Finite Group Databasebeta

← C23×#18321 #19165947

Subgroups — C23×#18321 #19165947

Order 5888.

982 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#539C211yesmaximal, characteristic
2C23×#3195C211yesmaximal, characteristic, complemented
3C23×#3193C211yesmaximal, characteristic
4C23×#2583C211yesmaximal, characteristic, complemented
5C23×#1609C214yesmaximal, complemented
6C23×#1603C214yesmaximal, complemented
7C23×#1508C211yesmaximal, characteristic, complemented
8C23×#1507C211yesmaximal, characteristic
9C23×#1463C211yesmaximal, characteristic, complemented
10C23×#539—22no—
11C23×#539—22no—
12C23×#539—22no—
13C23×#539C2214yes—
14C2×C23×#126—21no—
15C2×C23×#125—21no—
16C2×C23×#123—21no—
17C2×C23×#123—21no—
18C2×C23×C22⋊Q8—21no—
19C2×C23×C22⋊Q8C2211yescharacteristic
20C2×C23×C22⋊Q8C2211yescharacteristic
21C2×C23×#121C2211yescharacteristic
22C2×C4×D4×C23—21no—
23C22×C4⋊C4×C23C2211yescharacteristic
24C23×#484—21no—
25C23×#483—21no—
26C23×#483C2212yescomplemented
27C23×#480C2212yes—
28C23×#470—22no—
29C23×#468—22no—
30C23×#451—21no—
31C23×#451C2212yescomplemented
32C23×#450—21no—
33C23×#448C2212yes—
34C23×#407C2211yescharacteristic
35C23×#407C2211yescharacteristic
36C23×#398C2212yescomplemented
37C23×#395C2212yescomplemented
38C23×#389—22no—
39C23×#385—21no—
40C23×#385—21no—
41C4×C22⋊C4×C23C2211yescharacteristic
42C23×#328C2214yescomplemented
43C23×#327C2214yes—
44C23×#324C2214yescomplemented
45C22×Q8×C23D411yescharacteristic
46C22×Q8×C23C2311yescharacteristic
47C22×D4×C23—21no—
48C22×D4×C23D411yescharacteristic
49C23×C4×C23—21no—
50C23×C4×C23D411yescharacteristic
51C23×C4×C23D411yescharacteristic
52C23×C4×C23C2311yescharacteristic
53C2×Q16×C23—22no—
54C2×SD16×C23—22no—
55C2×M16×C23—21no—
56C2×M16×C23—21no—
57C23×#126—24no—
58C23×#126—24no—
59C23×#125—24no—
60C23×#125—24no—
61C23×#123—24no—
62C23×#123—24no—
63C23×#123—24no—
64C23×#123—24no—
65C23×C22⋊Q8—24no—
66C23×C22⋊Q8—24no—
67C23×C22⋊Q8—21no—
68C23×C22⋊Q8—21no—
69C23×C22⋊Q8—21no—
70C23×C22⋊Q8—21no—
71C23×C22⋊Q8—21no—
72C23×C22⋊Q8—21no—
73C23×C22⋊Q8D412yes—
74C23×C22⋊Q8C2312yes—
75C23×#121—21no—
76C23×#121—21no—
77C23×#121D412yes—
78C23×#121C2312yes—
79C4×D4×C23—24no—
80C4×D4×C23—24no—
81C2×C4⋊C4×C23—21no—
82C2×C4⋊C4×C23—21no—
83C2×C4⋊C4×C23—21no—
84C2×C4⋊C4×C23—21no—
85C2×C4⋊C4×C23—21no—
86C2×C4⋊C4×C23—21no—
87C2×C4⋊C4×C23—21no—
88C2×C4⋊C4×C23—21no—
89C2×C4⋊C4×C23—22no—
90C2×C4⋊C4×C23—21no—
91C2×C4⋊C4×C23—22no—
92C2×C4⋊C4×C23D412yes—
93C2×C4⋊C4×C23D411yescharacteristic
94C2×C4⋊C4×C23D411yescharacteristic
95C2×C4⋊C4×C23C2311yescharacteristic
96C2×C4⋊C4×C23C2312yes—
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—21no—
103C2×C22⋊C4×C23—21no—
104C2×C42×C23—21no—
105C2×C42×C23—21no—
106C23×#107—22no—
107C23×#106—22no—
108C23×#103—22no—
109C23×#103—22no—
110C23×#103—22no—
111C23×#102—22no—
