Finite Group Databasebeta

← C23×#18270 #19165896

Subgroups — C23×#18270 #19165896

Order 5888.

1000 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3536C211yesmaximal, characteristic
2C23×#3092C211yesmaximal, characteristic, complemented
3C2×C23×#480C211yesmaximal, characteristic, complemented
4C23×#2502C211yesmaximal, characteristic, complemented
5C23×#1610C214yesmaximal, complemented
6C23×#1609C214yesmaximal, complemented
7C23×#1507C211yesmaximal, characteristic, complemented
8C23×#1501C211yesmaximal, characteristic, complemented
9C23×#1472C211yesmaximal, characteristic, complemented
10C23×#562—22no—
11C23×#554—21no—
12C23×#548—21no—
13C23×#547—21no—
14C23×#546—21no—
15C23×#546—22no—
16C23×#543—21no—
17C23×#542—21no—
18C23×#539C2212yes—
19C23×#538—21no—
20C23×#537—21no—
21C23×#537C2212yes—
22C23×#525C2211yescharacteristic
23C2×C23×C22⋊Q8C2211yescharacteristic
24C2×C23×#121C2211yescharacteristic
25C22×C4⋊C4×C23C2211yescharacteristic
26C23×#484—21no—
27C23×#483C2212yescomplemented
28C23×#481—21no—
29C23×#480—22no—
30C23×#480C2214yes—
31C23×#480C2212yescomplemented
32C23×#428—21no—
33C2×C23×#107—21no—
34C23×#417—21no—
35C2×C23×#102—21no—
36C23×#407C2211yescharacteristic
37C2×C23×C22⋊C8C2211yescharacteristic
38C23×#398C2214yescomplemented
39C23×#391—21no—
40C23×#385—22no—
41C23×#384—21no—
42C4×C22⋊C4×C23C2211yescharacteristic
43C23×#328C2212yescomplemented
44C23×#327C2214yes—
45C23×#327C2212yescomplemented
46C23×#324C2214yescomplemented
47C2×#41×C23—21no—
48C2×#41×C23—22no—
49C2×#41×C23D411yescharacteristic
50C22×Q8×C23D411yescharacteristic
51C22×D4×C23—21no—
52C22×D4×C23D411yescharacteristic
53C23×C4×C23—21no—
54C23×C4×C23C2311yescharacteristic
55C2×M16×C23—21no—
56C22×C8×C23—21no—
57C23×C4⋊Q8—21no—
58C23×#126—22no—
59C23×#125—22no—
60C23×#124—22no—
61C23×#124—21no—
62C23×#124—22no—
63C23×#123—22no—
64C23×#123—22no—
65C23×#123—22no—
66C23×#123—22no—
67C23×#123—21no—
68C23×C22⋊Q8—22no—
69C23×C22⋊Q8—21no—
70C23×C22⋊Q8—22no—
71C23×C22⋊Q8—22no—
72C23×C22⋊Q8—22no—
73C23×C22⋊Q8—21no—
74C23×C22⋊Q8—21no—
75C23×C22⋊Q8—21no—
76C23×C22⋊Q8—21no—
77C23×C22⋊Q8—22no—
78C23×C22⋊Q8—22no—
79C23×C22⋊Q8D411yescharacteristic
80C23×C22⋊Q8C2311yescharacteristic
81C23×#121—21no—
82C23×#121—22no—
83C23×#121—22no—
84C23×#121—22no—
85C23×#121—22no—
86C23×#121—21no—
87C23×#121—22no—
88C23×#121—21no—
89C23×#121D412yes—
90C23×#121D411yescharacteristic
91C23×#121C2312yes—
92C23×#121C2311yescharacteristic
93C23×#120—22no—
94C23×#120—21no—
95C4×Q8×C23—21no—
96C4×D4×C23—21no—
97C4×D4×C23—22no—
98C4×D4×C23—21no—
99C4×D4×C23—22no—
100C4×D4×C23—21no—
101C23×#117D411yescharacteristic
102C2×C4⋊C4×C23—21no—
103C2×C4⋊C4×C23—21no—
104C2×C4⋊C4×C23—21no—
105C2×C4⋊C4×C23—21no—
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23—21no—
108C2×C4⋊C4×C23—24no—
109C2×C4⋊C4×C23D411yescharacteristic
110C2×C4⋊C4×C23D411yescharacteristic
111C2×C4⋊C4×C23C2312yes—
112C2×C4⋊C4×C23C2312yes—
113C2×C22⋊C4×C23—22no—
114C2×C22⋊C4×C23—21no—
115C2×C22⋊C4×C23—21no—
116C2×C22⋊C4×C23D412yes—
117C2×C42×C23—21no—
118C23×#107—24no—
119C23×#107—22no—
120C23×#106—22no—
121C23×#103—22no—
