Finite Group Databasebeta

← C23×#18423 #19166049

Subgroups — C23×#18423 #19166049

Order 5888.

934 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
1C23×#3538C211yesmaximal, characteristic
2C23×#3195C211yesmaximal, characteristic, complemented
3C23×#3190C211yesmaximal, characteristic
4C23×#2659C211yesmaximal, characteristic, complemented
5C23×#1611C212yesmaximal, complemented
6C23×#1609C212yesmaximal, complemented
7C23×#1605C212yesmaximal, complemented
8C23×#1603C212yesmaximal, complemented
9C23×#1504C211yesmaximal, characteristic
10C2×C23×#328C211yesmaximal, characteristic, complemented
11C23×#1472C211yesmaximal, characteristic, complemented
12C23×#563—22no—
13C23×#552—21no—
14C23×#552—21no—
15C23×#551—22no—
16D4×Q8×C23—21no—
17D4×Q8×C23—21no—
18C23×#544—21no—
19C23×#544C2212yes—
20C23×#541—21no—
21C23×#539—21no—
22C23×#539C2212yes—
23C23×#536—21no—
24C23×#525C2211yescharacteristic
25C2×C23×C22⋊Q8C2211yescharacteristic
26C2×C23×C22⋊Q8C2211yescharacteristic
27C2×C23×C22⋊Q8C2211yescharacteristic
28C23×#485—21no—
29C23×#483—21no—
30C23×#483C2211yescharacteristic
31C23×#483C2212yes—
32C23×#480C2211yescharacteristic
33C23×#470—22no—
34C23×#467—21no—
35C23×#465—21no—
36C23×#452—21no—
37C23×#451—21no—
38C23×#451C2211yescharacteristic
39C23×#451C2212yes—
40C23×#448C2211yescharacteristic
41C23×#407C2211yescharacteristic
42C2×C23×C22⋊C8C2211yescharacteristic
43C23×#398—21no—
44C23×#398C2212yes—
45C23×#395—21no—
46C23×#395C2212yes—
47C23×#380—22no—
48C2×C23×#95C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#328C2212yes—
52C23×#327C2212yes—
53C23×#324C2212yes—
54C23×#324C2212yes—
55C2×#41×C23D411yescharacteristic
56C22×Q8×C23—21no—
57C22×Q8×C23D411yescharacteristic
58C22×Q8×C23D411yescharacteristic
59C22×Q8×C23C2311yescharacteristic
60C23×C4×C23—21no—
61C23×C4×C23C2311yescharacteristic
62C2×Q16×C23—22no—
63C2×Q16×C23—21no—
64C2×SD16×C23—21no—
65C2×M16×C23—21no—
66C22×C8×C23—21no—
67C23×C4⋊Q8—21no—
68C23×C4⋊Q8—22no—
69C23×C4⋊Q8—22no—
70C23×C4⋊Q8—22no—
71C23×#126—22no—
72C23×#126—22no—
73C23×#126—22no—
74C23×#126—22no—
75C23×#125—21no—
76C23×#125—22no—
77C23×#125—22no—
78C23×#125—22no—
79C23×#125—22no—
80C23×#124—22no—
81C23×#123—22no—
82C23×#123—22no—
83C23×#123—22no—
84C23×#123—22no—
85C23×#123—21no—
86C23×C22⋊Q8—22no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—21no—
93C23×C22⋊Q8—21no—
94C23×C22⋊Q8—21no—
95C23×C22⋊Q8—21no—
96C23×C22⋊Q8—21no—
97C23×C22⋊Q8—21no—
98C23×C22⋊Q8—21no—
99C23×C22⋊Q8—21no—
100C23×C22⋊Q8—21no—
101C23×C22⋊Q8—21no—
102C23×C22⋊Q8D412yes—
103C23×C22⋊Q8D411yescharacteristic
104C23×C22⋊Q8C2312yes—
105C23×C22⋊Q8C2311yescharacteristic
106C23×#121D411yescharacteristic
107C23×#121C2311yescharacteristic
108C23×#120—21no—
109C4×Q8×C23—21no—
110C4×Q8×C23—21no—
111C4×D4×C23—22no—
