Finite Group Databasebeta

← C23×#18315 #19165941

Subgroups — C23×#18315 #19165941

Order 5888.

868 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3461C211yesmaximal, characteristic
2C23×#3232C212yesmaximal, complemented
3C23×#2634C211yesmaximal, characteristic, complemented
4C23×#1617C211yesmaximal, characteristic, complemented
5C23×#1616C211yesmaximal, characteristic
6C23×#1615C212yesmaximal, complemented
7C23×#1611C211yesmaximal, characteristic, complemented
8C23×#1610C211yesmaximal, characteristic
9C23×#1609C212yesmaximal, complemented
10C23×#1521C212yesmaximal, 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—
23C2×C23×#125C2211yescharacteristic
24C4×#41×C23—21no—
25C2×C4×Q8×C23C2211yescharacteristic
26C23×#499—22no—
27C23×#488—22no—
28C23×#484C2212yes—
29C23×#483C2212yes—
30C23×#481C2212yes—
31C23×#480C2212yes—
32C23×#479—22no—
33C23×#478—22no—
34C23×#431C2212yes—
35C23×#400—21no—
36C23×#400C2211yescharacteristic
37C23×#400C2211yescharacteristic
38C23×#398C2211yescharacteristic
39C23×#398C2211yescharacteristic
40C23×#395—21no—
41C23×#387—21no—
42C23×#387—21no—
43C23×#383C2211yescharacteristic
44C23×#328C2212yes—
45C23×#328C2212yes—
46C23×#327C2212yes—
47C23×#327C2212yes—
48C23×#324C2212yes—
49C23×#324C2211yescharacteristic
50C23×#324C2211yescharacteristic
51C2×#41×C23—21no—
52C22×Q8×C23D411yescharacteristic
53C23×C4⋊Q8—22no—
54C23×C4⋊Q8—22no—
55C23×#126—22no—
56C23×#126—22no—
57C23×#126—22no—
58C23×#126—22no—
59C23×#125—21no—
60C23×#125—22no—
61C23×#125—22no—
62C23×#125—22no—
63C23×#125—22no—
64C23×#125—22no—
65C23×#125D412yes—
66C23×#124—22no—
67C23×#124—21no—
68C23×#124D412yes—
69C23×#123—22no—
70C23×#123—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×C22⋊Q8—22no—
74C23×C22⋊Q8—22no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—21no—
78C23×C22⋊Q8—22no—
79C23×C22⋊Q8C2312yes—
80C23×#121—22no—
81C23×#121—21no—
82C23×#121C2312yes—
83C4×Q8×C23—21no—
84C4×Q8×C23—21no—
85C4×Q8×C23—21no—
86C4×Q8×C23—22no—
87C4×Q8×C23—22no—
88C4×Q8×C23D412yes—
89C4×D4×C23—21no—
90C4×D4×C23—21no—
91C4×D4×C23—21no—
92C4×D4×C23—22no—
93C4×D4×C23—22no—
94C4×D4×C23D412yes—
95C23×#117—22no—
96C23×#117—22no—
97C23×#117—21no—
98C23×#117—22no—
99C2×C4⋊C4×C23—21no—
100C2×C4⋊C4×C23—21no—
101C2×C4⋊C4×C23—21no—
102C2×C4⋊C4×C23—21no—
103C2×C4⋊C4×C23D411yescharacteristic
104C2×C4⋊C4×C23C2311yescharacteristic
105C2×C4⋊C4×C23C2311yescharacteristic
106C2×C4⋊C4×C23C2311yescharacteristic
107C2×C4⋊C4×C23C2311yescharacteristic
108C2×C42×C23—22no—
109C2×C42×C23D411yescharacteristic
110C2×C42×C23D411yescharacteristic
111C2×C42×C23C2311yescharacteristic
112C23×#107—22no—
113C23×#107—22no—
114C23×#106—22no—
115C23×#106—22no—
116C23×C4⋊C8—22no—
117C23×#103—22no—
118C23×#103—22no—
119C23×#102—22no—
120C23×#102—22no—
121C23×C22⋊C8C2312yes—
122C23×C22⋊C8C2312yes—
123C4×C8×C23—22no—
124C23×#95—21no—
125C23×#95—21no—
126C23×#95—21no—
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—41no—
141C2×Q8×C23C2×D412yes—
142C2×D4×C23—21no—
143C2×D4×C23—22no—
144C2×D4×C23—21no—
145C2×D4×C23C2×D412yes—
146C22×C4×C23—22no—
147C22×C4×C23—22no—
148C22×C4×C23—22no—
149C22×C4×C23—21no—
150C22×C4×C23—21no—
151C22×C4×C23—21no—
152C22×C4×C23—21no—
153C22×C4×C23C2×D411yescharacteristic
154C22×C4×C23C2×D411yescharacteristic
155C22×C4×C23C2×D411yescharacteristic
156C22×C4×C23C2×D411yescharacteristic
157C22×C4×C23C2×D411yescharacteristic
158C22×C4×C23C2×D411yescharacteristic
159C22×C4×C23C2411yescharacteristic
160C2×C8×C23—22no—
161C2×C8×C23—22no—
162C4⋊C4×C23—21no—
163C4⋊C4×C23—22no—
164C4⋊C4×C23—21no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—21no—
