Finite Group Databasebeta

← C23×#16514 #19164140

Subgroups — C23×#16514 #19164140

Order 5888.

1774 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#396C212yesmaximal, complemented
2C2×C23×#394C211yesmaximal, characteristic
3C2×C23×#394C212yesmaximal
4C2×C23×#388C212yesmaximal, complemented
5C2×C23×#384C211yesmaximal, characteristic, complemented
6C23×#1441C211yesmaximal, characteristic, complemented
7C23×#1441C212yesmaximal, complemented
8C23×#1439C212yesmaximal, complemented
9C23×#1439C212yesmaximal, complemented
10C23×D4×C23C2211yescharacteristic
11C22×C4⋊C4×C23C2212yes—
12C22×C4⋊C4×C23C2212yes—
13C22×C4⋊C4×C23C2212yes—
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C4⋊C4×C23C2211yescharacteristic
16C22×C4⋊C4×C23C2211yescharacteristic
17C22×C22⋊C4×C23C2212yes—
18C22×C22⋊C4×C23C2212yes—
19C22×C22⋊C4×C23C2212yes—
20C22×C22⋊C4×C23C2212yes—
21C22×C22⋊C4×C23C2212yes—
22C22×C22⋊C4×C23C2212yes—
23C22×C42×C23C2211yescharacteristic
24C23×#396—28no—
25C23×#394—24no—
26C23×#394—28no—
27C23×#388—28no—
28C23×#384—24no—
29C2×C23×#95C2212yes—
30C2×C23×#95C2212yes—
31C2×C23×#95C2212yes—
32C2×C23×#95C2212yes—
33C2×C23×#95C2211yescharacteristic
34C2×C23×#95C2211yescharacteristic
35C2×C23×#95C2212yes—
36C25×C23C2312yes—
37C22×D4×C23—24no—
38C22×D4×C23—22no—
39C22×D4×C23—42no—
40C22×D4×C23—41no—
41C22×D4×C23—41no—
42C23×C4×C23C2312yes—
43C23×C4×C23C2312yes—
44C23×C4×C23C2312yes—
45C23×C4×C23C2311yescharacteristic
46C23×C4×C23C2312yes—
47C23×C4×C23C2311yescharacteristic
48C23×C4×C23C2311yescharacteristic
49C23×C4×C23C2312yes—
50C2×C4⋊C4×C23—24no—
51C2×C4⋊C4×C23—24no—
52C2×C4⋊C4×C23—22no—
53C2×C4⋊C4×C23—24no—
54C2×C4⋊C4×C23—24no—
55C2×C4⋊C4×C23—24no—
56C2×C4⋊C4×C23—24no—
57C2×C4⋊C4×C23—24no—
58C2×C4⋊C4×C23—24no—
59C2×C4⋊C4×C23—41no—
60C2×C4⋊C4×C23—41no—
61C2×C4⋊C4×C23—42no—
62C2×C4⋊C4×C23—42no—
63C2×C4⋊C4×C23—41no—
64C2×C4⋊C4×C23—41no—
65C2×C4⋊C4×C23—41no—
66C2×C4⋊C4×C23D414yes—
67C2×C22⋊C4×C23—24no—
68C2×C22⋊C4×C23—24no—
69C2×C22⋊C4×C23—24no—
70C2×C22⋊C4×C23—24no—
71C2×C22⋊C4×C23—24no—
72C2×C22⋊C4×C23—24no—
73C2×C22⋊C4×C23—24no—
74C2×C22⋊C4×C23—24no—
75C2×C22⋊C4×C23—24no—
76C2×C22⋊C4×C23—24no—
77C2×C22⋊C4×C23—24no—
78C2×C22⋊C4×C23—24no—
79C2×C22⋊C4×C23—24no—
80C2×C22⋊C4×C23—42no—
81C2×C22⋊C4×C23—42no—
82C2×C22⋊C4×C23—42no—
83C2×C22⋊C4×C23—42no—
84C2×C22⋊C4×C23D418yes—
85C2×C42×C23—22no—
86C2×C42×C23—22no—
87C2×C42×C23D414yescomplemented
88C23×#95—24no—
89C23×#95—24no—
90C23×#95—24no—
91C23×#95—22no—
92C23×#95—24no—
93C23×#95—41no—
94C23×#95—42no—
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×C23—22no—
106C24×C23—22no—
107C24×C23—22no—
108C24×C23—22no—
109C24×C23C2×D414yes—
110C24×C23C2411yescharacteristic
111C2×D4×C23—48no—
112C2×D4×C23—44no—
113C2×D4×C23—44no—
114C2×D4×C23—48no—
115C2×D4×C23—44no—
116C22×C4×C23—22no—
117C22×C4×C23—22no—
118C22×C4×C23—21no—
