Finite Group Databasebeta

← C23×#16566 #19164192

Subgroups — C23×#16566 #19164192

Order 5888.

1454 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#393C211yesmaximal, characteristic
2C2×C23×#391C211yesmaximal, characteristic, complemented
3C2×C23×#388C211yesmaximal, characteristic
4C2×C23×#382C211yesmaximal, characteristic, complemented
5C23×#1446C211yesmaximal, characteristic, complemented
6C23×#1444C211yesmaximal, characteristic, complemented
7C23×#1442C212yesmaximal, complemented
8C23×#1441C212yesmaximal, complemented
9C23×#1439C212yesmaximal
10C23×#1439C212yesmaximal
11C23×#1439C211yesmaximal, characteristic
12C23×Q8×C23C2211yescharacteristic
13C22×C4⋊C4×C23C2212yes—
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C4⋊C4×C23C2211yescharacteristic
16C22×C4⋊C4×C23C2212yes—
17C22×C22⋊C4×C23C2211yescharacteristic
18C22×C22⋊C4×C23C2212yes—
19C22×C22⋊C4×C23C2211yescharacteristic
20C22×C22⋊C4×C23C2211yescharacteristic
21C22×C22⋊C4×C23C2212yes—
22C22×C42×C23C2211yescharacteristic
23C23×#393—24no—
24C23×#391—24no—
25C23×#388—24no—
26C23×#382—24no—
27C2×C23×#95C2212yes—
28C2×C23×#95C2212yes—
29C2×C23×#95C2212yes—
30C2×C23×#95C2212yes—
31C2×C23×#95C2211yescharacteristic
32C2×C23×#95C2211yescharacteristic
33C2×C23×#95C2212yes—
34C2×C23×#95C2211yescharacteristic
35C2×C23×#95C2212yes—
36C2×C23×#95C2212yes—
37C2×C23×#95C2212yes—
38C2×C23×#95C2211yescharacteristic
39C25×C23C2311yescharacteristic
40C22×Q8×C23—22no—
41C22×Q8×C23—22no—
42C22×Q8×C23—24no—
43C22×Q8×C23—22no—
44C22×Q8×C23—42no—
45C23×C4×C23C2311yescharacteristic
46C23×C4×C23C2312yes—
47C23×C4×C23C2311yescharacteristic
48C23×C4×C23C2312yes—
49C23×C4×C23C2311yescharacteristic
50C23×C4×C23C2312yes—
51C23×C4×C23C2311yescharacteristic
52C23×C4×C23C2312yes—
53C23×C4×C23C2311yescharacteristic
54C23×C4×C23C2311yescharacteristic
55C2×C4⋊C4×C23—24no—
56C2×C4⋊C4×C23—22no—
57C2×C4⋊C4×C23—22no—
58C2×C4⋊C4×C23—24no—
59C2×C4⋊C4×C23—42no—
60C2×C4⋊C4×C23—42no—
61C2×C4⋊C4×C23—42no—
62C2×C4⋊C4×C23—42no—
63C2×C4⋊C4×C23—42no—
64C2×C4⋊C4×C23—42no—
65C2×C22⋊C4×C23—22no—
66C2×C22⋊C4×C23—22no—
67C2×C22⋊C4×C23—22no—
68C2×C22⋊C4×C23—22no—
69C2×C22⋊C4×C23—24no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—24no—
72C2×C22⋊C4×C23—22no—
73C2×C22⋊C4×C23—42no—
74C2×C22⋊C4×C23—42no—
75C2×C22⋊C4×C23—42no—
76C2×C22⋊C4×C23—42no—
77C2×C22⋊C4×C23—42no—
78C2×C22⋊C4×C23Q814yescomplemented
79C2×C42×C23—22no—
80C2×C42×C23—22no—
81C2×C42×C23D414yes—
82C23×#95—24no—
83C23×#95—22no—
84C23×#95—22no—
85C23×#95—22no—
86C23×#95—24no—
87C23×#95—24no—
88C23×#95—24no—
89C23×#95—24no—
90C23×#95—42no—
91C23×#95—42no—
92C23×#95—42no—
93C23×#95D414yes—
94C24×C23—21no—
95C24×C23—21no—
96C24×C23—21no—
97C24×C23—21no—
98C24×C23—22no—
99C24×C23—22no—
100C24×C23—22no—
101C24×C23—21no—
102C24×C23—22no—
103C24×C23—21no—
104C24×C23C2×Q812yes—
105C24×C23C2411yescharacteristic
106C2×Q8×C23—28no—
107C2×Q8×C23—48no—
108C2×Q8×C23—48no—
109C2×Q8×C23—44no—
110C2×Q8×C23—44no—
111C22×C4×C23—21no—
112C22×C4×C23—21no—
113C22×C4×C23—22no—
114C22×C4×C23—21no—
115C22×C4×C23—22no—
116C22×C4×C23—21no—
117C22×C4×C23—21no—
118C22×C4×C23—21no—
119C22×C4×C23—22no—
120C22×C4×C23—22no—
121C22×C4×C23—21no—
122C22×C4×C23—21no—
123C22×C4×C23—22no—
