Finite Group Databasebeta

← C23×#18695 #19166321

Subgroups — C23×#18695 #19166321

Order 5888.

866 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3552C211yesmaximal, characteristic
2C23×#3238C211yesmaximal, characteristic, complemented
3C23×#3238C211yesmaximal, characteristic, complemented
4C2×C23×#487C211yesmaximal, characteristic, complemented
5C23×#1638C212yesmaximal, complemented
6C23×#1637C212yesmaximal, complemented
7C23×#1636C212yesmaximal, complemented
8C23×#1635C212yesmaximal, 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×#501—21no—
32C23×#489—21no—
33C23×#489—21no—
34C23×#487—22no—
35C23×#487C2212yes—
36C23×#487C2212yes—
37C23×#483—21no—
38C23×#483—21no—
39C23×#480—21no—
40C23×#480—21no—
41C23×#477C2212yes—
42C23×#477C2212yes—
43C23×#476C2212yes—
44C23×#476C2212yes—
45C23×#432C2211yescharacteristic
46C23×#432C2211yescharacteristic
47C2×C23×#103—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—
59C22×C8×C23—21no—
60C23×C4⋊Q8—22no—
61C23×C4⋊Q8—21no—
62C23×C4⋊Q8—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8—21no—
66C23×C4⋊Q8—21no—
67C23×C4⋊Q8—21no—
68C23×C4⋊Q8D412yes—
69C23×C4⋊Q8C2312yes—
70C23×C4⋊Q8C2311yescharacteristic
71C23×C4⋊Q8C2311yescharacteristic
72C23×#126—22no—
73C23×#126—22no—
74C23×#126—22no—
75C23×#126—21no—
76C23×#126—21no—
77C23×#125—22no—
78C23×#125—22no—
79C23×#125—21no—
80C23×#125—22no—
81C23×#125D411yescharacteristic
82C23×#125D411yescharacteristic
83C23×#124—22no—
84C23×#124—21no—
85C23×#124D411yescharacteristic
86C23×#124D411yescharacteristic
87C23×#123—22no—
88C23×#123—22no—
89C23×#123—21no—
90C23×#123—21no—
91C23×#123—22no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8—22no—
96C23×C22⋊Q8—22no—
97C23×C22⋊Q8—21no—
98C23×C22⋊Q8—21no—
99C23×C22⋊Q8—21no—
100C23×C22⋊Q8—22no—
101C23×C22⋊Q8D411yescharacteristic
102C23×C22⋊Q8D411yescharacteristic
103C23×#121—22no—
104C23×#121—21no—
105C23×#121D411yescharacteristic
106C23×#121D411yescharacteristic
107C4×Q8×C23—21no—
108C4×Q8×C23—21no—
109C4×Q8×C23—21no—
110C4×Q8×C23C2311yescharacteristic
111C4×Q8×C23C2311yescharacteristic
112C4×D4×C23—22no—
113C4×D4×C23—22no—
114C4×D4×C23—21no—
115C4×D4×C23—21no—
116C4×D4×C23—21no—
117C4×D4×C23C2311yescharacteristic
118C4×D4×C23C2311yescharacteristic
119C23×#117—21no—
120C23×#117—21no—
121C23×#117—21no—
122C2×C4⋊C4×C23—21no—
123C2×C4⋊C4×C23—21no—
124C2×C4⋊C4×C23—21no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23D412yes—
127C2×C42×C23C2311yescharacteristic
128C23×#107—22no—
129C23×#107—22no—
130C23×#106—22no—
131C23×#106—22no—
132C23×C4⋊C8C2312yes—
133C23×C4⋊C8C2312yes—
134C23×#103—22no—
135C23×#103—22no—
136C23×#103—24no—
137C23×#103—24no—
138C23×#102—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—44no—
153C2×Q8×C23—41no—
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
172C2×C8×C23—22no—
173C2×C8×C23—22no—
174C2×C8×C23—22no—
175C2×C8×C23—42no—
176C4⋊C4×C23—22no—
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—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—22no—
195C4⋊C4×C23—22no—
196C4⋊C4×C23—41no—
197C4⋊C4×C23—41no—
198C4⋊C4×C23C2×D412yes—
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—21no—
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—
218C42×C23—22no—
219C42×C23—22no—
220C42×C23—21no—
221C42×C23—21no—
222C42×C23C2×D411yescharacteristic
223C42×C23C2×D411yescharacteristic
224C42×C23C2×D411yescharacteristic
225C42×C23C2411yescharacteristic
226#18695C2311yesmaximal, characteristic, direct product factor, Sylow
227C23×C23—21no—
228C23×C23—21no—
229C23×C23#12011yescharacteristic
230Q8×C23—22no—
231Q8×C23—22no—
