Finite Group Databasebeta

← C23×#18469 #19166095

Subgroups — C23×#18469 #19166095

Order 5888.

814 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3563C211yesmaximal, characteristic
2C23×#3236C211yesmaximal, characteristic, complemented
3C23×#3232C211yesmaximal, characteristic
4C23×#2822C211yesmaximal, characteristic, complemented
5C23×#1617C212yesmaximal, complemented
6C23×#1615C212yesmaximal
7C23×#1611C212yesmaximal, complemented
8C23×#1609C212yesmaximal
9C23×#1518C211yesmaximal, characteristic, complemented
10C23×#1512C211yesmaximal, characteristic
11C23×#1470C211yesmaximal, characteristic, complemented
12C23×#564—21no—
13C23×#563—21no—
14C23×#557—21no—
15C23×#554—21no—
16C23×#552—21no—
17C23×#552—22no—
18C23×#552—21no—
19C23×#544—21no—
20C23×#544C2212yes—
21C23×#541—21no—
22C23×#541—22no—
23C23×#539C2212yes—
24C23×#533C2211yescharacteristic
25C23×#529C2211yescharacteristic
26C2×C23×#125C2211yescharacteristic
27C2×C23×#125C2211yescharacteristic
28C23×#501—21no—
29C23×#499—21no—
30C23×#491—21no—
31C23×#488—21no—
32C23×#484C2211yescharacteristic
33C23×#484C2212yes—
34C23×#483C2211yescharacteristic
35C23×#483C2212yes—
36C23×#481C2211yescharacteristic
37C23×#480C2211yescharacteristic
38C23×#479—21no—
39C23×#479—22no—
40C23×#478—21no—
41C23×#432C2211yescharacteristic
42C23×#431C2211yescharacteristic
43C23×#400—21no—
44C23×#400C2212yes—
45C23×#398—22no—
46C23×#398—21no—
47C23×#398C2212yes—
48C23×#382C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#328C2212yes—
52C23×#327C2212yes—
53C23×#324C2212yes—
54C23×#324C2212yes—
55C23×C4⋊Q8—21no—
56C23×C4⋊Q8—22no—
57C23×C4⋊Q8—22no—
58C23×C4⋊Q8D411yescharacteristic
59C23×C4⋊Q8D411yescharacteristic
60C23×#126—22no—
61C23×#126—22no—
62C23×#126—22no—
63C23×#126—22no—
64C23×#126—21no—
65C23×#125—22no—
66C23×#125—21no—
67C23×#125—22no—
68C23×#125—22no—
69C23×#125—22no—
70C23×#125—22no—
71C23×#125—22no—
72C23×#125—22no—
73C23×#125—22no—
74C23×#125—21no—
75C23×#125—21no—
76C23×#125—21no—
77C23×#125—21no—
78C23×#125—21no—
79C23×#125—21no—
80C23×#125—21no—
81C23×#125—21no—
82C23×#125—21no—
83C23×#125D411yescharacteristic
84C23×#124D411yescharacteristic
85C23×#123—22no—
86C23×#123—21no—
87C23×#123—22no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—21no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—21no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8C2311yescharacteristic
96C23×C22⋊Q8C2312yes—
97C23×#121C2311yescharacteristic
98C4×Q8×C23—21no—
99C4×Q8×C23—22no—
100C4×Q8×C23—21no—
101C4×Q8×C23—21no—
102C4×Q8×C23D411yescharacteristic
103C4×Q8×C23D412yes—
104C4×D4×C23—21no—
105C4×D4×C23—22no—
106C4×D4×C23D411yescharacteristic
107C23×#117—22no—
108C23×#117—21no—
109C23×#117—21no—
110C23×#117—22no—
111C23×#117—21no—
112C2×C4⋊C4×C23—21no—
113C2×C4⋊C4×C23—21no—
114C2×C4⋊C4×C23—21no—
115C2×C4⋊C4×C23—21no—
116C2×C4⋊C4×C23—22no—
117C2×C4⋊C4×C23D411yescharacteristic
