Finite Group Databasebeta

← C23×#16596 #19164222

Subgroups — C23×#16596 #19164222

Order 5888.

1590 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#398C211yesmaximal, characteristic, complemented
2C2×C23×#394C212yesmaximal
3C2×C23×#390C211yesmaximal, characteristic
4C23×#1441C211yesmaximal, characteristic, complemented
5C23×#1440C212yesmaximal, complemented
6C23×#1440C212yesmaximal, complemented
7C23×#1439C212yesmaximal, complemented
8C23×#1439C212yesmaximal, complemented
9C23×#1436C212yesmaximal, complemented
10C23×D4×C23C2211yescharacteristic
11C22×C4⋊C4×C23C2212yes—
12C22×C4⋊C4×C23C2211yescharacteristic
13C22×C4⋊C4×C23C2211yescharacteristic
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C22⋊C4×C23C2212yes—
16C22×C22⋊C4×C23C2212yes—
17C22×C22⋊C4×C23C2212yes—
18C22×C22⋊C4×C23C2212yes—
19C22×C22⋊C4×C23C2212yes—
20C22×C22⋊C4×C23C2212yes—
21C22×C42×C23C2211yescharacteristic
22C23×#398—24no—
23C23×#394—28no—
24C23×#390—24no—
25C2×C23×#95C2212yes—
26C2×C23×#95C2212yes—
27C2×C23×#95C2212yes—
28C2×C23×#95C2212yes—
29C2×C23×#95C2211yescharacteristic
30C2×C23×#95C2212yes—
31C2×C23×#95C2212yes—
32C2×C23×#95C2212yes—
33C2×C23×#95C2211yescharacteristic
34C25×C23C2312yes—
35C22×D4×C23—22no—
36C22×D4×C23—22no—
37C22×D4×C23—41no—
38C22×D4×C23—41no—
39C22×D4×C23—41no—
40C22×D4×C23—41no—
41C22×D4×C23D414yes—
42C23×C4×C23C2312yes—
43C23×C4×C23C2312yes—
44C23×C4×C23C2312yes—
45C23×C4×C23C2311yescharacteristic
46C23×C4×C23C2311yescharacteristic
47C23×C4×C23C2312yes—
48C23×C4×C23C2312yes—
49C23×C4×C23C2311yescharacteristic
50C2×C4⋊C4×C23—24no—
51C2×C4⋊C4×C23—22no—
52C2×C4⋊C4×C23—24no—
53C2×C4⋊C4×C23—24no—
54C2×C4⋊C4×C23—41no—
55C2×C4⋊C4×C23—41no—
56C2×C4⋊C4×C23—41no—
57C2×C4⋊C4×C23—41no—
58C2×C4⋊C4×C23—42no—
59C2×C4⋊C4×C23—41no—
60C2×C4⋊C4×C23D414yes—
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—24no—
69C2×C22⋊C4×C23—24no—
70C2×C22⋊C4×C23—24no—
71C2×C22⋊C4×C23—42no—
72C2×C22⋊C4×C23—42no—
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×C23—42no—
79C2×C42×C23—22no—
80C2×C42×C23—41no—
81C2×C42×C23—41no—
82C23×#95—24no—
83C23×#95—22no—
84C23×#95—24no—
85C23×#95—24no—
86C23×#95—24no—
87C23×#95—42no—
88C23×#95—42no—
89C23×#95—42no—
90C23×#95—41no—
91C24×C23—22no—
92C24×C23—22no—
93C24×C23—22no—
94C24×C23—22no—
95C24×C23—22no—
96C24×C23—22no—
97C24×C23—22no—
98C24×C23—22no—
99C24×C23—22no—
100C24×C23—22no—
101C24×C23—22no—
102C24×C23—22no—
103C24×C23—22no—
104C24×C23—22no—
105C24×C23C2×D414yes—
106C24×C23C2411yescharacteristic
107C2×D4×C23—28no—
108C2×D4×C23—44no—
109C2×D4×C23—44no—
110C2×D4×C23—44no—
111C2×D4×C23—44no—
112C2×D4×C23—44no—
113C2×D4×C23—44no—
114C22×C4×C23—21no—
115C22×C4×C23—21no—
116C22×C4×C23—21no—
117C22×C4×C23—21no—
118C22×C4×C23—22no—
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—21no—
133C22×C4×C23—21no—
134C22×C4×C23—21no—
135C22×C4×C23—21no—
136C22×C4×C23—22no—
137C22×C4×C23—22no—
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—22no—
156C22×C4×C23—22no—
157C22×C4×C23—22no—
158C22×C4×C23—22no—
159C22×C4×C23C2×D412yes—
160C22×C4×C23C2×D412yes—
161C22×C4×C23C2×D412yes—
162C22×C4×C23C2×D412yes—
163C22×C4×C23#4114yes—
164C22×C4×C23#4114yes—
165C22×C4×C23#4112yes—
166C4⋊C4×C23—44no—
167C4⋊C4×C23—44no—
168C4⋊C4×C23—44no—
169C4⋊C4×C23—48no—
170C22⋊C4×C23—48no—
171C22⋊C4×C23—48no—
172C22⋊C4×C23—48no—
173C22⋊C4×C23—48no—
174C22⋊C4×C23—48no—
175C22⋊C4×C23—48no—
176C42×C23—44no—
177#16596C2311yesmaximal, characteristic, direct product factor, Sylow
178C23×C23—24no—
179C23×C23—24no—
180C23×C23—24no—
181C23×C23—24no—
182C23×C23—24no—
183C23×C23—24no—
184C23×C23—24no—
185C23×C23—42no—
186C23×C23—42no—
187C23×C23—42no—
188C23×C23—42no—
189C23×C23—42no—
190C23×C23—42no—
191C23×C23—42no—
