Finite Group Databasebeta

← C23×#18427 #19166053

Subgroups — C23×#18427 #19166053

Order 5888.

996 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×#3558C211yesmaximal, characteristic
2C23×#3218C211yesmaximal, characteristic, complemented
3C23×#3206C211yesmaximal, characteristic, complemented
4C23×#2640C211yesmaximal, characteristic, complemented
5C23×#1606C214yesmaximal, complemented
6C23×#1601C214yesmaximal, complemented
7C23×#1505C211yesmaximal, characteristic, complemented
8C23×#1500C211yesmaximal, characteristic, complemented
9C23×#1469C211yesmaximal, characteristic, complemented
10C23×#562—21no—
11C23×H(3,𝔽4)—21no—
12C23×#559—21no—
13C23×#548—21no—
14C23×#546—21no—
15C23×#546—22no—
16D42×C23—21no—
17C23×#542—21no—
18C23×#538—22no—
19C23×#538C2212yes—
20C23×#535—21no—
21C23×#534C2212yes—
22C23×#532C2211yescharacteristic
23C2×C23×C4⋊Q8C2211yescharacteristic, complemented
24C23×#529C2211yescharacteristic
25C2×C23×#124C2211yescharacteristic
26C23×#497—21no—
27C23×#492—21no—
28C23×#491—21no—
29C23×#487—21no—
30C23×#462—21no—
31C23×#461—22no—
32C23×#461—21no—
33C23×#451C2212yescomplemented
34C23×#448C2212yes—
35C23×#448C2214yescomplemented
36C23×#432C2211yescharacteristic, complemented
37C23×#431C2211yescharacteristic, complemented
38C23×#397—22no—
39C23×#393C2214yescomplemented
40C23×#391—21no—
41C23×#390—21no—
42C23×#376C2211yescharacteristic, complemented
43C23×#328C2212yescomplemented
44C23×#327C2212yes—
45C23×#327C2214yescomplemented
46C23×#323C2214yescomplemented
47C2×#41×C23—21no—
48C2×#41×C23—21no—
49C22×Q8×C23D411yescharacteristic
50C22×Q8×C23C2312yes—
51C22×D4×C23—21no—
52C22×D4×C23—22no—
53C22×D4×C23D411yescharacteristic
54C2×Q16×C23—22no—
55C2×SD16×C23—22no—
56C2×SD16×C23—24no—
57C23×C4⋊Q8—21no—
58C23×C4⋊Q8—21no—
59C23×C4⋊Q8—21no—
60C23×C4⋊Q8—21no—
61C23×C4⋊Q8D411yescharacteristic
62C23×#127—21no—
63C23×#127D411yescharacteristic
64C23×#126—22no—
65C23×#126—21no—
66C23×#125D411yescharacteristic
67C23×#124—21no—
68C23×#124—22no—
69C23×#124—22no—
70C23×#124—22no—
71C23×#124—21no—
72C23×#124—21no—
73C23×#124—21no—
74C23×#124—21no—
75C23×#124—21no—
76C23×#124D411yescharacteristic
77C23×#123—22no—
78C23×#123—21no—
79C23×#123—22no—
80C23×#123—22no—
81C23×#123—22no—
82C23×C22⋊Q8—22no—
83C23×C22⋊Q8—21no—
84C23×C22⋊Q8—22no—
85C23×C22⋊Q8C2311yescharacteristic
86C23×#121—22no—
87C23×#121—22no—
88C23×#121—22no—
89C23×#121—21no—
90C23×#121—22no—
91C23×#121—21no—
92C23×#121—22no—
93C23×#121C2311yescharacteristic
94C23×#121C2312yes—
95C23×#120—22no—
96C23×#120—22no—
97C23×#120—22no—
98C23×#120—22no—
99C23×#120—22no—
100C4×Q8×C23—21no—
101C4×Q8×C23D411yescharacteristic
102C4×D4×C23—22no—
103C4×D4×C23—21no—
104C4×D4×C23—21no—
105C4×D4×C23—22no—
106C4×D4×C23—21no—
107C4×D4×C23D411yescharacteristic
108C4×D4×C23D412yes—
109C23×#117—21no—
110C2×C4⋊C4×C23—21no—
111C2×C4⋊C4×C23—21no—
112C2×C22⋊C4×C23—22no—
113C2×C22⋊C4×C23—24no—
114C2×C22⋊C4×C23D412yes—
115C2×C22⋊C4×C23C2312yes—
116C2×C42×C23—21no—
117C2×C42×C23C2311yescharacteristic
118C23×C4⋊C8—21no—
119C23×C4⋊C8—21no—
120C23×#103—22no—
121C23×#103—24no—
122C23×#103—22no—
123C23×C22⋊C8C2312yes—
124C23×C22⋊C8C2312yes—
125C23×#97—21no—
126C4×C8×C23—21no—
127C23×#95—21no—
