Finite Group Databasebeta

← C23×#14506 #19162132

Subgroups — C23×#14506 #19162132

Order 5888.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#123C211yesmaximal, characteristic
2C2×C23×#388C212yesmaximal, complemented
3C23×#1445C212yesmaximal, complemented
4C23×#1439C212yesmaximal, complemented
5C23×#1438C211yesmaximal, characteristic
6C23×#1438C211yesmaximal, characteristic
7C23×#1437C212yesmaximal, complemented
8C23×C4⋊#95C212yesmaximal, complemented
9C23×C22⋊#95C212yesmaximal, complemented
10C23×D4×C23C2211yescharacteristic
11C24×C4×C23C2211yescharacteristic
12C2×C23×#123—28no—
13C2×C23×#123C418yescomplemented
14C22×C4⋊C4×C23C2212yes—
15C22×C4⋊C4×C23C2212yes—
16C22×C22⋊C4×C23C2212yes—
17C22×C22⋊C4×C23C2212yes—
18C22×C22⋊C4×C23C2211yescharacteristic
19C22×C22⋊C4×C23C2212yes—
20C22×C22⋊C4×C23C2212yes—
21C22×C22⋊C4×C23C2211yescharacteristic
22C22×C22⋊C4×C23C2211yescharacteristic
23C22×C42×C23C2212yes—
24C23×#388—28no—
25C2×C23×#95C2212yes—
26C2×C23×#95C2212yes—
27C2×C23×#95C2212yes—
28C2×C23×#95C2212yes—
29C2×C23×#95C2212yes—
30C2×C23×#95C2212yes—
31C2×C23×#95C2211yescharacteristic
32C2×C23×#95C2212yes—
33C2×C23×#95C2211yescharacteristic
34C25×C23C2311yescharacteristic
35C25×C23C2311yescharacteristic
36C22×D4×C23—24no—
37C22×D4×C23—42no—
38C22×D4×C23—41no—
39C22×D4×C23—41no—
40C22×D4×C23C2×C414yes—
41C23×C4×C23—24no—
42C23×C4×C23—42no—
43C23×C4×C23—41no—
44C23×C4×C23—41no—
45C23×C4×C23C2×C414yes—
46C23×C4×C23C2312yes—
47C23×C4×C23C2312yes—
48C23×C4×C23C2312yes—
49C23×C4×C23C2311yescharacteristic
50C23×C4×C23C2312yes—
51C23×C4×C23C2312yes—
52C23×C4×C23C2312yes—
53C23×#123—232no—
54C2×C4⋊C4×C23—24no—
55C2×C4⋊C4×C23—24no—
56C2×C4⋊C4×C23—24no—
57C2×C4⋊C4×C23—24no—
58C2×C4⋊C4×C23—42no—
59C2×C4⋊C4×C23C2×C418yes—
60C2×C22⋊C4×C23—24no—
61C2×C22⋊C4×C23—24no—
62C2×C22⋊C4×C23—24no—
63C2×C22⋊C4×C23—24no—
64C2×C22⋊C4×C23—24no—
65C2×C22⋊C4×C23—24no—
66C2×C22⋊C4×C23—24no—
67C2×C22⋊C4×C23—24no—
68C2×C22⋊C4×C23—41no—
69C2×C22⋊C4×C23—42no—
70C2×C22⋊C4×C23—42no—
71C2×C22⋊C4×C23—42no—
72C2×C22⋊C4×C23—42no—
73C2×C22⋊C4×C23—42no—
74C2×C22⋊C4×C23—41no—
75C2×C22⋊C4×C23C2×C418yes—
76C2×C22⋊C4×C23C2×C414yes—
77C2×C22⋊C4×C23D418yes—
78C2×C42×C23—24no—
79C2×C42×C23—24no—
80C2×C42×C23—42no—
81C23×#95—24no—
82C23×#95—24no—
83C23×#95—42no—
84C23×#95—42no—
85C23×#95—42no—
86C23×#95—42no—
87C23×#95—41no—
88C23×#95—42no—
89C23×#95—41no—
90C24×C23—22no—
91C24×C23—22no—
92C24×C23—22no—
93C24×C23—22no—
94C24×C23—21no—
95C24×C23—21no—
96C24×C23—22no—
97C24×C23—22no—
98C24×C23—22no—
99C24×C23—22no—
100C24×C23—22no—
101C24×C23—22no—
102C24×C23—21no—
103C24×C23—21no—
104C24×C23—22no—
105C24×C23C22×C412yes—
106C24×C23C22×C412yes—
107C24×C23C2×D414yes—
108C24×C23C2411yescharacteristic
109C2×D4×C23—28no—
110C2×D4×C23—44no—
111C2×D4×C23—44no—
112C2×D4×C23—48no—
113C2×D4×C23—48no—
114C22×C4×C23—22no—
115C22×C4×C23—22no—
116C22×C4×C23—22no—
117C22×C4×C23—22no—
