Finite Group Databasebeta

← C23×#18972 #19166598

Subgroups — C23×#18972 #19166598

Order 5888.

818 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3555C211yesmaximal, characteristic
2C23×#3239C211yesmaximal, characteristic
3C23×#3229C211yesmaximal, characteristic, complemented
4C23×#3209C211yesmaximal, characteristic, complemented
5C23×#1653C212yesmaximal, complemented
6C23×#1652C212yesmaximal, complemented
7C23×#1645C212yesmaximal, complemented
8C23×#1644C212yesmaximal, complemented
9C23×#1554C211yesmaximal, characteristic, complemented
10C23×#1497C211yesmaximal, characteristic, complemented
11C23×#1482C211yesmaximal, characteristic
12C23×#563—22no—
13Q82×C23C2211yescharacteristic
14C23×#557—21no—
15C23×#557C2211yescharacteristic
16C23×#556—21no—
17C23×#556—21no—
18C23×#556C2211yescharacteristic
19C23×#554C2211yescharacteristic
20C23×#552—21no—
21C23×#544—21no—
22C23×#542—21no—
23C23×#541—21no—
24C23×#541—21no—
25C23×#541—22no—
26C23×#533C2211yescharacteristic
27C23×#533C2211yescharacteristic
28C2×C23×C4⋊Q8C2211yescharacteristic
29C23×#529C2211yescharacteristic
30C23×#501—21no—
31C23×#498C2212yes—
32C23×#494—21no—
33C23×#489—21no—
34C23×#487—21no—
35C23×#487C2212yes—
36C23×#484—21no—
37C23×#483—22no—
38C23×#481—21no—
39C23×#475—22no—
40C23×#475C2212yes—
41C23×#474C2212yes—
42C23×#462C2212yes—
43C23×#462C2212yes—
44C23×#451—21no—
45C23×#451—21no—
46C23×#432C2211yescharacteristic
47C23×#431C2211yescharacteristic
48C23×#431C2211yescharacteristic
49C23×#333C2212yes—
50C23×#332C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×Q8⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C22×Q8×C23—21no—
56C2×Q16×C23—22no—
57C2×Q16×C23—22no—
58C23×C4⋊Q8—22no—
59C23×C4⋊Q8—21no—
60C23×C4⋊Q8—22no—
61C23×C4⋊Q8—21no—
62C23×C4⋊Q8—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8—21no—
66C23×C4⋊Q8D411yescharacteristic
67C23×C4⋊Q8D411yescharacteristic
68C23×C4⋊Q8C2312yes—
69C23×C4⋊Q8C2311yescharacteristic
70C23×C4⋊Q8C2311yescharacteristic
71C23×#126—22no—
72C23×#126—22no—
73C23×#126—22no—
74C23×#126—22no—
75C23×#126—22no—
76C23×#126—22no—
77C23×#126—21no—
78C23×#125—22no—
79C23×#125—22no—
80C23×#125—22no—
81C23×#125—22no—
82C23×#125—21no—
83C23×#125—21no—
84C23×#125—22no—
85C23×#125—21no—
86C23×#125—21no—
87C23×#125D411yescharacteristic
88C23×#124—22no—
89C23×#124—21no—
90C23×#124D411yescharacteristic
91C23×#123—22no—
92C23×#123—21no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8—21no—
96C23×C22⋊Q8—21no—
97C23×C22⋊Q8—22no—
98C23×C22⋊Q8—21no—
99C23×C22⋊Q8—21no—
100C23×C22⋊Q8D411yescharacteristic
101C23×C22⋊Q8D411yescharacteristic
102C23×C22⋊Q8D412yes—
103C23×C22⋊Q8D411yescharacteristic
104C23×#121D411yescharacteristic
105C4×Q8×C23—22no—
106C4×Q8×C23—21no—
107C4×Q8×C23—21no—
108C4×Q8×C23—21no—
109C4×Q8×C23—21no—
110C4×Q8×C23—21no—
111C4×Q8×C23—21no—
112C4×Q8×C23—21no—
113C4×Q8×C23D412yes—
114C4×Q8×C23C2311yescharacteristic
115C4×Q8×C23C2311yescharacteristic
116C4×Q8×C23C2311yescharacteristic