128C23×#95—24no—
129C23×#95C2312yes—
130C24×C23—22no—
131C24×C23C2×D412yes—
132#41×C23—82no—
133#41×C23—82no—
134C2×Q8×C23—21no—
135C2×Q8×C23—21no—
136C2×Q8×C23—21no—
137C2×Q8×C23—22no—
138C2×Q8×C23—22no—
139C2×Q8×C23—42no—
140C2×Q8×C23—82no—
141C2×Q8×C23C2×D411yescharacteristic
142C2×D4×C23—22no—
143C2×D4×C23—21no—
144C2×D4×C23—22no—
145C2×D4×C23—22no—
146C2×D4×C23—22no—
147C2×D4×C23—22no—
148C2×D4×C23—22no—
149C2×D4×C23—21no—
150C2×D4×C23—21no—
151C2×D4×C23—21no—
152C2×D4×C23—21no—
153C2×D4×C23—42no—
154C2×D4×C23—82no—
155C2×D4×C23—84no—
156C2×D4×C23C2×D411yescharacteristic
157C2×D4×C23C2×D412yes—
158C22×C4×C23—22no—
159C22×C4×C23—22no—
160C22×C4×C23—21no—
161C22×C4×C23—22no—
162C22×C4×C23—22no—
163C22×C4×C23—22no—
164C22×C4×C23C2×D411yescharacteristic
165C22×C4×C23C2×D411yescharacteristic
166C22×C4×C23C2×D412yes—
167C22×C4×C23C2411yescharacteristic
168Q16×C23—82no—
169SD16×C23—84no—
170SD16×C23—82no—
171C2×C8×C23—22no—
172C2×C8×C23—22no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—41no—
181C4⋊C4×C23—41no—
182C4⋊C4×C23—41no—
183C4⋊C4×C23—41no—
184C4⋊C4×C23C2×D412yes—
185C4⋊C4×C23C2×D411yescharacteristic
186C4⋊C4×C23C2×D411yescharacteristic
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—21no—
193C22⋊C4×C23—21no—
194C22⋊C4×C23—22no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—21no—
197C22⋊C4×C23—21no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—21no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—84no—
204C42×C23—22no—
205C42×C23—21no—
206C42×C23—42no—
207C42×C23C2×D411yescharacteristic
208C42×C23C2×D411yescharacteristic
209C42×C23C2×D411yescharacteristic
210C42×C23C2×D411yescharacteristic
211#18427C2311yesmaximal, characteristic, direct product factor, Sylow
212C23×C23—22no—
213C23×C23—21no—
214C23×C23—22no—
215C23×C23—24no—
216C23×C23—42no—
217C23×C23—42no—
218C23×C23—82no—
219C23×C23—82no—
220C23×C23—82no—
221C23×C23C22×D411yescharacteristic
222Q8×C23—41no—
223Q8×C23—44no—
224Q8×C23—42no—
225Q8×C23—42no—
226Q8×C23—42no—
227Q8×C23—81no—
228Q8×C23—81no—
229D4×C23—42no—
230D4×C23—41no—
231D4×C23—42no—
232D4×C23—44no—
233D4×C23—42no—
234D4×C23—42no—
235D4×C23—81no—
236D4×C23—81no—
237D4×C23—82no—
238D4×C23—82no—
239D4×C23—82no—
240D4×C23—82no—
241D4×C23—82no—
242C2×C4×C23—22no—
243C2×C4×C23—22no—
244C2×C4×C23—21no—
245C2×C4×C23—22no—
246C2×C4×C23—22no—
247C2×C4×C23—21no—
248C2×C4×C23—21no—
249C2×C4×C23—41no—
250C2×C4×C23—42no—
251C2×C4×C23—42no—
252C2×C4×C23—42no—
253C2×C4×C23—42no—
254C2×C4×C23—44no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
257C2×C4×C23—82no—
258C2×C4×C23—82no—
259C2×C4×C23—82no—
260C2×C4×C23#12011yescharacteristic
261C2×C4×C23#12011yescharacteristic
262C2×C4×C23#12011yescharacteristic
263C2×C4×C23#12011yescharacteristic
264C2×C4×C23C22×D411yescharacteristic
265C2×C4×C23C22×D411yescharacteristic
266C8×C23—42no—
267C8×C23—42no—
268#3558C2×C2311yescharacteristic
269#3218C2×C2311yescharacteristic, complemented
270#3206C2×C2311yescharacteristic, complemented
271#2640C2×C2311yescharacteristic, complemented
272#1606C2×C2314yescomplemented
273#1601C2×C2314yescomplemented
274#1505C2×C2311yescharacteristic, complemented
275#1500C2×C2311yescharacteristic, complemented
