Finite Group Databasebeta

← C23×#14840 #19162466

Subgroups — C23×#14840 #19162466

Order 5888.

1622 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#398C211yesmaximal, characteristic, complemented
2C2×C23×#385C211yesmaximal, characteristic
3C2×C23×#380C211yesmaximal, characteristic
4C2×C4×C22⋊C4×C23C211yesmaximal, characteristic
5C23×#1445C211yesmaximal, characteristic, complemented
6C23×#1444C211yesmaximal, characteristic, complemented
7C23×#1441C211yesmaximal, characteristic, complemented
8C23×#1439C212yesmaximal
9C23×#1439C212yesmaximal
10C23×#1436C212yesmaximal, complemented
11C23×C4⋊#95C212yesmaximal, complemented
12C24×C4×C23C2211yescharacteristic
13C22×C4⋊C4×C23C2211yescharacteristic
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C4⋊C4×C23C2211yescharacteristic
16C22×C4⋊C4×C23C2212yes—
17C22×C4⋊C4×C23C2212yes—
18C22×C4⋊C4×C23C2211yescharacteristic
19C22×C22⋊C4×C23C2211yescharacteristic
20C22×C22⋊C4×C23C2212yes—
21C22×C22⋊C4×C23C2212yes—
22C22×C22⋊C4×C23C2211yescharacteristic
23C22×C42×C23C2212yes—
24C23×#398—24no—
25C23×#385—24no—
26C23×#380—24no—
27C4×C22⋊C4×C23—24no—
28C2×C23×#95C2211yescharacteristic
29C2×C23×#95C2212yes—
30C2×C23×#95C2212yes—
31C2×C23×#95C2212yes—
32C2×C23×#95C2212yes—
33C2×C23×#95C2211yescharacteristic
34C2×C23×#95C2211yescharacteristic
35C2×C23×#95C2212yes—
36C2×C23×#95C2212yes—
37C2×C23×#95C2212yes—
38C2×C23×#95C2211yescharacteristic
39C25×C23C2311yescharacteristic
40C23×C4×C23—22no—
41C23×C4×C23—22no—
42C23×C4×C23—24no—
43C23×C4×C23—42no—
44C23×C4×C23D414yes—
45C23×C4×C23C2311yescharacteristic
46C23×C4×C23C2312yes—
47C23×C4×C23C2312yes—
48C23×C4×C23C2311yescharacteristic
49C23×C4×C23C2312yes—
50C23×C4×C23C2311yescharacteristic
51C23×C4×C23C2311yescharacteristic
52C23×C4×C23C2311yescharacteristic
53C23×C4×C23C2312yes—
54C23×C4×C23C2311yescharacteristic
55C2×C4⋊C4×C23—22no—
56C2×C4⋊C4×C23—22no—
57C2×C4⋊C4×C23—22no—
58C2×C4⋊C4×C23—22no—
59C2×C4⋊C4×C23—24no—
60C2×C4⋊C4×C23—22no—
61C2×C4⋊C4×C23—22no—
62C2×C4⋊C4×C23—24no—
63C2×C4⋊C4×C23—24no—
64C2×C4⋊C4×C23—22no—
65C2×C4⋊C4×C23—22no—
66C2×C4⋊C4×C23—24no—
67C2×C4⋊C4×C23—42no—
68C2×C4⋊C4×C23—42no—
69C2×C4⋊C4×C23—42no—
70C2×C4⋊C4×C23—42no—
71C2×C22⋊C4×C23—22no—
72C2×C22⋊C4×C23—24no—
73C2×C22⋊C4×C23—22no—
74C2×C22⋊C4×C23—22no—
75C2×C22⋊C4×C23—24no—
76C2×C22⋊C4×C23—24no—
77C2×C22⋊C4×C23—22no—
78C2×C22⋊C4×C23—42no—
79C2×C22⋊C4×C23—42no—
80C2×C22⋊C4×C23—42no—
81C2×C22⋊C4×C23Q818yes—
82C2×C42×C23—24no—
83C2×C42×C23—24no—
84C2×C42×C23—42no—
85C23×#95—22no—
86C23×#95—24no—
87C23×#95—22no—
88C23×#95—22no—
89C23×#95—24no—
90C23×#95—24no—
91C23×#95—24no—
92C23×#95—42no—
93C23×#95—42no—
94C23×#95—42no—
95C23×#95D414yes—
96C24×C23—21no—
97C24×C23—22no—
98C24×C23—21no—
99C24×C23—21no—
100C24×C23—21no—
101C24×C23—22no—
102C24×C23—21no—
103C24×C23—21no—
104C24×C23—22no—
105C24×C23C2×D412yes—
106C24×C23C2×Q814yes—
107C24×C23C2411yescharacteristic
108C22×C4×C23—21no—
109C22×C4×C23—22no—
110C22×C4×C23—22no—
111C22×C4×C23—21no—
112C22×C4×C23—22no—
113C22×C4×C23—22no—
114C22×C4×C23—22no—
115C22×C4×C23—22no—
116C22×C4×C23—22no—
117C22×C4×C23—22no—
118C22×C4×C23—21no—
119C22×C4×C23—21no—
120C22×C4×C23—21no—
