Finite Group Databasebeta

← C23×#18690 #19166316

Subgroups — C23×#18690 #19166316

Order 5888.

878 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3544C211yesmaximal, characteristic
2C23×#3239C211yesmaximal, characteristic, complemented
3C23×#3218C211yesmaximal, characteristic, complemented
4C2×C23×#487C211yesmaximal, characteristic, complemented
5C23×#1646C212yesmaximal, complemented
6C23×#1645C212yesmaximal, complemented
7C23×#1644C212yesmaximal, complemented
8C23×#1643C212yesmaximal, complemented
9C23×#1558C211yesmaximal, characteristic, complemented
10C23×#1494C211yesmaximal, characteristic, complemented
11C23×#1494C211yesmaximal, characteristic, complemented
12C23×#563—22no—
13Q82×C23C2211yescharacteristic
14C23×#555C2211yescharacteristic
15C23×#554C2211yescharacteristic
16C23×#554C2211yescharacteristic
17C23×#552—21no—
18C23×#552—21no—
19C23×#547—21no—
20C23×#547—21no—
21C23×#544—21no—
22C23×#544—21no—
23C23×#542—21no—
24C23×#542—21no—
25C23×#536—22no—
26C2×C23×C4⋊Q8C2211yescharacteristic
27C2×C23×C4⋊Q8C2211yescharacteristic
28C23×#529C2211yescharacteristic
29C23×#529C2211yescharacteristic
30C23×#501—21no—
31C23×#497—21no—
32C23×#491—21no—
33C23×#489—21no—
34C23×#487—22no—
35C23×#487C2212yes—
36C23×#487C2212yes—
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×#103—22no—
48C2×C4×C8×C23C2211yescharacteristic
49C23×#333C2212yes—
50C23×#333C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×D4⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C2×#41×C23—22no—
56C22×Q8×C23—21no—
57C22×Q8×C23D412yes—
58C2×Q16×C23—22no—
59C2×SD16×C23—22no—
60C22×C8×C23—21no—
61C23×C4⋊Q8—22no—
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—21no—
77C23×#126—21no—
78C23×#125—22no—
79C23×#125—22no—
80C23×#125—21no—
81C23×#125—21no—
82C23×#125D411yescharacteristic
83C23×#125D411yescharacteristic
84C23×#124—22no—
85C23×#124—22no—
86C23×#124D411yescharacteristic
87C23×#124D411yescharacteristic
88C23×#123—22no—
89C23×#123—22no—
90C23×#123—22no—
91C23×#123—22no—
92C23×#123—21no—
93C23×#123—21no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8—22no—
96C23×C22⋊Q8—22no—
97C23×C22⋊Q8—22no—
98C23×C22⋊Q8—21no—
99C23×C22⋊Q8—21no—
100C23×C22⋊Q8—22no—
101C23×C22⋊Q8—22no—
102C23×C22⋊Q8D411yescharacteristic
103C23×C22⋊Q8D411yescharacteristic
104C23×#121—21no—
105C23×#121—21no—
106C23×#121D411yescharacteristic
107C23×#121D411yescharacteristic
108C4×Q8×C23—21no—
109C4×Q8×C23—21no—
110C4×Q8×C23C2311yescharacteristic
111C4×Q8×C23C2311yescharacteristic
112C4×D4×C23—21no—
113C4×D4×C23—21no—
114C4×D4×C23—21no—
115C4×D4×C23—21no—
116C4×D4×C23C2311yescharacteristic
117C4×D4×C23C2311yescharacteristic
118C23×#117—22no—
119C23×#117—21no—
120C23×#117—21no—
121C23×#117—21no—
122C23×#117—21no—
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×C4⋊C4×C23—21no—
129C2×C42×C23C2311yescharacteristic
130C23×#107—22no—
131C23×#106—22no—
132C23×C4⋊C8C2312yes—
133C23×C4⋊C8C2312yes—
134C23×#103—24no—
135C23×#103—24no—
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—84no—
147C2×Q8×C23—22no—
148C2×Q8×C23—22no—
149C2×Q8×C23—22no—
150C2×Q8×C23—44no—
151C2×Q8×C23—41no—
152C2×Q8×C23—82no—
153C2×Q8×C23C2×D411yescharacteristic
154C2×Q8×C23C2×D411yescharacteristic
155C2×Q8×C23C2×D412yes—
156C2×Q8×C23C2×D412yes—
157C2×D4×C23—21no—
158C2×D4×C23—21no—
159C2×D4×C23—22no—
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—41no—
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2×D411yescharacteristic
171Q16×C23—82no—
172SD16×C23—82no—
173C2×C8×C23—22no—
174C2×C8×C23—22no—
175C2×C8×C23—22no—
176C2×C8×C23—42no—
177C4⋊C4×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—21no—
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—21no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—22no—
199C4⋊C4×C23—41no—
200C4⋊C4×C23—41no—