112C4×D4×C23—22no—
113C4×D4×C23—21no—
114C4×D4×C23—21no—
115C23×#117—22no—
116C23×#117—21no—
117C23×#117—22no—
118C23×#117D411yescharacteristic
119C2×C4⋊C4×C23—21no—
120C2×C4⋊C4×C23—21no—
121C2×C4⋊C4×C23—22no—
122C2×C4⋊C4×C23—22no—
123C2×C4⋊C4×C23D411yescharacteristic
124C2×C4⋊C4×C23D411yescharacteristic
125C2×C4⋊C4×C23C2311yescharacteristic
126C2×C4⋊C4×C23C2312yes—
127C2×C22⋊C4×C23—21no—
128C2×C22⋊C4×C23—21no—
129C2×C22⋊C4×C23D412yes—
130C23×#107—21no—
131C23×#106—21no—
132C23×#106—22no—
133C23×#103—22no—
134C23×#103—21no—
135C23×#103—21no—
136C23×#103—21no—
137C23×#103—22no—
138C23×#102—21no—
139C23×C22⋊C8—21no—
140C23×C22⋊C8—21no—
141C23×C22⋊C8C2312yes—
142C23×C22⋊C8C2311yescharacteristic
143C23×C22⋊C8C2311yescharacteristic
144C23×#95—21no—
145C23×#95—22no—
146C23×#95—22no—
147C23×#95C2312yes—
148C24×C23C2×D411yescharacteristic
149#41×C23—81no—
150C2×Q8×C23—22no—
151C2×Q8×C23—21no—
152C2×Q8×C23—21no—
153C2×Q8×C23—21no—
154C2×Q8×C23—21no—
155C2×Q8×C23—21no—
156C2×Q8×C23—81no—
157C2×Q8×C23—81no—
158C2×Q8×C23—82no—
159C2×Q8×C23—81no—
160C2×Q8×C23C2×D412yes—
161C2×Q8×C23C2×D411yescharacteristic
162C2×D4×C23—21no—
163C2×D4×C23—21no—
164C2×D4×C23—21no—
165C2×D4×C23C2×D411yescharacteristic
166C22×C4×C23—21no—
167C22×C4×C23—22no—
168C22×C4×C23—22no—
169C22×C4×C23—24no—
170C22×C4×C23—21no—
171C22×C4×C23—21no—
172C22×C4×C23—21no—
173C22×C4×C23—21no—
174C22×C4×C23—81no—
175C22×C4×C23—81no—
176C22×C4×C23—81no—
177C22×C4×C23C2×D412yes—
178C22×C4×C23C2×D412yes—
179C22×C4×C23C2×D411yescharacteristic
180C22×C4×C23C2×D411yescharacteristic
181C22×C4×C23C2×D411yescharacteristic
182C22×C4×C23C2×D411yescharacteristic
183C22×C4×C23C2×D411yescharacteristic
184C22×C4×C23C2411yescharacteristic
185Q16×C23—81no—
186Q16×C23—82no—
187SD16×C23—81no—
188M16×C23—81no—
189C2×C8×C23—21no—
190C2×C8×C23—21no—
191C2×C8×C23—22no—
192C2×C8×C23—81no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23—22no—
195C4⋊C4×C23—22no—
196C4⋊C4×C23—22no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—22no—
199C4⋊C4×C23—22no—
200C4⋊C4×C23—22no—
201C4⋊C4×C23—22no—
202C4⋊C4×C23—21no—
203C4⋊C4×C23—21no—
204C4⋊C4×C23—21no—
205C4⋊C4×C23—21no—
206C4⋊C4×C23—22no—
207C4⋊C4×C23—22no—
208C4⋊C4×C23—22no—
209C4⋊C4×C23—21no—
210C4⋊C4×C23—21no—
211C4⋊C4×C23—21no—
212C4⋊C4×C23—41no—
213C4⋊C4×C23—41no—
214C4⋊C4×C23—82no—
215C4⋊C4×C23—82no—
216C4⋊C4×C23C2×D412yes—
217C4⋊C4×C23C2×D411yescharacteristic
218C4⋊C4×C23C2×D411yescharacteristic
219C22⋊C4×C23—22no—
220C22⋊C4×C23—22no—
221C22⋊C4×C23—22no—
222C22⋊C4×C23—22no—
223C22⋊C4×C23—22no—
224C22⋊C4×C23—22no—
225C22⋊C4×C23—21no—
226C22⋊C4×C23—21no—
227C22⋊C4×C23—22no—
228C22⋊C4×C23—21no—
229C22⋊C4×C23—21no—
230C22⋊C4×C23—21no—
231C22⋊C4×C23—41no—
232C22⋊C4×C23—41no—
233C42×C23—22no—
234C42×C23—22no—
