Finite Group Databasebeta

← C23×#18170 #19165796

Subgroups — C23×#18170 #19165796

Order 5888.

1188 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3490C211yesmaximal, characteristic
2C23×#3092C212yesmaximal, complemented
3C2×C23×#399C211yesmaximal, characteristic, complemented
4C23×#1618C214yesmaximal, complemented
5C23×#1614C212yesmaximal, complemented
6C23×#1613C212yesmaximal, complemented
7C23×#1504C212yesmaximal, complemented
8C23×#1462C211yesmaximal, characteristic, complemented
9C23×#551—22no—
10C23×#551—22no—
11C23×#546—22no—
12C23×#546—22no—
13C23×#543C2211yescharacteristic
14C23×#538—21no—
15C23×#538C2212yes—
16C23×#534C2211yescharacteristic
17C23×#525—21no—
18C23×#525C2212yes—
19C2×C23×#120—21no—
20C23×#518—21no—
21C22×C4⋊C4×C23—21no—
22C22×C4⋊C4×C23C2211yescharacteristic, complemented
23C22×C22⋊C4×C23—21no—
24C22×C22⋊C4×C23C2211yescharacteristic
25C23×#484C2214yescomplemented
26C23×#483—22no—
27C23×#481C2214yescomplemented
28C23×#480—22no—
29C23×#428—22no—
30C23×#417—22no—
31C23×#407C2212yescomplemented
32C23×#399—21no—
33C23×#399—21no—
34C23×#399C2212yes—
35C23×#399C2212yescomplemented
36C2×C23×#95C2211yescharacteristic, complemented
37C23×#328C2214yescomplemented
38C23×#327C2214yescomplemented
39C23×#323C2212yes—
40C23×#323C2212yescomplemented
41C25×C23D411yescharacteristic
42C2×#41×C23—21no—
43C2×#41×C23D412yes—
44C22×D4×C23—21no—
45C22×D4×C23—21no—
46C22×D4×C23—21no—
47C23×C4×C23—21no—
48C23×C4×C23—21no—
49C23×C4×C23D411yescharacteristic
50C23×C4×C23C2311yescharacteristic
51C2×M16×C23—22no—
52C23×#126—22no—
53C23×#126—22no—
54C23×#124—22no—
55C23×#124—22no—
56C23×#123—22no—
57C23×#123—22no—
58C23×#123—22no—
59C23×#123—22no—
60C23×#123—22no—
61C23×C22⋊Q8—21no—
62C23×C22⋊Q8—22no—
63C23×C22⋊Q8—21no—
64C23×C22⋊Q8—22no—
65C23×C22⋊Q8D412yes—
66C23×C22⋊Q8C2312yes—
67C23×#121—21no—
68C23×#121—22no—
69C23×#121—22no—
70C23×#121—21no—
71C23×#121D412yes—
72C23×#121C2312yes—
73C23×#120—22no—
74C23×#120—22no—
75C23×#120—22no—
76C23×#120—22no—
77C23×#120—22no—
78C4×D4×C23—22no—
79C4×D4×C23—22no—
80C4×D4×C23—22no—
81C4×D4×C23—22no—
82C4×D4×C23—22no—
83C4×D4×C23—22no—
84C23×#117—22no—
85C23×#117—21no—
86C23×#117D412yes—
87C2×C4⋊C4×C23—22no—
88C2×C4⋊C4×C23—22no—
89C2×C4⋊C4×C23—21no—
90C2×C4⋊C4×C23—22no—
91C2×C4⋊C4×C23—22no—
92C2×C4⋊C4×C23—44no—
93C2×C4⋊C4×C23C2312yes—
94C2×C22⋊C4×C23—21no—
95C2×C22⋊C4×C23—21no—
96C2×C22⋊C4×C23—22no—
97C2×C22⋊C4×C23—21no—
98C2×C22⋊C4×C23—22no—
99C2×C22⋊C4×C23—22no—
100C2×C22⋊C4×C23—22no—
101C2×C22⋊C4×C23—22no—
102C2×C22⋊C4×C23—44no—
103C2×C22⋊C4×C23D411yescharacteristic
104C2×C22⋊C4×C23D411yescharacteristic
105C2×C22⋊C4×C23C2311yescharacteristic
106C2×C22⋊C4×C23C2311yescharacteristic
107C23×#107—24no—
108C23×#106—24no—
109C23×#103—24no—
110C23×#102—24no—
111C23×C22⋊C8—22no—
112C23×C22⋊C8C2314yes—
113C23×#95—21no—
114C23×#95C2312yes—
115C24×C23—21no—
116C24×C23—22no—
117C24×C23—21no—
118C24×C23—21no—
119C24×C23—41no—
120C24×C23—42no—
121C24×C23—42no—
122C24×C23C2×D411yescharacteristic
123C24×C23C2×D411yescharacteristic
124C24×C23C2×D411yescharacteristic
