Finite Group Databasebeta

← C23×#18264 #19165890

Subgroups — C23×#18264 #19165890

Order 5888.

1068 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3537C211yesmaximal, characteristic
2C23×#3092C211yesmaximal, characteristic, complemented
3C2×C23×#480C211yesmaximal, characteristic, complemented
4C23×#2504C211yesmaximal, characteristic, complemented
5C23×#1612C214yesmaximal, complemented
6C23×#1607C214yesmaximal, complemented
7C23×#1507C211yesmaximal, characteristic, complemented
8C23×#1501C211yesmaximal, characteristic, complemented
9C23×#1471C211yesmaximal, characteristic, complemented
10C23×#560—22no—
11C23×#553—21no—
12C23×#553—21no—
13D42×C23—22no—
14D42×C23—21no—
15D42×C23—21no—
16C23×#540—21no—
17C23×#538—21no—
18C23×#538C2212yes—
19C23×#535—21no—
20C23×#534—21no—
21C23×#534C2212yes—
22C23×#525C2211yescharacteristic
23C2×C23×#121C2211yescharacteristic
24C2×C23×#121C2211yescharacteristic
25C22×C4⋊C4×C23C2211yescharacteristic, complemented
26C23×#484—21no—
27C23×#483C2212yescomplemented
28C23×#481—21no—
29C23×#480—22no—
30C23×#480C2214yescomplemented
31C23×#480C2212yescomplemented
32C23×#428—21no—
33C2×C23×#107—21no—
34C23×#417—21no—
35C2×C23×#102—21no—
36C23×#407C2211yescharacteristic, complemented
37C2×C23×C22⋊C8C2211yescharacteristic, complemented
38C23×#397C2214yescomplemented
39C23×#390—21no—
40C23×#386—22no—
41C23×#382—21no—
42C4×C22⋊C4×C23C2211yescharacteristic, complemented
43C23×#328C2212yescomplemented
44C23×#327C2212yescomplemented
45C23×#327C2214yescomplemented
46C23×#323C2214yescomplemented
47C2×#41×C23D411yescharacteristic
48C22×D4×C23—22no—
49C22×D4×C23—21no—
50C22×D4×C23—21no—
51C22×D4×C23—22no—
52C22×D4×C23D412yes—
53C22×D4×C23D411yescharacteristic
54C22×D4×C23D411yescharacteristic
55C23×C4×C23—21no—
56C23×C4×C23C2311yescharacteristic
57C2×M16×C23—21no—
58C22×C8×C23—21no—
59C23×#127—22no—
60C23×#127—21no—
61C23×#126—22no—
62C23×#126—22no—
63C23×#125—21no—
64C23×#124—22no—
65C23×#123—22no—
66C23×#123—21no—
67C23×#123—22no—
68C23×#123—22no—
69C23×#123—22no—
70C23×C22⋊Q8D411yescharacteristic
71C23×C22⋊Q8C2311yescharacteristic
72C23×#121—22no—
73C23×#121—21no—
74C23×#121—21no—
75C23×#121—22no—
76C23×#121—22no—
77C23×#121—22no—
78C23×#121—22no—
79C23×#121—21no—
80C23×#121—21no—
81C23×#121—22no—
82C23×#121—21no—
83C23×#121—21no—
84C23×#121—21no—
85C23×#121—21no—
86C23×#121D412yes—
87C23×#121D411yescharacteristic
88C23×#121C2312yes—
89C23×#121C2311yescharacteristic
90C23×#120—22no—
91C23×#120—22no—
92C23×#120—22no—
93C23×#120—21no—
94C23×#120—22no—
95C23×#120—22no—
96C23×#120—22no—
97C4×D4×C23—22no—
98C4×D4×C23—21no—
99C4×D4×C23—21no—
100C4×D4×C23—21no—
101C4×D4×C23—21no—
102C4×D4×C23—22no—
103C23×#117—21no—
104C23×#117D411yescharacteristic
105C2×C4⋊C4×C23—21no—
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23—21no—
108C2×C4⋊C4×C23—21no—
109C2×C4⋊C4×C23—21no—
110C2×C4⋊C4×C23D411yescharacteristic
111C2×C4⋊C4×C23D411yescharacteristic
112C2×C4⋊C4×C23C2312yes—
113C2×C22⋊C4×C23—21no—
114C2×C22⋊C4×C23—21no—
115C2×C22⋊C4×C23—24no—
116C2×C22⋊C4×C23C2312yes—
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—22no—
131C24×C23—21no—
132C24×C23C2×D411yescharacteristic
133C24×C23C2×D412yes—
134#41×C23—42no—
135C2×Q8×C23C2×D411yescharacteristic
136C2×D4×C23—22no—
137C2×D4×C23—21no—
138C2×D4×C23—21no—
139C2×D4×C23—22no—
