Finite Group Databasebeta

← C23×#18131 #19165757

Subgroups — C23×#18131 #19165757

Order 5888.

932 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3461C211yesmaximal, characteristic
2C23×#3108C212yesmaximal, complemented
3C23×#2399C211yesmaximal, characteristic, complemented
4C23×#1611C211yesmaximal, characteristic, complemented
5C23×#1610C211yesmaximal, characteristic
6C23×#1609C212yesmaximal, complemented
7C23×#1605C211yesmaximal, characteristic, complemented
8C23×#1604C211yesmaximal, characteristic
9C23×#1603C212yesmaximal, complemented
10C23×#1500C212yesmaximal, complemented
11C23×#1461C211yesmaximal, characteristic, complemented
12C23×#544C2211yescharacteristic
13C23×#542—22no—
14C23×#541—22no—
15C23×#539—21no—
16C23×#539C2212yes—
17C23×#537C2211yescharacteristic
18C23×#536—21no—
19C23×#533—22no—
20C23×#529—22no—
21C23×#529—21no—
22C23×#529C2212yes—
23C4×#41×C23—21no—
24C2×C4×Q8×C23C2211yescharacteristic
25C2×C4×Q8×C23C2211yescharacteristic
26C23×#483C2212yes—
27C23×#480C2212yes—
28C23×#479—22no—
29C23×#464—22no—
30C23×#451C2212yes—
31C23×#448C2212yes—
32C4×Q16×C23—22no—
33C4×SD16×C23—22no—
34C23×#431C2212yes—
35C23×#398—21no—
36C23×#398C2211yescharacteristic
37C23×#398C2211yescharacteristic
38C23×#395—21no—
39C23×#395C2211yescharacteristic
40C23×#395C2211yescharacteristic
41C4×C4⋊C4×C23—21no—
42C4×C4⋊C4×C23—21no—
43C23×#376C2211yescharacteristic
44C23×#328C2212yes—
45C23×#328C2212yes—
46C23×#327C2212yes—
47C23×#327C2212yes—
48C23×#324C2212yes—
49C23×#324C2211yescharacteristic
50C23×#324C2211yescharacteristic
51C2×#41×C23—21no—
52C22×Q8×C23D411yescharacteristic
53C22×Q8×C23C2311yescharacteristic
54C2×Q16×C23—22no—
55C2×SD16×C23—22no—
56C23×C4⋊Q8—22no—
57C23×C4⋊Q8—22no—
58C23×#126—22no—
59C23×#126—22no—
60C23×#126—22no—
61C23×#126—22no—
62C23×#125—21no—
63C23×#125—22no—
64C23×#125—22no—
65C23×#125—22no—
66C23×#125D412yes—
67C23×#124—22no—
68C23×#124—21no—
69C23×#124D412yes—
70C23×#123—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×#123—22no—
74C23×C22⋊Q8—22no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—22no—
78C23×C22⋊Q8—21no—
79C23×C22⋊Q8—22no—
80C23×C22⋊Q8C2312yes—
81C23×#121—22no—
82C23×#121—21no—
83C23×#121C2312yes—
84C4×Q8×C23—21no—
85C4×Q8×C23—21no—
86C4×Q8×C23—21no—
87C4×Q8×C23—22no—
88C4×Q8×C23—22no—
89C4×Q8×C23—22no—
90C4×Q8×C23—22no—
91C4×Q8×C23D412yes—
92C4×D4×C23—21no—
93C4×D4×C23—21no—
94C4×D4×C23—21no—
95C4×D4×C23—22no—
96C4×D4×C23—22no—
97C4×D4×C23D412yes—
98C23×#117—22no—
99C23×#117—22no—
100C23×#117—21no—
101C23×#117—22no—
102C2×C4⋊C4×C23—21no—
103C2×C4⋊C4×C23—21no—
104C2×C4⋊C4×C23—21no—
105C2×C4⋊C4×C23—21no—
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23D411yescharacteristic
108C2×C4⋊C4×C23C2311yescharacteristic
109C2×C4⋊C4×C23C2311yescharacteristic
110C2×C4⋊C4×C23C2311yescharacteristic
111C2×C42×C23—22no—
112C2×C42×C23—21no—
113C2×C42×C23—21no—
114C2×C42×C23D411yescharacteristic