232Q8×C23—44no—
233Q8×C23—42no—
234Q8×C23—42no—
235Q8×C23—82no—
236Q8×C23—82no—
237Q8×C23—82no—
238D4×C23—22no—
239D4×C23—22no—
240D4×C23—42no—
241D4×C23—42no—
242D4×C23—82no—
243C2×C4×C23—22no—
244C2×C4×C23—22no—
245C2×C4×C23—22no—
246C2×C4×C23—21no—
247C2×C4×C23—21no—
248C2×C4×C23—21no—
249C2×C4×C23—21no—
250C2×C4×C23—21no—
251C2×C4×C23—22no—
252C2×C4×C23—22no—
253C2×C4×C23—21no—
254C2×C4×C23—42no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
257C2×C4×C23—82no—
258C2×C4×C23—82no—
259C2×C4×C23—82no—
260C2×C4×C23—82no—
261C2×C4×C23#12011yescharacteristic
262C2×C4×C23#12011yescharacteristic
263C2×C4×C23#12011yescharacteristic
264C2×C4×C23C22×D411yescharacteristic
265C2×C4×C23C22×D411yescharacteristic
266C2×C4×C23C22×D411yescharacteristic
267C8×C23—42no—
268C8×C23—42no—
269C8×C23—42no—
270#3552C2×C2311yescharacteristic
271#3238C2×C2311yescharacteristic, complemented
272#3238C2×C2311yescharacteristic, complemented
273C2×#487C2×C2311yescharacteristic, complemented
274#1638C2×C2312yescomplemented
275#1637C2×C2312yescomplemented
276#1636C2×C2312yescomplemented
277#1635C2×C2312yescomplemented
278#1556C2×C2311yescharacteristic, complemented
279#1494C2×C2311yescharacteristic, complemented
280#1494C2×C2311yescharacteristic, complemented
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—41no—
284C22×C23—41no—
285C22×C23—41no—
286C22×C23—41no—
287C22×C23—41no—
288C22×C23—41no—
289C22×C23#45412yes—
290C22×C23C2×#12011yescharacteristic
291C4×C23—21no—
292C4×C23—21no—
293C4×C23—21no—
294C4×C23—21no—
295C4×C23—41no—
296C4×C23—42no—
297C4×C23—41no—
298C4×C23—42no—
299C4×C23—41no—
300C4×C23—41no—
301C4×C23—41no—
302C4×C23—41no—
303C4×C23—82no—
304C4×C23#57712yes—
305C4×C23#57812yes—
306#563—22no—
307#557C22×C2311yescharacteristic
308#556C22×C2311yescharacteristic
309#554—21no—
310#553—21no—
311D4×Q8—21no—
312D4×Q8C22×C2311yescharacteristic
313#548—21no—
314#547C22×C2311yescharacteristic
315#544—21no—
316#542—21no—
317#541—21no—
318#539—22no—
319#536—21no—
320C2×C4⋊Q8C22×C2311yescharacteristic
321C2×C4⋊Q8C22×C2311yescharacteristic
322#529C22×C2311yescharacteristic
323#529C22×C2311yescharacteristic
324#501—21no—
325#501—21no—
326#489—21no—
327#489—21no—
328#487—22no—
329#487C22×C2312yes—
330#487C22×C2312yes—
331#483—21no—
332#483—21no—
333#480—21no—
334#480—21no—
335#477C22×C2312yes—
336#477C22×C2312yes—
337#476C22×C2312yes—
338#476C22×C2312yes—
339#432C22×C2311yescharacteristic
340#432C22×C2311yescharacteristic
341C2×#103—22no—
342C2×C4×C8C22×C2311yescharacteristic
343#332C22×C2312yes—
344#332C22×C2312yes—
345Q8⋊C8C22×C2312yes—
346Q8⋊C8C22×C2312yes—
347D4⋊C8C22×C2312yes—
348D4⋊C8C22×C2312yes—
349C2×C23—22no—
350C2×C23—81no—
351C2×C23—81no—
352C2×C23#300911yescharacteristic
353C2×C23#301111yescharacteristic
354C2×C23C2×#45411yescharacteristic
355C2×#41—21no—
356C22×Q8—21no—
357C22×Q8—21no—
358C22×Q8—21no—
359C22×C8—21no—
360C4⋊Q8—22no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8—21no—
368C4⋊Q8D4×C2312yes—
369C4⋊Q8C23×C2312yes—
370C4⋊Q8C23×C2311yescharacteristic
371C4⋊Q8C23×C2311yescharacteristic
372#126—22no—
373#126—22no—
374#126—22no—
375#126—21no—
376#126—21no—
377#125—22no—
378#125—22no—
379#125—22no—
380#125—21no—
381#125D4×C2311yescharacteristic
382#125D4×C2311yescharacteristic
383#124—22no—
384#124—21no—
385#124D4×C2311yescharacteristic
386#124D4×C2311yescharacteristic
387#123—22no—
388#123—22no—
389#123—21no—
390#123—21no—
391#123—22no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—22no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8D4×C2311yescharacteristic
