Finite Group Databasebeta

← C23×#18959 #19166585

Subgroups — C23×#18959 #19166585

Order 5888.

874 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×#3554C211yesmaximal, characteristic
2C23×#3238C211yesmaximal, characteristic, complemented
3C23×#3230C211yesmaximal, characteristic, complemented
4C23×#3206C211yesmaximal, characteristic, complemented
5C23×#1637C212yesmaximal, complemented
6C23×#1636C212yesmaximal, complemented
7C23×#1631C212yesmaximal, complemented
8C23×#1630C212yesmaximal, complemented
9C23×#1553C211yesmaximal, characteristic, complemented
10C23×#1497C211yesmaximal, characteristic, complemented
11C23×#1482C211yesmaximal, characteristic, complemented
12C23×#562—22no—
13C23×#559—21no—
14C23×#559C2211yescharacteristic
15C23×#557C2211yescharacteristic
16C23×#554—21no—
17C23×#550C2211yescharacteristic
18D4×Q8×C23C2211yescharacteristic
19C23×#548—21no—
20C23×#547—21no—
21C23×#542—21no—
22C23×#542—21no—
23C23×#540—21no—
24C23×#539—22no—
25C23×#536—21no—
26C23×#533C2211yescharacteristic
27C23×#532C2211yescharacteristic
28C2×C23×C4⋊Q8C2211yescharacteristic
29C23×#529C2211yescharacteristic
30C23×#501—21no—
31C23×#500—21no—
32C23×#492C2212yes—
33C23×#489—21no—
34C23×#487—21no—
35C23×#487C2212yes—
36C23×#484—21no—
37C23×#484—21no—
38C23×#483—21no—
39C23×#480—21no—
40C23×#477C2212yes—
41C23×#477C2212yes—
42C23×#477C2212yes—
43C23×#476C2212yes—
44C23×#461—22no—
45C23×#448—22no—
46C23×#432C2211yescharacteristic
47C23×#431C2211yescharacteristic
48C23×#431C2211yescharacteristic
49C23×#332C2212yes—
50C23×#331C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×Q8⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C2×#41×C23—21no—
56C2×#41×C23—21no—
57C22×Q8×C23—21no—
58C22×Q8×C23—21no—
59C2×SD16×C23—24no—
60C23×C4⋊Q8—21no—
61C23×C4⋊Q8—21no—
62C23×C4⋊Q8—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8D411yescharacteristic
66C23×C4⋊Q8D411yescharacteristic
67C23×C4⋊Q8C2312yes—
68C23×C4⋊Q8C2311yescharacteristic
69C23×#127—21no—
70C23×#127C2311yescharacteristic
71C23×#126—22no—
72C23×#126—21no—
73C23×#125—21no—
74C23×#125—22no—
75C23×#125—21no—
76C23×#125D411yescharacteristic
77C23×#124—22no—
78C23×#124—22no—
79C23×#124—21no—
80C23×#124—21no—
81C23×#124—21no—
82C23×#124D411yescharacteristic
83C23×#123—22no—
84C23×#123—22no—
85C23×#123—22no—
86C23×#123—22no—
87C23×#123—22no—
88C23×#123—22no—
89C23×#123—21no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—21no—
95C23×C22⋊Q8—21no—
96C23×C22⋊Q8—22no—
97C23×C22⋊Q8D411yescharacteristic
98C23×C22⋊Q8D411yescharacteristic
99C23×C22⋊Q8D411yescharacteristic
100C23×#121—22no—
101C23×#121—22no—
102C23×#121—22no—
103C23×#121—21no—
104C23×#121—21no—
105C23×#121D412yes—
106C23×#121D411yescharacteristic
107C4×Q8×C23—21no—
108C4×Q8×C23—21no—
109C4×Q8×C23—21no—
110C4×Q8×C23C2311yescharacteristic
111C4×Q8×C23C2311yescharacteristic
112C4×Q8×C23C2311yescharacteristic
113C4×D4×C23—22no—
114C4×D4×C23—21no—
115C4×D4×C23—21no—
116C4×D4×C23—21no—
117C4×D4×C23—21no—
118C4×D4×C23—21no—
119C4×D4×C23—21no—
120C4×D4×C23D412yes—
121C4×D4×C23C2311yescharacteristic
122C23×#117—21no—
123C23×#117—21no—
124C2×C4⋊C4×C23—22no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23—21no—
127C2×C42×C23C2311yescharacteristic
128C23×#107—22no—
129C23×#107—22no—
130C23×#107—22no—
131C23×#106—22no—
132C23×C4⋊C8—21no—
133C23×C4⋊C8C2312yes—
134C23×C4⋊C8C2312yes—
135C23×#103—22no—
136C23×#103—22no—
137C23×#103—22no—
138C23×#103—24no—
139C23×#102—22no—
140C23×C22⋊C8—21no—
141C23×C22⋊C8—21no—
142C23×C22⋊C8—21no—
143C23×#97—21no—
144C4×C8×C23—21no—
145C4×C8×C23C2312yes—
146#41×C23—82no—
147#41×C23—82no—
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×C23C2×D411yescharacteristic
156C2×Q8×C23C2×D411yescharacteristic
157C2×Q8×C23C2×D411yescharacteristic
158C2×D4×C23—22no—
159C2×D4×C23—22no—
160C2×D4×C23—21no—
161C2×D4×C23—22no—
162C2×D4×C23—21no—
163C2×D4×C23—21no—
