Finite Group Databasebeta

← C23×#18312 #19165938

Subgroups — C23×#18312 #19165938

Order 5888.

930 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3455C211yesmaximal, characteristic
2C23×#3229C211yesmaximal, characteristic, complemented
3C23×#3226C211yesmaximal, characteristic, complemented
4C23×#2621C211yesmaximal, characteristic, complemented
5C23×#1612C214yesmaximal, complemented
6C23×#1609C214yesmaximal, complemented
7C23×#1518C211yesmaximal, characteristic, complemented
8C23×#1517C211yesmaximal, characteristic, complemented
9C23×#1458C211yesmaximal, characteristic, complemented
10C23×#542—21no—
11C23×#542—21no—
12C23×#541—21no—
13C23×#540—21no—
14C23×#539—21no—
15C23×#539C2212yes—
16C23×#538—21no—
17C23×#538C2212yes—
18C23×#533C2211yescharacteristic
19C23×#532C2211yescharacteristic
20C2×C23×C4⋊Q8C2211yescharacteristic
21C23×#529—22no—
22C23×#529—22no—
23C23×#525—21no—
24C2×C23×C22⋊Q8—21no—
25C4×#41×C23—21no—
26C2×C23×#117C2211yescharacteristic
27C23×#500—21no—
28C23×#498—21no—
29C23×#487—21no—
30C23×#486—21no—
31C23×#483C2214yes—
32C23×#480C2214yes—
33C23×#475—22no—
34C23×#474—22no—
35C23×#431C2211yescharacteristic
36C23×#431C2211yescharacteristic
37C23×#398—21no—
38C23×#398C2212yes—
39C23×#397—21no—
40C23×#397C2212yes—
41C23×#391—21no—
42C23×#382C2211yescharacteristic
43C23×#328C2214yes—
44C23×#327C2214yes—
45C23×#324C2212yes—
46C23×#323C2212yes—
47C2×#41×C23—21no—
48C22×Q8×C23—21no—
49C23×C4×C23D411yescharacteristic
50C23×C4⋊Q8—21no—
51C23×C4⋊Q8—21no—
52C23×C4⋊Q8—21no—
53C23×C4⋊Q8—21no—
54C23×C4⋊Q8D411yescharacteristic
55C23×C4⋊Q8D411yescharacteristic
56C23×C4⋊Q8D411yescharacteristic
57C23×#127D411yescharacteristic
58C23×#126—22no—
59C23×#126—22no—
60C23×#126—22no—
61C23×#126—22no—
62C23×#125—22no—
63C23×#125—22no—
64C23×#125—21no—
65C23×#124—22no—
66C23×#124—22no—
67C23×#124—21no—
68C23×#123—22no—
69C23×#123—22no—
70C23×#123—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×C22⋊Q8—21no—
74C23×C22⋊Q8—21no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—21no—
78C23×C22⋊Q8—44no—
79C23×C22⋊Q8C2312yes—
80C23×#121—21no—
81C23×#121—21no—
82C23×#121—22no—
83C23×#121—22no—
84C23×#121—21no—
85C23×#121C2312yes—
86C23×#120—22no—
87C4×Q8×C23—21no—
88C4×Q8×C23—21no—
89C4×Q8×C23—21no—
90C4×Q8×C23D412yes—
91C4×D4×C23—21no—
92C4×D4×C23—21no—
93C4×D4×C23—22no—
94C4×D4×C23—22no—
95C4×D4×C23—22no—
96C4×D4×C23—22no—
97C4×D4×C23—21no—
98C4×D4×C23D412yes—
99C23×#117—22no—
100C23×#117—21no—
101C23×#117—21no—
102C23×#117—21no—
103C23×#117—21no—
104C23×#117—21no—
105C2×C4⋊C4×C23—21no—
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23—22no—
108C2×C4⋊C4×C23D411yescharacteristic
109C2×C4⋊C4×C23C2312yes—
110C2×C4⋊C4×C23C2311yescharacteristic
111C2×C22⋊C4×C23—22no—
112C2×C22⋊C4×C23D411yescharacteristic
113C2×C22⋊C4×C23C2311yescharacteristic
114C2×C42×C23—22no—
115C2×C42×C23D411yescharacteristic
116C2×C42×C23C2311yescharacteristic
117C23×#107—24no—