140C2×D4×C23—22no—
141C2×D4×C23—22no—
142C2×D4×C23—22no—
143C2×D4×C23—22no—
144C2×D4×C23—22no—
145C2×D4×C23—21no—
146C2×D4×C23—22no—
147C2×D4×C23—22no—
148C2×D4×C23—21no—
149C2×D4×C23—21no—
150C2×D4×C23—21no—
151C2×D4×C23—42no—
152C2×D4×C23—42no—
153C2×D4×C23—44no—
154C2×D4×C23—82no—
155C2×D4×C23—82no—
156C2×D4×C23—84no—
157C2×D4×C23—84no—
158C2×D4×C23C2×D411yescharacteristic
159C2×D4×C23C2×D412yes—
160C22×C4×C23—22no—
161C22×C4×C23—22no—
162C22×C4×C23—21no—
163C22×C4×C23—21no—
164C22×C4×C23—22no—
165C22×C4×C23—22no—
166C22×C4×C23—42no—
167C22×C4×C23—41no—
168C22×C4×C23—41no—
169C22×C4×C23—82no—
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2×D411yescharacteristic
172C22×C4×C23C2×D411yescharacteristic
173C22×C4×C23C2×D412yes—
174C22×C4×C23C2×D411yescharacteristic
175C22×C4×C23C2×D411yescharacteristic
176C22×C4×C23C2411yescharacteristic
177M16×C23—42no—
178C2×C8×C23—22no—
179C2×C8×C23—22no—
180C2×C8×C23—42no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—21no—
184C4⋊C4×C23—21no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—41no—
191C4⋊C4×C23—41no—
192C4⋊C4×C23C2×D411yescharacteristic
193C4⋊C4×C23C2×D412yes—
194C4⋊C4×C23C2×D411yescharacteristic
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—21no—
202C22⋊C4×C23—21no—
203C22⋊C4×C23—22no—
204C22⋊C4×C23—21no—
205C22⋊C4×C23—22no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—21no—
209C22⋊C4×C23—42no—
210C22⋊C4×C23—84no—
211C42×C23—22no—
212C42×C23—21no—
213C42×C23—42no—
214#18264C2311yesmaximal, characteristic, direct product factor, Sylow
215C23×C23—22no—
216C23×C23—22no—
217C23×C23—22no—
218C23×C23—22no—
219C23×C23—21no—
220C23×C23—24no—
221C23×C23—41no—
222C23×C23—42no—
223C23×C23—42no—
224C23×C23—42no—
225C23×C23—81no—
226C23×C23—81no—
227C23×C23—82no—
228C23×C23—82no—
229C23×C23—82no—
230C23×C23#12011yescharacteristic
231C23×C23#12011yescharacteristic
232C23×C23C22×D411yescharacteristic
233Q8×C23—41no—
234D4×C23—42no—
235D4×C23—41no—
236D4×C23—44no—
237D4×C23—42no—
238D4×C23—44no—
239D4×C23—42no—
240D4×C23—42no—
241D4×C23—42no—
242D4×C23—81no—
243D4×C23—82no—
244D4×C23—82no—
245D4×C23—81no—
246D4×C23—82no—
247D4×C23—82no—
248D4×C23—82no—
249D4×C23—82no—
250D4×C23—82no—
251C2×C4×C23—22no—
252C2×C4×C23—22no—
253C2×C4×C23—22no—
254C2×C4×C23—21no—
255C2×C4×C23—22no—
256C2×C4×C23—22no—
257C2×C4×C23—21no—
258C2×C4×C23—21no—
259C2×C4×C23—41no—
260C2×C4×C23—41no—
261C2×C4×C23—41no—
262C2×C4×C23—42no—
263C2×C4×C23—42no—
264C2×C4×C23—42no—
265C2×C4×C23—42no—
266C2×C4×C23—42no—
267C2×C4×C23—42no—
268C2×C4×C23—44no—
269C2×C4×C23—81no—
270C2×C4×C23—82no—
271C2×C4×C23—82no—
272C2×C4×C23—82no—
273C2×C4×C23—82no—
274C2×C4×C23#12011yescharacteristic
275C2×C4×C23#12011yescharacteristic
276C2×C4×C23C22×D411yescharacteristic
277C2×C4×C23C22×D411yescharacteristic
278C8×C23—42no—
279C8×C23—42no—
280#3537C2×C2311yescharacteristic
281#3092C2×C2311yescharacteristic, complemented
282C2×#480C2×C2311yescharacteristic, complemented
283#2504C2×C2311yescharacteristic, complemented
284#1612C2×C2314yescomplemented
285#1607C2×C2314yescomplemented
286#1507C2×C2311yescharacteristic, complemented
287#1501C2×C2311yescharacteristic, complemented
288#1471C2×C2311yescharacteristic, complemented
289C22×C23—21no—
290C22×C23—21no—
291C22×C23—21no—
292C22×C23—21no—
293C22×C23—21no—
294C22×C23—21no—
295C22×C23—21no—
