Finite Group Databasebeta

← C23×#18421 #19166047

Subgroups — C23×#18421 #19166047

Order 5888.

994 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3536C211yesmaximal, characteristic
2C23×#3195C211yesmaximal, characteristic, complemented
3C23×#3190C211yesmaximal, characteristic, complemented
4C23×#2639C211yesmaximal, characteristic, complemented
5C23×#1612C212yesmaximal, complemented
6C23×#1608C212yesmaximal, complemented
7C23×#1606C212yesmaximal, complemented
8C23×#1602C212yesmaximal, complemented
9C23×#1504C211yesmaximal, characteristic, complemented
10C2×C23×#328C211yesmaximal, characteristic, complemented
11C23×#1471C211yesmaximal, characteristic, complemented
12C23×#562—22no—
13C23×#554—21no—
14C23×#548—21no—
15C23×#547—21no—
16C23×#546—22no—
17C23×#546—21no—
18C23×#543C2212yes—
19C23×#542—21no—
20C23×#539—21no—
21C23×#538C2212yes—
22C23×#537—21no—
23C23×#537—21no—
24C23×#525C2211yescharacteristic
25C2×C23×C22⋊Q8C2211yescharacteristic
26C2×C23×C22⋊Q8C2211yescharacteristic
27C2×C23×#121C2211yescharacteristic
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×#400—21no—
44C23×#397C2212yes—
45C23×#394—21no—
46C23×#393C2212yes—
47C23×#386—22no—
48C2×C23×#95C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#328C2212yes—
52C23×#327C2212yes—
53C23×#323C2212yes—
54C23×#323C2212yes—
55C2×#41×C23—22no—
56C2×#41×C23—21no—
57C2×#41×C23D411yescharacteristic
58C22×Q8×C23D411yescharacteristic
59C22×Q8×C23C2311yescharacteristic
60C22×D4×C23D412yes—
61C22×D4×C23D411yescharacteristic
62C23×C4×C23—21no—
63C23×C4×C23C2311yescharacteristic
64C2×Q16×C23—22no—
65C2×Q16×C23—21no—
66C2×SD16×C23—21no—
67C2×M16×C23—21no—
68C22×C8×C23—21no—
69C23×C4⋊Q8—21no—
70C23×#126—22no—
71C23×#125—22no—
72C23×#124—21no—
73C23×#124—22no—
74C23×#124—22no—
75C23×#123—22no—
76C23×#123—22no—
77C23×#123—22no—
78C23×#123—21no—
79C23×#123—22no—
80C23×C22⋊Q8—21no—
81C23×C22⋊Q8—22no—
82C23×C22⋊Q8—22no—
83C23×C22⋊Q8—22no—
84C23×C22⋊Q8—21no—
85C23×C22⋊Q8—22no—
86C23×C22⋊Q8—21no—
87C23×C22⋊Q8—22no—
88C23×C22⋊Q8—22no—
89C23×C22⋊Q8—21no—
90C23×C22⋊Q8—21no—
91C23×C22⋊Q8—21no—
92C23×C22⋊Q8—21no—
93C23×C22⋊Q8D412yes—
94C23×C22⋊Q8D411yescharacteristic
95C23×C22⋊Q8C2312yes—
96C23×C22⋊Q8C2311yescharacteristic
97C23×#121—21no—
98C23×#121—22no—
99C23×#121—22no—
100C23×#121—22no—
101C23×#121—22no—
102C23×#121—22no—
103C23×#121—21no—
104C23×#121—21no—
105C23×#121—21no—
106C23×#121—21no—
107C23×#121D411yescharacteristic
108C23×#121C2311yescharacteristic
109C23×#120—21no—
110C23×#120—22no—
111C4×Q8×C23—21no—
112C4×D4×C23—21no—
113C4×D4×C23—22no—
114C4×D4×C23—21no—
115C4×D4×C23—21no—
116C4×D4×C23—22no—
117C23×#117D411yescharacteristic
118C2×C4⋊C4×C23—21no—
119C2×C4⋊C4×C23—21no—
120C2×C4⋊C4×C23—21no—
121C2×C4⋊C4×C23D411yescharacteristic