402C22⋊Q8D4×C2311yescharacteristic
403#121—22no—
404#121—21no—
405#121D4×C2311yescharacteristic
406#121D4×C2311yescharacteristic
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8C23×C2311yescharacteristic
411C4×Q8C23×C2311yescharacteristic
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4C23×C2311yescharacteristic
418C4×D4C23×C2311yescharacteristic
419#117—21no—
420#117—21no—
421#117—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4D4×C2312yes—
427C2×C42C23×C2311yescharacteristic
428#107—22no—
429#107—22no—
430#106—22no—
431#106—22no—
432C4⋊C8C23×C2312yes—
433C4⋊C8C23×C2312yes—
434#103—22no—
435#103—22no—
436#103—24no—
437#103—24no—
438#102—22no—
439#102—22no—
440C22⋊C8—21no—
441C22⋊C8—21no—
442#97—21no—
443#97—21no—
444C4×C8—21no—
445C4×C8C23×C2312yes—
446C23#1869511yescharacteristic, direct product factor, Sylow
447#41—82no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—44no—
454C2×Q8—41no—
455C2×Q8—82no—
456C2×Q8—82no—
457C2×Q8C2×D4×C2311yescharacteristic
458C2×Q8C2×D4×C2311yescharacteristic
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4C2×D4×C2311yescharacteristic
463C2×D4C2×D4×C2311yescharacteristic
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470C22×C4C2×D4×C2311yescharacteristic
471C22×C4C2×D4×C2311yescharacteristic
472C22×C4C2×D4×C2311yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—42no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
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—22no—
496C4⋊C4—22no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4C2×D4×C2312yes—
500C4⋊C4C2×D4×C2312yes—
501C4⋊C4C2×D4×C2311yescharacteristic
502C4⋊C4C2×D4×C2311yescharacteristic
503C4⋊C4C2×D4×C2311yescharacteristic
504C4⋊C4C2×D4×C2311yescharacteristic
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—21no—
523C42C2×D4×C2311yescharacteristic
524C42C2×D4×C2311yescharacteristic
525C42C2×D4×C2311yescharacteristic
526C42C24×C2311yescharacteristic
527C23—21no—
528C23—21no—
529C23C23×#12011yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—44no—
533Q8—42no—
534Q8—42no—
535Q8—82no—
536Q8—82no—
537Q8—82no—
538D4—22no—
539D4—22no—
540D4—42no—
541D4—42no—
542D4—82no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4C23×#12011yescharacteristic
562C2×C4C23×#12011yescharacteristic
563C2×C4C23×#12011yescharacteristic
564C2×C4C22×D4×C2311yescharacteristic
565C2×C4C22×D4×C2311yescharacteristic
566C2×C4C22×D4×C2311yescharacteristic
567C8—42no—
568C8—42no—
569C8—42no—
570C22—21no—
571C22—21no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22C23×#45412yes—
579C22C2×C23×#12011yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—82no—
593C4C23×#57712yes—
594C4C23×#57812yes—
595C2—22no—
596C2—81no—
597C2—81no—
598C2C23×#300911yescharacteristic
599C2C23×#301111yescharacteristic
600C2C2×C23×#45411yescharacteristic
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×#3238C2semidirect
3. C23×#3238C2semidirect
4. C2×C23×#487C2semidirect
5. C23×#1638C2semidirect
6. C23×#1637C2semidirect
7. C23×#1636C2semidirect
8. C23×#1635C2semidirect
9. C23×#1556C2semidirect
10. C23×#1494C2semidirect
11. C23×#1494C2semidirect
226. #18695C23direct (N × H)
271. #3238C2×C23semidirect
272. #3238C2×C23semidirect
273. C2×#487C2×C23semidirect
274. #1638C2×C23semidirect
275. #1637C2×C23semidirect
276. #1636C2×C23semidirect
277. #1635C2×C23semidirect
278. #1556C2×C23semidirect
279. #1494C2×C23semidirect
280. #1494C2×C23semidirect
446. C23#18695direct (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×#32382
C2×C23×#4872
C23×#16382
C23×#16372
C23×#16362
C23×#16352
C23×#15562
C23×#14942
#1869523