296C22×C23—44no—
297C22×C23—42no—
298C22×C23—42no—
299C22×C23—41no—
300C22×C23—42no—
301C22×C23—42no—
302C22×C23—41no—
303C22×C23—41no—
304C22×C23—42no—
305C22×C23—42no—
306C22×C23—84no—
307C22×C23—162no—
308C22×C23—164no—
309C22×C23#45512yes—
310C22×C23C2×#12011yescharacteristic
311C22×C23#57612yes—
312C4×C23—21no—
313C4×C23—21no—
314C4×C23—21no—
315C4×C23—42no—
316C4×C23—41no—
317C4×C23—42no—
318C4×C23—41no—
319C4×C23—41no—
320C4×C23—82no—
321C4×C23—82no—
322C4×C23#45812yes—
323#560—22no—
324#553—21no—
325#553—21no—
326D42—22no—
327D42—21no—
328D42—21no—
329#540—21no—
330#538—21no—
331#538C22×C2312yes—
332#535—21no—
333#534—21no—
334#534C22×C2312yes—
335#525C22×C2311yescharacteristic
336C2×#121C22×C2311yescharacteristic
337C2×#121C22×C2311yescharacteristic
338C22×C4⋊C4C22×C2311yescharacteristic, complemented
339#484—21no—
340#483C22×C2312yescomplemented
341#481—21no—
342#480—22no—
343#480C22×C2314yescomplemented
344#480C22×C2312yescomplemented
345#428—21no—
346C2×#107—21no—
347#417—21no—
348C2×#102—21no—
349#407C22×C2311yescharacteristic, complemented
350C2×C22⋊C8C22×C2311yescharacteristic, complemented
351#397C22×C2314yescomplemented
352#390—21no—
353#386—22no—
354#382—21no—
355C4×C22⋊C4C22×C2311yescharacteristic, complemented
356#328C22×C2312yescomplemented
357#327C22×C2312yescomplemented
358#327C22×C2314yescomplemented
359#323C22×C2314yescomplemented
360C2×C23—22no—
361C2×C23—22no—
362C2×C23—41no—
363C2×C23—82no—
364C2×C23—82no—
365C2×C23—81no—
366C2×C23#300811yescharacteristic
367C2×C23C2×#45511yescharacteristic
368C2×C23C2×#45811yescharacteristic
369C2×#41D4×C2311yescharacteristic
370C22×D4—22no—
371C22×D4—21no—
372C22×D4—21no—
373C22×D4—22no—
374C22×D4D4×C2312yes—
375C22×D4D4×C2311yescharacteristic
376C22×D4D4×C2311yescharacteristic
377C23×C4—21no—
378C23×C4C23×C2311yescharacteristic
379C2×M16—21no—
380C22×C8—21no—
381#127—22no—
382#127—21no—
383#126—22no—
384#126—22no—
385#125—21no—
386#124—22no—
387#123—22no—
388#123—21no—
389#123—22no—
390#123—22no—
391#123—22no—
392C22⋊Q8D4×C2311yescharacteristic
393C22⋊Q8C23×C2311yescharacteristic
394#121—22no—
395#121—21no—
396#121—21no—
397#121—22no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121—21no—
402#121—21no—
403#121—21no—
404#121—21no—
405#121—21no—
406#121—21no—
407#121—22no—
408#121D4×C2312yes—
409#121D4×C2311yescharacteristic
410#121C23×C2312yes—
411#121C23×C2311yescharacteristic
412#120—22no—
413#120—22no—
414#120—22no—
415#120—21no—
416#120—22no—
417#120—22no—
418#120—22no—
419C4×D4—22no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—21no—
424C4×D4—22no—
425#117—21no—
426#117D4×C2311yescharacteristic
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4D4×C2311yescharacteristic
433C2×C4⋊C4D4×C2311yescharacteristic
434C2×C4⋊C4C23×C2312yes—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4C23×C2312yes—
439C2×C42—21no—
440#107—24no—
441#107—22no—
442#106—22no—
443#103—22no—
444#102—24no—
445#102—22no—
446C22⋊C8—21no—
447C22⋊C8—21no—
448C22⋊C8C23×C2312yes—
449C22⋊C8C23×C2312yes—
450#95—24no—
451#95C23×C2312yes—
452C23#1826411yescharacteristic, direct product factor, Sylow
453C24—22no—
454C24—21no—
455C24C2×D4×C2311yescharacteristic
456C24C2×D4×C2312yes—
457#41—42no—
458C2×Q8C2×D4×C2311yescharacteristic
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—44no—