125#41×C23—44no—
126#41×C23—82no—
127C2×Q8×C23—21no—
128C2×Q8×C23C2×D412yes—
129C2×D4×C23—22no—
130C2×D4×C23—22no—
131C2×D4×C23—22no—
132C2×D4×C23—22no—
133C2×D4×C23—21no—
134C2×D4×C23—22no—
135C2×D4×C23—22no—
136C2×D4×C23—42no—
137C2×D4×C23—42no—
138C2×D4×C23—82no—
139C2×D4×C23—82no—
140C2×D4×C23C2×D412yes—
141C22×C4×C23—21no—
142C22×C4×C23—24no—
143C22×C4×C23—21no—
144C22×C4×C23—22no—
145C22×C4×C23—21no—
146C22×C4×C23—21no—
147C22×C4×C23—22no—
148C22×C4×C23—21no—
149C22×C4×C23—22no—
150C22×C4×C23—22no—
151C22×C4×C23—42no—
152C22×C4×C23—42no—
153C22×C4×C23—42no—
154C22×C4×C23—42no—
155C22×C4×C23—42no—
156C22×C4×C23—42no—
157C22×C4×C23—41no—
158C22×C4×C23—41no—
159C22×C4×C23C2×D412yes—
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×D411yescharacteristic
162C22×C4×C23C2×D411yescharacteristic
163C22×C4×C23C2×D412yes—
164C22×C4×C23C2411yescharacteristic
165M16×C23—44no—
166C2×C8×C23—24no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—21no—
169C4⋊C4×C23—21no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—24no—
174C4⋊C4×C23—44no—
175C4⋊C4×C23—44no—
176C4⋊C4×C23—42no—
177C4⋊C4×C23C2×D412yes—
178C4⋊C4×C23C2×D412yes—
179C22⋊C4×C23—22no—
180C22⋊C4×C23—22no—
181C22⋊C4×C23—22no—
182C22⋊C4×C23—22no—
183C22⋊C4×C23—22no—
184C22⋊C4×C23—22no—
185C22⋊C4×C23—22no—
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—44no—
194C22⋊C4×C23—44no—
195C42×C23—22no—
196C42×C23—22no—
197#18170C2311yesmaximal, characteristic, direct product factor, Sylow
198C23×C23—21no—
199C23×C23—21no—
200C23×C23—22no—
201C23×C23—22no—
202C23×C23—21no—
203C23×C23—21no—
204C23×C23—21no—
205C23×C23—22no—
206C23×C23—21no—
207C23×C23—21no—
208C23×C23—42no—
209C23×C23—42no—
210C23×C23—41no—
211C23×C23—41no—
212C23×C23—41no—
213C23×C23—41no—
214C23×C23—48no—
215C23×C23—42no—
216C23×C23—42no—
217C23×C23—42no—
218C23×C23—81no—
219C23×C23—81no—
220C23×C23—84no—
221C23×C23—81no—
222C23×C23—81no—
223C23×C23#12011yescharacteristic
224C23×C23#12011yescharacteristic
225C23×C23#13712yes—
226C23×C23C22×D411yescharacteristic
227Q8×C23—42no—
228Q8×C23—81no—
229D4×C23—42no—
230D4×C23—44no—
231D4×C23—42no—
232D4×C23—42no—
233D4×C23—44no—
234D4×C23—44no—
235D4×C23—82no—
236D4×C23—82no—
237D4×C23—81no—
238C2×C4×C23—24no—
239C2×C4×C23—24no—
240C2×C4×C23—22no—
241C2×C4×C23—21no—
242C2×C4×C23—22no—
243C2×C4×C23—22no—
244C2×C4×C23—21no—
245C2×C4×C23—42no—
246C2×C4×C23—41no—
247C2×C4×C23—41no—
248C2×C4×C23—48no—
249C2×C4×C23—42no—
250C2×C4×C23—42no—
251C2×C4×C23—42no—
252C2×C4×C23—42no—
253C2×C4×C23—41no—
254C2×C4×C23—41no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
257C2×C4×C23—42no—
258C2×C4×C23—82no—
259C2×C4×C23—81no—
260C2×C4×C23—81no—
261C2×C4×C23—81no—
262C2×C4×C23—82no—
263C2×C4×C23—82no—
264C2×C4×C23—82no—
265C2×C4×C23#12012yes—
266C2×C4×C23C22×D412yes—
267C8×C23—44no—
268#3490C2×C2311yescharacteristic
269#3092C2×C2312yescomplemented
270C2×#399C2×C2311yescharacteristic, complemented
271#1618C2×C2314yescomplemented
272#1614C2×C2312yescomplemented
273#1613C2×C2312yescomplemented
274#1504C2×C2312yescomplemented