115C2×C42×C23D411yescharacteristic
116C2×C42×C23C2311yescharacteristic
117C23×#107—22no—
118C23×#106—22no—
119C23×C4⋊C8—22no—
120C23×#103—22no—
121C23×#103—22no—
122C23×#103—22no—
123C23×#102—22no—
124C23×C22⋊C8C2312yes—
125C23×C22⋊C8C2312yes—
126C4×C8×C23—22no—
127C23×#95—21no—
128C23×#95—21no—
129C23×#95—21no—
130C23×#95—21no—
131C23×#95—21no—
132C23×#95C2311yescharacteristic
133C23×#95C2311yescharacteristic
134#41×C23—42no—
135#41×C23—42no—
136C2×Q8×C23—21no—
137C2×Q8×C23—21no—
138C2×Q8×C23—22no—
139C2×Q8×C23—22no—
140C2×Q8×C23—22no—
141C2×Q8×C23—41no—
142C2×Q8×C23—42no—
143C2×Q8×C23C2×D412yes—
144C2×D4×C23—21no—
145C2×D4×C23—22no—
146C2×D4×C23—21no—
147C2×D4×C23C2×D412yes—
148C22×C4×C23—22no—
149C22×C4×C23—21no—
150C22×C4×C23—21no—
151C22×C4×C23—21no—
152C22×C4×C23—22no—
153C22×C4×C23—22no—
154C22×C4×C23—21no—
155C22×C4×C23C2×D411yescharacteristic
156C22×C4×C23C2×D411yescharacteristic
157C22×C4×C23C2×D411yescharacteristic
158C22×C4×C23C2×D411yescharacteristic
159C22×C4×C23C2×D411yescharacteristic
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2411yescharacteristic
162Q16×C23—44no—
163SD16×C23—44no—
164C2×C8×C23—22no—
165C2×C8×C23—22no—
166C4⋊C4×C23—21no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—21no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—21no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—42no—
183C4⋊C4×C23—42no—
184C4⋊C4×C23—42no—
185C4⋊C4×C23—42no—
186C4⋊C4×C23—42no—
187C4⋊C4×C23C2×D412yes—
188C4⋊C4×C23C2×D412yes—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—22no—
194C22⋊C4×C23—22no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—22no—
198C42×C23—24no—
199C42×C23—22no—
200C42×C23—22no—
201C42×C23—21no—
202C42×C23—21no—
203C42×C23—21no—
204C42×C23—22no—
205C42×C23—21no—
206C42×C23—41no—
207C42×C23—41no—
208C42×C23C2×D412yes—
209C42×C23C2×D412yes—
210#18131C2311yesmaximal, characteristic, direct product factor, Sylow
211C23×C23—22no—
212C23×C23C22×D411yescharacteristic
213Q8×C23—24no—
214Q8×C23—44no—
215Q8×C23—42no—
216Q8×C23—42no—
217Q8×C23—42no—
218Q8×C23—81no—
219D4×C23—42no—
220D4×C23—42no—
221D4×C23—42no—
222D4×C23—42no—
223D4×C23—81no—
224C2×C4×C23—21no—
225C2×C4×C23—21no—
226C2×C4×C23—21no—
227C2×C4×C23—21no—
228C2×C4×C23—22no—
229C2×C4×C23—22no—
230C2×C4×C23—22no—
231C2×C4×C23—22no—
232C2×C4×C23—21no—
233C2×C4×C23—22no—
234C2×C4×C23—22no—
235C2×C4×C23—21no—
236C2×C4×C23—21no—
237C2×C4×C23—22no—
238C2×C4×C23—21no—
239C2×C4×C23—21no—
240C2×C4×C23—41no—
241C2×C4×C23—44no—
242C2×C4×C23—41no—
243C2×C4×C23—41no—
244C2×C4×C23—41no—
245C2×C4×C23—41no—
246C2×C4×C23—41no—
247C2×C4×C23—41no—
248C2×C4×C23—44no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—41no—
254C2×C4×C23—82no—
255C2×C4×C23#12011yescharacteristic
256C2×C4×C23#12012yes—
257C2×C4×C23#12011yescharacteristic
258C2×C4×C23#13512yes—
259C2×C4×C23C22×D412yes—
260C8×C23—42no—
261C8×C23—42no—
262#3461C2×C2311yescharacteristic
263#3108C2×C2312yescomplemented