118C22×C4×C23—28no—
119C22×C4×C23—22no—
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—22no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
136C22×C4×C23—21no—
137C22×C4×C23—21no—
138C22×C4×C23—22no—
139C22×C4×C23—22no—
140C22×C4×C23—22no—
141C22×C4×C23—22no—
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—44no—
156C22×C4×C23—44no—
157C22×C4×C23—48no—
158C22×C4×C23—48no—
159C22×C4×C23C22×C414yes—
160C22×C4×C23C22×C412yes—
161C22×C4×C23C22×C414yes—
162C22×C4×C23C2×D414yes—
163C22×C4×C23C2×D414yes—
164C22×C4×C23#4114yes—
165C22×C4×C23#4114yes—
166C4⋊C4×C23—216no—
167C4⋊C4×C23—48no—
168C22⋊C4×C23—216no—
169C22⋊C4×C23—44no—
170C22⋊C4×C23—48no—
171C22⋊C4×C23—48no—
172C22⋊C4×C23—48no—
173C22⋊C4×C23—44no—
174C22⋊C4×C23C22⋊C4116yescomplemented
175C42×C23—48no—
176#14506C2311yesmaximal, characteristic, direct product factor, Sylow
177C23×C23—24no—
178C23×C23—22no—
179C23×C23—22no—
180C23×C23—24no—
181C23×C23—24no—
182C23×C23—24no—
183C23×C23—24no—
184C23×C23—24no—
185C23×C23—24no—
186C23×C23—24no—
187C23×C23—22no—
188C23×C23—24no—
189C23×C23—22no—
190C23×C23—22no—
191C23×C23—24no—
192C23×C23—42no—
193C23×C23—42no—
194C23×C23—42no—
195C23×C23—42no—
196C23×C23—41no—
197C23×C23—42no—
198C23×C23—41no—
199C23×C23—41no—
200C23×C23—42no—
201C23×C23—41no—
202C23×C23—42no—
203C23×C23—42no—
204C23×C23—42no—
205C23×C23—42no—
206C23×C23—42no—
207C23×C23—42no—
208C23×C23—42no—
209C23×C23—42no—
210C23×C23—42no—
211C23×C23—42no—
212C23×C23—41no—
213C23×C23—42no—
214C23×C23—41no—
215C23×C23—41no—
216C23×C23—42no—
217C23×C23—41no—
218C23×C23C2×C22⋊C414yes—
219C23×C23C23×C411yescharacteristic
220C23×C23C22×D412yes—
221C23×C23C2×#4112yes—
222C23×C23C2×#4112yes—
223C23×C23H(5,𝔽2)12yes—
224C23×C23H(5,𝔽2)11yescharacteristic
225C23×C23H(5,𝔽2)11yescharacteristic
226C23×C23H(5,𝔽2)12yes—
227C23×C232−1+412yes—
228D4×C23—416no—
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—24no—
236C2×C4×C23—22no—
237C2×C4×C23—24no—
238C2×C4×C23—24no—
239C2×C4×C23—22no—
240C2×C4×C23—22no—
241C2×C4×C23—24no—
242C2×C4×C23—24no—
243C2×C4×C23—24no—
244C2×C4×C23—24no—
245C2×C4×C23—24no—
246C2×C4×C23—24no—
247C2×C4×C23—416no—
248C2×C4×C23—42no—
249C2×C4×C23—42no—
250C2×C4×C23—42no—
251C2×C4×C23—42no—
252C2×C4×C23—42no—
253C2×C4×C23—42no—
254C2×C4×C23—42no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
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—42no—
273C2×C4×C23—42no—
274C2×C4×C23—42no—
275C2×C4×C23—42no—
276C2×C4×C23C2×C22⋊C418yes—
277C22×#123C2×C2311yescharacteristic
278C2×#388C2×C2312yescomplemented
279#1445C2×C2312yescomplemented
280#1439C2×C2312yescomplemented
281#1438C2×C2311yescharacteristic
282#1438C2×C2311yescharacteristic
283#1437C2×C2312yescomplemented
284C4⋊#95C2×C2312yescomplemented
285C22⋊#95C2×C2312yescomplemented
286C22×C23—24no—
287C22×C23—28no—
288C22×C23—44no—
289C22×C23—42no—
290C22×C23—42no—
291C22×C23—44no—