122C23×#102—24no—
123C23×#102—22no—
124C23×C22⋊C8—21no—
125C23×C22⋊C8—21no—
126C23×C22⋊C8C2312yes—
127C23×C22⋊C8C2312yes—
128C23×#95—24no—
129C23×#95C2312yes—
130C24×C23—21no—
131C24×C23C2×D411yescharacteristic
132#41×C23—42no—
133#41×C23—84no—
134#41×C23—82no—
135C2×Q8×C23—22no—
136C2×Q8×C23—21no—
137C2×Q8×C23—21no—
138C2×Q8×C23—42no—
139C2×Q8×C23C2×D411yescharacteristic
140C2×D4×C23—21no—
141C2×D4×C23—21no—
142C2×D4×C23—22no—
143C2×D4×C23—22no—
144C2×D4×C23—22no—
145C2×D4×C23—21no—
146C2×D4×C23—22no—
147C2×D4×C23—21no—
148C2×D4×C23—42no—
149C2×D4×C23—82no—
150C2×D4×C23C2×D411yescharacteristic
151C2×D4×C23C2×D412yes—
152C22×C4×C23—22no—
153C22×C4×C23—22no—
154C22×C4×C23—22no—
155C22×C4×C23—22no—
156C22×C4×C23—21no—
157C22×C4×C23—21no—
158C22×C4×C23—22no—
159C22×C4×C23—42no—
160C22×C4×C23—41no—
161C22×C4×C23—41no—
162C22×C4×C23—82no—
163C22×C4×C23C2×D412yes—
164C22×C4×C23C2×D411yescharacteristic
165C22×C4×C23C2×D411yescharacteristic
166C22×C4×C23C2×D411yescharacteristic
167C22×C4×C23C2×D412yes—
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2411yescharacteristic
171M16×C23—42no—
172C2×C8×C23—22no—
173C2×C8×C23—22no—
174C2×C8×C23—42no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—21no—
182C4⋊C4×C23—21no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—41no—
189C4⋊C4×C23—41no—
190C4⋊C4×C23—84no—
191C4⋊C4×C23C2×D411yescharacteristic
192C4⋊C4×C23C2×D412yes—
193C4⋊C4×C23C2×D411yescharacteristic
194C22⋊C4×C23—24no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—22no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—21no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—21no—
204C22⋊C4×C23—21no—
205C22⋊C4×C23—22no—
206C22⋊C4×C23—21no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—21no—
209C42×C23—22no—
210C42×C23—21no—
211C42×C23—42no—
212#18270C2311yesmaximal, characteristic, direct product factor, Sylow
213C23×C23—22no—
214C23×C23—22no—
215C23×C23—22no—
216C23×C23—21no—
217C23×C23—41no—
218C23×C23—42no—
219C23×C23—81no—
220C23×C23—81no—
221C23×C23#12011yescharacteristic
222C23×C23#12011yescharacteristic
223C23×C23C22×D411yescharacteristic
224Q8×C23—41no—
225Q8×C23—42no—
226Q8×C23—81no—
227Q8×C23—82no—
228D4×C23—42no—
229D4×C23—41no—
230D4×C23—44no—
231D4×C23—42no—
232D4×C23—42no—
233D4×C23—42no—
234D4×C23—82no—
235D4×C23—81no—
236D4×C23—82no—
237D4×C23—82no—
238C2×C4×C23—22no—
239C2×C4×C23—22no—
240C2×C4×C23—22no—
241C2×C4×C23—24no—
242C2×C4×C23—21no—
243C2×C4×C23—22no—
244C2×C4×C23—21no—
245C2×C4×C23—22no—
246C2×C4×C23—21no—
247C2×C4×C23—22no—
248C2×C4×C23—24no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—42no—
253C2×C4×C23—42no—
254C2×C4×C23—42no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
257C2×C4×C23—44no—
258C2×C4×C23—81no—
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×C23C22×D411yescharacteristic
267C2×C4×C23C22×D411yescharacteristic
268C8×C23—42no—
269C8×C23—42no—
270#3536C2×C2311yescharacteristic
271#3092C2×C2311yescharacteristic, complemented
272C2×#480C2×C2311yescharacteristic, complemented
273#2502C2×C2311yescharacteristic, complemented
274#1610C2×C2314yescomplemented