124C22×C4×C23—22no—
125C22×C4×C23—22no—
126C22×C4×C23—22no—
127C22×C4×C23—22no—
128C22×C4×C23—22no—
129C22×C4×C23—22no—
130C22×C4×C23—22no—
131C22×C4×C23—22no—
132C22×C4×C23—22no—
133C22×C4×C23—22no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
136C22×C4×C23—22no—
137C22×C4×C23—21no—
138C22×C4×C23—21no—
139C22×C4×C23—21no—
140C22×C4×C23—22no—
141C22×C4×C23—22no—
142C22×C4×C23—21no—
143C22×C4×C23—22no—
144C22×C4×C23—22no—
145C22×C4×C23—22no—
146C22×C4×C23—22no—
147C22×C4×C23—22no—
148C22×C4×C23—22no—
149C22×C4×C23—22no—
150C22×C4×C23—22no—
151C22×C4×C23—22no—
152C22×C4×C23—22no—
153C22×C4×C23—21no—
154C22×C4×C23—22no—
155C22×C4×C23—22no—
156C22×C4×C23—22no—
157C22×C4×C23—22no—
158C22×C4×C23—22no—
159C22×C4×C23—21no—
160C22×C4×C23—22no—
161C22×C4×C23C2×D412yes—
162C22×C4×C23C2×D414yes—
163C22×C4×C23C2×D412yes—
164C22×C4×C23C2×D412yes—
165C22×C4×C23C2×D412yes—
166C22×C4×C23C2×Q812yes—
167C22×C4×C23C2×Q812yes—
168C22×C4×C23#4114yes—
169C22×C4×C23#4112yes—
170C22×C4×C23#4112yes—
171C4⋊C4×C23—48no—
172C4⋊C4×C23—44no—
173C4⋊C4×C23—44no—
174C4⋊C4×C23—48no—
175C22⋊C4×C23—28no—
176C22⋊C4×C23—48no—
177C22⋊C4×C23—44no—
178C22⋊C4×C23—48no—
179C22⋊C4×C23—44no—
180C42×C23—28no—
181#16566C2311yesmaximal, characteristic, direct product factor, Sylow
182C23×C23—22no—
183C23×C23—24no—
184C23×C23—22no—
185C23×C23—24no—
186C23×C23—22no—
187C23×C23—41no—
188C23×C23—41no—
189C23×C23—41no—
190C23×C23—41no—
191C23×C23—41no—
192C23×C23—41no—
193C23×C23—42no—
194C23×C23—42no—
195C23×C23—42no—
196C23×C23—42no—
197C23×C23—42no—
198C23×C23—42no—
199C23×C23—41no—
200C23×C23—42no—
201C23×C23—41no—
202C23×C23—42no—
203C23×C23—42no—
204C23×C23—42no—
205C23×C23C22×D411yescharacteristic
206C23×C23C22×D411yescharacteristic
207C23×C23C22×Q811yescharacteristic
208C23×C23C2×#4111yescharacteristic
209C23×C23C2×#4111yescharacteristic
210C23×C23C2×#4112yes—
211C23×C23H(5,𝔽2)12yes—
212C23×C23H(5,𝔽2)12yes—
213C23×C23H(5,𝔽2)11yescharacteristic
214C23×C232−1+412yes—
215C23×C232−1+411yescharacteristic
216Q8×C23—416no—
217C2×C4×C23—22no—
218C2×C4×C23—22no—
219C2×C4×C23—22no—
220C2×C4×C23—22no—
221C2×C4×C23—22no—
222C2×C4×C23—24no—
223C2×C4×C23—22no—
224C2×C4×C23—24no—
225C2×C4×C23—22no—
226C2×C4×C23—22no—
227C2×C4×C23—22no—
228C2×C4×C23—22no—
229C2×C4×C23—24no—
230C2×C4×C23—22no—
231C2×C4×C23—22no—
232C2×C4×C23—24no—
233C2×C4×C23—24no—
234C2×C4×C23—24no—
235C2×C4×C23—24no—
236C2×C4×C23—24no—
237C2×C4×C23—24no—
238C2×C4×C23—42no—
239C2×C4×C23—41no—
240C2×C4×C23—41no—
241C2×C4×C23—42no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
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—42no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—42no—
254C2×C4×C23—41no—
255C2×C4×C23—42no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—42no—
260C2×C4×C23—42no—
261C2×C4×C23—42no—
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—42no—
269C2×C4×C23—42no—
270C2×C4×C23—42no—
271C2×C4×C23—42no—
272C2×C4×C23—42no—
273C2×C4×C23—42no—
274C2×C4×C23—42no—
275C2×C4×C23#12114yes—
276C2×C4×C23C22⋊Q814yes—
277C2×#393C2×C2311yescharacteristic
278C2×#391C2×C2311yescharacteristic, complemented
279C2×#388C2×C2311yescharacteristic
280C2×#382C2×C2311yescharacteristic, complemented
