Finite Group Databasebeta

← C23×#18243 #19165869

Subgroups — C23×#18243 #19165869

Order 5888.

928 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3561C211yesmaximal, characteristic
2C23×#3117C211yesmaximal, characteristic, complemented
3C23×#3104C211yesmaximal, characteristic, complemented
4C23×#2632C211yesmaximal, characteristic, complemented
5C23×#1618C212yesmaximal, complemented
6C23×#1614C212yesmaximal, complemented
7C23×#1606C212yesmaximal, complemented
8C23×#1602C212yesmaximal, complemented
9C23×#1521C211yesmaximal, characteristic, complemented
10C23×#1515C211yesmaximal, characteristic, complemented
11C23×#1469C211yesmaximal, characteristic, complemented
12C23×#563—21no—
13C23×#562—21no—
14C23×#556—21no—
15C23×#551—22no—
16D4×Q8×C23—21no—
17C23×#548—21no—
18C23×#546—21no—
19C23×#543C2212yes—
20C23×#542—21no—
21C23×#542—22no—
22C23×#538C2212yes—
23C23×#536—21no—
24C23×#533C2211yescharacteristic
25C23×#529C2211yescharacteristic
26C2×C23×#124C2211yescharacteristic
27C2×C4×Q8×C23C2211yescharacteristic
28C23×#484C2212yes—
29C23×#484C2211yescharacteristic
30C23×#481C2211yescharacteristic
31C23×#479—21no—
32C23×#477—21no—
33C23×#464—21no—
34C23×#462—21no—
35C23×#451C2212yes—
36C23×#451C2211yescharacteristic
37C23×#448C2211yescharacteristic
38C23×#441—21no—
39C23×#440—21no—
40C4×Q16×C23—22no—
41C23×#432C2211yescharacteristic
42C23×#431C2211yescharacteristic
43C23×#399C2212yes—
44C23×#395—21no—
45C23×#394—21no—
46C23×#393C2212yes—
47C23×#388—22no—
48C23×#383C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#328C2212yes—
52C23×#327C2212yes—
53C23×#323C2212yes—
54C23×#323C2212yes—
55C2×#41×C23—21no—
56C22×Q8×C23—21no—
57C22×Q8×C23D411yescharacteristic
58C22×Q8×C23C2311yescharacteristic
59C22×D4×C23D411yescharacteristic
60C2×Q16×C23—22no—
61C2×Q16×C23—21no—
62C2×SD16×C23—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—22no—
65C23×C4⋊Q8D411yescharacteristic
66C23×C4⋊Q8D411yescharacteristic
67C23×#126—22no—
68C23×#126—22no—
69C23×#126—22no—
70C23×#126—21no—
71C23×#125—21no—
72C23×#125D411yescharacteristic
73C23×#124—22no—
74C23×#124—21no—
75C23×#124—22no—
76C23×#124—21no—
77C23×#124—21no—
78C23×#124—21no—
79C23×#124—21no—
80C23×#124D411yescharacteristic
81C23×#123—22no—
82C23×#123—22no—
83C23×#123—22no—
84C23×#123—21no—
85C23×#123—22no—
86C23×C22⋊Q8—21no—
87C23×C22⋊Q8—22no—
88C23×C22⋊Q8—22no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—21no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8C2311yescharacteristic
96C23×C22⋊Q8C2312yes—
97C23×#121—21no—
98C23×#121—22no—
99C23×#121—22no—
100C23×#121C2311yescharacteristic
101C23×#120—22no—
102C4×Q8×C23—21no—
103C4×Q8×C23—21no—
104C4×Q8×C23—21no—
105C4×Q8×C23—21no—
106C4×Q8×C23—21no—
107C4×Q8×C23—21no—
108C4×Q8×C23D411yescharacteristic
109C4×Q8×C23D412yes—
110C4×D4×C23—22no—
111C4×D4×C23—21no—
112C4×D4×C23—22no—
113C4×D4×C23—21no—
114C4×D4×C23D411yescharacteristic
115C23×#117—22no—
116C23×#117—21no—
117C23×#117—21no—
118C2×C4⋊C4×C23—21no—
119C2×C4⋊C4×C23—22no—
120C2×C4⋊C4×C23C2311yescharacteristic
121C2×C22⋊C4×C23—22no—
122C2×C22⋊C4×C23—22no—
123C2×C22⋊C4×C23—22no—
124C2×C22⋊C4×C23D412yes—
125C2×C22⋊C4×C23C2312yes—
126C2×C42×C23—21no—
127C2×C42×C23C2311yescharacteristic
