Finite Group Databasebeta

← C23×#18297 #19165923

Subgroups — C23×#18297 #19165923

Order 5888.

1084 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#538C211yesmaximal, characteristic
2C23×#3189C211yesmaximal, characteristic, complemented
3C23×#3187C211yesmaximal, characteristic, complemented
4C23×#2626C211yesmaximal, characteristic, complemented
5C23×#1618C214yesmaximal, complemented
6C23×#1606C214yesmaximal, complemented
7C23×#1520C211yesmaximal, characteristic, complemented
8C23×#1519C211yesmaximal, characteristic, complemented
9C2×C23×#323C211yesmaximal, characteristic, complemented
10C23×#538—22no—
11C23×#538—22no—
12C23×#538—22no—
13C23×#538C2214yes—
14C2×C23×#126—21no—
15C2×C23×#124—21no—
16C2×C23×#123—21no—
17C2×C23×C22⋊Q8C2211yescharacteristic
18C2×C23×C22⋊Q8C2211yescharacteristic
19C2×C23×#121—21no—
20C2×C23×#121C2211yescharacteristic
21C2×C23×#120—21no—
22C2×C4×D4×C23—21no—
23C22×C22⋊C4×C23C2211yescharacteristic
24C23×#484—21no—
25C23×#484C2212yescomplemented
26C23×#483—21no—
27C23×#481C2212yescomplemented
28C23×#467—22no—
29C23×#465—22no—
30C23×#451—21no—
31C23×#451C2212yescomplemented
32C23×#450—21no—
33C23×#448C2212yescomplemented
34C2×C23×C22⋊C8C2211yescharacteristic
35C2×C23×C22⋊C8C2211yescharacteristic
36C23×#399C2212yescomplemented
37C23×#396—22no—
38C23×#393C2212yescomplemented
39C23×#385—21no—
40C23×#380C2211yescharacteristic
41C23×#379—21no—
42C23×#328C2214yescomplemented
43C23×#327C2214yescomplemented
44C23×#323C2214yescomplemented
45C25×C23D411yescharacteristic
46C22×Q8×C23D411yescharacteristic
47C22×Q8×C23C2311yescharacteristic
48C22×D4×C23—21no—
49C22×D4×C23—21no—
50C22×D4×C23—21no—
51C22×D4×C23D411yescharacteristic
52C23×C4×C23—21no—
53C23×C4×C23D411yescharacteristic
54C23×C4×C23C2311yescharacteristic
55C2×Q16×C23—22no—
56C2×SD16×C23—22no—
57C22×C8×C23—21no—
58C22×C8×C23—21no—
59C23×#126—24no—
60C23×#126—24no—
61C23×#124—24no—
62C23×#124—24no—
63C23×#123—24no—
64C23×#123—24no—
65C23×C22⋊Q8—21no—
66C23×C22⋊Q8—21no—
67C23×C22⋊Q8—21no—
68C23×C22⋊Q8—21no—
69C23×C22⋊Q8—21no—
70C23×C22⋊Q8—21no—
71C23×C22⋊Q8D412yes—
72C23×C22⋊Q8C2312yes—
73C23×#121—24no—
74C23×#121—24no—
75C23×#121—21no—
76C23×#121—21no—
77C23×#121D412yes—
78C23×#121C2312yes—
79C23×#120—24no—
80C23×#120—24no—
81C4×D4×C23—24no—
82C4×D4×C23—24no—
83C2×C4⋊C4×C23—21no—
84C2×C4⋊C4×C23—21no—
85C2×C4⋊C4×C23—21no—
86C2×C4⋊C4×C23—21no—
87C2×C4⋊C4×C23—22no—
88C2×C4⋊C4×C23D411yescharacteristic
89C2×C4⋊C4×C23D411yescharacteristic
90C2×C4⋊C4×C23C2311yescharacteristic
91C2×C22⋊C4×C23—21no—
92C2×C22⋊C4×C23—21no—
93C2×C22⋊C4×C23—21no—
94C2×C22⋊C4×C23—21no—
95C2×C22⋊C4×C23—21no—
96C2×C22⋊C4×C23—21no—
97C2×C22⋊C4×C23—21no—
98C2×C22⋊C4×C23—21no—
99C2×C22⋊C4×C23—21no—
100C2×C22⋊C4×C23—21no—
101C2×C22⋊C4×C23—21no—
102C2×C22⋊C4×C23—22no—
103C2×C22⋊C4×C23—22no—
104C2×C22⋊C4×C23D412yes—
105C2×C22⋊C4×C23C2312yes—
106C2×C42×C23—21no—
107C23×#107—22no—
108C23×#106—22no—
109C23×#103—22no—
110C23×#103—22no—
111C23×#103—22no—
112C23×#102—22no—
113C23×C22⋊C8—21no—
114C23×C22⋊C8—21no—
115C23×C22⋊C8C2312yes—
116C23×C22⋊C8C2312yes—
117C23×#95—22no—
118C23×#95C2312yes—
119C24×C23—21no—