281#1446C2×C2311yescharacteristic, complemented
282#1444C2×C2311yescharacteristic, complemented
283#1442C2×C2312yescomplemented
284#1441C2×C2312yescomplemented
285#1439C2×C2311yescharacteristic
286#1439C2×C2312yes—
287#1439C2×C2312yes—
288C22×C23—42no—
289C22×C23—42no—
290C22×C23—42no—
291C22×C23—44no—
292C22×C23—44no—
293C22×C23—81no—
294C22×C23—81no—
295C22×C23—81no—
296C22×C23—81no—
297C22×C23—82no—
298C22×C23—82no—
299C22×C23C2×#12111yescharacteristic
300C22×C23C2×C22⋊Q811yescharacteristic
301C22×C23#52511yescharacteristic
302C22×C23#53511yescharacteristic
303C22×C23#53511yescharacteristic
304C22×C23#53611yescharacteristic
305C22×C23#53711yescharacteristic
306C22×C23#53712yes—
307C22×C23#54112yes—
308C22×C23#54212yes—
309C22×C23#54212yes—
310C22×C23#54212yes—
311C22×C23#54412yes—
312C22×C23#54612yes—
313C22×C23#54611yescharacteristic
314C22×C23D4×Q811yescharacteristic
315C22×C23#55112yes—
316C22×C23#55211yescharacteristic
317C22×C23#55412yes—
318C22×C23#55411yescharacteristic
319C22×C23#55912yes—
320C22×C23H(3,𝔽4)12yes—
321C22×C23#56212yes—
322C4×C23—24no—
323C4×C23—24no—
324C4×C23—42no—
325C4×C23—42no—
326C4×C23—44no—
327C4×C23—44no—
328C4×C23—81no—
329C4×C23—82no—
330C4×C23—81no—
331C4×C23—82no—
332C23×Q8C22×C2311yescharacteristic
333C22×C4⋊C4C22×C2312yes—
334C22×C4⋊C4C22×C2311yescharacteristic
335C22×C4⋊C4C22×C2311yescharacteristic
336C22×C4⋊C4C22×C2312yes—
337C22×C22⋊C4C22×C2311yescharacteristic
338C22×C22⋊C4C22×C2312yes—
339C22×C22⋊C4C22×C2311yescharacteristic
340C22×C22⋊C4C22×C2311yescharacteristic
341C22×C22⋊C4C22×C2312yes—
342C22×C42C22×C2311yescharacteristic
343#393—24no—
344#391—24no—
345#388—24no—
346#382—24no—
347C2×#95C22×C2312yes—
348C2×#95C22×C2312yes—
349C2×#95C22×C2312yes—
350C2×#95C22×C2312yes—
351C2×#95C22×C2312yes—
352C2×#95C22×C2311yescharacteristic
353C2×#95C22×C2311yescharacteristic
354C2×#95C22×C2311yescharacteristic
355C2×#95C22×C2312yes—
356C2×#95C22×C2312yes—
357C2×#95C22×C2312yes—
358C2×#95C22×C2311yescharacteristic
359C2×C23—82no—
360C2×C23#241111yescharacteristic
361C2×C23#241711yescharacteristic
362C2×C23#258011yescharacteristic
363C2×C23#262011yescharacteristic
364C2×C23#262811yescharacteristic
365C2×C23#263711yescharacteristic
366C2×C23#270412yes—
367C2×C23#271212yes—
368C2×C23#275412yes—
369C2×C23#276312yes—
370C2×C23#281911yescharacteristic
371C25C23×C2311yescharacteristic
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—24no—
375C22×Q8—22no—
376C22×Q8—42no—
377C23×C4C23×C2311yescharacteristic
378C23×C4C23×C2312yes—
379C23×C4C23×C2311yescharacteristic
380C23×C4C23×C2312yes—
381C23×C4C23×C2312yes—
382C23×C4C23×C2311yescharacteristic
383C23×C4C23×C2311yescharacteristic
384C23×C4C23×C2312yes—
385C23×C4C23×C2311yescharacteristic
386C23×C4C23×C2311yescharacteristic
387C2×C4⋊C4—24no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—42no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4—42no—
394C2×C4⋊C4—42no—
395C2×C4⋊C4—42no—
396C2×C4⋊C4—42no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4—42no—
410C2×C22⋊C4Q8×C2314yescomplemented
411C2×C42—22no—
412C2×C42—22no—
413C2×C42D4×C2314yes—
414#95—24no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—42no—
423#95—42no—
424#95—42no—
425#95D4×C2314yes—