128C23×#107—22no—
129C23×#107—21no—
130C23×#106—21no—
131C23×C4⋊C8—21no—
132C23×C4⋊C8—21no—
133C23×#103—22no—
134C23×#103—21no—
135C23×#103—22no—
136C23×#103—21no—
137C23×#103—21no—
138C23×#102—21no—
139C23×C22⋊C8C2312yes—
140C23×C22⋊C8C2311yescharacteristic
141C23×C22⋊C8C2311yescharacteristic
142C23×#97—21no—
143C4×C8×C23—21no—
144C23×#95—21no—
145C23×#95—21no—
146C23×#95—22no—
147C23×#95C2312yes—
148C24×C23C2×D412yes—
149#41×C23—82no—
150C2×Q8×C23—22no—
151C2×Q8×C23—21no—
152C2×Q8×C23—21no—
153C2×Q8×C23—21no—
154C2×Q8×C23—21no—
155C2×Q8×C23—21no—
156C2×Q8×C23—42no—
157C2×Q8×C23—81no—
158C2×Q8×C23—82no—
159C2×Q8×C23C2×D411yescharacteristic
160C2×Q8×C23C2×D412yes—
161C2×D4×C23—21no—
162C2×D4×C23—22no—
163C2×D4×C23—22no—
164C2×D4×C23—21no—
165C2×D4×C23—21no—
166C2×D4×C23—81no—
167C2×D4×C23C2×D411yescharacteristic
168C22×C4×C23—21no—
169C22×C4×C23—22no—
170C22×C4×C23—22no—
171C22×C4×C23—22no—
172C22×C4×C23—21no—
173C22×C4×C23—21no—
174C22×C4×C23—22no—
175C22×C4×C23C2×D411yescharacteristic
176C22×C4×C23C2×D412yes—
177C22×C4×C23C2×D411yescharacteristic
178C22×C4×C23C2411yescharacteristic
179Q16×C23—42no—
180Q16×C23—44no—
181SD16×C23—42no—
182C2×C8×C23—21no—
183C2×C8×C23—21no—
184C2×C8×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—21no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—22no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—21no—
199C4⋊C4×C23—21no—
200C4⋊C4×C23—41no—
201C4⋊C4×C23—41no—
202C4⋊C4×C23—82no—
203C4⋊C4×C23C2×D411yescharacteristic
204C4⋊C4×C23C2×D411yescharacteristic
205C4⋊C4×C23C2×D412yes—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—22no—
213C22⋊C4×C23—22no—
214C22⋊C4×C23—22no—
215C22⋊C4×C23—21no—
216C22⋊C4×C23—21no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—22no—
219C22⋊C4×C23—22no—
220C22⋊C4×C23—21no—
221C22⋊C4×C23—44no—
222C22⋊C4×C23—82no—
223C42×C23—22no—
224C42×C23—22no—
225C42×C23—21no—
226C42×C23—41no—
227C42×C23—41no—
228C42×C23C2×D411yescharacteristic
229C42×C23C2×D411yescharacteristic
230C42×C23C2×D411yescharacteristic
231C42×C23C2×D411yescharacteristic
232#18243C2311yesmaximal, characteristic, direct product factor, Sylow
233C23×C23—22no—
234C23×C23—21no—
235C23×C23—24no—
236C23×C23—42no—
237C23×C23—42no—
238C23×C23C22×D411yescharacteristic
239Q8×C23—22no—
240Q8×C23—22no—
241Q8×C23—41no—
242Q8×C23—42no—
243Q8×C23—41no—
244Q8×C23—41no—
245Q8×C23—42no—
246Q8×C23—42no—
247Q8×C23—81no—
248Q8×C23—82no—
249D4×C23—41no—
250D4×C23—42no—
251D4×C23—42no—
252D4×C23—81no—
253D4×C23—82no—
254D4×C23—82no—
255C2×C4×C23—22no—
256C2×C4×C23—22no—
257C2×C4×C23—21no—
258C2×C4×C23—22no—
259C2×C4×C23—21no—
260C2×C4×C23—22no—
261C2×C4×C23—22no—
262C2×C4×C23—21no—
263C2×C4×C23—41no—
264C2×C4×C23—41no—
265C2×C4×C23—42no—
266C2×C4×C23—41no—
267C2×C4×C23—41no—
268C2×C4×C23—42no—
269C2×C4×C23—42no—
270C2×C4×C23—42no—
271C2×C4×C23—42no—
272C2×C4×C23—41no—
273C2×C4×C23—42no—
274C2×C4×C23—82no—
275C2×C4×C23—82no—
276C2×C4×C23—82no—
277C2×C4×C23#12011yescharacteristic
278C2×C4×C23#12011yescharacteristic
279C2×C4×C23#12011yescharacteristic
280C2×C4×C23#12011yescharacteristic
281C2×C4×C23C22×D411yescharacteristic
282C2×C4×C23C22×D411yescharacteristic
283C8×C23—41no—
284C8×C23—41no—
285C8×C23—42no—