121C22×C4×C23—22no—
122C22×C4×C23—22no—
123C22×C4×C23—21no—
124C22×C4×C23—21no—
125C22×C4×C23—22no—
126C22×C4×C23—21no—
127C22×C4×C23—22no—
128C22×C4×C23—22no—
129C22×C4×C23—22no—
130C22×C4×C23—21no—
131C22×C4×C23—21no—
132C22×C4×C23—22no—
133C22×C4×C23—28no—
134C22×C4×C23—28no—
135C22×C4×C23—21no—
136C22×C4×C23—22no—
137C22×C4×C23—22no—
138C22×C4×C23—21no—
139C22×C4×C23—22no—
140C22×C4×C23—21no—
141C22×C4×C23—21no—
142C22×C4×C23—22no—
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—22no—
154C22×C4×C23—22no—
155C22×C4×C23—22no—
156C22×C4×C23—21no—
157C22×C4×C23—22no—
158C22×C4×C23—21no—
159C22×C4×C23—44no—
160C22×C4×C23—48no—
161C22×C4×C23—48no—
162C22×C4×C23C2×D412yes—
163C22×C4×C23C2×D412yes—
164C22×C4×C23C2×D414yes—
165C22×C4×C23C2×D412yes—
166C22×C4×C23C2×Q814yes—
167C22×C4×C23C2×Q814yes—
168C22×C4×C23#4112yes—
169C22×C4×C23#4112yes—
170C22×C4×C23#4112yes—
171C22×C4×C23#4114yes—
172C22×C4×C23#4114yes—
173C4⋊C4×C23—48no—
174C4⋊C4×C23—44no—
175C4⋊C4×C23—48no—
176C4⋊C4×C23—44no—
177C4⋊C4×C23—44no—
178C4⋊C4×C23—44no—
179C22⋊C4×C23—28no—
180C22⋊C4×C23—216no—
181C22⋊C4×C23—44no—
182C22⋊C4×C23—48no—
183C42×C23—48no—
184#14840C2311yesmaximal, characteristic, direct product factor, Sylow
185C23×C23—22no—
186C23×C23—24no—
187C23×C23—22no—
188C23×C23—22no—
189C23×C23—22no—
190C23×C23—24no—
191C23×C23—22no—
192C23×C23—22no—
193C23×C23—24no—
194C23×C23—24no—
195C23×C23—24no—
196C23×C23—24no—
197C23×C23—42no—
198C23×C23—42no—
199C23×C23—42no—
200C23×C23—42no—
201C23×C23—42no—
202C23×C23—41no—
203C23×C23—41no—
204C23×C23—42no—
205C23×C23—41no—
206C23×C23—41no—
207C23×C23C4⋊Q814yes—
208C23×C23C22×D411yescharacteristic
209C23×C23C22×D411yescharacteristic
210C23×C23C22×Q812yes—
211C23×C23C2×#4111yescharacteristic
212C23×C23C2×#4112yes—
213C23×C23C2×#4112yes—
214C23×C23C2×#4111yescharacteristic
215C23×C23C2×#4111yescharacteristic
216C23×C23H(5,𝔽2)12yes—
217C23×C23H(5,𝔽2)11yescharacteristic
218C23×C23H(5,𝔽2)11yescharacteristic
219C2×C4×C23—22no—
220C2×C4×C23—22no—
221C2×C4×C23—24no—
222C2×C4×C23—22no—
223C2×C4×C23—22no—
224C2×C4×C23—24no—
225C2×C4×C23—24no—
226C2×C4×C23—24no—
227C2×C4×C23—24no—
228C2×C4×C23—24no—
229C2×C4×C23—24no—
230C2×C4×C23—24no—
231C2×C4×C23—24no—
232C2×C4×C23—24no—
233C2×C4×C23—24no—
234C2×C4×C23—22no—
235C2×C4×C23—22no—
236C2×C4×C23—24no—
237C2×C4×C23—22no—
238C2×C4×C23—22no—
239C2×C4×C23—24no—
240C2×C4×C23—24no—
241C2×C4×C23—416no—
242C2×C4×C23—41no—
243C2×C4×C23—41no—
244C2×C4×C23—42no—
245C2×C4×C23—42no—
246C2×C4×C23—41no—
247C2×C4×C23—41no—
248C2×C4×C23—42no—
249C2×C4×C23—42no—
250C2×C4×C23—41no—
251C2×C4×C23—42no—
252C2×C4×C23—41no—
253C2×C4×C23—41no—
254C2×C4×C23—42no—
255C2×C4×C23—41no—
256C2×C4×C23—42no—
257C2×C4×C23—42no—
258C2×C4×C23—41no—
259C2×C4×C23—41no—
260C2×C4×C23—41no—
261C2×C4×C23—42no—
262C2×C4×C23—42no—
263C2×C4×C23—42no—
264C2×C4×C23—41no—
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—42no—
276C2×C4×C23—42no—
277C2×C4×C23#12114yes—
278C2×C4×C23C22⋊Q818yes—
279C2×C4×C23#12314yes—
280C2×#398C2×C2311yescharacteristic, complemented
281C2×#385C2×C2311yescharacteristic
282C2×#380C2×C2311yescharacteristic
283C2×C4×C22⋊C4C2×C2311yescharacteristic