120C24×C23—21no—
121C24×C23—21no—
122C24×C23—22no—
123C24×C23—22no—
124C24×C23—81no—
125C24×C23—81no—
126C24×C23C2×D412yes—
127C24×C23C2×D411yescharacteristic
128C2×Q8×C23—21no—
129C2×Q8×C23—21no—
130C2×Q8×C23—21no—
131C2×Q8×C23—81no—
132C2×Q8×C23—81no—
133C2×Q8×C23C2×D412yes—
134C2×D4×C23—22no—
135C2×D4×C23—22no—
136C2×D4×C23—24no—
137C2×D4×C23—24no—
138C2×D4×C23—21no—
139C2×D4×C23—42no—
140C2×D4×C23—42no—
141C2×D4×C23—81no—
142C2×D4×C23—82no—
143C2×D4×C23—82no—
144C2×D4×C23C2×D412yes—
145C22×C4×C23—21no—
146C22×C4×C23—21no—
147C22×C4×C23—21no—
148C22×C4×C23—21no—
149C22×C4×C23—24no—
150C22×C4×C23—21no—
151C22×C4×C23—21no—
152C22×C4×C23—22no—
153C22×C4×C23—21no—
154C22×C4×C23—81no—
155C22×C4×C23—81no—
156C22×C4×C23—82no—
157C22×C4×C23C2×D412yes—
158C22×C4×C23C2×D411yescharacteristic
159C22×C4×C23C2×D411yescharacteristic
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×D411yescharacteristic
162C22×C4×C23C2×D411yescharacteristic
163C22×C4×C23C2411yescharacteristic
164Q16×C23—82no—
165SD16×C23—82no—
166C2×C8×C23—22no—
167C2×C8×C23—22no—
168C2×C8×C23—81no—
169C2×C8×C23—81no—
170C4⋊C4×C23—24no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—21no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—41no—
178C4⋊C4×C23—41no—
179C4⋊C4×C23—81no—
180C4⋊C4×C23—82no—
181C4⋊C4×C23C2×D412yes—
182C4⋊C4×C23C2×D412yes—
183C22⋊C4×C23—24no—
184C22⋊C4×C23—24no—
185C22⋊C4×C23—24no—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—22no—
194C22⋊C4×C23—41no—
195C22⋊C4×C23—41no—
196C22⋊C4×C23—42no—
197C22⋊C4×C23—82no—
198C22⋊C4×C23—82no—
199C42×C23—24no—
200#18297C2311yesmaximal, characteristic, direct product factor, Sylow
201C23×C23—22no—
202C23×C23—22no—
203C23×C23—22no—
204C23×C23—22no—
205C23×C23—41no—
206C23×C23—41no—
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—81no—
217C23×C23—81no—
218C23×C23—84no—
219C23×C23—84no—
220C23×C23#12011yescharacteristic
221C23×C23#12011yescharacteristic
222C23×C23#13512yes—
223C23×C23#13712yes—
224C23×C23C22×D411yescharacteristic
225Q8×C23—42no—
226Q8×C23—42no—
227Q8×C23—42no—
228Q8×C23—81no—
229D4×C23—42no—
230D4×C23—42no—
231D4×C23—42no—
232D4×C23—44no—
233D4×C23—81no—
234D4×C23—84no—
235D4×C23—84no—
236C2×C4×C23—22no—
237C2×C4×C23—22no—
238C2×C4×C23—22no—
239C2×C4×C23—22no—
240C2×C4×C23—41no—
241C2×C4×C23—41no—
242C2×C4×C23—41no—
243C2×C4×C23—41no—
244C2×C4×C23—41no—
245C2×C4×C23—41no—
246C2×C4×C23—42no—
247C2×C4×C23—42no—
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—41no—
256C2×C4×C23—41no—
257C2×C4×C23—42no—
258C2×C4×C23—81no—
259C2×C4×C23—84no—
260C2×C4×C23#12011yescharacteristic
261C2×C4×C23#12011yescharacteristic
262C2×C4×C23C22×D411yescharacteristic
263C2×C4×C23C22×D411yescharacteristic
264C8×C23—42no—
265C8×C23—42no—
266C2×#538C2×C2311yescharacteristic
267#3189C2×C2311yescharacteristic, complemented
268#3187C2×C2311yescharacteristic, complemented
269#2626C2×C2311yescharacteristic, complemented
270#1618C2×C2314yescomplemented
271#1606C2×C2314yescomplemented
272#1520C2×C2311yescharacteristic, complemented
273#1519C2×C2311yescharacteristic, complemented
274C2×#323C2×C2311yescharacteristic, complemented