164C2×D4×C23C2×D411yescharacteristic
165C2×D4×C23C2×D412yes—
166C22×C4×C23—22no—
167C22×C4×C23—22no—
168C22×C4×C23—22no—
169C22×C4×C23—21no—
170C22×C4×C23—22no—
171C22×C4×C23—41no—
172C22×C4×C23C2×D411yescharacteristic
173C22×C4×C23C2×D411yescharacteristic
174C22×C4×C23C2×D411yescharacteristic
175SD16×C23—84no—
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—21no—
184C4⋊C4×C23—21no—
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—22no—
196C4⋊C4×C23—22no—
197C4⋊C4×C23—41no—
198C4⋊C4×C23—41no—
199C4⋊C4×C23C2×D412yes—
200C4⋊C4×C23C2×D411yescharacteristic
201C4⋊C4×C23C2×D411yescharacteristic
202C4⋊C4×C23C2×D411yescharacteristic
203C4⋊C4×C23C2×D411yescharacteristic
204C22⋊C4×C23—22no—
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—22no—
216C22⋊C4×C23—22no—
217C22⋊C4×C23—21no—
218C42×C23—22no—
219C42×C23—21no—
220C42×C23—21no—
221C42×C23—21no—
222C42×C23C2×D411yescharacteristic
223C42×C23C2×D411yescharacteristic
224C42×C23C2×D411yescharacteristic
225C42×C23C2411yescharacteristic
226#18959C2311yesmaximal, characteristic, direct product factor, Sylow
227C23×C23—21no—
228C23×C23—22no—
229C23×C23#12011yescharacteristic
230Q8×C23—22no—
231Q8×C23—22no—
232Q8×C23—22no—
233Q8×C23—44no—
234Q8×C23—42no—
235Q8×C23—81no—
236Q8×C23—82no—
237Q8×C23—82no—
238D4×C23—22no—
239D4×C23—42no—
240D4×C23—42no—
241D4×C23—42no—
242D4×C23—81no—
243D4×C23—82no—
244D4×C23—82no—
245D4×C23—82no—
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—21no—
253C2×C4×C23—22no—
254C2×C4×C23—21no—
255C2×C4×C23—21no—
256C2×C4×C23—41no—
257C2×C4×C23—42no—
258C2×C4×C23—42no—
259C2×C4×C23—82no—
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—42no—
271C8×C23—42no—
272C8×C23—42no—
273#3554C2×C2311yescharacteristic
274#3238C2×C2311yescharacteristic, complemented
275#3230C2×C2311yescharacteristic, complemented
276#3206C2×C2311yescharacteristic, complemented
277#1637C2×C2312yescomplemented
278#1636C2×C2312yescomplemented
279#1631C2×C2312yescomplemented
280#1630C2×C2312yescomplemented
281#1553C2×C2311yescharacteristic, complemented
282#1497C2×C2311yescharacteristic, complemented
283#1482C2×C2311yescharacteristic, complemented
284C22×C23—21no—
285C22×C23—21no—
286C22×C23—21no—
287C22×C23—41no—
288C22×C23—42no—
289C22×C23—41no—
290C22×C23—42no—
291C22×C23—42no—
292C22×C23—41no—
293C22×C23C2×#12011yescharacteristic
294C4×C23—21no—
295C4×C23—21no—
296C4×C23—21no—
297C4×C23—21no—
298C4×C23—41no—
299C4×C23—41no—
300C4×C23—42no—
301C4×C23—41no—
302C4×C23—41no—
303C4×C23—41no—
304C4×C23—41no—
305C4×C23—41no—
306C4×C23—82no—
307C4×C23#57712yes—
308C4×C23#57812yes—
309#562—22no—
310#559—21no—
311#559C22×C2311yescharacteristic
312#557C22×C2311yescharacteristic
313#554—21no—
314#550C22×C2311yescharacteristic
315D4×Q8C22×C2311yescharacteristic
316#548—21no—
317#547—21no—
318#542—21no—
319#542—21no—
320#540—21no—
321#539—22no—
322#536—21no—
323#533C22×C2311yescharacteristic
324#532C22×C2311yescharacteristic
325C2×C4⋊Q8C22×C2311yescharacteristic
326#529C22×C2311yescharacteristic
327#501—21no—
328#500—21no—
329#492C22×C2312yes—
330#489—21no—
331#487—21no—
332#487C22×C2312yes—
333#484—21no—
334#484—21no—
335#483—21no—
336#480—21no—
337#477C22×C2312yes—
338#477C22×C2312yes—
339#477C22×C2312yes—
340#476C22×C2312yes—
341#461—22no—
342#448—22no—
343#432C22×C2311yescharacteristic
344#431C22×C2311yescharacteristic
345#431C22×C2311yescharacteristic
346#332C22×C2312yes—
347#331C22×C2312yes—
348Q8⋊C8C22×C2312yes—
349Q8⋊C8C22×C2312yes—
350Q8⋊C8C22×C2312yes—
351D4⋊C8C22×C2312yes—
352C2×C23—41no—
353C2×C23—81no—
354C2×C23—82no—
355C2×C23#300911yescharacteristic
356C2×C23#301111yescharacteristic
357C2×C23#302111yescharacteristic
358C2×#41—21no—
359C2×#41—21no—
360C22×Q8—21no—
361C22×Q8—21no—
362C2×SD16—24no—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8—21no—
368C4⋊Q8D4×C2311yescharacteristic
369C4⋊Q8D4×C2311yescharacteristic
370C4⋊Q8C23×C2312yes—