112C23×C22⋊C8—21no—
113C23×C22⋊C8—21no—
114C23×C22⋊C8C2312yes—
115C23×C22⋊C8C2312yes—
116C23×#95—22no—
117C23×#95—22no—
118C23×#95C2312yes—
119C24×C23—21no—
120C24×C23C2×D411yescharacteristic
121C2×Q8×C23—21no—
122C2×Q8×C23—21no—
123C2×Q8×C23—21no—
124C2×Q8×C23—81no—
125C2×Q8×C23—81no—
126C2×Q8×C23C2×D412yes—
127C2×D4×C23—22no—
128C2×D4×C23—22no—
129C2×D4×C23—21no—
130C2×D4×C23—44no—
131C2×D4×C23—81no—
132C2×D4×C23C2×D412yes—
133C22×C4×C23—21no—
134C22×C4×C23—21no—
135C22×C4×C23—21no—
136C22×C4×C23—21no—
137C22×C4×C23—21no—
138C22×C4×C23—21no—
139C22×C4×C23—22no—
140C22×C4×C23—24no—
141C22×C4×C23—21no—
142C22×C4×C23—21no—
143C22×C4×C23—21no—
144C22×C4×C23—42no—
145C22×C4×C23—81no—
146C22×C4×C23—81no—
147C22×C4×C23—82no—
148C22×C4×C23C2×D412yes—
149C22×C4×C23C2×D412yes—
150C22×C4×C23C2×D411yescharacteristic
151C22×C4×C23C2×D411yescharacteristic
152C22×C4×C23C2×D411yescharacteristic
153C22×C4×C23C2×D411yescharacteristic
154C22×C4×C23C2×D411yescharacteristic
155C22×C4×C23C2411yescharacteristic
156Q16×C23—82no—
157SD16×C23—82no—
158M16×C23—81no—
159M16×C23—81no—
160C2×C8×C23—22no—
161C2×C8×C23—22no—
162C4⋊C4×C23—24no—
163C4⋊C4×C23—24no—
164C4⋊C4×C23—24no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—21no—
169C4⋊C4×C23—21no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—21no—
175C4⋊C4×C23—44no—
176C4⋊C4×C23—41no—
177C4⋊C4×C23—41no—
178C4⋊C4×C23—82no—
179C4⋊C4×C23C2×D412yes—
180C4⋊C4×C23C2×D412yes—
181C22⋊C4×C23—24no—
182C22⋊C4×C23—24no—
183C22⋊C4×C23—24no—
184C22⋊C4×C23—24no—
185C22⋊C4×C23—22no—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—41no—
190C22⋊C4×C23—41no—
191C42×C23—24no—
192C42×C23—81no—
193#18321C2311yesmaximal, characteristic, direct product factor, Sylow
194C23×C23—22no—
195C23×C23—22no—
196C23×C23—22no—
197C23×C23—41no—
198C23×C23—42no—
199C23×C23—42no—
200C23×C23—42no—
201C23×C23#12011yescharacteristic
202C23×C23#12011yescharacteristic
203C23×C23C22×D411yescharacteristic
204Q8×C23—42no—
205Q8×C23—42no—
206Q8×C23—42no—
207Q8×C23—81no—
208D4×C23—42no—
209D4×C23—44no—
210D4×C23—44no—
211D4×C23—81no—
212C2×C4×C23—24no—
213C2×C4×C23—22no—
214C2×C4×C23—22no—
215C2×C4×C23—22no—
216C2×C4×C23—22no—
217C2×C4×C23—22no—
218C2×C4×C23—22no—
219C2×C4×C23—22no—
220C2×C4×C23—41no—
221C2×C4×C23—41no—
222C2×C4×C23—41no—
223C2×C4×C23—41no—
224C2×C4×C23—41no—
225C2×C4×C23—41no—
226C2×C4×C23—42no—
227C2×C4×C23—42no—
228C2×C4×C23—42no—
229C2×C4×C23—42no—
230C2×C4×C23—42no—
231C2×C4×C23—42no—
232C2×C4×C23—42no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—42no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—42no—
239C2×C4×C23—81no—
240C2×C4×C23—84no—
241C2×C4×C23#12011yescharacteristic
242C2×C4×C23#12011yescharacteristic
243C2×C4×C23C22×D411yescharacteristic
244C2×C4×C23C22×D411yescharacteristic
245C8×C23—42no—
246C8×C23—42no—