275#1462C2×C2311yescharacteristic, complemented
276C22×C23—24no—
277C22×C23—24no—
278C22×C23—21no—
279C22×C23—21no—
280C22×C23—21no—
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—21no—
284C22×C23—22no—
285C22×C23—42no—
286C22×C23—42no—
287C22×C23—41no—
288C22×C23—41no—
289C22×C23—42no—
290C22×C23—41no—
291C22×C23—42no—
292C22×C23—41no—
293C22×C23—42no—
294C22×C23—41no—
295C22×C23—81no—
296C22×C23—84no—
297C22×C23—84no—
298C22×C23—81no—
299C22×C23—162no—
300C22×C23#45312yes—
301C22×C23#45812yes—
302C22×C23C2×#12011yescharacteristic
303C22×C23C2×#13711yescharacteristic
304C22×C23#57511yescharacteristic
305C4×C23—22no—
306C4×C23—22no—
307C4×C23—44no—
308C4×C23—42no—
309C4×C23—41no—
310C4×C23—41no—
311C4×C23—42no—
312C4×C23—42no—
313C4×C23—82no—
314#551—22no—
315#551—22no—
316#546—22no—
317#546—22no—
318#543C22×C2311yescharacteristic
319#538—21no—
320#538C22×C2312yes—
321#534C22×C2311yescharacteristic
322#525—21no—
323#525C22×C2312yes—
324C2×#120—21no—
325#518—21no—
326C22×C4⋊C4—21no—
327C22×C4⋊C4C22×C2311yescharacteristic, complemented
328C22×C22⋊C4—21no—
329C22×C22⋊C4C22×C2311yescharacteristic
330#484C22×C2314yescomplemented
331#483—22no—
332#481C22×C2314yescomplemented
333#480—22no—
334#428—22no—
335#417—22no—
336#407C22×C2312yescomplemented
337#399—21no—
338#399—21no—
339#399C22×C2312yes—
340#399C22×C2312yescomplemented
341C2×#95C22×C2311yescharacteristic, complemented
342#328C22×C2314yescomplemented
343#327C22×C2314yescomplemented
344#323C22×C2312yes—
345#323C22×C2312yescomplemented
346C2×C23—21no—
347C2×C23—21no—
348C2×C23—22no—
349C2×C23—22no—
350C2×C23—42no—
351C2×C23—81no—
352C2×C23—82no—
353C2×C23#300711yescharacteristic
354C2×C23C2×#45311yescharacteristic
355C2×C23C2×#45811yescharacteristic
356C25D4×C2311yescharacteristic
357C2×#41—21no—
358C2×#41D4×C2312yes—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4—21no—
362C23×C4—21no—
363C23×C4—21no—
364C23×C4D4×C2311yescharacteristic
365C23×C4C23×C2311yescharacteristic
366C2×M16—22no—
367#126—22no—
368#126—22no—
369#124—22no—
370#124—22no—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375#123—22no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8D4×C2312yes—
381C22⋊Q8C23×C2312yes—
382#121—21no—
383#121—22no—
384#121—22no—
385#121—21no—
386#121D4×C2312yes—
387#121C23×C2312yes—
388#120—22no—
389#120—22no—
390#120—22no—
391#120—22no—
392#120—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399#117—22no—
400#117—21no—
401#117D4×C2312yes—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—44no—
408C2×C4⋊C4C23×C2312yes—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—44no—
418C2×C22⋊C4D4×C2311yescharacteristic
419C2×C22⋊C4D4×C2311yescharacteristic
420C2×C22⋊C4C23×C2311yescharacteristic
421C2×C22⋊C4C23×C2311yescharacteristic
422#107—24no—
423#106—24no—
424#103—24no—
425#102—24no—
426C22⋊C8—22no—
427C22⋊C8C23×C2314yes—
428#95—21no—
429#95C23×C2312yes—
430C23#1817011yescharacteristic, direct product factor, Sylow
431C24—21no—
432C24—22no—
433C24—21no—
434C24—21no—
435C24—41no—
436C24—42no—
437C24—42no—