235C42×C23—22no—
236C42×C23—21no—
237#18423C2311yesmaximal, characteristic, direct product factor, Sylow
238C23×C23—21no—
239C23×C23—21no—
240C23×C23—22no—
241C23×C23—21no—
242C23×C23—41no—
243C23×C23#12011yescharacteristic
244C23×C23#12011yescharacteristic
245C23×C23C22×D411yescharacteristic
246Q8×C23—42no—
247Q8×C23—41no—
248Q8×C23—42no—
249Q8×C23—42no—
250Q8×C23—41no—
251Q8×C23—41no—
252Q8×C23—42no—
253Q8×C23—81no—
254Q8×C23—82no—
255D4×C23—41no—
256D4×C23—41no—
257D4×C23—41no—
258D4×C23—42no—
259D4×C23—81no—
260C2×C4×C23—24no—
261C2×C4×C23—21no—
262C2×C4×C23—21no—
263C2×C4×C23—22no—
264C2×C4×C23—22no—
265C2×C4×C23—21no—
266C2×C4×C23—22no—
267C2×C4×C23—22no—
268C2×C4×C23—22no—
269C2×C4×C23—22no—
270C2×C4×C23—41no—
271C2×C4×C23—41no—
272C2×C4×C23—41no—
273C2×C4×C23—41no—
274C2×C4×C23—41no—
275C2×C4×C23—41no—
276C2×C4×C23—42no—
277C2×C4×C23—42no—
278C2×C4×C23—42no—
279C2×C4×C23—42no—
280C2×C4×C23—42no—
281C2×C4×C23—42no—
282C2×C4×C23—41no—
283C2×C4×C23—41no—
284C2×C4×C23—41no—
285C2×C4×C23—41no—
286C2×C4×C23—81no—
287C2×C4×C23—84no—
288C2×C4×C23—82no—
289C2×C4×C23#12011yescharacteristic
290C2×C4×C23#12011yescharacteristic
291C2×C4×C23C22×D411yescharacteristic
292C2×C4×C23C22×D411yescharacteristic
293C8×C23—41no—
294C8×C23—41no—
295C8×C23—42no—
296#3538C2×C2311yescharacteristic
297#3195C2×C2311yescharacteristic, complemented
298#3190C2×C2311yescharacteristic
299#2659C2×C2311yescharacteristic, complemented
300#1611C2×C2312yescomplemented
301#1609C2×C2312yescomplemented
302#1605C2×C2312yescomplemented
303#1603C2×C2312yescomplemented
304#1504C2×C2311yescharacteristic
305C2×#328C2×C2311yescharacteristic, complemented
306#1472C2×C2311yescharacteristic, complemented
307C22×C23—21no—
308C22×C23—21no—
309C22×C23—21no—
310C22×C23—21no—
311C22×C23—21no—
312C22×C23—21no—
313C22×C23—21no—
314C22×C23—42no—
315C22×C23—42no—
316C22×C23—41no—
317C22×C23—41no—
318C22×C23—41no—
319C22×C23#45612yes—
320C22×C23C2×#12011yescharacteristic
321C22×C23#57711yescharacteristic
322C22×C23#57811yescharacteristic
323C4×C23—21no—
324C4×C23—21no—
325C4×C23—21no—
326C4×C23—21no—
327C4×C23—44no—
328C4×C23—42no—
329C4×C23—41no—
330C4×C23—42no—
331C4×C23—41no—
332C4×C23—41no—
333C4×C23—42no—
334C4×C23—42no—
335C4×C23—81no—
336C4×C23—81no—
337#563—22no—
338#552—21no—
339#552—21no—
340#551—22no—
341D4×Q8—21no—
342D4×Q8—21no—
343#544—21no—
344#544C22×C2312yes—
345#541—21no—
346#539—21no—
347#539C22×C2312yes—
348#536—21no—
349#525C22×C2311yescharacteristic
350C2×C22⋊Q8C22×C2311yescharacteristic
351C2×C22⋊Q8C22×C2311yescharacteristic
352C2×C22⋊Q8C22×C2311yescharacteristic
353#485—21no—
354#483—21no—
355#483C22×C2312yes—
356#483C22×C2311yescharacteristic
357#480C22×C2311yescharacteristic
358#470—22no—
359#467—21no—
360#465—21no—
361#452—21no—