276#1469C2×C2311yescharacteristic, complemented
277C22×C23—21no—
278C22×C23—21no—
279C22×C23—42no—
280C22×C23—42no—
281C22×C23—41no—
282C22×C23—41no—
283C22×C23—42no—
284C22×C23—42no—
285C22×C23—42no—
286C22×C23—41no—
287C22×C23—42no—
288C22×C23—84no—
289C22×C23—164no—
290C22×C23C2×#12011yescharacteristic
291C22×C23#57712yes—
292C4×C23—21no—
293C4×C23—21no—
294C4×C23—21no—
295C4×C23—21no—
296C4×C23—21no—
297C4×C23—41no—
298C4×C23—42no—
299C4×C23—41no—
300C4×C23—42no—
301C4×C23—41no—
302C4×C23—82no—
303C4×C23—82no—
304C4×C23#45512yes—
305#562—21no—
306H(3,𝔽4)—21no—
307#559—21no—
308#548—21no—
309#546—21no—
310#546—22no—
311D42—21no—
312#542—21no—
313#538—22no—
314#538C22×C2312yes—
315#535—21no—
316#534C22×C2312yes—
317#532C22×C2311yescharacteristic
318C2×C4⋊Q8C22×C2311yescharacteristic, complemented
319#529C22×C2311yescharacteristic
320C2×#124C22×C2311yescharacteristic
321#497—21no—
322#492—21no—
323#491—21no—
324#487—21no—
325#462—21no—
326#461—22no—
327#461—21no—
328#451C22×C2312yescomplemented
329#448C22×C2312yes—
330#448C22×C2314yescomplemented
331#432C22×C2311yescharacteristic, complemented
332#431C22×C2311yescharacteristic, complemented
333#397—22no—
334#393C22×C2314yescomplemented
335#391—21no—
336#390—21no—
337#376C22×C2311yescharacteristic, complemented
338#328C22×C2312yescomplemented
339#327C22×C2312yes—
340#327C22×C2314yescomplemented
341#323C22×C2314yescomplemented
342C2×C23—22no—
343C2×C23—82no—
344C2×C23—82no—
345C2×C23—81no—
346C2×C23#300911yescharacteristic
347C2×C23C2×#45511yescharacteristic
348C2×C23#302711yescharacteristic
349C2×#41—21no—
350C2×#41—21no—
351C22×Q8D4×C2311yescharacteristic
352C22×Q8C23×C2312yes—
353C22×D4—21no—
354C22×D4—22no—
355C22×D4D4×C2311yescharacteristic
356C2×Q16—22no—
357C2×SD16—22no—
358C2×SD16—24no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8D4×C2311yescharacteristic
364#127—21no—
365#127D4×C2311yescharacteristic
366#126—22no—
367#126—21no—
368#125D4×C2311yescharacteristic
369#124—21no—
370#124—22no—
371#124—22no—
372#124—22no—
373#124—21no—
374#124—21no—
375#124—21no—
376#124—21no—
377#124—21no—
378#124D4×C2311yescharacteristic
379#123—22no—
380#123—21no—
381#123—22no—
382#123—22no—
383#123—22no—
384C22⋊Q8—22no—
385C22⋊Q8—21no—
386C22⋊Q8—22no—
387C22⋊Q8C23×C2311yescharacteristic
388#121—22no—
389#121—22no—
390#121—22no—
391#121—21no—
392#121—22no—
393#121—21no—
394#121—22no—
395#121C23×C2311yescharacteristic
396#121C23×C2312yes—
397#120—22no—
398#120—22no—
399#120—22no—
400#120—22no—
401#120—22no—
402C4×Q8—21no—
403C4×Q8D4×C2311yescharacteristic
404C4×D4—22no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—22no—
408C4×D4—21no—
409C4×D4D4×C2311yescharacteristic
410C4×D4D4×C2312yes—
411#117—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—24no—
416C2×C22⋊C4D4×C2312yes—
417C2×C22⋊C4C23×C2312yes—
418C2×C42—21no—
419C2×C42C23×C2311yescharacteristic
420C4⋊C8—21no—
421C4⋊C8—21no—
422#103—22no—
423#103—24no—
424#103—22no—
425C22⋊C8C23×C2312yes—
426C22⋊C8C23×C2312yes—
427#97—21no—
428C4×C8—21no—
429#95—21no—
430#95—24no—
431#95C23×C2312yes—
432C23#1842711yescharacteristic, direct product factor, Sylow
433C24—22no—
434C24C2×D4×C2312yes—
435#41—82no—