247C2×#539C2×C2311yescharacteristic
248#3195C2×C2311yescharacteristic, complemented
249#3193C2×C2311yescharacteristic
250#2583C2×C2311yescharacteristic, complemented
251#1609C2×C2314yescomplemented
252#1603C2×C2314yescomplemented
253#1508C2×C2311yescharacteristic, complemented
254#1507C2×C2311yescharacteristic
255#1463C2×C2311yescharacteristic, complemented
256C22×C23—21no—
257C22×C23—21no—
258C22×C23—21no—
259C22×C23—21no—
260C22×C23—21no—
261C22×C23—21no—
262C22×C23—21no—
263C22×C23—44no—
264C22×C23—42no—
265C22×C23—42no—
266C22×C23—42no—
267C22×C23—84no—
268C22×C23#45612yes—
269C22×C23C2×#12011yescharacteristic
270C22×C23#57512yes—
271C4×C23—21no—
272C4×C23—21no—
273C4×C23—21no—
274C4×C23—21no—
275C4×C23—44no—
276C4×C23—42no—
277C4×C23—42no—
278C4×C23—42no—
279C4×C23—42no—
280C4×C23—42no—
281C4×C23—81no—
282C4×C23—81no—
283#539—22no—
284#539—22no—
285#539—22no—
286#539C22×C2314yes—
287C2×#126—21no—
288C2×#125—21no—
289C2×#123—21no—
290C2×#123—21no—
291C2×C22⋊Q8—21no—
292C2×C22⋊Q8C22×C2311yescharacteristic
293C2×C22⋊Q8C22×C2311yescharacteristic
294C2×#121C22×C2311yescharacteristic
295C2×C4×D4—21no—
296C22×C4⋊C4C22×C2311yescharacteristic
297#484—21no—
298#483—21no—
299#483C22×C2312yescomplemented
300#480C22×C2312yes—
301#470—22no—
302#468—22no—
303#451—21no—
304#451C22×C2312yescomplemented
305#450—21no—
306#448C22×C2312yes—
307#407C22×C2311yescharacteristic
308#407C22×C2311yescharacteristic
309#398C22×C2312yescomplemented
310#395C22×C2312yescomplemented
311#389—22no—
312#385—21no—
313#385—21no—
314C4×C22⋊C4C22×C2311yescharacteristic
315#328C22×C2314yescomplemented
316#327C22×C2314yes—
317#324C22×C2314yescomplemented
318C2×C23—22no—
319C2×C23—22no—
320C2×C23—41no—
321C2×C23—82no—
322C2×C23#300511yescharacteristic
323C2×C23C2×#45611yescharacteristic
324C2×C23#302611yescharacteristic
325C22×Q8D4×C2311yescharacteristic
326C22×Q8C23×C2311yescharacteristic
327C22×D4—21no—
328C22×D4D4×C2311yescharacteristic
329C23×C4—21no—
330C23×C4D4×C2311yescharacteristic
331C23×C4D4×C2311yescharacteristic
332C23×C4C23×C2311yescharacteristic
333C2×Q16—22no—
334C2×SD16—22no—
335C2×M16—21no—
336C2×M16—21no—
337#126—24no—
338#126—24no—
339#125—24no—
340#125—24no—
341#123—24no—
342#123—24no—
343#123—24no—
344#123—24no—
345C22⋊Q8—24no—
346C22⋊Q8—24no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353C22⋊Q8D4×C2312yes—
354C22⋊Q8C23×C2312yes—
355#121—21no—
356#121—21no—
357#121D4×C2312yes—
358#121C23×C2312yes—
359C4×D4—24no—
360C4×D4—24no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4D4×C2312yes—
373C2×C4⋊C4D4×C2311yescharacteristic
374C2×C4⋊C4D4×C2311yescharacteristic
375C2×C4⋊C4C23×C2311yescharacteristic
376C2×C4⋊C4C23×C2312yes—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C42—21no—
385C2×C42—21no—
386#107—22no—
387#106—22no—
388#103—22no—
389#103—22no—
390#103—22no—
391#102—22no—