117C4×D4×C23—21no—
118C4×D4×C23—21no—
119C4×D4×C23C2311yescharacteristic
120C23×#117—22no—
121C23×#117—21no—
122C23×#117—21no—
123C23×#117—21no—
124C23×#117—21no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23—21no—
127C2×C4⋊C4×C23—21no—
128C2×C42×C23C2311yescharacteristic
129C23×#107—22no—
130C23×#106—24no—
131C23×#106—22no—
132C23×C4⋊C8—21no—
133C23×C4⋊C8C2312yes—
134C23×C4⋊C8C2312yes—
135C23×#103—24no—
136C23×#103—22no—
137C23×#103—22no—
138C23×#103—22no—
139C23×#102—22no—
140C23×C22⋊C8—21no—
141C23×C22⋊C8—21no—
142C23×C22⋊C8—21no—
143C23×#97—21no—
144C4×C8×C23—21no—
145C4×C8×C23C2312yes—
146C2×Q8×C23—22no—
147C2×Q8×C23—21no—
148C2×Q8×C23—21no—
149C2×Q8×C23—22no—
150C2×Q8×C23—41no—
151C2×Q8×C23—82no—
152C2×Q8×C23C2×D411yescharacteristic
153C2×Q8×C23C2×D411yescharacteristic
154C2×Q8×C23C2×D411yescharacteristic
155C2×Q8×C23C2×D412yes—
156C2×D4×C23—21no—
157C2×D4×C23C2×D411yescharacteristic
158C22×C4×C23—22no—
159C22×C4×C23—21no—
160C22×C4×C23—41no—
161C22×C4×C23C2×D411yescharacteristic
162C22×C4×C23C2×D411yescharacteristic
163C22×C4×C23C2×D411yescharacteristic
164Q16×C23—82no—
165Q16×C23—82no—
166C2×C8×C23—22no—
167C2×C8×C23—22no—
168C2×C8×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—22no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—21no—
182C4⋊C4×C23—21no—
183C4⋊C4×C23—21no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—21no—
189C4⋊C4×C23—21no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23—22no—
195C4⋊C4×C23—41no—
196C4⋊C4×C23—41no—
197C4⋊C4×C23C2×D411yescharacteristic
198C4⋊C4×C23C2×D411yescharacteristic
199C4⋊C4×C23C2×D411yescharacteristic
200C4⋊C4×C23C2×D412yes—
201C4⋊C4×C23C2×D411yescharacteristic
202C22⋊C4×C23—22no—
203C22⋊C4×C23—22no—
204C22⋊C4×C23—22no—
205C22⋊C4×C23—21no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—21no—
208C22⋊C4×C23—21no—
209C22⋊C4×C23—21no—
210C22⋊C4×C23—21no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—21no—
213C42×C23—22no—
214C42×C23—22no—
215C42×C23—22no—
216C42×C23—22no—
217C42×C23—21no—
218C42×C23—21no—
219C42×C23—21no—
220C42×C23C2×D411yescharacteristic
221C42×C23C2×D411yescharacteristic
222C42×C23C2×D411yescharacteristic
223C42×C23C2411yescharacteristic
224#18972C2311yesmaximal, characteristic, direct product factor, Sylow
225C23×C23—21no—
226C23×C23#12011yescharacteristic
227Q8×C23—22no—
228Q8×C23—22no—
229Q8×C23—22no—
230Q8×C23—42no—
231Q8×C23—42no—
232Q8×C23—42no—
233Q8×C23—42no—
234Q8×C23—81no—
235Q8×C23—82no—
236Q8×C23—82no—
237D4×C23—22no—
238D4×C23—81no—
239C2×C4×C23—22no—
240C2×C4×C23—22no—
241C2×C4×C23—21no—
242C2×C4×C23—21no—
243C2×C4×C23—21no—
244C2×C4×C23—22no—
245C2×C4×C23—21no—
246C2×C4×C23—21no—
247C2×C4×C23—21no—
248C2×C4×C23—22no—
249C2×C4×C23—21no—
250C2×C4×C23—41no—
251C2×C4×C23—42no—
252C2×C4×C23—42no—
253C2×C4×C23—82no—
254C2×C4×C23—82no—
255C2×C4×C23#12011yescharacteristic
256C2×C4×C23#12011yescharacteristic
257C2×C4×C23#12011yescharacteristic
258C2×C4×C23C22×D411yescharacteristic