436#41—82no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—42no—
443C2×Q8—82no—
444C2×Q8C2×D4×C2311yescharacteristic
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—82no—
458C2×D4—84no—
459C2×D4C2×D4×C2311yescharacteristic
460C2×D4C2×D4×C2312yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4C2×D4×C2311yescharacteristic
468C22×C4C2×D4×C2311yescharacteristic
469C22×C4C2×D4×C2312yes—
470C22×C4C24×C2311yescharacteristic
471Q16—82no—
472SD16—84no—
473SD16—82no—
474C2×C8—22no—
475C2×C8—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4C2×D4×C2312yes—
488C4⋊C4C2×D4×C2311yescharacteristic
489C4⋊C4C2×D4×C2311yescharacteristic
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—21no—
505C22⋊C4—22no—
506C22⋊C4—84no—
507C42—22no—
508C42—21no—
509C42—42no—
510C42C2×D4×C2311yescharacteristic
511C42C2×D4×C2311yescharacteristic
512C42C2×D4×C2311yescharacteristic
513C42C2×D4×C2311yescharacteristic
514C23—22no—
515C23—21no—
516C23—22no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23C22×D4×C2311yescharacteristic
524Q8—41no—
525Q8—44no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—81no—
530Q8—81no—
531D4—42no—
532D4—41no—
533D4—42no—
534D4—44no—
535D4—42no—
536D4—42no—
537D4—81no—
538D4—81no—
539D4—82no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4C23×#12011yescharacteristic
563C2×C4C23×#12011yescharacteristic
564C2×C4C23×#12011yescharacteristic
565C2×C4C23×#12011yescharacteristic
566C2×C4C22×D4×C2311yescharacteristic
567C2×C4C22×D4×C2311yescharacteristic
568C8—42no—
569C8—42no—
570C22—21no—
571C22—21no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—42no—
581C22—84no—
582C22—164no—
583C22C2×C23×#12011yescharacteristic
584C22C23×#57712yes—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—82no—
596C4—82no—
597C4C23×#45512yes—
598C2—22no—
599C2—82no—
600C2—81no—
601C2—82no—
602C2C23×#300911yescharacteristic
603C2C2×C23×#45511yescharacteristic
604C2C23×#302711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3558C2
LCSG ▷ C22×C4 ▷ 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×#3218C2semidirect
3. C23×#3206C2semidirect
4. C23×#2640C2semidirect
5. C23×#1606C2semidirect
6. C23×#1601C2semidirect
7. C23×#1505C2semidirect
8. C23×#1500C2semidirect
9. C23×#1469C2semidirect
23. C2×C23×C4⋊Q8C22semidirect
33. C23×#451C22semidirect
35. C23×#448C22semidirect
36. C23×#432C22semidirect
37. C23×#431C22semidirect
39. C23×#393C22semidirect
42. C23×#376C22semidirect
43. C23×#328C22semidirect
45. C23×#327C22semidirect
46. C23×#323C22semidirect
211. #18427C23direct (N × H)
269. #3218C2×C23semidirect
270. #3206C2×C23semidirect
271. #2640C2×C23semidirect
272. #1606C2×C23semidirect
273. #1601C2×C23semidirect
274. #1505C2×C23semidirect
275. #1500C2×C23semidirect
276. #1469C2×C23semidirect
318. C2×C4⋊Q8C22×C23semidirect
328. #451C22×C23semidirect
330. #448C22×C23semidirect
331. #432C22×C23semidirect
332. #431C22×C23semidirect
334. #393C22×C23semidirect
337. #376C22×C23semidirect
338. #328C22×C23semidirect
340. #327C22×C23semidirect
341. #323C22×C23semidirect
432. C23#18427direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35582
C23×#32182
C23×#32062
C23×#26402
C23×#16062
C23×#16012
C23×#15052
C23×#15002
C23×#14692
#1842723