Finite Group Databasebeta

← C23×#16518 #19164144

Subgroups — C23×#16518 #19164144

Order 5888.

1430 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#397C211yesmaximal, characteristic
2C2×C23×#396C212yesmaximal
3C2×C23×#389C211yesmaximal, characteristic, complemented
4C2×C23×#388C211yesmaximal, characteristic
5C2×C23×#382C211yesmaximal, characteristic, complemented
6C23×#1445C211yesmaximal, characteristic, complemented
7C23×#1444C211yesmaximal, characteristic, complemented
8C23×#1441C212yesmaximal, complemented
9C23×#1441C212yesmaximal, complemented
10C23×#1440C212yesmaximal
11C23×#1439C211yesmaximal, characteristic
12C22×C4⋊C4×C23C2211yescharacteristic
13C22×C4⋊C4×C23C2212yes—
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C4⋊C4×C23C2212yes—
16C22×C4⋊C4×C23C2212yes—
17C22×C4⋊C4×C23C2211yescharacteristic
18C22×C22⋊C4×C23C2211yescharacteristic
19C22×C22⋊C4×C23C2212yes—
20C22×C22⋊C4×C23C2212yes—
21C22×C22⋊C4×C23C2211yescharacteristic
22C22×C22⋊C4×C23C2211yescharacteristic
23C22×C42×C23C2211yescharacteristic
24C23×#397—24no—
25C23×#396—28no—
26C23×#389—24no—
27C23×#388—24no—
28C23×#382—24no—
29C2×C23×#95C2211yescharacteristic
30C2×C23×#95C2212yes—
31C2×C23×#95C2212yes—
32C2×C23×#95C2211yescharacteristic
33C2×C23×#95C2212yes—
34C2×C23×#95C2212yes—
35C2×C23×#95C2211yescharacteristic
36C2×C23×#95C2212yes—
37C2×C23×#95C2212yes—
38C2×C23×#95C2212yes—
39C2×C23×#95C2211yescharacteristic
40C25×C23C2311yescharacteristic
41C23×C4×C23C2311yescharacteristic
42C23×C4×C23C2311yescharacteristic
43C23×C4×C23C2312yes—
44C23×C4×C23C2312yes—
45C23×C4×C23C2311yescharacteristic
46C23×C4×C23C2312yes—
47C23×C4×C23C2311yescharacteristic
48C23×C4×C23C2312yes—
49C23×C4×C23C2311yescharacteristic
50C23×C4×C23C2311yescharacteristic
51C2×C4⋊C4×C23—22no—
52C2×C4⋊C4×C23—24no—
53C2×C4⋊C4×C23—24no—
54C2×C4⋊C4×C23—22no—
55C2×C4⋊C4×C23—24no—
56C2×C4⋊C4×C23—22no—
57C2×C4⋊C4×C23—24no—
58C2×C4⋊C4×C23—24no—
59C2×C4⋊C4×C23—24no—
60C2×C4⋊C4×C23—22no—
61C2×C4⋊C4×C23—24no—
62C2×C4⋊C4×C23—24no—
63C2×C4⋊C4×C23—22no—
64C2×C4⋊C4×C23—42no—
65C2×C4⋊C4×C23—42no—
66C2×C4⋊C4×C23—42no—
67C2×C22⋊C4×C23—22no—
68C2×C22⋊C4×C23—22no—
69C2×C22⋊C4×C23—24no—
70C2×C22⋊C4×C23—24no—
71C2×C22⋊C4×C23—22no—
72C2×C22⋊C4×C23—22no—
73C2×C22⋊C4×C23—24no—
74C2×C22⋊C4×C23—22no—
75C2×C22⋊C4×C23—24no—
76C2×C22⋊C4×C23—22no—
77C2×C22⋊C4×C23—42no—
78C2×C22⋊C4×C23—42no—
79C2×C22⋊C4×C23—42no—
80C2×C22⋊C4×C23D414yes—
81C2×C42×C23—22no—
82C2×C42×C23—22no—
83C2×C42×C23D414yes—
84C23×#95—24no—
85C23×#95—22no—
86C23×#95—22no—
87C23×#95—22no—
88C23×#95—24no—
89C23×#95—24no—
90C23×#95—24no—
91C23×#95—42no—
92C23×#95—42no—
93C23×#95—42no—
94C23×#95D414yes—
95C24×C23—21no—
96C24×C23—21no—
97C24×C23—21no—
98C24×C23—22no—
99C24×C23—21no—
100C24×C23—21no—
101C24×C23—22no—
102C24×C23—22no—
103C24×C23—22no—
104C24×C23—21no—
105C24×C23C2×D412yes—
106C24×C23C2411yescharacteristic
107C22×C4×C23—21no—
108C22×C4×C23—21no—
109C22×C4×C23—22no—
110C22×C4×C23—21no—
111C22×C4×C23—22no—
112C22×C4×C23—21no—