286#3561C2×C2311yescharacteristic
287#3117C2×C2311yescharacteristic, complemented
288#3104C2×C2311yescharacteristic, complemented
289#2632C2×C2311yescharacteristic, complemented
290#1618C2×C2312yescomplemented
291#1614C2×C2312yescomplemented
292#1606C2×C2312yescomplemented
293#1602C2×C2312yescomplemented
294#1521C2×C2311yescharacteristic, complemented
295#1515C2×C2311yescharacteristic, complemented
296#1469C2×C2311yescharacteristic, complemented
297C22×C23—21no—
298C22×C23—21no—
299C22×C23—41no—
300C22×C23—42no—
301C22×C23—41no—
302C22×C23—42no—
303C22×C23—42no—
304C22×C23—41no—
305C22×C23—84no—
306C22×C23C2×#12011yescharacteristic
307C22×C23#57812yes—
308C4×C23—21no—
309C4×C23—21no—
310C4×C23—21no—
311C4×C23—21no—
312C4×C23—21no—
313C4×C23—21no—
314C4×C23—41no—
315C4×C23—41no—
316C4×C23—42no—
317C4×C23—42no—
318C4×C23—42no—
319C4×C23—41no—
320C4×C23—42no—
321C4×C23—82no—
322C4×C23—81no—
323#563—21no—
324#562—21no—
325#556—21no—
326#551—22no—
327D4×Q8—21no—
328#548—21no—
329#546—21no—
330#543C22×C2312yes—
331#542—21no—
332#542—22no—
333#538C22×C2312yes—
334#536—21no—
335#533C22×C2311yescharacteristic
336#529C22×C2311yescharacteristic
337C2×#124C22×C2311yescharacteristic
338C2×C4×Q8C22×C2311yescharacteristic
339#484C22×C2312yes—
340#484C22×C2311yescharacteristic
341#481C22×C2311yescharacteristic
342#479—21no—
343#477—21no—
344#464—21no—
345#462—21no—
346#451C22×C2312yes—
347#451C22×C2311yescharacteristic
348#448C22×C2311yescharacteristic
349#441—21no—
350#440—21no—
351C4×Q16—22no—
352#432C22×C2311yescharacteristic
353#431C22×C2311yescharacteristic
354#399C22×C2312yes—
355#395—21no—
356#394—21no—
357#393C22×C2312yes—
358#388—22no—
359#383C22×C2311yescharacteristic
360#328C22×C2312yes—
361#328C22×C2312yes—
362#328C22×C2312yes—
363#327C22×C2312yes—
364#323C22×C2312yes—
365#323C22×C2312yes—
366C2×C23—22no—
367C2×C23—81no—
368C2×C23—82no—
369C2×C23#301111yescharacteristic
370C2×C23#301911yescharacteristic
371C2×C23#302511yescharacteristic
372C2×#41—21no—
373C22×Q8—21no—
374C22×Q8D4×C2311yescharacteristic
375C22×Q8C23×C2311yescharacteristic
376C22×D4D4×C2311yescharacteristic
377C2×Q16—22no—
378C2×Q16—21no—
379C2×SD16—21no—
380C4⋊Q8—21no—
381C4⋊Q8—22no—
382C4⋊Q8D4×C2311yescharacteristic
383C4⋊Q8D4×C2311yescharacteristic
384#126—22no—
385#126—22no—
386#126—22no—
387#126—21no—
388#125—21no—
389#125D4×C2311yescharacteristic
390#124—21no—
391#124—22no—
392#124—22no—
393#124—21no—
394#124—21no—
395#124—21no—
396#124—21no—
397#124D4×C2311yescharacteristic
398#123—22no—
399#123—22no—
400#123—22no—
401#123—21no—
402#123—22no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408C22⋊Q8—21no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8C23×C2311yescharacteristic
413C22⋊Q8C23×C2312yes—
414#121—21no—
415#121—22no—
416#121—22no—
417#121C23×C2311yescharacteristic
418#120—22no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8D4×C2311yescharacteristic
426C4×Q8D4×C2312yes—
427C4×D4—21no—
428C4×D4—22no—
429C4×D4—21no—
430C4×D4—22no—
431C4×D4D4×C2311yescharacteristic
432#117—22no—
433#117—21no—
434#117—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—22no—
437C2×C4⋊C4C23×C2311yescharacteristic