292C22×C23—44no—
293C22×C23—44no—
294C22×C23—44no—
295C22×C23—42no—
296C22×C23—44no—
297C22×C23—44no—
298C22×C23—42no—
299C22×C23—42no—
300C22×C23—82no—
301C22×C23—82no—
302C22×C23—82no—
303C22×C23—82no—
304C22×C23C22×C22⋊C411yescharacteristic
305C22×C23C4×#4112yes—
306C22×C23#51811yescharacteristic
307C22×C23#51811yescharacteristic
308C22×C23#52012yes—
309C22×C23#53512yes—
310C22×C23#53512yes—
311C22×C23#53812yes—
312C22×C23#53912yes—
313C22×C23#54012yes—
314C22×C23#54212yes—
315C22×C23#54612yes—
316C22×C23#54612yes—
317C22×C23#54712yes—
318C22×C23#54712yes—
319C22×C23#55111yescharacteristic
320C22×C23#55312yes—
321C22×C23#55911yescharacteristic
322C22×C23#56011yescharacteristic
323C22×C23#56212yes—
324C22×C23#56311yescharacteristic
325C4×C23—28no—
326C4×C23—44no—
327C4×C23—42no—
328C4×C23—44no—
329C4×C23—44no—
330C4×C23—82no—
331C4×C23—82no—
332C23×D4C22×C2311yescharacteristic
333C24×C4C22×C2311yescharacteristic
334C2×#123—28no—
335C2×#123C4×C2318yescomplemented
336C22×C4⋊C4C22×C2312yes—
337C22×C4⋊C4C22×C2312yes—
338C22×C22⋊C4C22×C2312yes—
339C22×C22⋊C4C22×C2312yes—
340C22×C22⋊C4C22×C2311yescharacteristic
341C22×C22⋊C4C22×C2312yes—
342C22×C22⋊C4C22×C2312yes—
343C22×C22⋊C4C22×C2311yescharacteristic
344C22×C22⋊C4C22×C2311yescharacteristic
345C22×C42C22×C2312yes—
346#388—28no—
347C2×#95C22×C2312yes—
348C2×#95C22×C2312yes—
349C2×#95C22×C2312yes—
350C2×#95C22×C2312yes—
351C2×#95C22×C2312yes—
352C2×#95C22×C2312yes—
353C2×#95C22×C2311yescharacteristic
354C2×#95C22×C2312yes—
355C2×#95C22×C2311yescharacteristic
356C2×C23—44no—
357C2×C23—82no—
358C2×C23#230911yescharacteristic
359C2×C23#233512yes—
360C2×C23#233912yes—
361C2×C23#237411yescharacteristic
362C2×C23#237611yescharacteristic
363C2×C23#270312yes—
364C2×C23#270612yes—
365C2×C23#271012yes—
366C2×C23#271812yes—
367C25C23×C2311yescharacteristic
368C25C23×C2311yescharacteristic
369C22×D4—24no—
370C22×D4—41no—
371C22×D4—41no—
372C22×D4—42no—
373C22×D4C2×C4×C2314yes—
374C23×C4—24no—
375C23×C4—41no—
376C23×C4—41no—
377C23×C4—42no—
378C23×C4C2×C4×C2314yes—
379C23×C4C23×C2312yes—
380C23×C4C23×C2312yes—
381C23×C4C23×C2312yes—
382C23×C4C23×C2311yescharacteristic
383C23×C4C23×C2312yes—
384C23×C4C23×C2312yes—
385C23×C4C23×C2312yes—
386#123—232no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—42no—
392C2×C4⋊C4C2×C4×C2318yes—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—41no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—41no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4C2×C4×C2318yes—
409C2×C22⋊C4C2×C4×C2314yes—
410C2×C22⋊C4D4×C2318yes—
411C2×C42—24no—
412C2×C42—24no—
413C2×C42—42no—
414#95—24no—
415#95—24no—
416#95—41no—
417#95—42no—
418#95—42no—
419#95—42no—
420#95—42no—
421#95—42no—
422#95—41no—
423C23#1450611yescharacteristic, direct product factor, Sylow
424C24—22no—
425C24—22no—
426C24—22no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24—22no—
