Finite Group Databasebeta

← C23×#18429 #19166055

Subgroups — C23×#18429 #19166055

Order 5888.

932 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3561C211yesmaximal, characteristic
2C23×#3218C211yesmaximal, characteristic, complemented
3C23×#3209C211yesmaximal, characteristic, complemented
4C23×#2640C211yesmaximal, characteristic, complemented
5C23×#1606C214yesmaximal, complemented
6C23×#1602C214yesmaximal, complemented
7C23×#1506C211yesmaximal, characteristic, complemented
8C23×#1500C211yesmaximal, characteristic, complemented
9C23×#1469C211yesmaximal, characteristic, complemented
10C23×#563—21no—
11C23×#562—21no—
12C23×#556—21no—
13C23×#551—22no—
14D4×Q8×C23—21no—
15C23×#548—21no—
16C23×#546—21no—
17C23×#543C2212yes—
18C23×#542—21no—
19C23×#542—22no—
20C23×#538C2212yes—
21C23×#536—21no—
22C23×#533C2211yescharacteristic
23C2×C23×C4⋊Q8C2211yescharacteristic
24C23×#529C2211yescharacteristic
25C2×C23×#124C2211yescharacteristic
26C23×#497—21no—
27C23×#494—21no—
28C23×#491—21no—
29C23×#487—21no—
30C23×#462—22no—
31C23×#462—21no—
32C23×#461—21no—
33C23×#451C2214yes—
34C23×#451C2212yes—
35C23×#448C2212yes—
36C23×#432C2211yescharacteristic
37C23×#431C2211yescharacteristic
38C23×#397—22no—
39C23×#393C2214yes—
40C23×#391—21no—
41C23×#390—21no—
42C23×#376C2211yescharacteristic
43C23×#328C2212yes—
44C23×#328C2214yes—
45C23×#327C2212yes—
46C23×#323C2214yes—
47C2×#41×C23—21no—
48C22×Q8×C23—21no—
49C22×Q8×C23D411yescharacteristic
50C22×Q8×C23C2312yes—
51C22×D4×C23D411yescharacteristic
52C2×Q16×C23—24no—
53C2×Q16×C23—22no—
54C2×SD16×C23—22no—
55C23×C4⋊Q8—21no—
56C23×C4⋊Q8—22no—
57C23×C4⋊Q8—21no—
58C23×C4⋊Q8—21no—
59C23×C4⋊Q8—21no—
60C23×C4⋊Q8—21no—
61C23×C4⋊Q8D411yescharacteristic
62C23×C4⋊Q8D411yescharacteristic
63C23×#126—22no—
64C23×#126—22no—
65C23×#126—22no—
66C23×#126—21no—
67C23×#125—21no—
68C23×#125D411yescharacteristic
69C23×#124—22no—
70C23×#124—21no—
71C23×#124—22no—
72C23×#124—21no—
73C23×#124—21no—
74C23×#124—21no—
75C23×#124—21no—
76C23×#124D411yescharacteristic
77C23×#123—22no—
78C23×#123—22no—
79C23×#123—22no—
80C23×#123—21no—
81C23×#123—22no—
82C23×C22⋊Q8—21no—
83C23×C22⋊Q8—22no—
84C23×C22⋊Q8—22no—
85C23×C22⋊Q8—22no—
86C23×C22⋊Q8—22no—
87C23×C22⋊Q8—22no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8C2311yescharacteristic
92C23×C22⋊Q8C2312yes—
93C23×#121—21no—
94C23×#121—22no—
95C23×#121—22no—
96C23×#121C2311yescharacteristic
97C23×#120—22no—
98C4×Q8×C23—21no—
99C4×Q8×C23—21no—
100C4×Q8×C23D411yescharacteristic
101C4×Q8×C23D412yes—
102C4×D4×C23—22no—
103C4×D4×C23—21no—
104C4×D4×C23—22no—
105C4×D4×C23—21no—
106C4×D4×C23D411yescharacteristic
107C23×#117—22no—
108C23×#117—21no—
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—22no—
122C23×#103—24no—
123C23×C22⋊C8C2312yes—
124C23×C22⋊C8C2312yes—
125C23×#97—21no—
126C4×C8×C23—21no—
127C23×#95—21no—
128C23×#95—24no—