118C23×#106—24no—
119C23×C4⋊C8—21no—
120C23×C4⋊C8—21no—
121C23×#103—24no—
122C23×#102—24no—
123C23×C22⋊C8C2312yes—
124C23×C22⋊C8C2312yes—
125C4×C8×C23—21no—
126C4×C8×C23—21no—
127C23×#95—21no—
128C23×#95—22no—
129C23×#95—22no—
130C23×#95C2312yes—
131C24×C23C2×D411yescharacteristic
132#41×C23—42no—
133#41×C23—42no—
134C2×Q8×C23—21no—
135C2×Q8×C23—21no—
136C2×Q8×C23—41no—
137C2×Q8×C23—41no—
138C2×Q8×C23—42no—
139C2×Q8×C23—42no—
140C2×Q8×C23C2×D412yes—
141C2×D4×C23—22no—
142C2×D4×C23—22no—
143C2×D4×C23—22no—
144C2×D4×C23—21no—
145C2×D4×C23—21no—
146C2×D4×C23C2×D412yes—
147C22×C4×C23—22no—
148C22×C4×C23—22no—
149C22×C4×C23—22no—
150C22×C4×C23—21no—
151C22×C4×C23—21no—
152C22×C4×C23—22no—
153C22×C4×C23—22no—
154C22×C4×C23—22no—
155C22×C4×C23—22no—
156C22×C4×C23C2×D411yescharacteristic
157C22×C4×C23C2×D411yescharacteristic
158C22×C4×C23C2×D411yescharacteristic
159C22×C4×C23C2×D411yescharacteristic
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2411yescharacteristic
162C2×C8×C23—22no—
163C2×C8×C23—22no—
164C4⋊C4×C23—22no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—21no—
169C4⋊C4×C23—21no—
170C4⋊C4×C23—21no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—21no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—44no—
181C4⋊C4×C23—41no—
182C4⋊C4×C23—41no—
183C4⋊C4×C23—42no—
184C4⋊C4×C23C2×D412yes—
185C4⋊C4×C23C2×D412yes—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—21no—
194C22⋊C4×C23—21no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—21no—
197C22⋊C4×C23—21no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—44no—
200C42×C23—22no—
201C42×C23—22no—
202C42×C23—21no—
203C42×C23—22no—
204C42×C23—21no—
205C42×C23C2×D411yescharacteristic
206C42×C23C2×D411yescharacteristic
207C42×C23C2×D411yescharacteristic
208C42×C23C2×D411yescharacteristic
209#18312C2311yesmaximal, characteristic, direct product factor, Sylow
210C23×C23—21no—
211C23×C23—22no—
212C23×C23—21no—
213C23×C23—21no—
214C23×C23—41no—
215C23×C23—41no—
216C23×C23C22×D411yescharacteristic
217Q8×C23—42no—
218Q8×C23—42no—
219Q8×C23—81no—
220Q8×C23—82no—
221Q8×C23—82no—
222D4×C23—42no—
223D4×C23—42no—
224D4×C23—42no—
225D4×C23—42no—
226D4×C23—81no—
227D4×C23—82no—
228D4×C23—82no—
229C2×C4×C23—21no—
230C2×C4×C23—21no—
231C2×C4×C23—22no—
232C2×C4×C23—22no—
233C2×C4×C23—22no—
234C2×C4×C23—21no—
235C2×C4×C23—21no—
236C2×C4×C23—21no—
237C2×C4×C23—41no—
238C2×C4×C23—41no—
239C2×C4×C23—42no—
240C2×C4×C23—42no—
241C2×C4×C23—44no—
242C2×C4×C23—41no—
243C2×C4×C23—41no—
244C2×C4×C23—42no—
245C2×C4×C23—42no—
246C2×C4×C23—42no—
247C2×C4×C23—42no—
248C2×C4×C23—42no—
249C2×C4×C23—42no—
250C2×C4×C23—82no—
251C2×C4×C23—82no—
252C2×C4×C23#12011yescharacteristic
253C2×C4×C23#12011yescharacteristic
254C2×C4×C23#12011yescharacteristic
255C2×C4×C23#12011yescharacteristic
256C2×C4×C23#13512yes—
257C2×C4×C23AGL(1,ℤ/8)12yes—
258C2×C4×C23C22×D411yescharacteristic
259C2×C4×C23C22×D411yescharacteristic