168C4⋊C4×C23—21no—
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—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—42no—
180C4⋊C4×C23—41no—
181C4⋊C4×C23—41no—
182C4⋊C4×C23—42no—
183C4⋊C4×C23—42no—
184C4⋊C4×C23C2×D412yes—
185C4⋊C4×C23C2×D412yes—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—22no—
194C22⋊C4×C23—22no—
195C42×C23—22no—
196C42×C23—22no—
197C42×C23—21no—
198C42×C23—22no—
199C42×C23—21no—
200C42×C23—21no—
201C42×C23—21no—
202C42×C23C2×D412yes—
203C42×C23C2×D412yes—
204#18315C2311yesmaximal, characteristic, direct product factor, Sylow
205C23×C23—22no—
206C23×C23C22×D411yescharacteristic
207Q8×C23—44no—
208Q8×C23—42no—
209Q8×C23—42no—
210Q8×C23—81no—
211D4×C23—42no—
212D4×C23—42no—
213D4×C23—42no—
214D4×C23—42no—
215D4×C23—81no—
216C2×C4×C23—21no—
217C2×C4×C23—21no—
218C2×C4×C23—21no—
219C2×C4×C23—21no—
220C2×C4×C23—21no—
221C2×C4×C23—22no—
222C2×C4×C23—22no—
223C2×C4×C23—21no—
224C2×C4×C23—21no—
225C2×C4×C23—22no—
226C2×C4×C23—21no—
227C2×C4×C23—21no—
228C2×C4×C23—41no—
229C2×C4×C23—41no—
230C2×C4×C23—44no—
231C2×C4×C23—44no—
232C2×C4×C23—41no—
233C2×C4×C23—41no—
234C2×C4×C23—41no—
235C2×C4×C23—41no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—42no—
239C2×C4×C23—41no—
240C2×C4×C23—41no—
241C2×C4×C23—41no—
242C2×C4×C23—41no—
243C2×C4×C23—41no—
244C2×C4×C23—41no—
245C2×C4×C23—82no—
246C2×C4×C23#12011yescharacteristic
247C2×C4×C23#12011yescharacteristic
248C2×C4×C23#12012yes—
249C2×C4×C23#13512yes—
250C2×C4×C23C22×D412yes—
251C8×C23—42no—
252C8×C23—42no—
253#3461C2×C2311yescharacteristic
254#3232C2×C2312yescomplemented
255#2634C2×C2311yescharacteristic, complemented
256#1617C2×C2311yescharacteristic, complemented
257#1616C2×C2311yescharacteristic
258#1615C2×C2312yescomplemented
259#1611C2×C2311yescharacteristic, complemented
260#1610C2×C2311yescharacteristic
261#1609C2×C2312yescomplemented
262#1521C2×C2312yescomplemented
263#1461C2×C2311yescharacteristic, complemented
264C22×C23—21no—
265C22×C23—21no—
266C22×C23—42no—
267C22×C23—42no—
268C22×C23—42no—
269C22×C23C2×#12011yescharacteristic
270C22×C23C2×#13511yescharacteristic
271C22×C23#57511yescharacteristic
272C4×C23—22no—
273C4×C23—22no—
274C4×C23—21no—
275C4×C23—21no—
276C4×C23—22no—
277C4×C23—41no—
278C4×C23—42no—
279C4×C23—42no—
280C4×C23—41no—
281C4×C23—41no—
282C4×C23—42no—
283C4×C23—81no—
284C4×C23—81no—
285C4×C23—81no—
286C4×C23—81no—
287#544C22×C2311yescharacteristic
288#542—22no—
289#541—22no—
290#539—21no—
291#539C22×C2312yes—
292#537C22×C2311yescharacteristic
293#536—21no—
294#533—22no—
295#529—22no—
296#529—21no—
297#529C22×C2312yes—
298C2×#125C22×C2311yescharacteristic
299C4×#41—21no—
300C2×C4×Q8C22×C2311yescharacteristic
301#499—22no—
302#488—22no—
303#484C22×C2312yes—
304#483C22×C2312yes—
305#481C22×C2312yes—
306#480C22×C2312yes—
307#479—22no—
308#478—22no—
309#431C22×C2312yes—
310#400—21no—
311#400C22×C2311yescharacteristic
312#400C22×C2311yescharacteristic
313#398C22×C2311yescharacteristic
314#398C22×C2311yescharacteristic
315#395—21no—
316#387—21no—
317#387—21no—
318#383C22×C2311yescharacteristic
319#328C22×C2312yes—
320#328C22×C2312yes—
321#327C22×C2312yes—
322#327C22×C2312yes—
323#324C22×C2312yes—
324#324C22×C2311yescharacteristic
325#324C22×C2311yescharacteristic
326C2×C23—21no—
327C2×C23—21no—
328C2×C23—82no—
329C2×C23#300211yescharacteristic
330C2×C23#302011yescharacteristic
331C2×C23#302811yescharacteristic
332C2×#41—21no—
333C22×Q8D4×C2311yescharacteristic
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336#126—22no—
337#126—22no—
338#126—22no—