113C22×C4×C23—21no—
114C22×C4×C23—22no—
115C22×C4×C23—22no—
116C22×C4×C23—21no—
117C22×C4×C23—22no—
118C22×C4×C23—21no—
119C22×C4×C23—21no—
120C22×C4×C23—22no—
121C22×C4×C23—22no—
122C22×C4×C23—22no—
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—21no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
136C22×C4×C23—22no—
137C22×C4×C23—21no—
138C22×C4×C23—21no—
139C22×C4×C23—22no—
140C22×C4×C23—22no—
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—21no—
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×C23C2×D412yes—
158C22×C4×C23C2×D414yes—
159C22×C4×C23C2×D412yes—
160C22×C4×C23C2×D412yes—
161C22×C4×C23C2×D412yes—
162C22×C4×C23C2×D412yes—
163C22×C4×C23C2×D412yes—
164C22×C4×C23#4114yes—
165C22×C4×C23#4112yes—
166C22×C4×C23#4112yes—
167C4⋊C4×C23—28no—
168C4⋊C4×C23—48no—
169C4⋊C4×C23—48no—
170C4⋊C4×C23—44no—
171C4⋊C4×C23—48no—
172C4⋊C4×C23—44no—
173C22⋊C4×C23—28no—
174C22⋊C4×C23—48no—
175C22⋊C4×C23—48no—
176C22⋊C4×C23—44no—
177C22⋊C4×C23—44no—
178C42×C23—28no—
179#16518C2311yesmaximal, characteristic, direct product factor, Sylow
180C23×C23—22no—
181C23×C23—24no—
182C23×C23—22no—
183C23×C23—24no—
184C23×C23—22no—
185C23×C23—41no—
186C23×C23—41no—
187C23×C23—41no—
188C23×C23—41no—
189C23×C23—41no—
190C23×C23—41no—
191C23×C23—42no—
192C23×C23—42no—
193C23×C23—42no—
194C23×C23—42no—
195C23×C23—42no—
196C23×C23—42no—
197C23×C23—41no—
198C23×C23—42no—
199C23×C23—41no—
200C23×C23—42no—
201C23×C23—42no—
202C23×C23—42no—
203C23×C23C22×D411yescharacteristic
204C23×C23C22×D411yescharacteristic
205C23×C23C22×D411yescharacteristic
206C23×C23C2×#4111yescharacteristic
207C23×C23C2×#4111yescharacteristic
208C23×C23C2×#4112yes—
209C23×C23H(5,𝔽2)12yes—
210C23×C23H(5,𝔽2)11yescharacteristic
211C23×C232−1+412yes—
212C23×C232−1+412yes—
213C23×C232−1+411yescharacteristic
214C2×C4×C23—22no—
215C2×C4×C23—22no—
216C2×C4×C23—22no—
217C2×C4×C23—22no—
218C2×C4×C23—22no—
219C2×C4×C23—24no—
220C2×C4×C23—22no—
221C2×C4×C23—22no—
222C2×C4×C23—24no—
223C2×C4×C23—22no—
224C2×C4×C23—22no—
225C2×C4×C23—22no—
226C2×C4×C23—24no—
227C2×C4×C23—22no—
228C2×C4×C23—22no—
229C2×C4×C23—24no—
230C2×C4×C23—24no—
231C2×C4×C23—24no—
232C2×C4×C23—24no—
233C2×C4×C23—24no—
234C2×C4×C23—24no—
235C2×C4×C23—42no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—42no—
239C2×C4×C23—42no—
240C2×C4×C23—42no—
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—41no—
249C2×C4×C23—42no—
250C2×C4×C23—42no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—41no—
254C2×C4×C23—41no—
255C2×C4×C23—42no—
256C2×C4×C23—41no—
257C2×C4×C23—42no—
258C2×C4×C23—42no—
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#12114yes—
273C2×C4×C23#12114yes—
274C2×#397C2×C2311yescharacteristic
275C2×#396C2×C2312yes—
276C2×#389C2×C2311yescharacteristic, complemented
277C2×#388C2×C2311yescharacteristic
278C2×#382C2×C2311yescharacteristic, complemented
279#1445C2×C2311yescharacteristic, complemented
280#1444C2×C2311yescharacteristic, complemented
281#1441C2×C2312yescomplemented