438C2×C22⋊C4—22no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4D4×C2312yes—
442C2×C22⋊C4C23×C2312yes—
443C2×C42—21no—
444C2×C42C23×C2311yescharacteristic
445#107—22no—
446#107—21no—
447#106—21no—
448C4⋊C8—21no—
449C4⋊C8—21no—
450#103—22no—
451#103—21no—
452#103—22no—
453#103—21no—
454#103—21no—
455#102—21no—
456C22⋊C8C23×C2312yes—
457C22⋊C8C23×C2311yescharacteristic
458C22⋊C8C23×C2311yescharacteristic
459#97—21no—
460C4×C8—21no—
461#95—21no—
462#95—21no—
463#95—22no—
464#95C23×C2312yes—
465C23#1824311yescharacteristic, direct product factor, Sylow
466C24C2×D4×C2312yes—
467#41—82no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—42no—
475C2×Q8—81no—
476C2×Q8—82no—
477C2×Q8C2×D4×C2311yescharacteristic
478C2×Q8C2×D4×C2312yes—
479C2×D4—21no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—81no—
485C2×D4C2×D4×C2311yescharacteristic
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4C2×D4×C2311yescharacteristic
494C22×C4C2×D4×C2312yes—
495C22×C4C2×D4×C2311yescharacteristic
496C22×C4C24×C2311yescharacteristic
497Q16—42no—
498Q16—44no—
499SD16—42no—
500C2×C8—22no—
501C2×C8—21no—
502C2×C8—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—82no—
521C4⋊C4C2×D4×C2311yescharacteristic
522C4⋊C4C2×D4×C2311yescharacteristic
523C4⋊C4C2×D4×C2312yes—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C42—22no—
542C42—22no—
543C42—21no—
544C42—41no—
545C42—41no—
546C42C2×D4×C2311yescharacteristic
547C42C2×D4×C2311yescharacteristic
548C42C2×D4×C2311yescharacteristic
549C42C2×D4×C2311yescharacteristic
550C23—22no—
551C23—21no—
552C23—24no—
553C23—42no—
554C23—42no—
555C23C22×D4×C2311yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—42no—
560Q8—41no—
561Q8—41no—
562Q8—42no—
563Q8—42no—
564Q8—81no—
565Q8—82no—
566D4—41no—
567D4—42no—
568D4—42no—
569D4—81no—
570D4—82no—
571D4—82no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4C23×#12011yescharacteristic
595C2×C4C23×#12011yescharacteristic
596C2×C4C23×#12011yescharacteristic
597C2×C4C23×#12011yescharacteristic
598C2×C4C22×D4×C2311yescharacteristic
599C2×C4C22×D4×C2311yescharacteristic
600C8—41no—
601C8—41no—
602C8—42no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—84no—
612C22C2×C23×#12011yescharacteristic
613C22C23×#57812yes—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—82no—
628C4—81no—
629C2—22no—
630C2—82no—
631C2—81no—
632C2C23×#301111yescharacteristic
633C2C23×#301911yescharacteristic
634C2C23×#302511yescharacteristic
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×#3117C2semidirect
3. C23×#3104C2semidirect
4. C23×#2632C2semidirect
5. C23×#1618C2semidirect
6. C23×#1614C2semidirect
7. C23×#1606C2semidirect
8. C23×#1602C2semidirect
9. C23×#1521C2semidirect
10. C23×#1515C2semidirect
11. C23×#1469C2semidirect
232. #18243C23direct (N × H)
287. #3117C2×C23semidirect
288. #3104C2×C23semidirect
289. #2632C2×C23semidirect
290. #1618C2×C23semidirect
291. #1614C2×C23semidirect
292. #1606C2×C23semidirect
293. #1602C2×C23semidirect
294. #1521C2×C23semidirect
295. #1515C2×C23semidirect
296. #1469C2×C23semidirect
465. C23#18243direct (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×#31172
C23×#31042
C23×#26322
C23×#16182
C23×#16142
C23×#16062
C23×#16022
C23×#15212
C23×#15152
C23×#14692
#1824323