284#1445C2×C2311yescharacteristic, complemented
285#1444C2×C2311yescharacteristic, complemented
286#1441C2×C2311yescharacteristic, complemented
287#1439C2×C2312yes—
288#1439C2×C2312yes—
289#1436C2×C2312yescomplemented
290C4⋊#95C2×C2312yescomplemented
291C22×C23—24no—
292C22×C23—24no—
293C22×C23—24no—
294C22×C23—44no—
295C22×C23—42no—
296C22×C23—42no—
297C22×C23—42no—
298C22×C23—42no—
299C22×C23—44no—
300C22×C23—44no—
301C22×C23—44no—
302C22×C23C2×#12111yescharacteristic
303C22×C23C2×C22⋊Q812yes—
304C22×C23C2×#12311yescharacteristic
305C22×C23#52511yescharacteristic
306C22×C23#52912yes—
307C22×C23#52912yes—
308C22×C23C2×C4⋊Q811yescharacteristic
309C22×C23#53211yescharacteristic
310C22×C23#53312yes—
311C22×C23#53411yescharacteristic
312C22×C23#53511yescharacteristic
313C22×C23#53812yes—
314C22×C23#54012yes—
315C22×C23#54612yes—
316C22×C23#54611yescharacteristic
317C22×C23#55011yescharacteristic
318C22×C23#55112yes—
319C22×C23#55311yescharacteristic
320C22×C23#55412yes—
321C22×C23#55412yes—
322C22×C23#55412yes—
323C22×C23#55412yes—
324C22×C23#55511yescharacteristic
325C4×C23—24no—
326C4×C23—24no—
327C4×C23—28no—
328C4×C23—42no—
329C4×C23—42no—
330C4×C23—44no—
331C4×C23—81no—
332C4×C23—82no—
333C4×C23—81no—
334C4×C23—82no—
335C24×C4C22×C2311yescharacteristic
336C22×C4⋊C4C22×C2311yescharacteristic
337C22×C4⋊C4C22×C2311yescharacteristic
338C22×C4⋊C4C22×C2311yescharacteristic
339C22×C4⋊C4C22×C2312yes—
340C22×C4⋊C4C22×C2312yes—
341C22×C4⋊C4C22×C2311yescharacteristic
342C22×C22⋊C4C22×C2311yescharacteristic
343C22×C22⋊C4C22×C2312yes—
344C22×C22⋊C4C22×C2312yes—
345C22×C22⋊C4C22×C2311yescharacteristic
346C22×C42C22×C2312yes—
347#398—24no—
348#385—24no—
349#380—24no—
350C4×C22⋊C4—24no—
351C2×#95C22×C2312yes—
352C2×#95C22×C2311yescharacteristic
353C2×#95C22×C2312yes—
354C2×#95C22×C2312yes—
355C2×#95C22×C2312yes—
356C2×#95C22×C2311yescharacteristic
357C2×#95C22×C2311yescharacteristic
358C2×#95C22×C2312yes—
359C2×#95C22×C2312yes—
360C2×#95C22×C2312yes—
361C2×#95C22×C2311yescharacteristic
362C2×C23—44no—
363C2×C23#239511yescharacteristic
364C2×C23#241211yescharacteristic
365C2×C23#241411yescharacteristic
366C2×C23#245012yes—
367C2×C23#245412yes—
368C2×C23#250211yescharacteristic
369C2×C23#252212yes—
370C2×C23#257412yes—
371C2×C23#258311yescharacteristic
372C2×C23#266111yescharacteristic
373C2×C23#285711yescharacteristic
374C25C23×C2311yescharacteristic
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—24no—
378C23×C4—42no—
379C23×C4D4×C2314yes—
380C23×C4C23×C2311yescharacteristic
381C23×C4C23×C2311yescharacteristic
382C23×C4C23×C2312yes—
383C23×C4C23×C2312yes—
384C23×C4C23×C2312yes—
385C23×C4C23×C2311yescharacteristic
386C23×C4C23×C2311yescharacteristic
387C23×C4C23×C2311yescharacteristic
388C23×C4C23×C2312yes—
389C23×C4C23×C2311yescharacteristic
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—42no—
403C2×C4⋊C4—42no—
404C2×C4⋊C4—42no—
405C2×C4⋊C4—42no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4Q8×C2318yes—
417C2×C42—24no—
418C2×C42—24no—
419C2×C42—42no—
420#95—22no—
421#95—24no—
422#95—22no—
423#95—22no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—42no—