282#1441C2×C2312yescomplemented
283#1440C2×C2312yes—
284#1439C2×C2311yescharacteristic
285C22×C23—42no—
286C22×C23—42no—
287C22×C23—44no—
288C22×C23—44no—
289C22×C23—42no—
290C22×C23—81no—
291C22×C23—81no—
292C22×C23—81no—
293C22×C23—81no—
294C22×C23—82no—
295C22×C23—82no—
296C22×C23C2×#12111yescharacteristic
297C22×C23C2×#12111yescharacteristic
298C22×C23#52511yescharacteristic
299C22×C23#53511yescharacteristic
300C22×C23#53611yescharacteristic
301C22×C23#53611yescharacteristic
302C22×C23#53612yes—
303C22×C23#53711yescharacteristic
304C22×C23#53712yes—
305C22×C23#54112yes—
306C22×C23#54112yes—
307C22×C23#54212yes—
308C22×C23#54212yes—
309C22×C23D4211yescharacteristic
310C22×C23#54612yes—
311C22×C23#54712yes—
312C22×C23#54711yescharacteristic
313C22×C23#54811yescharacteristic
314C22×C23#55212yes—
315C22×C23#55311yescharacteristic
316C22×C23#55312yes—
317C22×C23#56212yes—
318C22×C23#56312yes—
319C4×C23—24no—
320C4×C23—24no—
321C4×C23—42no—
322C4×C23—44no—
323C4×C23—42no—
324C4×C23—44no—
325C4×C23—81no—
326C4×C23—81no—
327C4×C23—82no—
328C4×C23—82no—
329C22×C4⋊C4C22×C2312yes—
330C22×C4⋊C4C22×C2311yescharacteristic
331C22×C4⋊C4C22×C2311yescharacteristic
332C22×C4⋊C4C22×C2312yes—
333C22×C4⋊C4C22×C2312yes—
334C22×C4⋊C4C22×C2311yescharacteristic
335C22×C22⋊C4C22×C2311yescharacteristic
336C22×C22⋊C4C22×C2312yes—
337C22×C22⋊C4C22×C2312yes—
338C22×C22⋊C4C22×C2311yescharacteristic
339C22×C22⋊C4C22×C2311yescharacteristic
340C22×C42C22×C2311yescharacteristic
341#397—24no—
342#396—28no—
343#389—24no—
344#388—24no—
345#382—24no—
346C2×#95C22×C2312yes—
347C2×#95C22×C2312yes—
348C2×#95C22×C2311yescharacteristic
349C2×#95C22×C2312yes—
350C2×#95C22×C2312yes—
351C2×#95C22×C2311yescharacteristic
352C2×#95C22×C2311yescharacteristic
353C2×#95C22×C2312yes—
354C2×#95C22×C2312yes—
355C2×#95C22×C2312yes—
356C2×#95C22×C2311yescharacteristic
357C2×C23—82no—
358C2×C23#241311yescharacteristic
359C2×C23#241611yescharacteristic
360C2×C23#253511yescharacteristic
361C2×C23#262011yescharacteristic
362C2×C23#262111yescharacteristic
363C2×C23#263711yescharacteristic
364C2×C23#267212yes—
365C2×C23#269812yes—
366C2×C23#272512yes—
367C2×C23#277212yes—
368C2×C23#282011yescharacteristic
369C25C23×C2311yescharacteristic
370C23×C4C23×C2312yes—
371C23×C4C23×C2311yescharacteristic
372C23×C4C23×C2312yes—
373C23×C4C23×C2311yescharacteristic
374C23×C4C23×C2311yescharacteristic
375C23×C4C23×C2312yes—
376C23×C4C23×C2311yescharacteristic
377C23×C4C23×C2312yes—
378C23×C4C23×C2311yescharacteristic
379C23×C4C23×C2311yescharacteristic
380C2×C4⋊C4—22no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—42no—
394C2×C4⋊C4—42no—
395C2×C4⋊C4—42no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4D4×C2314yes—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42D4×C2314yes—
413#95—24no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95D4×C2314yes—
424C23#1651811yescharacteristic, direct product factor, Sylow
425C24—21no—
426C24—21no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—21no—