122C2×C4⋊C4×C23D411yescharacteristic
123C2×C4⋊C4×C23C2311yescharacteristic
124C2×C22⋊C4×C23—21no—
125C2×C22⋊C4×C23—22no—
126C2×C22⋊C4×C23—21no—
127C2×C22⋊C4×C23—22no—
128C2×C22⋊C4×C23—22no—
129C2×C22⋊C4×C23C2312yes—
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
149C24×C23C2×D412yes—
150#41×C23—81no—
151#41×C23—84no—
152#41×C23—82no—
153C2×Q8×C23—22no—
154C2×Q8×C23—21no—
155C2×Q8×C23—21no—
156C2×Q8×C23—21no—
157C2×Q8×C23—81no—
158C2×Q8×C23—81no—
159C2×Q8×C23C2×D411yescharacteristic
160C2×Q8×C23C2×D412yes—
161C2×D4×C23—21no—
162C2×D4×C23—22no—
163C2×D4×C23—21no—
164C2×D4×C23—22no—
165C2×D4×C23—22no—
166C2×D4×C23—21no—
167C2×D4×C23—22no—
168C2×D4×C23—21no—
169C2×D4×C23—21no—
170C2×D4×C23—44no—
171C2×D4×C23—81no—
172C2×D4×C23C2×D411yescharacteristic
173C22×C4×C23—21no—
174C22×C4×C23—22no—
175C22×C4×C23—22no—
176C22×C4×C23—22no—
177C22×C4×C23—21no—
178C22×C4×C23—21no—
179C22×C4×C23—21no—
180C22×C4×C23—21no—
181C22×C4×C23—22no—
182C22×C4×C23—42no—
183C22×C4×C23—81no—
184C22×C4×C23—81no—
185C22×C4×C23—81no—
186C22×C4×C23C2×D412yes—
187C22×C4×C23C2×D411yescharacteristic
188C22×C4×C23C2×D411yescharacteristic
189C22×C4×C23C2×D411yescharacteristic
190C22×C4×C23C2×D411yescharacteristic
191C22×C4×C23C2×D411yescharacteristic
192C22×C4×C23C2411yescharacteristic
193Q16×C23—81no—
194Q16×C23—82no—
195SD16×C23—81no—
196M16×C23—81no—
197C2×C8×C23—21no—
198C2×C8×C23—21no—
199C2×C8×C23—22no—
200C2×C8×C23—81no—
201C4⋊C4×C23—22no—
202C4⋊C4×C23—22no—
203C4⋊C4×C23—22no—
204C4⋊C4×C23—22no—
205C4⋊C4×C23—21no—
206C4⋊C4×C23—21no—
207C4⋊C4×C23—22no—
208C4⋊C4×C23—21no—
209C4⋊C4×C23—21no—
210C4⋊C4×C23—22no—
211C4⋊C4×C23—21no—
212C4⋊C4×C23—21no—
213C4⋊C4×C23—21no—
214C4⋊C4×C23—41no—
215C4⋊C4×C23—41no—
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—22no—
231C22⋊C4×C23—22no—
232C22⋊C4×C23—21no—
233C22⋊C4×C23—41no—
234C22⋊C4×C23—41no—
235C22⋊C4×C23—82no—
236C22⋊C4×C23—82no—
237C42×C23—22no—
238C42×C23—21no—
239#18421C2311yesmaximal, characteristic, direct product factor, Sylow
240C23×C23—21no—
241C23×C23—21no—
242C23×C23—22no—
243C23×C23—22no—
244C23×C23—21no—
245C23×C23—22no—
246C23×C23—22no—
247C23×C23—41no—
248C23×C23—42no—
249C23×C23—42no—
250C23×C23#12011yescharacteristic
251C23×C23#12011yescharacteristic
252C23×C23C22×D411yescharacteristic
253Q8×C23—42no—
254Q8×C23—41no—
255Q8×C23—42no—
256Q8×C23—41no—
257Q8×C23—41no—
258Q8×C23—42no—
259Q8×C23—82no—
260D4×C23—41no—
261D4×C23—42no—
262D4×C23—41no—
263D4×C23—41no—
264D4×C23—44no—
265D4×C23—42no—
266D4×C23—81no—
267D4×C23—82no—
268D4×C23—81no—
269D4×C23—82no—
270D4×C23—82no—
271C2×C4×C23—21no—
272C2×C4×C23—22no—
273C2×C4×C23—21no—
274C2×C4×C23—22no—
275C2×C4×C23—21no—
276C2×C4×C23—22no—
277C2×C4×C23—41no—
278C2×C4×C23—41no—
279C2×C4×C23—41no—