259C2×C4×C23C22×D411yescharacteristic
260C2×C4×C23C22×D411yescharacteristic
261C8×C23—42no—
262C8×C23—42no—
263C8×C23—42no—
264#3555C2×C2311yescharacteristic
265#3239C2×C2311yescharacteristic
266#3229C2×C2311yescharacteristic, complemented
267#3209C2×C2311yescharacteristic, complemented
268#1653C2×C2312yescomplemented
269#1652C2×C2312yescomplemented
270#1645C2×C2312yescomplemented
271#1644C2×C2312yescomplemented
272#1554C2×C2311yescharacteristic, complemented
273#1497C2×C2311yescharacteristic, complemented
274#1482C2×C2311yescharacteristic
275C22×C23—21no—
276C22×C23—21no—
277C22×C23—21no—
278C22×C23—41no—
279C22×C23—41no—
280C22×C23—41no—
281C22×C23C2×#12011yescharacteristic
282C4×C23—21no—
283C4×C23—21no—
284C4×C23—21no—
285C4×C23—21no—
286C4×C23—41no—
287C4×C23—41no—
288C4×C23—41no—
289C4×C23—41no—
290C4×C23—42no—
291C4×C23—41no—
292C4×C23—42no—
293C4×C23—41no—
294C4×C23—41no—
295C4×C23—82no—
296C4×C23#57712yes—
297C4×C23#57812yes—
298#563—22no—
299Q82C22×C2311yescharacteristic
300#557—21no—
301#557C22×C2311yescharacteristic
302#556—21no—
303#556—21no—
304#556C22×C2311yescharacteristic
305#554C22×C2311yescharacteristic
306#552—21no—
307#544—21no—
308#542—21no—
309#541—21no—
310#541—21no—
311#541—22no—
312#533C22×C2311yescharacteristic
313#533C22×C2311yescharacteristic
314C2×C4⋊Q8C22×C2311yescharacteristic
315#529C22×C2311yescharacteristic
316#501—21no—
317#498C22×C2312yes—
318#494—21no—
319#489—21no—
320#487—21no—
321#487C22×C2312yes—
322#484—21no—
323#483—22no—
324#481—21no—
325#475—22no—
326#475C22×C2312yes—
327#474C22×C2312yes—
328#462C22×C2312yes—
329#462C22×C2312yes—
330#451—21no—
331#451—21no—
332#432C22×C2311yescharacteristic
333#431C22×C2311yescharacteristic
334#431C22×C2311yescharacteristic
335#333C22×C2312yes—
336#332C22×C2312yes—
337Q8⋊C8C22×C2312yes—
338Q8⋊C8C22×C2312yes—
339Q8⋊C8C22×C2312yes—
340D4⋊C8C22×C2312yes—
341C2×C23—41no—
342C2×C23—81no—
343C2×C23#301011yescharacteristic
344C2×C23#301111yescharacteristic
345C2×C23#302211yescharacteristic
346C22×Q8—21no—
347C2×Q16—22no—
348C2×Q16—22no—
349C4⋊Q8—22no—
350C4⋊Q8—21no—
351C4⋊Q8—22no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8D4×C2311yescharacteristic
358C4⋊Q8D4×C2311yescharacteristic
359C4⋊Q8C23×C2312yes—
360C4⋊Q8C23×C2311yescharacteristic
361C4⋊Q8C23×C2311yescharacteristic
362#126—22no—
363#126—22no—
364#126—22no—
365#126—22no—
366#126—22no—
367#126—22no—
368#126—21no—
369#125—22no—
370#125—22no—
371#125—22no—
372#125—22no—
373#125—21no—
374#125—22no—
375#125—21no—
376#125—21no—
377#125—21no—
378#125D4×C2311yescharacteristic
379#124—22no—
380#124—21no—
381#124D4×C2311yescharacteristic
382#123—22no—
383#123—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8—22no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8D4×C2311yescharacteristic
392C22⋊Q8D4×C2311yescharacteristic
393C22⋊Q8D4×C2311yescharacteristic
394C22⋊Q8D4×C2312yes—
395#121D4×C2311yescharacteristic
396C4×Q8—22no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8D4×C2312yes—