119C22×C4×C23—21no—
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—21no—
133C22×C4×C23—21no—
134C22×C4×C23—21no—
135C22×C4×C23—22no—
136C22×C4×C23—21no—
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×D414yes—
161C22×C4×C23C2×D412yes—
162C22×C4×C23C2×D412yes—
163C22×C4×C23C2×D414yes—
164C22×C4×C23C2×D412yes—
165C22×C4×C23C2×D414yes—
166C22×C4×C23#4112yes—
167C22×C4×C23#4114yes—
168C4⋊C4×C23—44no—
169C4⋊C4×C23—48no—
170C4⋊C4×C23—48no—
171C4⋊C4×C23—44no—
172C4⋊C4×C23—44no—
173C4⋊C4×C23—48no—
174C22⋊C4×C23—216no—
175C22⋊C4×C23—48no—
176C22⋊C4×C23—48no—
177C22⋊C4×C23—48no—
178C22⋊C4×C23—48no—
179C22⋊C4×C23—48no—
180C42×C23—28no—
181#16514C2311yesmaximal, characteristic, direct product factor, Sylow
182C23×C23—24no—
183C23×C23—24no—
184C23×C23—24no—
185C23×C23—24no—
186C23×C23—24no—
187C23×C23—24no—
188C23×C23—24no—
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×C23—42no—
214C23×C23—42no—
215C23×C23—42no—
216C23×C23—42no—
217C23×C23C22×D411yescharacteristic
218C23×C23C22×D411yescharacteristic
219C23×C23C22×D412yes—
220C23×C23C2×#4111yescharacteristic
221C23×C23C2×#4112yes—
222C23×C23H(5,𝔽2)11yescharacteristic
223C23×C23H(5,𝔽2)11yescharacteristic
224C23×C23H(5,𝔽2)12yes—
225C23×C232−1+411yescharacteristic
226C23×C232−1+411yescharacteristic
227C23×C232−1+412yes—
228D4×C23—416no—
229C2×C4×C23—24no—
230C2×C4×C23—22no—
231C2×C4×C23—22no—
232C2×C4×C23—22no—
233C2×C4×C23—22no—
234C2×C4×C23—24no—
235C2×C4×C23—22no—
236C2×C4×C23—22no—
237C2×C4×C23—24no—
238C2×C4×C23—24no—
239C2×C4×C23—24no—
240C2×C4×C23—24no—
241C2×C4×C23—24no—
242C2×C4×C23—22no—
243C2×C4×C23—22no—
244C2×C4×C23—24no—
245C2×C4×C23—24no—
246C2×C4×C23—24no—
247C2×C4×C23—24no—
248C2×C4×C23—42no—
249C2×C4×C23—42no—
250C2×C4×C23—42no—
251C2×C4×C23—42no—
252C2×C4×C23—41no—
253C2×C4×C23—41no—
254C2×C4×C23—42no—
255C2×C4×C23—42no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
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—42no—
277C2×C4×C23—42no—
278C2×C4×C23—42no—
279C2×C4×C23#12114yes—
280C2×C4×C23#12118yes—
281C2×#396C2×C2312yescomplemented
282C2×#394C2×C2311yescharacteristic
283C2×#394C2×C2312yes—
284C2×#388C2×C2312yescomplemented
285C2×#384C2×C2311yescharacteristic, complemented
286#1441C2×C2312yescomplemented
287#1441C2×C2311yescharacteristic, complemented
288#1439C2×C2312yescomplemented
289#1439C2×C2312yescomplemented
290C22×C23—44no—
291C22×C23—44no—
292C22×C23—44no—
293C22×C23—44no—
294C22×C23—44no—
295C22×C23—44no—
296C22×C23—44no—
297C22×C23—82no—
298C22×C23—82no—
299C22×C23—82no—
300C22×C23—82no—
301C22×C23—82no—
302C22×C23—82no—
303C22×C23—82no—
304C22×C23—82no—
305C22×C23C2×#12111yescharacteristic
306C22×C23C2×#12112yes—
307C22×C23#53411yescharacteristic
308C22×C23#53512yes—
309C22×C23#53611yescharacteristic
310C22×C23#53612yes—
311C22×C23#53711yescharacteristic
312C22×C23#53712yes—