477C2×D4—82no—
478C2×D4—82no—
479C2×D4—84no—
480C2×D4—84no—
481C2×D4C2×D4×C2311yescharacteristic
482C2×D4C2×D4×C2312yes—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—42no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—82no—
493C22×C4C2×D4×C2311yescharacteristic
494C22×C4C2×D4×C2311yescharacteristic
495C22×C4C2×D4×C2311yescharacteristic
496C22×C4C2×D4×C2312yes—
497C22×C4C2×D4×C2311yescharacteristic
498C22×C4C2×D4×C2311yescharacteristic
499C22×C4C24×C2311yescharacteristic
500M16—42no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—42no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4C2×D4×C2311yescharacteristic
516C4⋊C4C2×D4×C2312yes—
517C4⋊C4C2×D4×C2311yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—42no—
533C22⋊C4—84no—
534C42—22no—
535C42—21no—
536C42—42no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—21no—
542C23—24no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—81no—
548C23—81no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23C23×#12011yescharacteristic
553C23C23×#12011yescharacteristic
554C23C22×D4×C2311yescharacteristic
555Q8—41no—
556D4—42no—
557D4—41no—
558D4—44no—
559D4—42no—
560D4—44no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—81no—
565D4—82no—
566D4—82no—
567D4—81no—
568D4—82no—
569D4—82no—
570D4—82no—
571D4—82no—
572D4—82no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4C23×#12011yescharacteristic
597C2×C4C23×#12011yescharacteristic
598C2×C4C22×D4×C2311yescharacteristic
599C2×C4C22×D4×C2311yescharacteristic
600C8—42no—
601C8—42no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—84no—
620C22—162no—
621C22—164no—
622C22C23×#45512yes—
623C22C2×C23×#12011yescharacteristic
624C22C23×#57612yes—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—82no—
634C4—82no—
635C4C23×#45812yes—
636C2—22no—
637C2—22no—
638C2—41no—
639C2—82no—
640C2—82no—
641C2—81no—
642C2C23×#300811yescharacteristic
643C2C2×C23×#45511yescharacteristic
644C2C2×C23×#45811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3537C2
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×#2504C2semidirect
5. C23×#1612C2semidirect
6. C23×#1607C2semidirect
7. C23×#1507C2semidirect
8. C23×#1501C2semidirect
9. C23×#1471C2semidirect
25. C22×C4⋊C4×C23C22semidirect
27. C23×#483C22semidirect
30. C23×#480C22semidirect
31. C23×#480C22semidirect
36. C23×#407C22semidirect
37. C2×C23×C22⋊C8C22semidirect
38. C23×#397C22semidirect
42. C4×C22⋊C4×C23C22semidirect
43. C23×#328C22semidirect
44. C23×#327C22semidirect
45. C23×#327C22semidirect
46. C23×#323C22semidirect
214. #18264C23direct (N × H)
281. #3092C2×C23semidirect
282. C2×#480C2×C23semidirect
283. #2504C2×C23semidirect
284. #1612C2×C23semidirect
285. #1607C2×C23semidirect
286. #1507C2×C23semidirect
287. #1501C2×C23semidirect
288. #1471C2×C23semidirect
338. C22×C4⋊C4C22×C23semidirect
340. #483C22×C23semidirect
343. #480C22×C23semidirect
344. #480C22×C23semidirect
349. #407C22×C23semidirect
350. C2×C22⋊C8C22×C23semidirect
351. #397C22×C23semidirect
355. C4×C22⋊C4C22×C23semidirect
356. #328C22×C23semidirect
357. #327C22×C23semidirect
358. #327C22×C23semidirect
359. #323C22×C23semidirect
452. C23#18264direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35372
C23×#30922
C2×C23×#4802
C23×#25042
C23×#16122
C23×#16072
C23×#15072
C23×#15012
C23×#14712
#1826423