201C4⋊C4×C23C2×D411yescharacteristic
202C4⋊C4×C23C2×D411yescharacteristic
203C4⋊C4×C23C2×D411yescharacteristic
204C4⋊C4×C23C2×D411yescharacteristic
205C22⋊C4×C23—22no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—21no—
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—
219C42×C23—22no—
220C42×C23—22no—
221C42×C23—21no—
222C42×C23—21no—
223C42×C23C2×D411yescharacteristic
224C42×C23C2×D411yescharacteristic
225C42×C23C2×D411yescharacteristic
226C42×C23C2411yescharacteristic
227#18690C2311yesmaximal, characteristic, direct product factor, Sylow
228C23×C23—21no—
229C23×C23—21no—
230C23×C23#12011yescharacteristic
231Q8×C23—22no—
232Q8×C23—22no—
233Q8×C23—42no—
234Q8×C23—42no—
235Q8×C23—42no—
236Q8×C23—42no—
237Q8×C23—82no—
238Q8×C23—82no—
239Q8×C23—82no—
240D4×C23—22no—
241D4×C23—22no—
242D4×C23—42no—
243D4×C23—42no—
244D4×C23—82no—
245C2×C4×C23—22no—
246C2×C4×C23—22no—
247C2×C4×C23—22no—
248C2×C4×C23—21no—
249C2×C4×C23—21no—
250C2×C4×C23—21no—
251C2×C4×C23—21no—
252C2×C4×C23—22no—
253C2×C4×C23—21no—
254C2×C4×C23—21no—
255C2×C4×C23—22no—
256C2×C4×C23—44no—
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#12011yescharacteristic
264C2×C4×C23#12011yescharacteristic
265C2×C4×C23#12011yescharacteristic
266C2×C4×C23C22×D411yescharacteristic
267C2×C4×C23C22×D411yescharacteristic
268C2×C4×C23C22×D411yescharacteristic
269C8×C23—42no—
270C8×C23—42no—
271C8×C23—42no—
272#3544C2×C2311yescharacteristic
273#3239C2×C2311yescharacteristic, complemented
274#3218C2×C2311yescharacteristic, complemented
275C2×#487C2×C2311yescharacteristic, complemented
276#1646C2×C2312yescomplemented
277#1645C2×C2312yescomplemented
278#1644C2×C2312yescomplemented
279#1643C2×C2312yescomplemented
280#1558C2×C2311yescharacteristic, complemented
281#1494C2×C2311yescharacteristic, complemented
282#1494C2×C2311yescharacteristic, complemented
283C22×C23—21no—
284C22×C23—21no—
285C22×C23—41no—
286C22×C23—41no—
287C22×C23—41no—
288C22×C23—41no—
289C22×C23—41no—
290C22×C23—41no—
291C22×C23#45512yes—
292C22×C23C2×#12011yescharacteristic
293C4×C23—21no—
294C4×C23—21no—
295C4×C23—21no—
296C4×C23—21no—
297C4×C23—41no—
298C4×C23—41no—
299C4×C23—41no—
300C4×C23—41no—
301C4×C23—42no—
302C4×C23—42no—
303C4×C23—41no—
304C4×C23—41no—
305C4×C23—82no—
306C4×C23#57712yes—
307C4×C23#57812yes—
308#563—22no—
309Q82C22×C2311yescharacteristic
310#555C22×C2311yescharacteristic
311#554C22×C2311yescharacteristic
312#554C22×C2311yescharacteristic
313#552—21no—
314#552—21no—
315#547—21no—
316#547—21no—
317#544—21no—
318#544—21no—
319#542—21no—
320#542—21no—
321#536—22no—
322C2×C4⋊Q8C22×C2311yescharacteristic
323C2×C4⋊Q8C22×C2311yescharacteristic
324#529C22×C2311yescharacteristic
325#529C22×C2311yescharacteristic
326#501—21no—
327#497—21no—
328#491—21no—
329#489—21no—
330#487—22no—
331#487C22×C2312yes—
332#487C22×C2312yes—
333#484—21no—
334#481—21no—
335#475C22×C2312yes—
336#474C22×C2312yes—
337#462C22×C2312yes—
338#461C22×C2312yes—
339#451—21no—
340#448—21no—
341#432C22×C2311yescharacteristic
342#432C22×C2311yescharacteristic
343C2×#103—22no—
344C2×C4×C8C22×C2311yescharacteristic
345#333C22×C2312yes—
346#333C22×C2312yes—
347Q8⋊C8C22×C2312yes—
348Q8⋊C8C22×C2312yes—
349D4⋊C8C22×C2312yes—
350D4⋊C8C22×C2312yes—
351C2×C23—22no—
352C2×C23—81no—
353C2×C23—81no—
354C2×C23#301011yescharacteristic
355C2×C23#301111yescharacteristic
356C2×C23C2×#45511yescharacteristic
357C2×#41—22no—
358C22×Q8—21no—
359C22×Q8D4×C2312yes—
360C2×Q16—22no—
361C2×SD16—22no—
362C22×C8—21no—
363C4⋊Q8—22no—
364C4⋊Q8—22no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8—21no—
368C4⋊Q8—21no—
369C4⋊Q8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8D4×C2312yes—
373C4⋊Q8C23×C2312yes—
374C4⋊Q8C23×C2311yescharacteristic
375C4⋊Q8C23×C2311yescharacteristic
376#126—22no—
