Finite Group Databasebeta

← C23×#18336 #19165962

Subgroups — C23×#18336 #19165962

Order 5888.

1002 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3459C211yesmaximal, characteristic
2C23×#3218C212yesmaximal, complemented
3C23×#2563C211yesmaximal, characteristic, complemented
4C23×#1606C214yesmaximal, complemented
5C23×#1602C212yesmaximal, complemented
6C23×#1601C212yesmaximal, complemented
7C23×#1509C212yesmaximal, complemented
8C23×#1466C211yesmaximal, characteristic, complemented
9C23×#543C2211yescharacteristic
10C23×#539—22no—
11C23×#539—21no—
12C23×#538—22no—
13C23×#538C2212yes—
14C23×#534—21no—
15C23×#534C2211yescharacteristic
16C2×C23×C4⋊Q8C2211yescharacteristic, complemented
17C23×#529—22no—
18C23×#529C2212yes—
19C23×#525—21no—
20C23×#525—22no—
21C2×C4×D4×C23—21no—
22C2×C4×D4×C23C2211yescharacteristic
23C23×#497—22no—
24C23×#491—22no—
25C23×#462—22no—
26C23×#461—22no—
27C23×#451C2214yescomplemented
28C23×#448C2214yes—
29C23×#432C2212yescomplemented
30C23×#393C2212yes—
31C23×#393C2212yescomplemented
32C23×#390—21no—
33C23×#382—21no—
34C23×#380—21no—
35C23×#380—21no—
36C4×C4⋊C4×C23C2211yescharacteristic, complemented
37C23×#328C2214yescomplemented
38C23×#327C2214yes—
39C23×#323C2212yes—
40C23×#323C2212yescomplemented
41C2×#41×C23—21no—
42C22×Q8×C23C2312yes—
43C22×D4×C23—21no—
44C22×D4×C23D411yescharacteristic
45C23×C4×C23—21no—
46C23×C4×C23D411yescharacteristic
47C23×C4×C23D411yescharacteristic
48C2×Q16×C23—24no—
49C2×SD16×C23—24no—
50C23×C4⋊Q8—22no—
51C23×C4⋊Q8—22no—
52C23×#126—22no—
53C23×#126—22no—
54C23×#126—22no—
55C23×#125—22no—
56C23×#125D412yes—
57C23×#124—22no—
58C23×#124D412yes—
59C23×#123—22no—
60C23×#123—22no—
61C23×#123—22no—
62C23×#123—22no—
63C23×#123—22no—
64C23×#123—22no—
65C23×C22⋊Q8—21no—
66C23×C22⋊Q8—22no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8—21no—
69C23×C22⋊Q8—22no—
70C23×C22⋊Q8C2312yes—
71C23×#121—22no—
72C23×#121—21no—
73C23×#121—22no—
74C23×#121—22no—
75C23×#121—21no—
76C23×#121C2312yes—
77C23×#120—22no—
78C23×#120—22no—
79C23×#120—22no—
80C4×Q8×C23D412yes—
81C4×D4×C23—22no—
82C4×D4×C23—22no—
83C4×D4×C23—22no—
84C4×D4×C23—22no—
85C4×D4×C23—22no—
86C4×D4×C23—22no—
87C4×D4×C23—22no—
88C4×D4×C23D412yes—
89C23×#117—21no—
90C23×#117—22no—
91C2×C4⋊C4×C23—21no—
92C2×C4⋊C4×C23—21no—
93C2×C4⋊C4×C23—21no—
94C2×C4⋊C4×C23—21no—
95C2×C4⋊C4×C23D411yescharacteristic
96C2×C22⋊C4×C23—21no—
97C2×C22⋊C4×C23—21no—
98C2×C22⋊C4×C23—22no—
99C2×C22⋊C4×C23—22no—
100C2×C22⋊C4×C23C2311yescharacteristic
101C2×C22⋊C4×C23C2311yescharacteristic
102C2×C42×C23—21no—
103C2×C42×C23—21no—
104C2×C42×C23C2311yescharacteristic
105C23×C4⋊C8—22no—
106C23×#103—24no—
107C23×#103—24no—
108C23×C22⋊C8C2314yes—
109C23×#97—22no—
110C23×#95—22no—
111C23×#95—22no—
112C23×#95C2312yes—
113C24×C23—21no—
114C24×C23C2×D411yescharacteristic
115C24×C23C2×D411yescharacteristic
116#41×C23—82no—