260C8×C23—42no—
261C8×C23—42no—
262#3455C2×C2311yescharacteristic
263#3229C2×C2311yescharacteristic, complemented
264#3226C2×C2311yescharacteristic, complemented
265#2621C2×C2311yescharacteristic, complemented
266#1612C2×C2314yescomplemented
267#1609C2×C2314yescomplemented
268#1518C2×C2311yescharacteristic, complemented
269#1517C2×C2311yescharacteristic, complemented
270#1458C2×C2311yescharacteristic, complemented
271C22×C23—21no—
272C22×C23—21no—
273C22×C23—42no—
274C22×C23—41no—
275C22×C23—42no—
276C22×C23—42no—
277C22×C23—41no—
278C22×C23—42no—
279C22×C23—41no—
280C22×C23—81no—
281C22×C23C2×#12011yescharacteristic
282C22×C23C2×#13511yescharacteristic
283C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C23—21no—
285C4×C23—21no—
286C4×C23—21no—
287C4×C23—21no—
288C4×C23—41no—
289C4×C23—42no—
290C4×C23—41no—
291C4×C23—42no—
292C4×C23—41no—
293C4×C23—42no—
294C4×C23—82no—
295C4×C23—82no—
296C4×C23#44912yes—
297C4×C23#44912yes—
298#542—21no—
299#542—21no—
300#541—21no—
301#540—21no—
302#539—21no—
303#539C22×C2312yes—
304#538—21no—
305#538C22×C2312yes—
306#533C22×C2311yescharacteristic
307#532C22×C2311yescharacteristic
308C2×C4⋊Q8C22×C2311yescharacteristic
309#529—22no—
310#529—22no—
311#525—21no—
312C2×C22⋊Q8—21no—
313C4×#41—21no—
314C2×#117C22×C2311yescharacteristic
315#500—21no—
316#498—21no—
317#487—21no—
318#486—21no—
319#483C22×C2314yes—
320#480C22×C2314yes—
321#475—22no—
322#474—22no—
323#431C22×C2311yescharacteristic
324#431C22×C2311yescharacteristic
325#398—21no—
326#398C22×C2312yes—
327#397—21no—
328#397C22×C2312yes—
329#391—21no—
330#382C22×C2311yescharacteristic
331#328C22×C2314yes—
332#327C22×C2314yes—
333#324C22×C2312yes—
334#323C22×C2312yes—
335C2×C23—21no—
336C2×C23—21no—
337C2×C23—82no—
338C2×C23—81no—
339C2×C23C2×#44911yescharacteristic
340C2×C23#302211yescharacteristic
341C2×C23#302811yescharacteristic
342C2×#41—21no—
343C22×Q8—21no—
344C23×C4D4×C2311yescharacteristic
345C4⋊Q8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8D4×C2311yescharacteristic
350C4⋊Q8D4×C2311yescharacteristic
351C4⋊Q8D4×C2311yescharacteristic
352#127D4×C2311yescharacteristic
353#126—22no—
354#126—22no—
355#126—22no—
356#126—22no—
357#125—22no—
358#125—22no—
359#125—21no—
360#124—22no—
361#124—22no—
362#124—21no—
363#123—22no—
364#123—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—21no—
373C22⋊Q8—44no—
374C22⋊Q8C23×C2312yes—
375#121—21no—
376#121—21no—
377#121—22no—
378#121—22no—
379#121—21no—
380#121C23×C2312yes—
381#120—22no—
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8D4×C2312yes—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—21no—
393C4×D4D4×C2312yes—
394#117—22no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4D4×C2311yescharacteristic
404C2×C4⋊C4C23×C2312yes—
405C2×C4⋊C4C23×C2311yescharacteristic
406C2×C22⋊C4—22no—
407C2×C22⋊C4D4×C2311yescharacteristic
408C2×C22⋊C4C23×C2311yescharacteristic
409C2×C42—22no—
410C2×C42D4×C2311yescharacteristic