280C2×C4×C23—41no—
281C2×C4×C23—41no—
282C2×C4×C23—41no—
283C2×C4×C23—42no—
284C2×C4×C23—42no—
285C2×C4×C23—42no—
286C2×C4×C23—42no—
287C2×C4×C23—42no—
288C2×C4×C23—42no—
289C2×C4×C23—41no—
290C2×C4×C23—41no—
291C2×C4×C23—41no—
292C2×C4×C23—42no—
293C2×C4×C23—41no—
294C2×C4×C23—81no—
295C2×C4×C23—82no—
296C2×C4×C23—82no—
297C2×C4×C23—82no—
298C2×C4×C23—84no—
299C2×C4×C23#12011yescharacteristic
300C2×C4×C23#12011yescharacteristic
301C2×C4×C23C22×D411yescharacteristic
302C2×C4×C23C22×D411yescharacteristic
303C8×C23—41no—
304C8×C23—41no—
305C8×C23—42no—
306#3536C2×C2311yescharacteristic
307#3195C2×C2311yescharacteristic, complemented
308#3190C2×C2311yescharacteristic, complemented
309#2639C2×C2311yescharacteristic, complemented
310#1612C2×C2312yescomplemented
311#1608C2×C2312yescomplemented
312#1606C2×C2312yescomplemented
313#1602C2×C2312yescomplemented
314#1504C2×C2311yescharacteristic, complemented
315C2×#328C2×C2311yescharacteristic, complemented
316#1471C2×C2311yescharacteristic, complemented
317C22×C23—21no—
318C22×C23—21no—
319C22×C23—21no—
320C22×C23—21no—
321C22×C23—21no—
322C22×C23—21no—
323C22×C23—21no—
324C22×C23—42no—
325C22×C23—42no—
326C22×C23—41no—
327C22×C23—42no—
328C22×C23—41no—
329C22×C23—42no—
330C22×C23—41no—
331C22×C23—42no—
332C22×C23—82no—
333C22×C23—82no—
334C22×C23#45512yes—
335C22×C23C2×#12011yescharacteristic
336C22×C23#57711yescharacteristic
337C22×C23#57811yescharacteristic
338C4×C23—21no—
339C4×C23—21no—
340C4×C23—21no—
341C4×C23—21no—
342C4×C23—41no—
343C4×C23—42no—
344C4×C23—42no—
345C4×C23—41no—
346C4×C23—41no—
347C4×C23—42no—
348C4×C23—82no—
349C4×C23—81no—
350C4×C23—81no—
351#562—22no—
352#554—21no—
353#548—21no—
354#547—21no—
355#546—22no—
356#546—21no—
357#543C22×C2312yes—
358#542—21no—
359#539—21no—
360#538C22×C2312yes—
361#537—21no—
362#537—21no—
363#525C22×C2311yescharacteristic
364C2×C22⋊Q8C22×C2311yescharacteristic
365C2×C22⋊Q8C22×C2311yescharacteristic
366C2×#121C22×C2311yescharacteristic
367#485—21no—
368#483—21no—
369#483C22×C2311yescharacteristic
370#483C22×C2312yes—
371#480C22×C2311yescharacteristic
372#470—22no—
373#467—21no—
374#465—21no—
375#452—21no—
376#451—21no—
377#451C22×C2311yescharacteristic
378#451C22×C2312yes—
379#448C22×C2311yescharacteristic
380#407C22×C2311yescharacteristic
381C2×C22⋊C8C22×C2311yescharacteristic
382#400—21no—
383#397C22×C2312yes—
384#394—21no—
385#393C22×C2312yes—
386#386—22no—
387C2×#95C22×C2311yescharacteristic
388#328C22×C2312yes—
389#328C22×C2312yes—
390#328C22×C2312yes—
391#327C22×C2312yes—
392#323C22×C2312yes—
393#323C22×C2312yes—
394C2×C23—21no—
395C2×C23—21no—
396C2×C23—22no—
397C2×C23—41no—
398C2×C23—81no—
399C2×C23—82no—
400C2×C23#301311yescharacteristic
401C2×C23C2×#45511yescharacteristic