117C2×Q8×C23—21no—
118C2×Q8×C23—24no—
119C2×Q8×C23—82no—
120C2×Q8×C23C2×D412yes—
121C2×D4×C23—22no—
122C2×D4×C23—22no—
123C2×D4×C23—22no—
124C2×D4×C23—22no—
125C2×D4×C23—21no—
126C2×D4×C23—22no—
127C2×D4×C23—22no—
128C2×D4×C23—41no—
129C2×D4×C23—42no—
130C2×D4×C23—42no—
131C2×D4×C23C2×D412yes—
132C22×C4×C23—21no—
133C22×C4×C23—21no—
134C22×C4×C23—21no—
135C22×C4×C23—22no—
136C22×C4×C23—21no—
137C22×C4×C23—21no—
138C22×C4×C23—22no—
139C22×C4×C23—21no—
140C22×C4×C23—21no—
141C22×C4×C23—22no—
142C22×C4×C23—22no—
143C22×C4×C23—42no—
144C22×C4×C23—42no—
145C22×C4×C23—82no—
146C22×C4×C23C2×D411yescharacteristic
147C22×C4×C23C2×D411yescharacteristic
148C22×C4×C23C2×D411yescharacteristic
149C22×C4×C23C2×D411yescharacteristic
150C22×C4×C23C2411yescharacteristic
151Q16×C23—84no—
152SD16×C23—84no—
153C2×C8×C23—24no—
154C4⋊C4×C23—22no—
155C4⋊C4×C23—21no—
156C4⋊C4×C23—21no—
157C4⋊C4×C23—22no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—22no—
161C4⋊C4×C23—22no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—42no—
164C4⋊C4×C23—42no—
165C4⋊C4×C23—41no—
166C4⋊C4×C23—41no—
167C4⋊C4×C23C2×D412yes—
168C4⋊C4×C23C2×D412yes—
169C22⋊C4×C23—22no—
170C22⋊C4×C23—22no—
171C22⋊C4×C23—22no—
172C22⋊C4×C23—22no—
173C22⋊C4×C23—22no—
174C22⋊C4×C23—22no—
175C22⋊C4×C23—22no—
176C22⋊C4×C23—22no—
177C22⋊C4×C23—22no—
178C22⋊C4×C23—22no—
179C22⋊C4×C23—22no—
180C22⋊C4×C23—22no—
181C22⋊C4×C23—82no—
182C22⋊C4×C23—82no—
183C42×C23—22no—
184C42×C23—22no—
185C42×C23—42no—
186C42×C23C2×D412yes—
187C42×C23C2×D412yes—
188#18336C2311yesmaximal, characteristic, direct product factor, Sylow
189C23×C23—21no—
190C23×C23—21no—
191C23×C23—21no—
192C23×C23—22no—
193C23×C23—21no—
194C23×C23—21no—
195C23×C23—41no—
196C23×C23—41no—
197C23×C23—41no—
198C23×C23—41no—
199C23×C23—42no—
200C23×C23—81no—
201C23×C23—81no—
202C23×C23#13712yes—
203C23×C23C22×D411yescharacteristic
204Q8×C23—42no—
205Q8×C23—44no—
206Q8×C23—44no—
207Q8×C23—81no—
208D4×C23—44no—
209D4×C23—42no—
210D4×C23—44no—
211D4×C23—42no—
212D4×C23—42no—
213D4×C23—82no—
214D4×C23—81no—
215D4×C23—82no—
216C2×C4×C23—21no—
217C2×C4×C23—21no—
218C2×C4×C23—22no—
219C2×C4×C23—22no—
220C2×C4×C23—22no—
221C2×C4×C23—21no—
222C2×C4×C23—21no—
223C2×C4×C23—22no—
224C2×C4×C23—41no—
225C2×C4×C23—42no—
226C2×C4×C23—42no—
227C2×C4×C23—42no—
228C2×C4×C23—44no—
229C2×C4×C23—41no—
230C2×C4×C23—41no—
231C2×C4×C23—42no—
232C2×C4×C23—42no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—42no—
236C2×C4×C23—42no—
237C2×C4×C23—81no—
238C2×C4×C23—81no—
239C2×C4×C23—81no—
240C2×C4×C23—82no—
241C2×C4×C23—81no—
242C2×C4×C23#12012yes—
243C2×C4×C23#12011yescharacteristic
244C2×C4×C23#12011yescharacteristic
245C2×C4×C23C22×D412yes—
246C8×C23—44no—
247#3459C2×C2311yescharacteristic
248#3218C2×C2312yescomplemented
249#2563C2×C2311yescharacteristic, complemented
250#1606C2×C2314yescomplemented