129C23×#95C2312yes—
130C24×C23C2×D412yes—
131#41×C23—82no—
132C2×Q8×C23—22no—
133C2×Q8×C23—21no—
134C2×Q8×C23—21no—
135C2×Q8×C23—21no—
136C2×Q8×C23—22no—
137C2×Q8×C23—22no—
138C2×Q8×C23—42no—
139C2×Q8×C23—82no—
140C2×Q8×C23—82no—
141C2×Q8×C23C2×D411yescharacteristic
142C2×Q8×C23C2×D412yes—
143C2×D4×C23—21no—
144C2×D4×C23—22no—
145C2×D4×C23—22no—
146C2×D4×C23—21no—
147C2×D4×C23—21no—
148C2×D4×C23—42no—
149C2×D4×C23C2×D411yescharacteristic
150C22×C4×C23—22no—
151C22×C4×C23—22no—
152C22×C4×C23—22no—
153C22×C4×C23—21no—
154C22×C4×C23—22no—
155C22×C4×C23—22no—
156C22×C4×C23C2×D411yescharacteristic
157C22×C4×C23C2×D411yescharacteristic
158C22×C4×C23C2×D412yes—
159C22×C4×C23C2411yescharacteristic
160Q16×C23—82no—
161Q16×C23—84no—
162SD16×C23—82no—
163C2×C8×C23—22no—
164C2×C8×C23—22no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—21no—
170C4⋊C4×C23—21no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—21no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—21no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—41no—
179C4⋊C4×C23—41no—
180C4⋊C4×C23—41no—
181C4⋊C4×C23—41no—
182C4⋊C4×C23C2×D411yescharacteristic
183C4⋊C4×C23C2×D411yescharacteristic
184C4⋊C4×C23C2×D412yes—
185C22⋊C4×C23—22no—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—21no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—22no—
194C22⋊C4×C23—21no—
195C22⋊C4×C23—21no—
196C22⋊C4×C23—21no—
197C22⋊C4×C23—22no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—21no—
200C22⋊C4×C23—84no—
201C42×C23—22no—
202C42×C23—22no—
203C42×C23—21no—
204C42×C23—42no—
205C42×C23C2×D411yescharacteristic
206C42×C23C2×D411yescharacteristic
207C42×C23C2×D411yescharacteristic
208C42×C23C2×D411yescharacteristic
209#18429C2311yesmaximal, characteristic, direct product factor, Sylow
210C23×C23—22no—
211C23×C23—21no—
212C23×C23—24no—
213C23×C23—42no—
214C23×C23—42no—
215C23×C23C22×D411yescharacteristic
216Q8×C23—41no—
217Q8×C23—42no—
218Q8×C23—42no—
219Q8×C23—42no—
220Q8×C23—44no—
221Q8×C23—42no—
222Q8×C23—42no—
223Q8×C23—81no—
224Q8×C23—82no—
225D4×C23—41no—
226D4×C23—42no—
227D4×C23—42no—
228D4×C23—81no—
229D4×C23—82no—
230D4×C23—82no—
231C2×C4×C23—22no—
232C2×C4×C23—22no—
233C2×C4×C23—21no—
234C2×C4×C23—22no—
235C2×C4×C23—21no—
236C2×C4×C23—22no—
237C2×C4×C23—22no—
238C2×C4×C23—21no—
239C2×C4×C23—41no—
240C2×C4×C23—42no—
241C2×C4×C23—42no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—44no—
245C2×C4×C23—42no—
246C2×C4×C23—42no—
247C2×C4×C23—82no—
248C2×C4×C23—82no—
249C2×C4×C23—82no—
250C2×C4×C23#12011yescharacteristic
251C2×C4×C23#12011yescharacteristic
252C2×C4×C23#12011yescharacteristic
253C2×C4×C23#12011yescharacteristic
254C2×C4×C23C22×D411yescharacteristic
255C2×C4×C23C22×D411yescharacteristic
256C8×C23—42no—
257C8×C23—42no—
258#3561C2×C2311yescharacteristic
259#3218C2×C2311yescharacteristic, complemented