431C24—22no—
432C24—21no—
433C24—22no—
434C24—22no—
435C24C2×D4×C2312yes—
436C24C24×C2311yescharacteristic
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4C2×D4×C2314yes—
488C22×C4C2×D4×C2312yes—
489C22×C4C2×D4×C2312yes—
490C22×C4C2×D4×C2312yes—
491C22×C4C2×D4×C2312yes—
492C22×C4C2×D4×C2312yes—
493C22×C4C2×D4×C2312yes—
494C22×C4#41×C2314yes—
495C22×C4#41×C2312yes—
496C22×C4#41×C2312yes—
497C4⋊C4—28no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C22⋊C4—28no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C42—28no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—22no—
513C23—24no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C22×D4×C2311yescharacteristic
533C23C22×D4×C2311yescharacteristic
534C23C22×D4×C2311yescharacteristic
535C23C2×#41×C2311yescharacteristic
536C23C2×#41×C2312yes—
537C23C2×#41×C2311yescharacteristic
538C23C23×H(5,𝔽2)11yescharacteristic
539C23C23×H(5,𝔽2)12yes—
540C23C23×2−1+411yescharacteristic
541C23C23×2−1+412yes—
542C23C23×2−1+412yes—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
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—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4C23×#12114yes—
602C2×C4C23×#12114yes—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—82no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C2×C23×#12111yescharacteristic
615C22C2×C23×#12111yescharacteristic
616C22C23×#52511yescharacteristic
617C22C23×#53511yescharacteristic
618C22C23×#53611yescharacteristic
619C22C23×#53612yes—
620C22C23×#53611yescharacteristic
621C22C23×#53711yescharacteristic
622C22C23×#53712yes—
623C22C23×#54112yes—
624C22C23×#54112yes—
625C22C23×#54212yes—
626C22C23×#54212yes—
627C22D42×C2311yescharacteristic
628C22C23×#54612yes—
629C22C23×#54712yes—
630C22C23×#54711yescharacteristic
631C22C23×#54811yescharacteristic
632C22C23×#55212yes—
633C22C23×#55311yescharacteristic
634C22C23×#55312yes—
635C22C23×#56212yes—
636C22C23×#56312yes—
637C4—24no—
638C4—24no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—82no—
645C4—81no—
646C4—81no—
647C2—82no—
648C2C23×#241311yescharacteristic
649C2C23×#241611yescharacteristic
650C2C23×#253511yescharacteristic
651C2C23×#262011yescharacteristic
652C2C23×#262111yescharacteristic
653C2C23×#263711yescharacteristic
654C2C23×#267212yes—
655C2C23×#269812yes—
656C2C23×#272512yes—
657C2C23×#277212yes—
658C2C23×#282011yescharacteristic
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
3. C2×C23×#389C2semidirect
5. C2×C23×#382C2semidirect
6. C23×#1445C2semidirect
7. C23×#1444C2semidirect
8. C23×#1441C2semidirect
9. C23×#1441C2semidirect
179. #16518C23direct (N × H)
276. C2×#389C2×C23semidirect
278. C2×#382C2×C23semidirect
279. #1445C2×C23semidirect
280. #1444C2×C23semidirect
281. #1441C2×C23semidirect
282. #1441C2×C23semidirect
424. C23#16518direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3972
C2×C23×#3962
C2×C23×#3892
C2×C23×#3882
C2×C23×#3822
C23×#14452
C23×#14442
C23×#14412
C23×#14402
C23×#14392
#1651823