251#1602C2×C2312yescomplemented
252#1601C2×C2312yescomplemented
253#1509C2×C2312yescomplemented
254#1466C2×C2311yescharacteristic, complemented
255C22×C23—21no—
256C22×C23—21no—
257C22×C23—22no—
258C22×C23—42no—
259C22×C23—42no—
260C22×C23—41no—
261C22×C23—41no—
262C22×C23—41no—
263C22×C23—42no—
264C22×C23—41no—
265C22×C23—42no—
266C22×C23—42no—
267C22×C23—41no—
268C22×C23—81no—
269C22×C23—81no—
270C22×C23—84no—
271C22×C23C2×#12011yescharacteristic
272C22×C23C2×#13711yescharacteristic
273C22×C23#57511yescharacteristic
274C4×C23—22no—
275C4×C23—21no—
276C4×C23—22no—
277C4×C23—42no—
278C4×C23—41no—
279C4×C23—41no—
280C4×C23—42no—
281C4×C23—42no—
282C4×C23—82no—
283C4×C23—82no—
284C4×C23#45512yes—
285#543C22×C2311yescharacteristic
286#539—22no—
287#539—21no—
288#538—22no—
289#538C22×C2312yes—
290#534—21no—
291#534C22×C2311yescharacteristic
292C2×C4⋊Q8C22×C2311yescharacteristic, complemented
293#529—22no—
294#529C22×C2312yes—
295#525—21no—
296#525—22no—
297C2×C4×D4—21no—
298C2×C4×D4C22×C2311yescharacteristic
299#497—22no—
300#491—22no—
301#462—22no—
302#461—22no—
303#451C22×C2314yescomplemented
304#448C22×C2314yes—
305#432C22×C2312yescomplemented
306#393C22×C2312yes—
307#393C22×C2312yescomplemented
308#390—21no—
309#382—21no—
310#380—21no—
311#380—21no—
312C4×C4⋊C4C22×C2311yescharacteristic, complemented
313#328C22×C2314yescomplemented
314#327C22×C2314yes—
315#323C22×C2312yes—
316#323C22×C2312yescomplemented
317C2×C23—21no—
318C2×C23—21no—
319C2×C23—42no—
320C2×C23—81no—
321C2×C23—82no—
322C2×C23#300711yescharacteristic
323C2×C23C2×#45511yescharacteristic
324C2×C23#302711yescharacteristic
325C2×#41—21no—
326C22×Q8C23×C2312yes—
327C22×D4—21no—
328C22×D4D4×C2311yescharacteristic
329C23×C4—21no—
330C23×C4D4×C2311yescharacteristic
331C23×C4D4×C2311yescharacteristic
332C2×Q16—24no—
333C2×SD16—24no—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336#126—22no—
337#126—22no—
338#126—22no—
339#125—22no—
340#125D4×C2312yes—
341#124—22no—
342#124D4×C2312yes—
343#123—22no—
344#123—22no—
345#123—22no—
346#123—22no—
347#123—22no—
348#123—22no—
349C22⋊Q8—21no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—21no—
353C22⋊Q8—22no—
354C22⋊Q8C23×C2312yes—
355#121—22no—
356#121—21no—
357#121—22no—
358#121—22no—
359#121—21no—
360#121C23×C2312yes—
361#120—22no—
362#120—22no—
363#120—22no—
364C4×Q8D4×C2312yes—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4D4×C2312yes—
373#117—21no—
374#117—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4D4×C2311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4C23×C2311yescharacteristic
385C2×C22⋊C4C23×C2311yescharacteristic
386C2×C42—21no—
387C2×C42—21no—
388C2×C42C23×C2311yescharacteristic
389C4⋊C8—22no—
390#103—24no—
391#103—24no—
392C22⋊C8C23×C2314yes—
393#97—22no—
394#95—22no—
395#95—22no—
396#95C23×C2312yes—
397C23#1833611yescharacteristic, direct product factor, Sylow