405C4×Q8C23×C2311yescharacteristic
406C4×Q8C23×C2311yescharacteristic
407C4×Q8C23×C2311yescharacteristic
408C4×D4—21no—
409C4×D4—21no—
410C4×D4C23×C2311yescharacteristic
411#117—22no—
412#117—21no—
413#117—21no—
414#117—21no—
415#117—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C42C23×C2311yescharacteristic
420#107—22no—
421#106—24no—
422#106—22no—
423C4⋊C8—21no—
424C4⋊C8C23×C2312yes—
425C4⋊C8C23×C2312yes—
426#103—24no—
427#103—22no—
428#103—22no—
429#103—22no—
430#102—22no—
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8—21no—
434#97—21no—
435C4×C8—21no—
436C4×C8C23×C2312yes—
437C23#1897211yescharacteristic, direct product factor, Sylow
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—82no—
444C2×Q8C2×D4×C2311yescharacteristic
445C2×Q8C2×D4×C2311yescharacteristic
446C2×Q8C2×D4×C2311yescharacteristic
447C2×Q8C2×D4×C2312yes—
448C2×D4—21no—
449C2×D4C2×D4×C2311yescharacteristic
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4C2×D4×C2311yescharacteristic
454C22×C4C2×D4×C2311yescharacteristic
455C22×C4C2×D4×C2311yescharacteristic
456Q16—82no—
457Q16—82no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C2311yescharacteristic
490C4⋊C4C2×D4×C2311yescharacteristic
491C4⋊C4C2×D4×C2311yescharacteristic
492C4⋊C4C2×D4×C2312yes—
493C4⋊C4C2×D4×C2311yescharacteristic
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42C2×D4×C2311yescharacteristic
513C42C2×D4×C2311yescharacteristic
514C42C2×D4×C2311yescharacteristic
515C42C24×C2311yescharacteristic
516C23—21no—
517C23C23×#12011yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—22no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—81no—
526Q8—82no—
527Q8—82no—
528D4—22no—
529D4—81no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4C23×#12011yescharacteristic
547C2×C4C23×#12011yescharacteristic
548C2×C4C23×#12011yescharacteristic
549C2×C4C22×D4×C2311yescharacteristic
550C2×C4C22×D4×C2311yescharacteristic
551C2×C4C22×D4×C2311yescharacteristic
552C8—42no—
553C8—42no—
554C8—42no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22C2×C23×#12011yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—82no—
576C4C23×#57712yes—
577C4C23×#57812yes—
578C2—41no—
579C2—81no—
580C2C23×#301011yescharacteristic
581C2C23×#301111yescharacteristic
582C2C23×#302211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#3229C2semidirect
4. C23×#3209C2semidirect
5. C23×#1653C2semidirect
6. C23×#1652C2semidirect
7. C23×#1645C2semidirect
8. C23×#1644C2semidirect
9. C23×#1554C2semidirect
10. C23×#1497C2semidirect
224. #18972C23direct (N × H)
266. #3229C2×C23semidirect
267. #3209C2×C23semidirect
268. #1653C2×C23semidirect
269. #1652C2×C23semidirect
270. #1645C2×C23semidirect
271. #1644C2×C23semidirect
272. #1554C2×C23semidirect
273. #1497C2×C23semidirect
437. C23#18972direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35552
C23×#32392
C23×#32292
C23×#32092
C23×#16532
C23×#16522
C23×#16452
C23×#16442
C23×#15542
C23×#14972
C23×#14822
#1897223