339#126—22no—
340#125—21no—
341#125—22no—
342#125—22no—
343#125—22no—
344#125—22no—
345#125—22no—
346#125D4×C2312yes—
347#124—22no—
348#124—21no—
349#124D4×C2312yes—
350#123—22no—
351#123—22no—
352#123—22no—
353#123—22no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—21no—
359C22⋊Q8—22no—
360C22⋊Q8C23×C2312yes—
361#121—22no—
362#121—21no—
363#121C23×C2312yes—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8D4×C2312yes—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4D4×C2312yes—
376#117—22no—
377#117—22no—
378#117—21no—
379#117—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4D4×C2311yescharacteristic
385C2×C4⋊C4C23×C2311yescharacteristic
386C2×C4⋊C4C23×C2311yescharacteristic
387C2×C4⋊C4C23×C2311yescharacteristic
388C2×C4⋊C4C23×C2311yescharacteristic
389C2×C42—22no—
390C2×C42D4×C2311yescharacteristic
391C2×C42D4×C2311yescharacteristic
392C2×C42C23×C2311yescharacteristic
393#107—22no—
394#107—22no—
395#106—22no—
396#106—22no—
397C4⋊C8—22no—
398#103—22no—
399#103—22no—
400#102—22no—
401#102—22no—
402C22⋊C8C23×C2312yes—
403C22⋊C8C23×C2312yes—
404C4×C8—22no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95C23×C2311yescharacteristic
414#95C23×C2311yescharacteristic
415C23#1831511yescharacteristic, direct product factor, Sylow
416#41—42no—
417#41—42no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—41no—
423C2×Q8C2×D4×C2312yes—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4C2×D4×C2312yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4C2×D4×C2311yescharacteristic
436C22×C4C2×D4×C2311yescharacteristic
437C22×C4C2×D4×C2311yescharacteristic
438C22×C4C2×D4×C2311yescharacteristic
439C22×C4C2×D4×C2311yescharacteristic
440C22×C4C2×D4×C2311yescharacteristic
441C22×C4C24×C2311yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—42no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4C2×D4×C2312yes—
467C4⋊C4C2×D4×C2312yes—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42C2×D4×C2312yes—
485C42C2×D4×C2312yes—
486C23—22no—
487C23C22×D4×C2311yescharacteristic
488Q8—44no—
489Q8—42no—
490Q8—42no—
491Q8—81no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—81no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—82no—
527C2×C4C23×#12011yescharacteristic
528C2×C4C23×#12011yescharacteristic
529C2×C4C23×#12012yes—
530C2×C4C23×#13512yes—
531C2×C4C22×D4×C2312yes—
532C8—42no—
533C8—42no—
534C22—21no—
535C22—21no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22C2×C23×#12011yescharacteristic
540C22C2×C23×#13511yescharacteristic
541C22C23×#57511yescharacteristic
542C4—22no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—41no—
551C4—41no—
552C4—42no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—81no—
557C2—21no—
558C2—21no—
559C2—82no—
560C2C23×#300211yescharacteristic
561C2C23×#302011yescharacteristic
562C2C23×#302811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#537C22
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×#3232C2semidirect
3. C23×#2634C2semidirect
4. C23×#1617C2semidirect
6. C23×#1615C2semidirect
7. C23×#1611C2semidirect
9. C23×#1609C2semidirect
10. C23×#1521C2semidirect
11. C23×#1461C2semidirect
204. #18315C23direct (N × H)
254. #3232C2×C23semidirect
255. #2634C2×C23semidirect
256. #1617C2×C23semidirect
258. #1615C2×C23semidirect
259. #1611C2×C23semidirect
261. #1609C2×C23semidirect
262. #1521C2×C23semidirect
263. #1461C2×C23semidirect
415. C23#18315direct (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×#32322
C23×#26342
C23×#16172
C23×#16162
C23×#16152
C23×#16112
C23×#16102
C23×#16092
C23×#15212
C23×#14612
#1831523