192C23×C23—42no—
193C23×C23—42no—
194C23×C23—42no—
195C23×C23—42no—
196C23×C23—42no—
197C23×C23—42no—
198C23×C23—42no—
199C23×C23—42no—
200C23×C23—42no—
201C23×C23—42no—
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—42no—
213C23×C23C22×D411yescharacteristic
214C23×C23C22×D411yescharacteristic
215C23×C23C2×#4111yescharacteristic
216C23×C23C2×#4112yes—
217C23×C23C2×#4112yes—
218C23×C23H(5,𝔽2)11yescharacteristic
219C23×C23H(5,𝔽2)11yescharacteristic
220C23×C23H(5,𝔽2)11yescharacteristic
221C23×C23H(5,𝔽2)11yescharacteristic
222C23×C23H(5,𝔽2)12yes—
223C23×C232−1+412yes—
224D4×C23—416no—
225C2×C4×C23—22no—
226C2×C4×C23—22no—
227C2×C4×C23—22no—
228C2×C4×C23—22no—
229C2×C4×C23—22no—
230C2×C4×C23—22no—
231C2×C4×C23—24no—
232C2×C4×C23—22no—
233C2×C4×C23—22no—
234C2×C4×C23—24no—
235C2×C4×C23—24no—
236C2×C4×C23—24no—
237C2×C4×C23—24no—
238C2×C4×C23—24no—
239C2×C4×C23—24no—
240C2×C4×C23—24no—
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—41no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—41no—
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×C23#12114yes—
277C2×#398C2×C2311yescharacteristic, complemented
278C2×#394C2×C2312yes—
279C2×#390C2×C2311yescharacteristic
280#1441C2×C2311yescharacteristic, complemented
281#1440C2×C2312yescomplemented
282#1440C2×C2312yescomplemented
283#1439C2×C2312yescomplemented
284#1439C2×C2312yescomplemented
285#1436C2×C2312yescomplemented
286C22×C23—44no—
287C22×C23—44no—
288C22×C23—44no—
289C22×C23—44no—
290C22×C23—44no—
291C22×C23—44no—
292C22×C23—44no—
293C22×C23—82no—
294C22×C23—82no—
295C22×C23—82no—
296C22×C23—82no—
297C22×C23—82no—
298C22×C23—82no—
299C22×C23—82no—
300C22×C23—82no—
301C22×C23C2×#12111yescharacteristic
302C22×C23#52912yes—
303C22×C23#53411yescharacteristic
304C22×C23#53411yescharacteristic
305C22×C23#53411yescharacteristic
306C22×C23#53511yescharacteristic
307C22×C23#53811yescharacteristic
308C22×C23#53811yescharacteristic
309C22×C23#53912yes—
310C22×C23#54012yes—
311C22×C23#54012yes—
312C22×C23#54212yes—
313C22×C23#54212yes—
314C22×C23#54612yes—
315C22×C23#54612yes—
316C22×C23#54712yes—
317C22×C23#54812yes—
318C22×C23#55312yes—
319C22×C23#55511yescharacteristic
320C22×C23#55911yescharacteristic
321C22×C23#56011yescharacteristic
322C22×C23#56212yes—
323C22×C23#56311yescharacteristic
324C4×C23—24no—
325C4×C23—44no—
326C4×C23—42no—
327C4×C23—42no—
328C4×C23—44no—
329C4×C23—82no—
330C4×C23—82no—
331C4×C23—82no—
332C23×D4C22×C2311yescharacteristic
333C22×C4⋊C4C22×C2312yes—
334C22×C4⋊C4C22×C2311yescharacteristic
335C22×C4⋊C4C22×C2311yescharacteristic
336C22×C4⋊C4C22×C2311yescharacteristic
337C22×C22⋊C4C22×C2312yes—
338C22×C22⋊C4C22×C2312yes—
339C22×C22⋊C4C22×C2312yes—
340C22×C22⋊C4C22×C2312yes—
341C22×C22⋊C4C22×C2312yes—
342C22×C22⋊C4C22×C2312yes—
343C22×C42C22×C2311yescharacteristic
344#398—24no—
345#394—28no—
346#390—24no—
347C2×#95C22×C2312yes—
348C2×#95C22×C2312yes—
349C2×#95C22×C2311yescharacteristic
350C2×#95C22×C2312yes—
351C2×#95C22×C2312yes—
352C2×#95C22×C2312yes—
353C2×#95C22×C2312yes—
354C2×#95C22×C2312yes—
355C2×#95C22×C2311yescharacteristic
356C2×C23—84no—
357C2×C23#242712yes—
358C2×C23#259311yescharacteristic
359C2×C23#261711yescharacteristic
360C2×C23#265112yes—
361C2×C23#269911yescharacteristic
362C2×C23#270112yes—
363C2×C23#270911yescharacteristic
364C2×C23#271112yes—
365C2×C23#272411yescharacteristic
366C2×C23#273111yescharacteristic
367C2×C23#283011yescharacteristic
368C25C23×C2312yes—
369C22×D4—22no—
370C22×D4—22no—
371C22×D4—41no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4—41no—
375C22×D4D4×C2314yes—
376C23×C4C23×C2312yes—
377C23×C4C23×C2312yes—
378C23×C4C23×C2312yes—
379C23×C4C23×C2311yescharacteristic
380C23×C4C23×C2311yescharacteristic