264#2399C2×C2311yescharacteristic, complemented
265#1611C2×C2311yescharacteristic, complemented
266#1610C2×C2311yescharacteristic
267#1609C2×C2312yescomplemented
268#1605C2×C2311yescharacteristic, complemented
269#1604C2×C2311yescharacteristic
270#1603C2×C2312yescomplemented
271#1500C2×C2312yescomplemented
272#1461C2×C2311yescharacteristic, complemented
273C22×C23—21no—
274C22×C23—21no—
275C22×C23—42no—
276C22×C23—42no—
277C22×C23—42no—
278C22×C23C2×#12011yescharacteristic
279C22×C23C2×#13511yescharacteristic
280C22×C23#57511yescharacteristic
281C4×C23—22no—
282C4×C23—22no—
283C4×C23—21no—
284C4×C23—22no—
285C4×C23—42no—
286C4×C23—42no—
287C4×C23—41no—
288C4×C23—42no—
289C4×C23—42no—
290C4×C23—42no—
291C4×C23—41no—
292C4×C23—41no—
293C4×C23—42no—
294C4×C23—81no—
295C4×C23#45612yes—
296#544C22×C2311yescharacteristic
297#542—22no—
298#541—22no—
299#539—21no—
300#539C22×C2312yes—
301#537C22×C2311yescharacteristic
302#536—21no—
303#533—22no—
304#529—22no—
305#529—21no—
306#529C22×C2312yes—
307C4×#41—21no—
308C2×C4×Q8C22×C2311yescharacteristic
309C2×C4×Q8C22×C2311yescharacteristic
310#483C22×C2312yes—
311#480C22×C2312yes—
312#479—22no—
313#464—22no—
314#451C22×C2312yes—
315#448C22×C2312yes—
316C4×Q16—22no—
317C4×SD16—22no—
318#431C22×C2312yes—
319#398—21no—
320#398C22×C2311yescharacteristic
321#398C22×C2311yescharacteristic
322#395—21no—
323#395C22×C2311yescharacteristic
324#395C22×C2311yescharacteristic
325C4×C4⋊C4—21no—
326C4×C4⋊C4—21no—
327#376C22×C2311yescharacteristic
328#328C22×C2312yes—
329#328C22×C2312yes—
330#327C22×C2312yes—
331#327C22×C2312yes—
332#324C22×C2312yes—
333#324C22×C2311yescharacteristic
334#324C22×C2311yescharacteristic
335C2×C23—21no—
336C2×C23—21no—
337C2×C23—82no—
338C2×C23#300211yescharacteristic
339C2×C23C2×#45611yescharacteristic
340C2×C23#302511yescharacteristic
341C2×#41—21no—
342C22×Q8D4×C2311yescharacteristic
343C22×Q8C23×C2311yescharacteristic
344C2×Q16—22no—
345C2×SD16—22no—
346C4⋊Q8—22no—
347C4⋊Q8—22no—
348#126—22no—
349#126—22no—
350#126—22no—
351#126—22no—
352#125—21no—
353#125—22no—
354#125—22no—
355#125—22no—
356#125D4×C2312yes—
357#124—22no—
358#124—21no—
359#124D4×C2312yes—
360#123—22no—
361#123—22no—
362#123—22no—
363#123—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8C23×C2312yes—
371#121—22no—
372#121—21no—
373#121C23×C2312yes—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—21no—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×Q8D4×C2312yes—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4D4×C2312yes—
388#117—22no—
389#117—22no—
390#117—21no—
391#117—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4D4×C2311yescharacteristic
398C2×C4⋊C4C23×C2311yescharacteristic
399C2×C4⋊C4C23×C2311yescharacteristic
400C2×C4⋊C4C23×C2311yescharacteristic
401C2×C42—22no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42D4×C2311yescharacteristic
405C2×C42D4×C2311yescharacteristic