275#1609C2×C2314yescomplemented
276#1507C2×C2311yescharacteristic, complemented
277#1501C2×C2311yescharacteristic, complemented
278#1472C2×C2311yescharacteristic, complemented
279C22×C23—21no—
280C22×C23—21no—
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—21no—
284C22×C23—21no—
285C22×C23—21no—
286C22×C23—44no—
287C22×C23—41no—
288C22×C23—42no—
289C22×C23—41no—
290C22×C23—42no—
291C22×C23—41no—
292C22×C23—42no—
293C22×C23—162no—
294C22×C23#45612yes—
295C22×C23C2×#12011yescharacteristic
296C22×C23#57612yes—
297C4×C23—21no—
298C4×C23—21no—
299C4×C23—21no—
300C4×C23—44no—
301C4×C23—42no—
302C4×C23—42no—
303C4×C23—41no—
304C4×C23—41no—
305C4×C23—41no—
306C4×C23—42no—
307C4×C23—82no—
308C4×C23#45812yes—
309#562—22no—
310#554—21no—
311#548—21no—
312#547—21no—
313#546—21no—
314#546—22no—
315#543—21no—
316#542—21no—
317#539C22×C2312yes—
318#538—21no—
319#537—21no—
320#537C22×C2312yes—
321#525C22×C2311yescharacteristic
322C2×C22⋊Q8C22×C2311yescharacteristic
323C2×#121C22×C2311yescharacteristic
324C22×C4⋊C4C22×C2311yescharacteristic
325#484—21no—
326#483C22×C2312yescomplemented
327#481—21no—
328#480—22no—
329#480C22×C2314yes—
330#480C22×C2312yescomplemented
331#428—21no—
332C2×#107—21no—
333#417—21no—
334C2×#102—21no—
335#407C22×C2311yescharacteristic
336C2×C22⋊C8C22×C2311yescharacteristic
337#398C22×C2314yescomplemented
338#391—21no—
339#385—22no—
340#384—21no—
341C4×C22⋊C4C22×C2311yescharacteristic
342#328C22×C2312yescomplemented
343#327C22×C2314yes—
344#327C22×C2312yescomplemented
345#324C22×C2314yescomplemented
346C2×C23—22no—
347C2×C23—22no—
348C2×C23—41no—
349C2×C23—81no—
350C2×C23—82no—
351C2×C23#300811yescharacteristic
352C2×C23C2×#45611yescharacteristic
353C2×C23C2×#45811yescharacteristic
354C2×#41—21no—
355C2×#41—22no—
356C2×#41D4×C2311yescharacteristic
357C22×Q8D4×C2311yescharacteristic
358C22×D4—21no—
359C22×D4D4×C2311yescharacteristic
360C23×C4—21no—
361C23×C4C23×C2311yescharacteristic
362C2×M16—21no—
363C22×C8—21no—
364C4⋊Q8—21no—
365#126—22no—
366#125—22no—
367#124—22no—
368#124—21no—
369#124—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373#123—21no—
374#123—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8D4×C2311yescharacteristic
387C22⋊Q8C23×C2311yescharacteristic
388#121—21no—
389#121—22no—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—21no—
394#121—21no—
395#121—22no—
396#121D4×C2312yes—
397#121D4×C2311yescharacteristic
398#121C23×C2312yes—
399#121C23×C2311yescharacteristic
400#120—22no—
401#120—21no—
402C4×Q8—21no—
403C4×D4—22no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—22no—
408#117D4×C2311yescharacteristic
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4D4×C2311yescharacteristic
417C2×C4⋊C4D4×C2311yescharacteristic
418C2×C4⋊C4C23×C2312yes—
419C2×C4⋊C4C23×C2312yes—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4D4×C2312yes—
424C2×C42—21no—
425#107—24no—
426#107—22no—
427#106—22no—
428#103—22no—
429#102—24no—
430#102—22no—
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8C23×C2312yes—
434C22⋊C8C23×C2312yes—
435#95—24no—
436#95C23×C2312yes—
437C23#1827011yescharacteristic, direct product factor, Sylow