275C22×C23—21no—
276C22×C23—21no—
277C22×C23—21no—
278C22×C23—21no—
279C22×C23—21no—
280C22×C23—21no—
281C22×C23—42no—
282C22×C23—42no—
283C22×C23—44no—
284C22×C23—42no—
285C22×C23—42no—
286C22×C23—42no—
287C22×C23—42no—
288C22×C23—42no—
289C22×C23—42no—
290C22×C23—82no—
291C22×C23—84no—
292C22×C23—84no—
293C22×C23—84no—
294C22×C23#45212yes—
295C22×C23#45212yes—
296C22×C23C2×#12011yescharacteristic
297C22×C23C2×#13511yescharacteristic
298C22×C23C2×#13711yescharacteristic
299C4×C23—21no—
300C4×C23—21no—
301C4×C23—21no—
302C4×C23—21no—
303C4×C23—42no—
304C4×C23—42no—
305C4×C23—42no—
306C4×C23—42no—
307C4×C23—82no—
308C4×C23—81no—
309C4×C23—81no—
310#538—22no—
311#538—22no—
312#538—22no—
313#538C22×C2314yes—
314C2×#126—21no—
315C2×#124—21no—
316C2×#123—21no—
317C2×C22⋊Q8C22×C2311yescharacteristic
318C2×C22⋊Q8C22×C2311yescharacteristic
319C2×#121—21no—
320C2×#121C22×C2311yescharacteristic
321C2×#120—21no—
322C2×C4×D4—21no—
323C22×C22⋊C4C22×C2311yescharacteristic
324#484—21no—
325#484C22×C2312yescomplemented
326#483—21no—
327#481C22×C2312yescomplemented
328#467—22no—
329#465—22no—
330#451—21no—
331#451C22×C2312yescomplemented
332#450—21no—
333#448C22×C2312yescomplemented
334C2×C22⋊C8C22×C2311yescharacteristic
335C2×C22⋊C8C22×C2311yescharacteristic
336#399C22×C2312yescomplemented
337#396—22no—
338#393C22×C2312yescomplemented
339#385—21no—
340#380C22×C2311yescharacteristic
341#379—21no—
342#328C22×C2314yescomplemented
343#327C22×C2314yescomplemented
344#323C22×C2314yescomplemented
345C2×C23—21no—
346C2×C23—21no—
347C2×C23—22no—
348C2×C23—22no—
349C2×C23—82no—
350C2×C23—82no—
351C2×C23C2×#45211yescharacteristic
352C2×C23#301911yescharacteristic
353C2×C23#302611yescharacteristic
354C25D4×C2311yescharacteristic
355C22×Q8D4×C2311yescharacteristic
356C22×Q8C23×C2311yescharacteristic
357C22×D4—21no—
358C22×D4—21no—
359C22×D4—21no—
360C22×D4D4×C2311yescharacteristic
361C23×C4—21no—
362C23×C4D4×C2311yescharacteristic
363C23×C4C23×C2311yescharacteristic
364C2×Q16—22no—
365C2×SD16—22no—
366C22×C8—21no—
367C22×C8—21no—
368#126—24no—
369#126—24no—
370#124—24no—
371#124—24no—
372#123—24no—
373#123—24no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8D4×C2312yes—
381C22⋊Q8C23×C2312yes—
382#121—24no—
383#121—24no—
384#121—21no—
385#121—21no—
386#121D4×C2312yes—
387#121C23×C2312yes—
388#120—24no—
389#120—24no—
390C4×D4—24no—
391C4×D4—24no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4D4×C2311yescharacteristic
398C2×C4⋊C4D4×C2311yescharacteristic
399C2×C4⋊C4C23×C2311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4D4×C2312yes—
414C2×C22⋊C4C23×C2312yes—
415C2×C42—21no—
416#107—22no—
417#106—22no—
418#103—22no—
419#103—22no—
420#103—22no—
421#102—22no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424C22⋊C8C23×C2312yes—
425C22⋊C8C23×C2312yes—
426#95—22no—
427#95C23×C2312yes—
428C23#1829711yescharacteristic, direct product factor, Sylow
429C24—21no—
430C24—21no—
431C24—21no—
432C24—22no—
433C24—22no—
434C24—81no—
435C24—81no—
436C24C2×D4×C2312yes—
437C24C2×D4×C2311yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—81no—