313C22×C23#53711yescharacteristic
314C22×C23#53911yescharacteristic
315C22×C23#53911yescharacteristic
316C22×C23#54212yes—
317C22×C23#54212yes—
318C22×C23D4212yes—
319C22×C23D4211yescharacteristic
320C22×C23#54712yes—
321C22×C23#54712yes—
322C22×C23#54812yes—
323C22×C23#55312yes—
324C22×C23#55511yescharacteristic
325C22×C23#56011yescharacteristic
326C22×C23#56312yes—
327C22×C23#56311yescharacteristic
328C4×C23—24no—
329C4×C23—28no—
330C4×C23—42no—
331C4×C23—42no—
332C4×C23—44no—
333C4×C23—82no—
334C4×C23—82no—
335C4×C23—82no—
336C23×D4C22×C2311yescharacteristic
337C22×C4⋊C4C22×C2312yes—
338C22×C4⋊C4C22×C2312yes—
339C22×C4⋊C4C22×C2311yescharacteristic
340C22×C4⋊C4C22×C2312yes—
341C22×C4⋊C4C22×C2311yescharacteristic
342C22×C4⋊C4C22×C2311yescharacteristic
343C22×C22⋊C4C22×C2312yes—
344C22×C22⋊C4C22×C2312yes—
345C22×C22⋊C4C22×C2312yes—
346C22×C22⋊C4C22×C2312yes—
347C22×C22⋊C4C22×C2312yes—
348C22×C22⋊C4C22×C2312yes—
349C22×C42C22×C2311yescharacteristic
350#396—28no—
351#394—24no—
352#394—28no—
353#388—28no—
354#384—24no—
355C2×#95C22×C2311yescharacteristic
356C2×#95C22×C2312yes—
357C2×#95C22×C2312yes—
358C2×#95C22×C2312yes—
359C2×#95C22×C2312yes—
360C2×#95C22×C2312yes—
361C2×#95C22×C2311yescharacteristic
362C2×C23—84no—
363C2×C23#241612yes—
364C2×C23#253711yescharacteristic
365C2×C23#261811yescharacteristic
366C2×C23#262012yes—
367C2×C23#266911yescharacteristic
368C2×C23#267412yes—
369C2×C23#268011yescharacteristic
370C2×C23#271812yes—
371C2×C23#273111yescharacteristic
372C2×C23#279511yescharacteristic
373C2×C23#282311yescharacteristic
374C25C23×C2312yes—
375C22×D4—24no—
376C22×D4—22no—
377C22×D4—42no—
378C22×D4—41no—
379C22×D4—41no—
380C23×C4C23×C2312yes—
381C23×C4C23×C2312yes—
382C23×C4C23×C2312yes—
383C23×C4C23×C2311yescharacteristic
384C23×C4C23×C2312yes—
385C23×C4C23×C2311yescharacteristic
386C23×C4C23×C2312yes—
387C23×C4C23×C2311yescharacteristic
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—41no—
399C2×C4⋊C4—42no—
400C2×C4⋊C4—42no—
401C2×C4⋊C4—41no—
402C2×C4⋊C4—41no—
403C2×C4⋊C4—41no—
404C2×C4⋊C4D4×C2314yes—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—24no—
416C2×C22⋊C4—24no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4—42no—
419C2×C22⋊C4—42no—
420C2×C22⋊C4—42no—
421C2×C22⋊C4—42no—
422C2×C22⋊C4D4×C2318yes—
423C2×C42—22no—
424C2×C42—22no—
425C2×C42D4×C2314yescomplemented
426#95—24no—
427#95—24no—
428#95—24no—
429#95—22no—
430#95—24no—
431#95—41no—
432#95—42no—
433C23#1651411yescharacteristic, direct product factor, Sylow
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24C2×D4×C2314yes—
449C24C24×C2311yescharacteristic
450C2×D4—48no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—48no—
454C2×D4—44no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
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×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4C2×D4×C2314yes—