362#451—21no—
363#451C22×C2312yes—
364#451C22×C2311yescharacteristic
365#448C22×C2311yescharacteristic
366#407C22×C2311yescharacteristic
367C2×C22⋊C8C22×C2311yescharacteristic
368#398—21no—
369#398C22×C2312yes—
370#395—21no—
371#395C22×C2312yes—
372#380—22no—
373C2×#95C22×C2311yescharacteristic
374#328C22×C2312yes—
375#328C22×C2312yes—
376#328C22×C2312yes—
377#327C22×C2312yes—
378#324C22×C2312yes—
379#324C22×C2312yes—
380C2×C23—21no—
381C2×C23—21no—
382C2×C23—22no—
383C2×C23—41no—
384C2×C23—81no—
385C2×C23#301311yescharacteristic
386C2×C23C2×#45611yescharacteristic
387C2×C23#302611yescharacteristic
388C2×#41D4×C2311yescharacteristic
389C22×Q8—21no—
390C22×Q8D4×C2311yescharacteristic
391C22×Q8D4×C2311yescharacteristic
392C22×Q8C23×C2311yescharacteristic
393C23×C4—21no—
394C23×C4C23×C2311yescharacteristic
395C2×Q16—22no—
396C2×Q16—21no—
397C2×SD16—21no—
398C2×M16—21no—
399C22×C8—21no—
400C4⋊Q8—21no—
401C4⋊Q8—22no—
402C4⋊Q8—22no—
403C4⋊Q8—22no—
404#126—22no—
405#126—22no—
406#126—22no—
407#126—22no—
408#125—21no—
409#125—22no—
410#125—22no—
411#125—22no—
412#125—22no—
413#124—22no—
414#123—22no—
415#123—22no—
416#123—22no—
417#123—21no—
418#123—22no—
419C22⋊Q8—21no—
420C22⋊Q8—21no—
421C22⋊Q8—22no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8—21no—
425C22⋊Q8—21no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C22⋊Q8—21no—
430C22⋊Q8—22no—
431C22⋊Q8—21no—
432C22⋊Q8—21no—
433C22⋊Q8—21no—
434C22⋊Q8—21no—
435C22⋊Q8D4×C2312yes—
436C22⋊Q8D4×C2311yescharacteristic
437C22⋊Q8C23×C2312yes—
438C22⋊Q8C23×C2311yescharacteristic
439#121D4×C2311yescharacteristic
440#121C23×C2311yescharacteristic
441#120—21no—
442C4×Q8—21no—
443C4×Q8—21no—
444C4×D4—22no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—22no—
448#117—21no—
449#117—22no—
450#117—22no—
451#117D4×C2311yescharacteristic
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—22no—
455C2×C4⋊C4—22no—
456C2×C4⋊C4D4×C2311yescharacteristic
457C2×C4⋊C4D4×C2311yescharacteristic
458C2×C4⋊C4C23×C2311yescharacteristic
459C2×C4⋊C4C23×C2312yes—
460C2×C22⋊C4—21no—
461C2×C22⋊C4—21no—
462C2×C22⋊C4D4×C2312yes—
463#107—21no—
464#106—21no—
465#106—22no—
466#103—22no—
467#103—21no—
468#103—21no—
469#103—21no—
470#103—22no—
471#102—21no—
472C22⋊C8—21no—
473C22⋊C8—21no—
474C22⋊C8C23×C2312yes—
475C22⋊C8C23×C2311yescharacteristic
476C22⋊C8C23×C2311yescharacteristic
477#95—21no—
478#95—22no—
479#95—22no—
480#95C23×C2312yes—
481C23#1842311yescharacteristic, direct product factor, Sylow
482C24C2×D4×C2311yescharacteristic
483#41—81no—
484C2×Q8—22no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—81no—
491C2×Q8—81no—
492C2×Q8—82no—
493C2×Q8—81no—
494C2×Q8C2×D4×C2312yes—
495C2×Q8C2×D4×C2311yescharacteristic
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4C2×D4×C2311yescharacteristic