426C23#1656611yescharacteristic, direct product factor, Sylow
427C24—21no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—21no—
432C24—22no—
433C24—21no—
434C24—22no—
435C24—21no—
436C24—22no—
437C24C2×Q8×C2312yes—
438C24C24×C2311yescharacteristic
439C2×Q8—28no—
440C2×Q8—48no—
441C2×Q8—48no—
442C2×Q8—44no—
443C2×Q8—44no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4C2×D4×C2314yes—
495C22×C4C2×D4×C2312yes—
496C22×C4C2×D4×C2312yes—
497C22×C4C2×D4×C2312yes—
498C22×C4C2×D4×C2312yes—
499C22×C4C2×Q8×C2312yes—
500C22×C4C2×Q8×C2312yes—
501C22×C4#41×C2314yes—
502C22×C4#41×C2312yes—
503C22×C4#41×C2312yes—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C22⋊C4—28no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C42—28no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—24no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23C22×D4×C2311yescharacteristic
538C23C22×D4×C2311yescharacteristic
539C23C22×Q8×C2311yescharacteristic
540C23C2×#41×C2311yescharacteristic
541C23C2×#41×C2312yes—
542C23C2×#41×C2311yescharacteristic
543C23C23×H(5,𝔽2)12yes—
544C23C23×H(5,𝔽2)12yes—
545C23C23×H(5,𝔽2)11yescharacteristic
546C23C23×2−1+411yescharacteristic
547C23C23×2−1+412yes—
548Q8—416no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4C23×#12114yes—
608C2×C4C23×C22⋊Q814yes—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—82no—
616C22—82no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22C2×C23×#12111yescharacteristic
621C22C2×C23×C22⋊Q811yescharacteristic
622C22C23×#52511yescharacteristic
623C22C23×#53511yescharacteristic
624C22C23×#53511yescharacteristic
625C22C23×#53611yescharacteristic
626C22C23×#53711yescharacteristic
627C22C23×#53712yes—
628C22C23×#54112yes—
629C22C23×#54212yes—
630C22C23×#54212yes—
631C22C23×#54212yes—
632C22C23×#54412yes—
633C22C23×#54612yes—
634C22C23×#54611yescharacteristic
635C22D4×Q8×C2311yescharacteristic
636C22C23×#55112yes—
637C22C23×#55211yescharacteristic
638C22C23×#55411yescharacteristic
639C22C23×#55412yes—
640C22C23×#55912yes—
641C22C23×H(3,𝔽4)12yes—
642C22C23×#56212yes—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—82no—
651C4—81no—
652C4—81no—
653C2—82no—
654C2C23×#241111yescharacteristic
655C2C23×#241711yescharacteristic
656C2C23×#258011yescharacteristic
657C2C23×#262011yescharacteristic
658C2C23×#262811yescharacteristic
659C2C23×#263711yescharacteristic
660C2C23×#270412yes—
661C2C23×#271212yes—
662C2C23×#275412yes—
663C2C23×#276312yes—
664C2C23×#281911yescharacteristic
subgroupG/H
center, socleC24×C23C24
commutator, Frattini, cosocleC24C24×C23
prime radicalC25×C23C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#391C2semidirect
4. C2×C23×#382C2semidirect
5. C23×#1446C2semidirect
6. C23×#1444C2semidirect
7. C23×#1442C2semidirect
8. C23×#1441C2semidirect
78. C2×C22⋊C4×C23Q8semidirect
181. #16566C23direct (N × H)
278. C2×#391C2×C23semidirect
280. C2×#382C2×C23semidirect
281. #1446C2×C23semidirect
282. #1444C2×C23semidirect
283. #1442C2×C23semidirect
284. #1441C2×C23semidirect
410. C2×C22⋊C4Q8×C23semidirect
426. C23#16566direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3932
C2×C23×#3912
C2×C23×#3882
C2×C23×#3822
C23×#14462
C23×#14442
C23×#14422
C23×#14412
C23×#14392
#1656623