377#126—22no—
378#126—21no—
379#126—21no—
380#125—22no—
381#125—22no—
382#125—21no—
383#125—21no—
384#125D4×C2311yescharacteristic
385#125D4×C2311yescharacteristic
386#124—22no—
387#124—22no—
388#124D4×C2311yescharacteristic
389#124D4×C2311yescharacteristic
390#123—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394#123—21no—
395#123—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8D4×C2311yescharacteristic
405C22⋊Q8D4×C2311yescharacteristic
406#121—21no—
407#121—21no—
408#121D4×C2311yescharacteristic
409#121D4×C2311yescharacteristic
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8C23×C2311yescharacteristic
413C4×Q8C23×C2311yescharacteristic
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4C23×C2311yescharacteristic
419C4×D4C23×C2311yescharacteristic
420#117—22no—
421#117—21no—
422#117—21no—
423#117—21no—
424#117—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C42C23×C2311yescharacteristic
432#107—22no—
433#106—22no—
434C4⋊C8C23×C2312yes—
435C4⋊C8C23×C2312yes—
436#103—24no—
437#103—24no—
438#103—22no—
439#103—22no—
440#103—22no—
441#102—22no—
442C22⋊C8—21no—
443C22⋊C8—21no—
444#97—21no—
445#97—21no—
446C4×C8—21no—
447C4×C8C23×C2312yes—
448C23#1869011yescharacteristic, direct product factor, Sylow
449#41—84no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—44no—
454C2×Q8—41no—
455C2×Q8—82no—
456C2×Q8C2×D4×C2311yescharacteristic
457C2×Q8C2×D4×C2311yescharacteristic
458C2×Q8C2×D4×C2312yes—
459C2×Q8C2×D4×C2312yes—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4C2×D4×C2311yescharacteristic
464C2×D4C2×D4×C2311yescharacteristic
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—41no—
471C22×C4C2×D4×C2311yescharacteristic
472C22×C4C2×D4×C2311yescharacteristic
473C22×C4C2×D4×C2311yescharacteristic
474Q16—82no—
475SD16—82no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—42no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
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
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—21no—
526C42C2×D4×C2311yescharacteristic
527C42C2×D4×C2311yescharacteristic
528C42C2×D4×C2311yescharacteristic
529C42C24×C2311yescharacteristic
530C23—21no—
531C23—21no—
532C23C23×#12011yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—82no—
540Q8—82no—
541Q8—82no—
542D4—22no—
543D4—22no—
544D4—42no—
545D4—42no—
546D4—82no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4C23×#12011yescharacteristic
566C2×C4C23×#12011yescharacteristic
567C2×C4C23×#12011yescharacteristic
568C2×C4C22×D4×C2311yescharacteristic
569C2×C4C22×D4×C2311yescharacteristic
570C2×C4C22×D4×C2311yescharacteristic
571C8—42no—
572C8—42no—
573C8—42no—
574C22—21no—
575C22—21no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22C23×#45512yes—
583C22C2×C23×#12011yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—82no—
597C4C23×#57712yes—
598C4C23×#57812yes—
599C2—22no—
600C2—81no—
601C2—81no—
602C2C23×#301011yescharacteristic
603C2C23×#301111yescharacteristic
604C2C2×C23×#45511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3544C2
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×#487C2semidirect
5. C23×#1646C2semidirect
6. C23×#1645C2semidirect
7. C23×#1644C2semidirect
8. C23×#1643C2semidirect
9. C23×#1558C2semidirect
10. C23×#1494C2semidirect
11. C23×#1494C2semidirect
227. #18690C23direct (N × H)
273. #3239C2×C23semidirect
274. #3218C2×C23semidirect
275. C2×#487C2×C23semidirect
276. #1646C2×C23semidirect
277. #1645C2×C23semidirect
278. #1644C2×C23semidirect
279. #1643C2×C23semidirect
280. #1558C2×C23semidirect
281. #1494C2×C23semidirect
282. #1494C2×C23semidirect
448. C23#18690direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35442
C23×#32392
C23×#32182
C2×C23×#4872
C23×#16462
C23×#16452
C23×#16442
C23×#16432
C23×#15582
C23×#14942
#1869023