428#95—42no—
429#95—42no—
430#95D4×C2314yes—
431C23#1484011yescharacteristic, direct product factor, Sylow
432C24—21no—
433C24—21no—
434C24—21no—
435C24—22no—
436C24—22no—
437C24—21no—
438C24—21no—
439C24—21no—
440C24—22no—
441C24C2×D4×C2312yes—
442C24C2×Q8×C2314yes—
443C24C24×C2311yescharacteristic
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—44no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4C2×D4×C2312yes—
499C22×C4C2×D4×C2312yes—
500C22×C4C2×D4×C2314yes—
501C22×C4C2×D4×C2312yes—
502C22×C4C2×Q8×C2314yes—
503C22×C4C2×Q8×C2314yes—
504C22×C4#41×C2312yes—
505C22×C4#41×C2312yes—
506C22×C4#41×C2312yes—
507C22×C4#41×C2314yes—
508C22×C4#41×C2314yes—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C22⋊C4—28no—
516C22⋊C4—216no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C42—48no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23C23×C4⋊Q814yes—
543C23C22×D4×C2311yescharacteristic
544C23C22×D4×C2311yescharacteristic
545C23C22×Q8×C2312yes—
546C23C2×#41×C2311yescharacteristic
547C23C2×#41×C2311yescharacteristic
548C23C2×#41×C2312yes—
549C23C2×#41×C2311yescharacteristic
550C23C2×#41×C2312yes—
551C23C23×H(5,𝔽2)11yescharacteristic
552C23C23×H(5,𝔽2)12yes—
553C23C23×H(5,𝔽2)11yescharacteristic
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—416no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4C23×#12114yes—
613C2×C4C23×C22⋊Q818yes—
614C2×C4C23×#12314yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—42no—
626C22C2×C23×#12111yescharacteristic
627C22C2×C23×C22⋊Q812yes—
628C22C2×C23×#12311yescharacteristic
629C22C23×#52511yescharacteristic
630C22C23×#52912yes—
631C22C23×#52912yes—
632C22C2×C23×C4⋊Q811yescharacteristic
633C22C23×#53211yescharacteristic
634C22C23×#53312yes—
635C22C23×#53411yescharacteristic
636C22C23×#53511yescharacteristic
637C22C23×#53812yes—
638C22C23×#54012yes—
639C22C23×#54612yes—
640C22C23×#54611yescharacteristic
641C22C23×#55011yescharacteristic
642C22C23×#55112yes—
643C22C23×#55311yescharacteristic
644C22C23×#55412yes—
645C22C23×#55412yes—
646C22C23×#55412yes—
647C22C23×#55412yes—
648C22C23×#55511yescharacteristic
649C4—24no—
650C4—24no—
651C4—28no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—82no—
659C2—44no—
660C2C23×#239511yescharacteristic
661C2C23×#241211yescharacteristic
662C2C23×#241411yescharacteristic
663C2C23×#245012yes—
664C2C23×#245412yes—
665C2C23×#250211yescharacteristic
666C2C23×#252212yes—
667C2C23×#257412yes—
668C2C23×#258311yescharacteristic
669C2C23×#266111yescharacteristic
670C2C23×#285711yescharacteristic
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
1. C2×C23×#398C2semidirect
5. C23×#1445C2semidirect
6. C23×#1444C2semidirect
7. C23×#1441C2semidirect
10. C23×#1436C2semidirect
11. C23×C4⋊#95C2semidirect
184. #14840C23direct (N × H)
280. C2×#398C2×C23semidirect
284. #1445C2×C23semidirect
285. #1444C2×C23semidirect
286. #1441C2×C23semidirect
289. #1436C2×C23semidirect
290. C4⋊#95C2×C23semidirect
431. C23#14840direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3982
C2×C23×#3852
C2×C23×#3802
C2×C4×C22⋊C4×C232
C23×#14452
C23×#14442
C23×#14412
C23×#14392
C23×#14362
C23×C4⋊#952
#1484023