381C23×C4C23×C2312yes—
382C23×C4C23×C2312yes—
383C23×C4C23×C2311yescharacteristic
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—41no—
389C2×C4⋊C4—41no—
390C2×C4⋊C4—41no—
391C2×C4⋊C4—41no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4—41no—
394C2×C4⋊C4D4×C2314yes—
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—24no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4—42no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4—42no—
412C2×C22⋊C4—42no—
413C2×C42—22no—
414C2×C42—41no—
415C2×C42—41no—
416#95—24no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—42no—
422#95—42no—
423#95—42no—
424#95—41no—
425C23#1659611yescharacteristic, direct product factor, Sylow
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24C2×D4×C2314yes—
441C24C24×C2311yescharacteristic
442C2×D4—28no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—44no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
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—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
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—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4C2×D4×C2312yes—
495C22×C4C2×D4×C2312yes—
496C22×C4C2×D4×C2312yes—
497C22×C4C2×D4×C2312yes—
498C22×C4#41×C2314yes—
499C22×C4#41×C2314yes—
500C22×C4#41×C2312yes—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C42—44no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23C22×D4×C2311yescharacteristic
548C23C22×D4×C2311yescharacteristic
549C23C2×#41×C2312yes—
550C23C2×#41×C2312yes—
551C23C2×#41×C2311yescharacteristic
552C23C23×H(5,𝔽2)11yescharacteristic
553C23C23×H(5,𝔽2)11yescharacteristic
554C23C23×H(5,𝔽2)12yes—
555C23C23×H(5,𝔽2)11yescharacteristic
556C23C23×H(5,𝔽2)11yescharacteristic
557C23C23×2−1+412yes—
558D4—416no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
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×C4—42no—
610C2×C4C23×#12114yes—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C2×C23×#12111yescharacteristic
627C22C23×#52912yes—
628C22C23×#53411yescharacteristic
629C22C23×#53411yescharacteristic
630C22C23×#53411yescharacteristic
631C22C23×#53511yescharacteristic
632C22C23×#53811yescharacteristic
633C22C23×#53811yescharacteristic
634C22C23×#53912yes—
635C22C23×#54012yes—
636C22C23×#54012yes—
637C22C23×#54212yes—
638C22C23×#54212yes—
639C22C23×#54612yes—
640C22C23×#54612yes—
641C22C23×#54712yes—
642C22C23×#54812yes—
643C22C23×#55312yes—
644C22C23×#55511yescharacteristic
645C22C23×#55911yescharacteristic
646C22C23×#56011yescharacteristic
647C22C23×#56212yes—
648C22C23×#56311yescharacteristic
649C4—24no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—82no—
657C2—84no—
658C2C23×#242712yes—
659C2C23×#259311yescharacteristic
660C2C23×#261711yescharacteristic
661C2C23×#265112yes—
662C2C23×#269911yescharacteristic
663C2C23×#270112yes—
664C2C23×#270911yescharacteristic
665C2C23×#271112yes—
666C2C23×#272411yescharacteristic
667C2C23×#273111yescharacteristic
668C2C23×#283011yescharacteristic
subgroupG/H
center, socleC24×C23C24
commutator, Frattini, cosocleC24C24×C23
prime radicalC23×D4×C23C22
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
4. C23×#1441C2semidirect
5. C23×#1440C2semidirect
6. C23×#1440C2semidirect
7. C23×#1439C2semidirect
8. C23×#1439C2semidirect
9. C23×#1436C2semidirect
177. #16596C23direct (N × H)
277. C2×#398C2×C23semidirect
280. #1441C2×C23semidirect
281. #1440C2×C23semidirect
282. #1440C2×C23semidirect
283. #1439C2×C23semidirect
284. #1439C2×C23semidirect
285. #1436C2×C23semidirect
425. C23#16596direct (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×#3942
C2×C23×#3902
C23×#14412
C23×#14402
C23×#14392
C23×#14362
#1659623