118C2×C4⋊C4×C23D412yes—
119C2×C4⋊C4×C23D411yescharacteristic
120C2×C4⋊C4×C23C2311yescharacteristic
121C2×C4⋊C4×C23C2311yescharacteristic
122C2×C4⋊C4×C23C2312yes—
123C2×C42×C23C2311yescharacteristic
124C23×#107—21no—
125C23×#107—21no—
126C23×#107—22no—
127C23×#106—21no—
128C23×#106—21no—
129C23×#106—22no—
130C23×C4⋊C8—21no—
131C23×C4⋊C8—21no—
132C23×#103—21no—
133C23×#103—22no—
134C23×#103—21no—
135C23×#103—22no—
136C23×#102—21no—
137C23×#102—21no—
138C23×C22⋊C8C2312yes—
139C23×C22⋊C8C2311yescharacteristic
140C23×C22⋊C8C2311yescharacteristic
141C23×#97—21no—
142C4×C8×C23—21no—
143C23×#95—21no—
144C23×#95—22no—
145C23×#95—22no—
146C23×#95—22no—
147C23×#95C2312yes—
148C2×Q8×C23—22no—
149C2×Q8×C23—21no—
150C2×Q8×C23C2×D412yes—
151C2×Q8×C23C2×D411yescharacteristic
152C2×D4×C23—21no—
153C2×D4×C23C2×D411yescharacteristic
154C22×C4×C23—22no—
155C22×C4×C23—21no—
156C22×C4×C23—21no—
157C22×C4×C23—21no—
158C22×C4×C23—22no—
159C22×C4×C23C2×D412yes—
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×D412yes—
162C22×C4×C23C2×D411yescharacteristic
163C22×C4×C23C2411yescharacteristic
164C2×C8×C23—21no—
165C2×C8×C23—21no—
166C2×C8×C23—22no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—21no—
182C4⋊C4×C23—21no—
183C4⋊C4×C23—21no—
184C4⋊C4×C23—21no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—41no—
197C4⋊C4×C23—41no—
198C4⋊C4×C23—41no—
199C4⋊C4×C23—41no—
200C4⋊C4×C23—81no—
201C4⋊C4×C23—82no—
202C4⋊C4×C23C2×D411yescharacteristic
203C4⋊C4×C23C2×D411yescharacteristic
204C4⋊C4×C23C2×D412yes—
205C22⋊C4×C23—22no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—21no—
210C22⋊C4×C23—21no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—21no—
213C22⋊C4×C23—21no—
214C42×C23—22no—
215C42×C23—22no—
216C42×C23—22no—
217C42×C23—22no—
218C42×C23—21no—
219C42×C23C2×D411yescharacteristic
220C42×C23C2×D411yescharacteristic
221C42×C23C2×D411yescharacteristic
222C42×C23C2×D411yescharacteristic
223#18469C2311yesmaximal, characteristic, direct product factor, Sylow
224C23×C23—21no—
225C23×C23C22×D411yescharacteristic
226Q8×C23—41no—
227Q8×C23—42no—
228Q8×C23—42no—
229Q8×C23—82no—
230D4×C23—41no—
231D4×C23—41no—
232D4×C23—41no—
233C2×C4×C23—21no—
234C2×C4×C23—21no—
235C2×C4×C23—22no—
236C2×C4×C23—21no—
237C2×C4×C23—21no—
238C2×C4×C23—22no—
239C2×C4×C23—21no—
240C2×C4×C23—22no—
241C2×C4×C23—22no—
242C2×C4×C23—22no—
243C2×C4×C23—22no—
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—41no—
252C2×C4×C23—41no—
253C2×C4×C23—42no—
254C2×C4×C23—41no—
255C2×C4×C23—41no—
256C2×C4×C23—82no—
257C2×C4×C23#12011yescharacteristic
258C2×C4×C23#12011yescharacteristic
259C2×C4×C23#12011yescharacteristic
260C2×C4×C23#12011yescharacteristic
261C2×C4×C23C22×D411yescharacteristic
262C2×C4×C23C22×D411yescharacteristic
263C8×C23—41no—
264C8×C23—41no—
265C8×C23—42no—
266#3563C2×C2311yescharacteristic