398C24—21no—
399C24C2×D4×C2311yescharacteristic
400C24C2×D4×C2311yescharacteristic
401#41—82no—
402C2×Q8—21no—
403C2×Q8—24no—
404C2×Q8—82no—
405C2×Q8C2×D4×C2312yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—41no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4C2×D4×C2312yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4C2×D4×C2311yescharacteristic
432C22×C4C2×D4×C2311yescharacteristic
433C22×C4C2×D4×C2311yescharacteristic
434C22×C4C2×D4×C2311yescharacteristic
435C22×C4C24×C2311yescharacteristic
436Q16—84no—
437SD16—84no—
438C2×C8—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4C2×D4×C2312yes—
453C4⋊C4C2×D4×C2312yes—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C42—22no—
469C42—22no—
470C42—42no—
471C42C2×D4×C2312yes—
472C42C2×D4×C2312yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—21no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—81no—
485C23—81no—
486C23C23×#13712yes—
487C23C22×D4×C2311yescharacteristic
488Q8—42no—
489Q8—44no—
490Q8—44no—
491Q8—81no—
492D4—44no—
493D4—42no—
494D4—44no—
495D4—42no—
496D4—42no—
497D4—81no—
498D4—82no—
499D4—82no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4C23×#12012yes—
527C2×C4C23×#12011yescharacteristic
528C2×C4C23×#12011yescharacteristic
529C2×C4C22×D4×C2312yes—
530C8—44no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—42no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22—84no—
547C22C2×C23×#12011yescharacteristic
548C22C2×C23×#13711yescharacteristic
549C22C23×#57511yescharacteristic
550C4—22no—
551C4—21no—
552C4—22no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4C23×#45512yes—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—81no—
565C2—82no—
566C2C23×#300711yescharacteristic
567C2C2×C23×#45511yescharacteristic
568C2C23×#302711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3459C2
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×#3218C2semidirect
3. C23×#2563C2semidirect
4. C23×#1606C2semidirect
5. C23×#1602C2semidirect
6. C23×#1601C2semidirect
7. C23×#1509C2semidirect
8. C23×#1466C2semidirect
16. C2×C23×C4⋊Q8C22semidirect
27. C23×#451C22semidirect
29. C23×#432C22semidirect
31. C23×#393C22semidirect
36. C4×C4⋊C4×C23C22semidirect
37. C23×#328C22semidirect
40. C23×#323C22semidirect
188. #18336C23direct (N × H)
248. #3218C2×C23semidirect
249. #2563C2×C23semidirect
250. #1606C2×C23semidirect
251. #1602C2×C23semidirect
252. #1601C2×C23semidirect
253. #1509C2×C23semidirect
254. #1466C2×C23semidirect
292. C2×C4⋊Q8C22×C23semidirect
303. #451C22×C23semidirect
305. #432C22×C23semidirect
307. #393C22×C23semidirect
312. C4×C4⋊C4C22×C23semidirect
313. #328C22×C23semidirect
316. #323C22×C23semidirect
397. C23#18336direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34592
C23×#32182
C23×#25632
C23×#16062
C23×#16022
C23×#16012
C23×#15092
C23×#14662
#1833623