392C22⋊C8—21no—
393C22⋊C8—21no—
394C22⋊C8C23×C2312yes—
395C22⋊C8C23×C2312yes—
396#95—22no—
397#95—22no—
398#95C23×C2312yes—
399C23#1832111yescharacteristic, direct product factor, Sylow
400C24—21no—
401C24C2×D4×C2311yescharacteristic
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8C2×D4×C2312yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—44no—
412C2×D4—81no—
413C2×D4C2×D4×C2312yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—82no—
429C22×C4C2×D4×C2312yes—
430C22×C4C2×D4×C2312yes—
431C22×C4C2×D4×C2311yescharacteristic
432C22×C4C2×D4×C2311yescharacteristic
433C22×C4C2×D4×C2311yescharacteristic
434C22×C4C2×D4×C2311yescharacteristic
435C22×C4C2×D4×C2311yescharacteristic
436C22×C4C24×C2311yescharacteristic
437Q16—82no—
438SD16—82no—
439M16—81no—
440M16—81no—
441C2×C8—22no—
442C2×C8—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—44no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—82no—
460C4⋊C4C2×D4×C2312yes—
461C4⋊C4C2×D4×C2312yes—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C42—24no—
473C42—81no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23C23×#12011yescharacteristic
482C23C23×#12011yescharacteristic
483C23C22×D4×C2311yescharacteristic
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—81no—
488D4—42no—
489D4—44no—
490D4—44no—
491D4—81no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—81no—
520C2×C4—84no—
521C2×C4C23×#12011yescharacteristic
522C2×C4C23×#12011yescharacteristic
523C2×C4C22×D4×C2311yescharacteristic
524C2×C4C22×D4×C2311yescharacteristic
525C8—42no—
526C8—42no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—84no—
539C22C23×#45612yes—
540C22C2×C23×#12011yescharacteristic
541C22C23×#57512yes—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—81no—
553C4—81no—
554C2—22no—
555C2—22no—
556C2—41no—
557C2—82no—
558C2C23×#300511yescharacteristic
559C2C2×C23×#45611yescharacteristic
560C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC2×C23×#121C22
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×#2583C2semidirect
5. C23×#1609C2semidirect
6. C23×#1603C2semidirect
7. C23×#1508C2semidirect
9. C23×#1463C2semidirect
26. C23×#483C22semidirect
31. C23×#451C22semidirect
36. C23×#398C22semidirect
37. C23×#395C22semidirect
42. C23×#328C22semidirect
44. C23×#324C22semidirect
193. #18321C23direct (N × H)
248. #3195C2×C23semidirect
250. #2583C2×C23semidirect
251. #1609C2×C23semidirect
252. #1603C2×C23semidirect
253. #1508C2×C23semidirect
255. #1463C2×C23semidirect
299. #483C22×C23semidirect
304. #451C22×C23semidirect
309. #398C22×C23semidirect
310. #395C22×C23semidirect
315. #328C22×C23semidirect
317. #324C22×C23semidirect
399. C23#18321direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5392
C23×#31952
C23×#31932
C23×#25832
C23×#16092
C23×#16032
C23×#15082
C23×#15072
C23×#14632
#1832123