438C24—21no—
439C24C2×D4×C2311yescharacteristic
440#41—42no—
441#41—84no—
442#41—82no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—42no—
447C2×Q8C2×D4×C2311yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—82no—
458C2×D4C2×D4×C2311yescharacteristic
459C2×D4C2×D4×C2312yes—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—82no—
471C22×C4C2×D4×C2312yes—
472C22×C4C2×D4×C2311yescharacteristic
473C22×C4C2×D4×C2311yescharacteristic
474C22×C4C2×D4×C2311yescharacteristic
475C22×C4C2×D4×C2312yes—
476C22×C4C2×D4×C2311yescharacteristic
477C22×C4C2×D4×C2311yescharacteristic
478C22×C4C24×C2311yescharacteristic
479M16—42no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—42no—
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—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—84no—
499C4⋊C4C2×D4×C2311yescharacteristic
500C4⋊C4C2×D4×C2312yes—
501C4⋊C4C2×D4×C2311yescharacteristic
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—21no—
517C42—22no—
518C42—21no—
519C42—42no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—21no—
524C23—41no—
525C23—42no—
526C23—81no—
527C23—81no—
528C23C23×#12011yescharacteristic
529C23C23×#12011yescharacteristic
530C23C22×D4×C2311yescharacteristic
531Q8—41no—
532Q8—42no—
533Q8—81no—
534Q8—82no—
535D4—42no—
536D4—41no—
537D4—44no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—82no—
542D4—81no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4C23×#12011yescharacteristic
572C2×C4C23×#12011yescharacteristic
573C2×C4C22×D4×C2311yescharacteristic
574C2×C4C22×D4×C2311yescharacteristic
575C8—42no—
576C8—42no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—44no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—162no—
592C22C23×#45612yes—
593C22C2×C23×#12011yescharacteristic
594C22C23×#57612yes—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—82no—
606C4C23×#45812yes—
607C2—22no—
608C2—22no—
609C2—41no—
610C2—82no—
611C2—81no—
612C2C23×#300811yescharacteristic
613C2C2×C23×#45611yescharacteristic
614C2C2×C23×#45811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3536C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C23×#3092C2semidirect
3. C2×C23×#480C2semidirect
4. C23×#2502C2semidirect
5. C23×#1610C2semidirect
6. C23×#1609C2semidirect
7. C23×#1507C2semidirect
8. C23×#1501C2semidirect
9. C23×#1472C2semidirect
27. C23×#483C22semidirect
31. C23×#480C22semidirect
38. C23×#398C22semidirect
43. C23×#328C22semidirect
45. C23×#327C22semidirect
46. C23×#324C22semidirect
212. #18270C23direct (N × H)
271. #3092C2×C23semidirect
272. C2×#480C2×C23semidirect
273. #2502C2×C23semidirect
274. #1610C2×C23semidirect
275. #1609C2×C23semidirect
276. #1507C2×C23semidirect
277. #1501C2×C23semidirect
278. #1472C2×C23semidirect
326. #483C22×C23semidirect
330. #480C22×C23semidirect
337. #398C22×C23semidirect
342. #328C22×C23semidirect
344. #327C22×C23semidirect
345. #324C22×C23semidirect
437. C23#18270direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35362
C23×#30922
C2×C23×#4802
C23×#25022
C23×#16102
C23×#16092
C23×#15072
C23×#15012
C23×#14722
#1827023