438C24C2×D4×C2311yescharacteristic
439C24C2×D4×C2311yescharacteristic
440C24C2×D4×C2311yescharacteristic
441#41—44no—
442#41—82no—
443C2×Q8—21no—
444C2×Q8C2×D4×C2312yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4C2×D4×C2312yes—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4C2×D4×C2312yes—
476C22×C4C2×D4×C2311yescharacteristic
477C22×C4C2×D4×C2311yescharacteristic
478C22×C4C2×D4×C2311yescharacteristic
479C22×C4C2×D4×C2312yes—
480C22×C4C24×C2311yescharacteristic
481M16—44no—
482C2×C8—24no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4C2×D4×C2312yes—
494C4⋊C4C2×D4×C2312yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C42—22no—
512C42—22no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—22no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—48no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—81no—
534C23—81no—
535C23—84no—
536C23—81no—
537C23—81no—
538C23C23×#12011yescharacteristic
539C23C23×#12011yescharacteristic
540C23C23×#13712yes—
541C23C22×D4×C2311yescharacteristic
542Q8—42no—
543Q8—81no—
544D4—42no—
545D4—44no—
546D4—42no—
547D4—42no—
548D4—44no—
549D4—44no—
550D4—82no—
551D4—82no—
552D4—81no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4C23×#12012yes—
581C2×C4C22×D4×C2312yes—
582C8—44no—
583C22—24no—
584C22—24no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—22no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—41no—
600C22—42no—
601C22—41no—
602C22—81no—
603C22—84no—
604C22—84no—
605C22—81no—
606C22—162no—
607C22C23×#45312yes—
608C22C23×#45812yes—
609C22C2×C23×#12011yescharacteristic
610C22C2×C23×#13711yescharacteristic
611C22C23×#57511yescharacteristic
612C4—22no—
613C4—22no—
614C4—44no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—82no—
621C2—21no—
622C2—21no—
623C2—22no—
624C2—22no—
625C2—42no—
626C2—81no—
627C2—82no—
628C2C23×#300711yescharacteristic
629C2C2×C23×#45311yescharacteristic
630C2C2×C23×#45811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3490C2
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×#399C2semidirect
4. C23×#1618C2semidirect
5. C23×#1614C2semidirect
6. C23×#1613C2semidirect
7. C23×#1504C2semidirect
8. C23×#1462C2semidirect
22. C22×C4⋊C4×C23C22semidirect
25. C23×#484C22semidirect
27. C23×#481C22semidirect
31. C23×#407C22semidirect
35. C23×#399C22semidirect
36. C2×C23×#95C22semidirect
37. C23×#328C22semidirect
38. C23×#327C22semidirect
40. C23×#323C22semidirect
197. #18170C23direct (N × H)
269. #3092C2×C23semidirect
270. C2×#399C2×C23semidirect
271. #1618C2×C23semidirect
272. #1614C2×C23semidirect
273. #1613C2×C23semidirect
274. #1504C2×C23semidirect
275. #1462C2×C23semidirect
327. C22×C4⋊C4C22×C23semidirect
330. #484C22×C23semidirect
332. #481C22×C23semidirect
336. #407C22×C23semidirect
340. #399C22×C23semidirect
341. C2×#95C22×C23semidirect
342. #328C22×C23semidirect
343. #327C22×C23semidirect
345. #323C22×C23semidirect
430. C23#18170direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34902
C23×#30922
C2×C23×#3992
C23×#16182
C23×#16142
C23×#16132
C23×#15042
C23×#14622
#1817023