438C24—22no—
439C24C22×C4×C2312yes—
440C24C22×C4×C2312yes—
441C24C2×D4×C2314yes—
442C24C24×C2311yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
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—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
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—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4C22×C4×C2314yes—
494C22×C4C22×C4×C2312yes—
495C22×C4C22×C4×C2314yes—
496C22×C4C2×D4×C2314yes—
497C22×C4C2×D4×C2314yes—
498C22×C4#41×C2314yes—
499C22×C4#41×C2314yes—
500C4⋊C4—216no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4C22⋊C4×C23116yescomplemented
509C42—48no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23C2×C22⋊C4×C2314yes—
552C23C23×C4×C2311yescharacteristic
553C23C22×D4×C2312yes—
554C23C2×#41×C2312yes—
555C23C2×#41×C2312yes—
556C23C23×H(5,𝔽2)11yescharacteristic
557C23C23×H(5,𝔽2)11yescharacteristic
558C23C23×H(5,𝔽2)12yes—
559C23C23×H(5,𝔽2)12yes—
560C23C23×2−1+412yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
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—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4C2×C22⋊C4×C2318yes—
610C22—24no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊C4×C2311yescharacteristic
629C22C4×#41×C2312yes—
630C22C23×#51811yescharacteristic
631C22C23×#51811yescharacteristic
632C22C23×#52012yes—
633C22C23×#53512yes—
634C22C23×#53512yes—
635C22C23×#53812yes—
636C22C23×#53912yes—
637C22C23×#54012yes—
638C22C23×#54212yes—
639C22C23×#54612yes—
640C22C23×#54612yes—
641C22C23×#54712yes—
642C22C23×#54712yes—
643C22C23×#55111yescharacteristic
644C22C23×#55312yes—
645C22C23×#55911yescharacteristic
646C22C23×#56011yescharacteristic
647C22C23×#56212yes—
648C22C23×#56311yescharacteristic
649C4—28no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—82no—
655C4—82no—
656C2—44no—
657C2—82no—
658C2C23×#230911yescharacteristic
659C2C23×#233512yes—
660C2C23×#233912yes—
661C2C23×#237411yescharacteristic
662C2C23×#237611yescharacteristic
663C2C23×#270312yes—
664C2C23×#270612yes—
665C2C23×#271012yes—
666C2C23×#271812yes—
subgroupG/H
center, socleC24×C23C24
commutatorC23C23×C4×C23
Frattini, cosocleC24C24×C23
prime radicalC23×D4×C23C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#388C2semidirect
3. C23×#1445C2semidirect
4. C23×#1439C2semidirect
7. C23×#1437C2semidirect
8. C23×C4⋊#95C2semidirect
9. C23×C22⋊#95C2semidirect
13. C2×C23×#123C4semidirect
174. C22⋊C4×C23C22⋊C4semidirect
176. #14506C23direct (N × H)
278. C2×#388C2×C23semidirect
279. #1445C2×C23semidirect
280. #1439C2×C23semidirect
283. #1437C2×C23semidirect
284. C4⋊#95C2×C23semidirect
285. C22⋊#95C2×C23semidirect
335. C2×#123C4×C23semidirect
423. C23#14506direct (N × H)
508. C22⋊C4C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C22×C23×#1232
C2×C23×#3882
C23×#14452
C23×#14392
C23×#14382
C23×#14372
C23×C4⋊#952
C23×C22⋊#952
#1450623