402C2×C23#302611yescharacteristic
403C2×#41—22no—
404C2×#41—21no—
405C2×#41D4×C2311yescharacteristic
406C22×Q8D4×C2311yescharacteristic
407C22×Q8C23×C2311yescharacteristic
408C22×D4D4×C2312yes—
409C22×D4D4×C2311yescharacteristic
410C23×C4—21no—
411C23×C4C23×C2311yescharacteristic
412C2×Q16—22no—
413C2×Q16—21no—
414C2×SD16—21no—
415C2×M16—21no—
416C22×C8—21no—
417C4⋊Q8—21no—
418#126—22no—
419#125—22no—
420#124—21no—
421#124—22no—
422#124—22no—
423#123—22no—
424#123—22no—
425#123—22no—
426#123—21no—
427#123—22no—
428C22⋊Q8—21no—
429C22⋊Q8—22no—
430C22⋊Q8—22no—
431C22⋊Q8—22no—
432C22⋊Q8—21no—
433C22⋊Q8—21no—
434C22⋊Q8—22no—
435C22⋊Q8—22no—
436C22⋊Q8—22no—
437C22⋊Q8—21no—
438C22⋊Q8—21no—
439C22⋊Q8—21no—
440C22⋊Q8—21no—
441C22⋊Q8D4×C2312yes—
442C22⋊Q8D4×C2311yescharacteristic
443C22⋊Q8C23×C2312yes—
444C22⋊Q8C23×C2311yescharacteristic
445#121—21no—
446#121—22no—
447#121—22no—
448#121—22no—
449#121—22no—
450#121—22no—
451#121—21no—
452#121—21no—
453#121—21no—
454#121—21no—
455#121D4×C2311yescharacteristic
456#121C23×C2311yescharacteristic
457#120—21no—
458#120—22no—
459C4×Q8—21no—
460C4×D4—22no—
461C4×D4—21no—
462C4×D4—21no—
463C4×D4—21no—
464C4×D4—22no—
465#117D4×C2311yescharacteristic
466C2×C4⋊C4—21no—
467C2×C4⋊C4—21no—
468C2×C4⋊C4—21no—
469C2×C4⋊C4D4×C2311yescharacteristic
470C2×C4⋊C4D4×C2311yescharacteristic
471C2×C4⋊C4C23×C2311yescharacteristic
472C2×C22⋊C4—21no—
473C2×C22⋊C4—22no—
474C2×C22⋊C4—21no—
475C2×C22⋊C4—22no—
476C2×C22⋊C4—22no—
477C2×C22⋊C4C23×C2312yes—
478#107—21no—
479#106—21no—
480#106—22no—
481#103—22no—
482#103—21no—
483#103—21no—
484#103—21no—
485#103—22no—
486#102—21no—
487C22⋊C8—21no—
488C22⋊C8—21no—
489C22⋊C8C23×C2312yes—
490C22⋊C8C23×C2311yescharacteristic
491C22⋊C8C23×C2311yescharacteristic
492#95—21no—
493#95—22no—
494#95—22no—
495#95C23×C2312yes—
496C23#1842111yescharacteristic, direct product factor, Sylow
497C24C2×D4×C2311yescharacteristic
498C24C2×D4×C2312yes—
499#41—81no—
500#41—84no—
501#41—82no—
502C2×Q8—22no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—81no—
507C2×Q8—81no—
508C2×Q8C2×D4×C2311yescharacteristic
509C2×Q8C2×D4×C2312yes—
510C2×D4—21no—
511C2×D4—22no—
512C2×D4—21no—
513C2×D4—22no—
514C2×D4—22no—
515C2×D4—21no—
516C2×D4—22no—
517C2×D4—21no—
518C2×D4—21no—
519C2×D4—44no—
520C2×D4—81no—
521C2×D4C2×D4×C2311yescharacteristic
522C22×C4—21no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—42no—
532C22×C4—81no—
533C22×C4—81no—
534C22×C4—81no—
535C22×C4C2×D4×C2312yes—
536C22×C4C2×D4×C2311yescharacteristic
537C22×C4C2×D4×C2311yescharacteristic
538C22×C4C2×D4×C2311yescharacteristic
539C22×C4C2×D4×C2311yescharacteristic
540C22×C4C2×D4×C2311yescharacteristic
541C22×C4C24×C2311yescharacteristic
542Q16—81no—
543Q16—82no—
544SD16—81no—