267#3236C2×C2311yescharacteristic, complemented
268#3232C2×C2311yescharacteristic
269#2822C2×C2311yescharacteristic, complemented
270#1617C2×C2312yescomplemented
271#1615C2×C2312yes—
272#1611C2×C2312yescomplemented
273#1609C2×C2312yes—
274#1518C2×C2311yescharacteristic, complemented
275#1512C2×C2311yescharacteristic
276#1470C2×C2311yescharacteristic, complemented
277C22×C23—21no—
278C22×C23—21no—
279C22×C23—41no—
280C22×C23—41no—
281C22×C23—41no—
282C22×C23C2×#12011yescharacteristic
283C22×C23#57711yescharacteristic
284C22×C23#57811yescharacteristic
285C4×C23—21no—
286C4×C23—21no—
287C4×C23—21no—
288C4×C23—21no—
289C4×C23—21no—
290C4×C23—21no—
291C4×C23—42no—
292C4×C23—41no—
293C4×C23—41no—
294C4×C23—42no—
295C4×C23—42no—
296C4×C23—42no—
297C4×C23—41no—
298C4×C23—82no—
299C4×C23—81no—
300C4×C23—81no—
301#564—21no—
302#563—21no—
303#557—21no—
304#554—21no—
305#552—21no—
306#552—22no—
307#552—21no—
308#544—21no—
309#544C22×C2312yes—
310#541—21no—
311#541—22no—
312#539C22×C2312yes—
313#533C22×C2311yescharacteristic
314#529C22×C2311yescharacteristic
315C2×#125C22×C2311yescharacteristic
316C2×#125C22×C2311yescharacteristic
317#501—21no—
318#499—21no—
319#491—21no—
320#488—21no—
321#484C22×C2312yes—
322#484C22×C2311yescharacteristic
323#483C22×C2312yes—
324#483C22×C2311yescharacteristic
325#481C22×C2311yescharacteristic
326#480C22×C2311yescharacteristic
327#479—21no—
328#479—22no—
329#478—21no—
330#432C22×C2311yescharacteristic
331#431C22×C2311yescharacteristic
332#400—21no—
333#400C22×C2312yes—
334#398—21no—
335#398—22no—
336#398C22×C2312yes—
337#382C22×C2311yescharacteristic
338#328C22×C2312yes—
339#328C22×C2312yes—
340#328C22×C2312yes—
341#327C22×C2312yes—
342#324C22×C2312yes—
343#324C22×C2312yes—
344C2×C23—21no—
345C2×C23—21no—
346C2×C23—81no—
347C2×C23#301311yescharacteristic
348C2×C23#302011yescharacteristic
349C2×C23#302811yescharacteristic
350C4⋊Q8—21no—
351C4⋊Q8—22no—
352C4⋊Q8—22no—
353C4⋊Q8D4×C2311yescharacteristic
354C4⋊Q8D4×C2311yescharacteristic
355#126—22no—
356#126—22no—
357#126—22no—
358#126—22no—
359#126—21no—
360#125—22no—
361#125—22no—
362#125—22no—
363#125—22no—
364#125—21no—
365#125—22no—
366#125—22no—
367#125—22no—
368#125—22no—
369#125—21no—
370#125—21no—
371#125—21no—
372#125—21no—
373#125—21no—
374#125—21no—
375#125—21no—
376#125—21no—
377#125—21no—
378#125D4×C2311yescharacteristic
379#124D4×C2311yescharacteristic
380#123—22no—
381#123—21no—
382#123—22no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—21no—
389C22⋊Q8—22no—
390C22⋊Q8C23×C2311yescharacteristic
391C22⋊Q8C23×C2312yes—
392#121C23×C2311yescharacteristic
393C4×Q8—21no—
394C4×Q8—22no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8D4×C2311yescharacteristic
398C4×Q8D4×C2312yes—
399C4×D4—21no—
400C4×D4—22no—
401C4×D4D4×C2311yescharacteristic
402#117—22no—
403#117—21no—
404#117—21no—
405#117—22no—