406C2×C42C23×C2311yescharacteristic
407#107—22no—
408#106—22no—
409C4⋊C8—22no—
410#103—22no—
411#103—22no—
412#103—22no—
413#102—22no—
414C22⋊C8C23×C2312yes—
415C22⋊C8C23×C2312yes—
416C4×C8—22no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95C23×C2311yescharacteristic
423#95C23×C2311yescharacteristic
424C23#1813111yescharacteristic, direct product factor, Sylow
425#41—42no—
426#41—42no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—41no—
433C2×Q8—42no—
434C2×Q8C2×D4×C2312yes—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4C2×D4×C2312yes—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4C2×D4×C2311yescharacteristic
447C22×C4C2×D4×C2311yescharacteristic
448C22×C4C2×D4×C2311yescharacteristic
449C22×C4C2×D4×C2311yescharacteristic
450C22×C4C2×D4×C2311yescharacteristic
451C22×C4C2×D4×C2311yescharacteristic
452C22×C4C24×C2311yescharacteristic
453Q16—44no—
454SD16—44no—
455C2×C8—22no—
456C2×C8—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4C2×D4×C2312yes—
479C4⋊C4C2×D4×C2312yes—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C42—22no—
490C42—22no—
491C42—24no—
492C42—21no—
493C42—22no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—41no—
498C42—41no—
499C42C2×D4×C2312yes—
500C42C2×D4×C2312yes—
501C23—22no—
502C23C22×D4×C2311yescharacteristic
503Q8—24no—
504Q8—44no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—81no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—81no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—82no—
545C2×C4C23×#12011yescharacteristic
546C2×C4C23×#12012yes—
547C2×C4C23×#12011yescharacteristic
548C2×C4C23×#13512yes—
549C2×C4C22×D4×C2312yes—
550C8—42no—
551C8—42no—
552C22—21no—
553C22—21no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22C2×C23×#12011yescharacteristic
558C22C2×C23×#13511yescharacteristic
559C22C23×#57511yescharacteristic
560C4—22no—
561C4—22no—
562C4—21no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—81no—
574C4C23×#45612yes—
575C2—21no—
576C2—21no—
577C2—82no—
578C2C23×#300211yescharacteristic
579C2C2×C23×#45611yescharacteristic
580C2C23×#302511yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutator, Frattini, cosocleC22×C4C24×C23
socleC22×C23C2×#120
prime radicalC23×#537C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3108C2semidirect
3. C23×#2399C2semidirect
4. C23×#1611C2semidirect
6. C23×#1609C2semidirect
7. C23×#1605C2semidirect
9. C23×#1603C2semidirect
10. C23×#1500C2semidirect
11. C23×#1461C2semidirect
210. #18131C23direct (N × H)
263. #3108C2×C23semidirect
264. #2399C2×C23semidirect
265. #1611C2×C23semidirect
267. #1609C2×C23semidirect
268. #1605C2×C23semidirect
270. #1603C2×C23semidirect
271. #1500C2×C23semidirect
272. #1461C2×C23semidirect
424. C23#18131direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34612
C23×#31082
C23×#23992
C23×#16112
C23×#16102
C23×#16092
C23×#16052
C23×#16042
C23×#16032
C23×#15002
C23×#14612
#1813123