260#3209C2×C2311yescharacteristic, complemented
261#2640C2×C2311yescharacteristic, complemented
262#1606C2×C2314yescomplemented
263#1602C2×C2314yescomplemented
264#1506C2×C2311yescharacteristic, complemented
265#1500C2×C2311yescharacteristic, complemented
266#1469C2×C2311yescharacteristic, complemented
267C22×C23—21no—
268C22×C23—21no—
269C22×C23—41no—
270C22×C23—42no—
271C22×C23—41no—
272C22×C23—42no—
273C22×C23—42no—
274C22×C23—41no—
275C22×C23—84no—
276C22×C23C2×#12011yescharacteristic
277C22×C23#57812yes—
278C4×C23—21no—
279C4×C23—21no—
280C4×C23—21no—
281C4×C23—21no—
282C4×C23—21no—
283C4×C23—41no—
284C4×C23—41no—
285C4×C23—42no—
286C4×C23—42no—
287C4×C23—41no—
288C4×C23—42no—
289C4×C23—82no—
290C4×C23—82no—
291C4×C23#45512yes—
292#563—21no—
293#562—21no—
294#556—21no—
295#551—22no—
296D4×Q8—21no—
297#548—21no—
298#546—21no—
299#543C22×C2312yes—
300#542—21no—
301#542—22no—
302#538C22×C2312yes—
303#536—21no—
304#533C22×C2311yescharacteristic
305C2×C4⋊Q8C22×C2311yescharacteristic
306#529C22×C2311yescharacteristic
307C2×#124C22×C2311yescharacteristic
308#497—21no—
309#494—21no—
310#491—21no—
311#487—21no—
312#462—22no—
313#462—21no—
314#461—21no—
315#451C22×C2312yes—
316#451C22×C2314yes—
317#448C22×C2312yes—
318#432C22×C2311yescharacteristic
319#431C22×C2311yescharacteristic
320#397—22no—
321#393C22×C2314yes—
322#391—21no—
323#390—21no—
324#376C22×C2311yescharacteristic
325#328C22×C2312yes—
326#328C22×C2314yes—
327#327C22×C2312yes—
328#323C22×C2314yes—
329C2×C23—22no—
330C2×C23—81no—
331C2×C23—82no—
332C2×C23#301111yescharacteristic
333C2×C23C2×#45511yescharacteristic
334C2×C23#302711yescharacteristic
335C2×#41—21no—
336C22×Q8—21no—
337C22×Q8D4×C2311yescharacteristic
338C22×Q8C23×C2312yes—
339C22×D4D4×C2311yescharacteristic
340C2×Q16—24no—
341C2×Q16—22no—
342C2×SD16—22no—
343C4⋊Q8—21no—
344C4⋊Q8—22no—
345C4⋊Q8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8D4×C2311yescharacteristic
350C4⋊Q8D4×C2311yescharacteristic
351#126—22no—
352#126—22no—
353#126—22no—
354#126—21no—
355#125—21no—
356#125D4×C2311yescharacteristic
357#124—21no—
358#124—22no—
359#124—22no—
360#124—21no—
361#124—21no—
362#124—21no—
363#124—21no—
364#124D4×C2311yescharacteristic
365#123—22no—
366#123—22no—
367#123—22no—
368#123—21no—
369#123—22no—
370C22⋊Q8—22no—
371C22⋊Q8—21no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8C23×C2311yescharacteristic
380C22⋊Q8C23×C2312yes—
381#121—21no—
382#121—22no—
383#121—22no—
384#121C23×C2311yescharacteristic
385#120—22no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8D4×C2311yescharacteristic
389C4×Q8D4×C2312yes—
390C4×D4—21no—
391C4×D4—22no—
392C4×D4—21no—
393C4×D4—22no—
394C4×D4D4×C2311yescharacteristic
395#117—22no—
396#117—21no—
397#117—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4D4×C2312yes—
403C2×C22⋊C4C23×C2312yes—
404C2×C42—21no—