545M16—81no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—22no—
549C2×C8—81no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—41no—
564C4⋊C4—41no—
565C4⋊C4C2×D4×C2312yes—
566C4⋊C4C2×D4×C2311yescharacteristic
567C4⋊C4C2×D4×C2311yescharacteristic
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—21no—
575C22⋊C4—21no—
576C22⋊C4—22no—
577C22⋊C4—21no—
578C22⋊C4—21no—
579C22⋊C4—22no—
580C22⋊C4—22no—
581C22⋊C4—21no—
582C22⋊C4—41no—
583C22⋊C4—41no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C42—22no—
587C42—21no—
588C23—21no—
589C23—21no—
590C23—22no—
591C23—22no—
592C23—21no—
593C23—22no—
594C23—22no—
595C23—41no—
596C23—42no—
597C23—42no—
598C23C23×#12011yescharacteristic
599C23C23×#12011yescharacteristic
600C23C22×D4×C2311yescharacteristic
601Q8—42no—
602Q8—41no—
603Q8—42no—
604Q8—41no—
605Q8—41no—
606Q8—42no—
607Q8—82no—
608D4—41no—
609D4—42no—
610D4—41no—
611D4—41no—
612D4—44no—
613D4—42no—
614D4—81no—
615D4—82no—
616D4—81no—
617D4—82no—
618D4—82no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—81no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—84no—
647C2×C4C23×#12011yescharacteristic
648C2×C4C23×#12011yescharacteristic
649C2×C4C22×D4×C2311yescharacteristic
650C2×C4C22×D4×C2311yescharacteristic
651C8—41no—
652C8—41no—
653C8—42no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—42no—
662C22—42no—
663C22—41no—
664C22—42no—
665C22—41no—
666C22—42no—
667C22—41no—
668C22—42no—
669C22—82no—
670C22—82no—
671C22C23×#45512yes—
672C22C2×C23×#12011yescharacteristic
673C22C23×#57711yescharacteristic
674C22C23×#57811yescharacteristic
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—41no—
680C4—42no—
681C4—42no—
682C4—41no—
683C4—41no—
684C4—42no—
685C4—82no—
686C4—81no—
687C4—81no—
688C2—21no—
689C2—21no—
690C2—22no—
691C2—41no—
692C2—82no—
693C2—81no—
694C2C23×#301311yescharacteristic
695C2C2×C23×#45511yescharacteristic
696C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3536C2
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
3. C23×#3190C2semidirect
4. C23×#2639C2semidirect
5. C23×#1612C2semidirect
6. C23×#1608C2semidirect
7. C23×#1606C2semidirect
8. C23×#1602C2semidirect
9. C23×#1504C2semidirect
10. C2×C23×#328C2semidirect
11. C23×#1471C2semidirect
239. #18421C23direct (N × H)
307. #3195C2×C23semidirect
308. #3190C2×C23semidirect
309. #2639C2×C23semidirect
310. #1612C2×C23semidirect
311. #1608C2×C23semidirect
312. #1606C2×C23semidirect
313. #1602C2×C23semidirect
314. #1504C2×C23semidirect
315. C2×#328C2×C23semidirect
316. #1471C2×C23semidirect
496. C23#18421direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35362
C23×#31952
C23×#31902
C23×#26392
C23×#16122
C23×#16082
C23×#16062
C23×#16022
C23×#15042
C2×C23×#3282
C23×#14712
#1842123