—
448C4×C8C23×C2312yes—
449C23#1869911yescharacteristic, direct product factor, Sylow
450#41—82no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—41no—
457C2×Q8—82no—
458C2×Q8—82no—
459C2×Q8—82no—
460C2×Q8C2×D4×C2311yescharacteristic
461C2×Q8C2×D4×C2311yescharacteristic
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4C2×D4×C2311yescharacteristic
466C2×D4C2×D4×C2311yescharacteristic
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—41no—
473C22×C4C2×D4×C2311yescharacteristic
474C22×C4C2×D4×C2311yescharacteristic
475C22×C4C2×D4×C2311yescharacteristic
476Q16—82no—
477SD16—82no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4C2×D4×C2311yescharacteristic
505C4⋊C4C2×D4×C2311yescharacteristic
506C4⋊C4C2×D4×C2311yescharacteristic
507C4⋊C4C2×D4×C2311yescharacteristic
508C4⋊C4C2×D4×C2312yes—
509C4⋊C4C2×D4×C2312yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C42—22no—
525C42—22no—
526C42—21no—
527C42—21no—
528C42C2×D4×C2311yescharacteristic
529C42C2×D4×C2311yescharacteristic
530C42C2×D4×C2311yescharacteristic
531C42C24×C2311yescharacteristic
532C23—21no—
533C23—21no—
534C23C23×#12011yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—82no—
542Q8—82no—
543Q8—82no—
544D4—22no—
545D4—22no—
546D4—42no—
547D4—42no—
548D4—82no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4C23×#12011yescharacteristic
568C2×C4C23×#12011yescharacteristic
569C2×C4C23×#12011yescharacteristic
570C2×C4C22×D4×C2311yescharacteristic
571C2×C4C22×D4×C2311yescharacteristic
572C2×C4C22×D4×C2311yescharacteristic
573C8—24no—
574C8—42no—
575C8—42no—
576C22—21no—
577C22—21no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22C23×#45612yes—
585C22C2×C23×#12011yescharacteristic
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—82no—
599C4C23×#57612yes—
600C4C23×#57812yes—
601C2—22no—
602C2—81no—
603C2—81no—
604C2C23×#300811yescharacteristic
605C2C23×#301111yescharacteristic
606C2C2×C23×#45611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3552C2
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×#3239C2semidirect
3. C23×#3218C2semidirect
4. C2×C23×#500C2semidirect
5. C23×#1650C2semidirect
6. C23×#1649C2semidirect
7. C23×#1648C2semidirect
8. C23×#1647C2semidirect
9. C23×#1556C2semidirect
10. C23×#1494C2semidirect
11. C23×#1494C2semidirect
228. #18699C23direct (N × H)
274. #3239C2×C23semidirect
275. #3218C2×C23semidirect
276. C2×#500C2×C23semidirect
277. #1650C2×C23semidirect
278. #1649C2×C23semidirect
279. #1648C2×C23semidirect
280. #1647C2×C23semidirect
281. #1556C2×C23semidirect
282. #1494C2×C23semidirect
283. #1494C2×C23semidirect
449. C23#18699direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35522
C23×#32392
C23×#32182
C2×C23×#5002
C23×#16502
C23×#16492
C23×#16482
C23×#16472
C23×#15562
C23×#14942
#1869923