411C2×C42C23×C2311yescharacteristic
412#107—24no—
413#106—24no—
414C4⋊C8—21no—
415C4⋊C8—21no—
416#103—24no—
417#102—24no—
418C22⋊C8C23×C2312yes—
419C22⋊C8C23×C2312yes—
420C4×C8—21no—
421C4×C8—21no—
422#95—21no—
423#95—22no—
424#95—22no—
425#95C23×C2312yes—
426C23#1831211yescharacteristic, direct product factor, Sylow
427C24C2×D4×C2311yescharacteristic
428#41—42no—
429#41—42no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8C2×D4×C2312yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×D4×C2312yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4C2×D4×C2311yescharacteristic
453C22×C4C2×D4×C2311yescharacteristic
454C22×C4C2×D4×C2311yescharacteristic
455C22×C4C2×D4×C2311yescharacteristic
456C22×C4C2×D4×C2311yescharacteristic
457C22×C4C24×C2311yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4C2×D4×C2312yes—
481C4⋊C4C2×D4×C2312yes—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—44no—
496C42—22no—
497C42—22no—
498C42—21no—
499C42—22no—
500C42—21no—
501C42C2×D4×C2311yescharacteristic
502C42C2×D4×C2311yescharacteristic
503C42C2×D4×C2311yescharacteristic
504C42C2×D4×C2311yescharacteristic
505C23—21no—
506C23—21no—
507C23—22no—
508C23—21no—
509C23—41no—
510C23—41no—
511C23C22×D4×C2311yescharacteristic
512Q8—42no—
513Q8—42no—
514Q8—81no—
515Q8—82no—
516Q8—82no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—81no—
522D4—82no—
523D4—82no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4C23×#12011yescharacteristic
548C2×C4C23×#12011yescharacteristic
549C2×C4C23×#12011yescharacteristic
550C2×C4C23×#12011yescharacteristic
551C2×C4C23×#13512yes—
552C2×C4C23×AGL(1,ℤ/8)12yes—
553C2×C4C22×D4×C2311yescharacteristic
554C2×C4C22×D4×C2311yescharacteristic
555C8—42no—
556C8—42no—
557C22—21no—
558C22—21no—
559C22—42no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—42no—
565C22—41no—
566C22—81no—
567C22C2×C23×#12011yescharacteristic
568C22C2×C23×#13511yescharacteristic
569C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—82no—
581C4—82no—
582C4C23×#44912yes—
583C4C23×#44912yes—
584C2—21no—
585C2—21no—
586C2—81no—
587C2—82no—
588C2C2×C23×#44911yescharacteristic
589C2C23×#302211yescharacteristic
590C2C23×#302811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3455C2
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×#3229C2semidirect
3. C23×#3226C2semidirect
4. C23×#2621C2semidirect
5. C23×#1612C2semidirect
6. C23×#1609C2semidirect
7. C23×#1518C2semidirect
8. C23×#1517C2semidirect
9. C23×#1458C2semidirect
209. #18312C23direct (N × H)
263. #3229C2×C23semidirect
264. #3226C2×C23semidirect
265. #2621C2×C23semidirect
266. #1612C2×C23semidirect
267. #1609C2×C23semidirect
268. #1518C2×C23semidirect
269. #1517C2×C23semidirect
270. #1458C2×C23semidirect
426. C23#18312direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34552
C23×#32292
C23×#32262
C23×#26212
C23×#16122
C23×#16092
C23×#15182
C23×#15172
C23×#14582
#1831223