405C2×C42C23×C2311yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—21no—
408#103—22no—
409#103—22no—
410#103—24no—
411C22⋊C8C23×C2312yes—
412C22⋊C8C23×C2312yes—
413#97—21no—
414C4×C8—21no—
415#95—21no—
416#95—24no—
417#95C23×C2312yes—
418C23#1842911yescharacteristic, direct product factor, Sylow
419C24C2×D4×C2312yes—
420#41—82no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—42no—
428C2×Q8—82no—
429C2×Q8—82no—
430C2×Q8C2×D4×C2311yescharacteristic
431C2×Q8C2×D4×C2312yes—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4C2×D4×C2311yescharacteristic
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4C2×D4×C2311yescharacteristic
446C22×C4C2×D4×C2311yescharacteristic
447C22×C4C2×D4×C2312yes—
448C22×C4C24×C2311yescharacteristic
449Q16—82no—
450Q16—84no—
451SD16—82no—
452C2×C8—22no—
453C2×C8—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4C2×D4×C2311yescharacteristic
472C4⋊C4C2×D4×C2311yescharacteristic
473C4⋊C4C2×D4×C2312yes—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—84no—
490C42—22no—
491C42—22no—
492C42—21no—
493C42—42no—
494C42C2×D4×C2311yescharacteristic
495C42C2×D4×C2311yescharacteristic
496C42C2×D4×C2311yescharacteristic
497C42C2×D4×C2311yescharacteristic
498C23—22no—
499C23—21no—
500C23—24no—
501C23—42no—
502C23—42no—
503C23C22×D4×C2311yescharacteristic
504Q8—41no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—44no—
509Q8—42no—
510Q8—42no—
511Q8—81no—
512Q8—82no—
513D4—41no—
514D4—42no—
515D4—42no—
516D4—81no—
517D4—82no—
518D4—82no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4C23×#12011yescharacteristic
539C2×C4C23×#12011yescharacteristic
540C2×C4C23×#12011yescharacteristic
541C2×C4C23×#12011yescharacteristic
542C2×C4C22×D4×C2311yescharacteristic
543C2×C4C22×D4×C2311yescharacteristic
544C8—42no—
545C8—42no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—42no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—84no—
555C22C2×C23×#12011yescharacteristic
556C22C23×#57812yes—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—82no—
569C4—82no—
570C4C23×#45512yes—
571C2—22no—
572C2—82no—
573C2—81no—
574C2C23×#301111yescharacteristic
575C2C2×C23×#45511yescharacteristic
576C2C23×#302711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3561C2
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×#3209C2semidirect
4. C23×#2640C2semidirect
5. C23×#1606C2semidirect
6. C23×#1602C2semidirect
7. C23×#1506C2semidirect
8. C23×#1500C2semidirect
9. C23×#1469C2semidirect
209. #18429C23direct (N × H)
259. #3218C2×C23semidirect
260. #3209C2×C23semidirect
261. #2640C2×C23semidirect
262. #1606C2×C23semidirect
263. #1602C2×C23semidirect
264. #1506C2×C23semidirect
265. #1500C2×C23semidirect
266. #1469C2×C23semidirect
418. C23#18429direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35612
C23×#32182
C23×#32092
C23×#26402
C23×#16062
C23×#16022
C23×#15062
C23×#15002
C23×#14692
#1842923