442C2×Q8—81no—
443C2×Q8C2×D4×C2312yes—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—81no—
452C2×D4—82no—
453C2×D4—82no—
454C2×D4C2×D4×C2312yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4—82no—
467C22×C4C2×D4×C2312yes—
468C22×C4C2×D4×C2311yescharacteristic
469C22×C4C2×D4×C2311yescharacteristic
470C22×C4C2×D4×C2311yescharacteristic
471C22×C4C2×D4×C2311yescharacteristic
472C22×C4C2×D4×C2311yescharacteristic
473C22×C4C24×C2311yescharacteristic
474Q16—82no—
475SD16—82no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—81no—
479C2×C8—81no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4C2×D4×C2312yes—
492C4⋊C4C2×D4×C2312yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—42no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C42—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—81no—
526C23—81no—
527C23—84no—
528C23—84no—
529C23C23×#12011yescharacteristic
530C23C23×#12011yescharacteristic
531C23C23×#13512yes—
532C23C23×#13712yes—
533C23C22×D4×C2311yescharacteristic
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—81no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—44no—
542D4—81no—
543D4—84no—
544D4—84no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—81no—
568C2×C4—84no—
569C2×C4C23×#12011yescharacteristic
570C2×C4C23×#12011yescharacteristic
571C2×C4C22×D4×C2311yescharacteristic
572C2×C4C22×D4×C2311yescharacteristic
573C8—42no—
574C8—42no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—82no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22C23×#45212yes—
595C22C23×#45212yes—
596C22C2×C23×#12011yescharacteristic
597C22C2×C23×#13511yescharacteristic
598C22C2×C23×#13711yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—82no—
608C4—81no—
609C4—81no—
610C2—21no—
611C2—21no—
612C2—22no—
613C2—22no—
614C2—82no—
615C2—82no—
616C2C2×C23×#45211yescharacteristic
617C2C23×#301911yescharacteristic
618C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC2×C23×#538C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3189C2semidirect
3. C23×#3187C2semidirect
4. C23×#2626C2semidirect
5. C23×#1618C2semidirect
6. C23×#1606C2semidirect
7. C23×#1520C2semidirect
8. C23×#1519C2semidirect
9. C2×C23×#323C2semidirect
25. C23×#484C22semidirect
27. C23×#481C22semidirect
31. C23×#451C22semidirect
33. C23×#448C22semidirect
36. C23×#399C22semidirect
38. C23×#393C22semidirect
42. C23×#328C22semidirect
43. C23×#327C22semidirect
44. C23×#323C22semidirect
200. #18297C23direct (N × H)
267. #3189C2×C23semidirect
268. #3187C2×C23semidirect
269. #2626C2×C23semidirect
270. #1618C2×C23semidirect
271. #1606C2×C23semidirect
272. #1520C2×C23semidirect
273. #1519C2×C23semidirect
274. C2×#323C2×C23semidirect
325. #484C22×C23semidirect
327. #481C22×C23semidirect
331. #451C22×C23semidirect
333. #448C22×C23semidirect
336. #399C22×C23semidirect
338. #393C22×C23semidirect
342. #328C22×C23semidirect
343. #327C22×C23semidirect
344. #323C22×C23semidirect
428. C23#18297direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5382
C23×#31892
C23×#31872
C23×#26262
C23×#16182
C23×#16062
C23×#15202
C23×#15192
C2×C23×#3232
#1829723