371C4⋊Q8C23×C2311yescharacteristic
372#127—21no—
373#127C23×C2311yescharacteristic
374#126—22no—
375#126—21no—
376#125—22no—
377#125—21no—
378#125—21no—
379#125D4×C2311yescharacteristic
380#124—22no—
381#124—22no—
382#124—21no—
383#124—21no—
384#124—21no—
385#124D4×C2311yescharacteristic
386#123—22no—
387#123—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—22no—
399C22⋊Q8—21no—
400C22⋊Q8D4×C2311yescharacteristic
401C22⋊Q8D4×C2311yescharacteristic
402C22⋊Q8D4×C2311yescharacteristic
403#121—22no—
404#121—22no—
405#121—22no—
406#121—21no—
407#121—21no—
408#121D4×C2311yescharacteristic
409#121D4×C2312yes—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8C23×C2311yescharacteristic
414C4×Q8C23×C2311yescharacteristic
415C4×Q8C23×C2311yescharacteristic
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4D4×C2312yes—
424C4×D4C23×C2311yescharacteristic
425#117—21no—
426#117—21no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C42C23×C2311yescharacteristic
431#107—22no—
432#107—22no—
433#107—22no—
434#106—22no—
435C4⋊C8—21no—
436C4⋊C8C23×C2312yes—
437C4⋊C8C23×C2312yes—
438#103—22no—
439#103—22no—
440#103—22no—
441#103—24no—
442#102—22no—
443C22⋊C8—21no—
444C22⋊C8—21no—
445C22⋊C8—21no—
446#97—21no—
447C4×C8—21no—
448C4×C8C23×C2312yes—
449C23#1895911yescharacteristic, direct product factor, Sylow
450#41—82no—
451#41—82no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—41no—
457C2×Q8—82no—
458C2×Q8—82no—
459C2×Q8C2×D4×C2311yescharacteristic
460C2×Q8C2×D4×C2311yescharacteristic
461C2×Q8C2×D4×C2311yescharacteristic
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×D4×C2311yescharacteristic
469C2×D4C2×D4×C2312yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4C2×D4×C2311yescharacteristic
477C22×C4C2×D4×C2311yescharacteristic
478C22×C4C2×D4×C2311yescharacteristic
479SD16—84no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
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—22no—
500C4⋊C4—22no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4C2×D4×C2312yes—
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—22no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42C2×D4×C2311yescharacteristic
527C42C2×D4×C2311yescharacteristic
528C42C2×D4×C2311yescharacteristic
529C42C24×C2311yescharacteristic
530C23—21no—
531C23—22no—
532C23C23×#12011yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—44no—
537Q8—42no—
538Q8—81no—
539Q8—82no—
540Q8—82no—
541D4—22no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—82no—
546D4—81no—
547D4—82no—
548D4—82no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—82no—
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—42no—
574C8—42no—
575C8—42no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—41no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22C2×C23×#12011yescharacteristic
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—82no—
599C4C23×#57712yes—
600C4C23×#57812yes—
601C2—41no—
602C2—81no—
603C2—82no—
604C2C23×#300911yescharacteristic
605C2C23×#301111yescharacteristic
606C2C23×#302111yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3554C2
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×#3238C2semidirect
3. C23×#3230C2semidirect
4. C23×#3206C2semidirect
5. C23×#1637C2semidirect
6. C23×#1636C2semidirect
7. C23×#1631C2semidirect
8. C23×#1630C2semidirect
9. C23×#1553C2semidirect
10. C23×#1497C2semidirect
11. C23×#1482C2semidirect
226. #18959C23direct (N × H)
274. #3238C2×C23semidirect
275. #3230C2×C23semidirect
276. #3206C2×C23semidirect
277. #1637C2×C23semidirect
278. #1636C2×C23semidirect
279. #1631C2×C23semidirect
280. #1630C2×C23semidirect
281. #1553C2×C23semidirect
282. #1497C2×C23semidirect
283. #1482C2×C23semidirect
449. C23#18959direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35542
C23×#32382
C23×#32302
C23×#32062
C23×#16372
C23×#16362
C23×#16312
C23×#16302
C23×#15532
C23×#14972
C23×#14822
#1895923