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—24no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4—81no—
511C22×C4C2×D4×C2312yes—
512C22×C4C2×D4×C2312yes—
513C22×C4C2×D4×C2311yescharacteristic
514C22×C4C2×D4×C2311yescharacteristic
515C22×C4C2×D4×C2311yescharacteristic
516C22×C4C2×D4×C2311yescharacteristic
517C22×C4C2×D4×C2311yescharacteristic
518C22×C4C24×C2311yescharacteristic
519Q16—81no—
520Q16—82no—
521SD16—81no—
522M16—81no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C2×C8—81no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4C2×D4×C2312yes—
551C4⋊C4C2×D4×C2311yescharacteristic
552C4⋊C4C2×D4×C2311yescharacteristic
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—22no—
562C22⋊C4—21no—
563C22⋊C4—21no—
564C22⋊C4—21no—
565C22⋊C4—41no—
566C22⋊C4—41no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—21no—
571C23—21no—
572C23—21no—
573C23—22no—
574C23—21no—
575C23—41no—
576C23C23×#12011yescharacteristic
577C23C23×#12011yescharacteristic
578C23C22×D4×C2311yescharacteristic
579Q8—42no—
580Q8—41no—
581Q8—42no—
582Q8—42no—
583Q8—41no—
584Q8—41no—
585Q8—42no—
586Q8—81no—
587Q8—82no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—42no—
592D4—81no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—81no—
620C2×C4—84no—
621C2×C4—82no—
622C2×C4C23×#12011yescharacteristic
623C2×C4C23×#12011yescharacteristic
624C2×C4C22×D4×C2311yescharacteristic
625C2×C4C22×D4×C2311yescharacteristic
626C8—41no—
627C8—41no—
628C8—42no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—42no—
637C22—42no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22C23×#45612yes—
642C22C2×C23×#12011yescharacteristic
643C22C23×#57711yescharacteristic
644C22C23×#57811yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—44no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—81no—
658C4—81no—
659C2—21no—
660C2—21no—
661C2—22no—
662C2—41no—
663C2—81no—
664C2C23×#301311yescharacteristic
665C2C2×C23×#45611yescharacteristic
666C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#525C22
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×#3195C2semidirect
4. C23×#2659C2semidirect
5. C23×#1611C2semidirect
6. C23×#1609C2semidirect
7. C23×#1605C2semidirect
8. C23×#1603C2semidirect
10. C2×C23×#328C2semidirect
11. C23×#1472C2semidirect
237. #18423C23direct (N × H)
297. #3195C2×C23semidirect
299. #2659C2×C23semidirect
300. #1611C2×C23semidirect
301. #1609C2×C23semidirect
302. #1605C2×C23semidirect
303. #1603C2×C23semidirect
305. C2×#328C2×C23semidirect
306. #1472C2×C23semidirect
481. C23#18423direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35382
C23×#31952
C23×#31902
C23×#26592
C23×#16112
C23×#16092
C23×#16052
C23×#16032
C23×#15042
C2×C23×#3282
C23×#14722
#1842323