499C22×C4C2×D4×C2312yes—
500C22×C4C2×D4×C2312yes—
501C22×C4C2×D4×C2312yes—
502C22×C4C2×D4×C2314yes—
503C22×C4C2×D4×C2312yes—
504C22×C4C2×D4×C2314yes—
505C22×C4#41×C2312yes—
506C22×C4#41×C2314yes—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C22⋊C4—216no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C42—28no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
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—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23C22×D4×C2312yes—
556C23C22×D4×C2311yescharacteristic
557C23C22×D4×C2311yescharacteristic
558C23C2×#41×C2311yescharacteristic
559C23C2×#41×C2312yes—
560C23C23×H(5,𝔽2)12yes—
561C23C23×H(5,𝔽2)11yescharacteristic
562C23C23×H(5,𝔽2)11yescharacteristic
563C23C23×2−1+411yescharacteristic
564C23C23×2−1+412yes—
565C23C23×2−1+411yescharacteristic
566D4—416no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
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×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4C23×#12114yes—
618C2×C4C23×#12118yes—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×C23×#12111yescharacteristic
635C22C2×C23×#12112yes—
636C22C23×#53411yescharacteristic
637C22C23×#53512yes—
638C22C23×#53611yescharacteristic
639C22C23×#53612yes—
640C22C23×#53712yes—
641C22C23×#53711yescharacteristic
642C22C23×#53711yescharacteristic
643C22C23×#53911yescharacteristic
644C22C23×#53911yescharacteristic
645C22C23×#54212yes—
646C22C23×#54212yes—
647C22D42×C2312yes—
648C22D42×C2311yescharacteristic
649C22C23×#54712yes—
650C22C23×#54712yes—
651C22C23×#54812yes—
652C22C23×#55312yes—
653C22C23×#55511yescharacteristic
654C22C23×#56011yescharacteristic
655C22C23×#56312yes—
656C22C23×#56311yescharacteristic
657C4—28no—
658C4—24no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C2—84no—
666C2C23×#241612yes—
667C2C23×#253711yescharacteristic
668C2C23×#261811yescharacteristic
669C2C23×#262012yes—
670C2C23×#266911yescharacteristic
671C2C23×#267412yes—
672C2C23×#268011yescharacteristic
673C2C23×#271812yes—
674C2C23×#273111yescharacteristic
675C2C23×#279511yescharacteristic
676C2C23×#282311yescharacteristic
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×#396C2semidirect
4. C2×C23×#388C2semidirect
5. C2×C23×#384C2semidirect
6. C23×#1441C2semidirect
7. C23×#1441C2semidirect
8. C23×#1439C2semidirect
9. C23×#1439C2semidirect
87. C2×C42×C23D4semidirect
181. #16514C23direct (N × H)
281. C2×#396C2×C23semidirect
284. C2×#388C2×C23semidirect
285. C2×#384C2×C23semidirect
286. #1441C2×C23semidirect
287. #1441C2×C23semidirect
288. #1439C2×C23semidirect
289. #1439C2×C23semidirect
425. C2×C42D4×C23semidirect
433. C23#16514direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3962
C2×C23×#3942
C2×C23×#3882
C2×C23×#3842
C23×#14412
C23×#14392
#1651423