406#117—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4D4×C2311yescharacteristic
413C2×C4⋊C4D4×C2312yes—
414C2×C4⋊C4D4×C2311yescharacteristic
415C2×C4⋊C4C23×C2311yescharacteristic
416C2×C4⋊C4C23×C2311yescharacteristic
417C2×C4⋊C4C23×C2312yes—
418C2×C42C23×C2311yescharacteristic
419#107—21no—
420#107—21no—
421#107—22no—
422#106—21no—
423#106—21no—
424#106—22no—
425C4⋊C8—21no—
426C4⋊C8—21no—
427#103—21no—
428#103—22no—
429#103—21no—
430#103—22no—
431#102—21no—
432#102—21no—
433C22⋊C8C23×C2312yes—
434C22⋊C8C23×C2311yescharacteristic
435C22⋊C8C23×C2311yescharacteristic
436#97—21no—
437C4×C8—21no—
438#95—21no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95C23×C2312yes—
443C23#1846911yescharacteristic, direct product factor, Sylow
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8C2×D4×C2312yes—
447C2×Q8C2×D4×C2311yescharacteristic
448C2×D4—21no—
449C2×D4C2×D4×C2311yescharacteristic
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4C2×D4×C2312yes—
456C22×C4C2×D4×C2311yescharacteristic
457C22×C4C2×D4×C2312yes—
458C22×C4C2×D4×C2311yescharacteristic
459C22×C4C24×C2311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4C2×D4×C2311yescharacteristic
499C4⋊C4C2×D4×C2311yescharacteristic
500C4⋊C4C2×D4×C2312yes—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42C2×D4×C2311yescharacteristic
516C42C2×D4×C2311yescharacteristic
517C42C2×D4×C2311yescharacteristic
518C42C2×D4×C2311yescharacteristic
519C23—21no—
520C23C22×D4×C2311yescharacteristic
521Q8—41no—
522Q8—42no—
523Q8—42no—
524Q8—82no—
525D4—41no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—82no—
552C2×C4C23×#12011yescharacteristic
553C2×C4C23×#12011yescharacteristic
554C2×C4C23×#12011yescharacteristic
555C2×C4C23×#12011yescharacteristic
556C2×C4C22×D4×C2311yescharacteristic
557C2×C4C22×D4×C2311yescharacteristic
558C8—41no—
559C8—41no—
560C8—42no—
561C22—21no—
562C22—21no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C2×C23×#12011yescharacteristic
567C22C23×#57711yescharacteristic
568C22C23×#57811yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—41no—
577C4—41no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—41no—
582C4—82no—
583C4—81no—
584C4—81no—
585C2—21no—
586C2—21no—
587C2—81no—
588C2C23×#301311yescharacteristic
589C2C23×#302011yescharacteristic
590C2C23×#302811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3236C2semidirect
4. C23×#2822C2semidirect
5. C23×#1617C2semidirect
7. C23×#1611C2semidirect
9. C23×#1518C2semidirect
11. C23×#1470C2semidirect
223. #18469C23direct (N × H)
267. #3236C2×C23semidirect
269. #2822C2×C23semidirect
270. #1617C2×C23semidirect
272. #1611C2×C23semidirect
274. #1518C2×C23semidirect
276. #1470C2×C23semidirect
443. C23#18469direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35632
C23×#32362
C23×#32322
C23×#28222
C23×#16172
C23×#16152
C23×#16112
C23×#16092
C23×#15182
C23×#15122
C23×#14702
#1846923