Finite Group Databasebeta

← C23×#18130 #19165756

Subgroups — C23×#18130 #19165756

Order 5888.

964 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3461C211yesmaximal, characteristic, complemented
2C23×#3105C212yesmaximal, complemented
3C23×#2398C211yesmaximal, characteristic, complemented
4C23×#1617C211yesmaximal, characteristic, complemented
5C23×#1616C211yesmaximal, characteristic, complemented
6C23×#1615C212yesmaximal, complemented
7C23×#1599C211yesmaximal, characteristic, complemented
8C23×#1598C211yesmaximal, characteristic
9C23×#1597C212yesmaximal, 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×D4×C23C2211yescharacteristic
26C23×#484C2212yes—
27C23×#481C2212yes—
28C23×#478—22no—
29C23×#463—22no—
30C23×#450C2212yes—
31C23×#447C2212yes—
32C4×SD16×C23—22no—
33C4×D8×C23—22no—
34C23×#431C2212yes—
35C23×#400C2211yescharacteristic
36C23×#400C2211yescharacteristic
37C23×#397—21no—
38C23×#394C2211yescharacteristic
39C23×#394C2211yescharacteristic
40C23×#393—21no—
41C4×C22⋊C4×C23—21no—
42C4×C22⋊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×D4×C23C2311yescharacteristic
54C23×C4×C23—21no—
55C2×SD16×C23—22no—
56C2×D8×C23—22no—
57C23×C4⋊Q8—22no—
58C23×C4⋊Q8—22no—
59C23×#126—22no—
60C23×#126—22no—
61C23×#126—22no—
62C23×#126—22no—
63C23×#125—21no—
64C23×#125—22no—
65C23×#125—22no—
66C23×#125—22no—
67C23×#125D412yes—
68C23×#124—22no—
69C23×#124—21no—
70C23×#124D412yes—
71C23×#123—22no—
72C23×#123—22no—
73C23×#123—22no—
74C23×#123—22no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—22no—
78C23×C22⋊Q8—22no—
79C23×C22⋊Q8—21no—
80C23×C22⋊Q8—22no—
81C23×C22⋊Q8C2312yes—
82C23×#121—22no—
83C23×#121—21no—
84C23×#121C2312yes—
85C4×Q8×C23—21no—
86C4×Q8×C23—21no—
87C4×Q8×C23—21no—
88C4×Q8×C23—22no—
89C4×Q8×C23—22no—
90C4×Q8×C23D412yes—
91C4×D4×C23—21no—
92C4×D4×C23—21no—
93C4×D4×C23—21no—
94C4×D4×C23—22no—
95C4×D4×C23—22no—
96C4×D4×C23—22no—
97C4×D4×C23—22no—
98C4×D4×C23D412yes—
99C23×#117—22no—
100C23×#117—22no—
101C23×#117—21no—
102C23×#117—22no—
103C2×C4⋊C4×C23D411yescharacteristic
104C2×C4⋊C4×C23C2311yescharacteristic
105C2×C4⋊C4×C23C2311yescharacteristic
106C2×C4⋊C4×C23C2311yescharacteristic
107C2×C22⋊C4×C23—21no—
108C2×C22⋊C4×C23—21no—
109C2×C22⋊C4×C23—21no—
110C2×C22⋊C4×C23—21no—
111C2×C22⋊C4×C23—21no—
112C2×C42×C23—22no—
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×#102—22no—
122C23×#102—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
134C24×C23—21no—
135#41×C23—42no—
136#41×C23—42no—
137C2×Q8×C23—21no—
138C2×Q8×C23—21no—
139C2×Q8×C23—22no—
140C2×Q8×C23—22no—
141C2×Q8×C23—41no—
142C2×Q8×C23C2×D412yes—
143C2×D4×C23—21no—
144C2×D4×C23—22no—
145C2×D4×C23—21no—
146C2×D4×C23—22no—
147C2×D4×C23—42no—
148C2×D4×C23C2×D412yes—
149C22×C4×C23—22no—
150C22×C4×C23—22no—
151C22×C4×C23—22no—
152C22×C4×C23—21no—
153C22×C4×C23—21no—
154C22×C4×C23—21no—
155C22×C4×C23—42no—
156C22×C4×C23—42no—
157C22×C4×C23—41no—
158C22×C4×C23—41no—
159C22×C4×C23C2×D411yescharacteristic
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×D411yescharacteristic
162C22×C4×C23C2×D411yescharacteristic
163C22×C4×C23C2×D411yescharacteristic
164C22×C4×C23C2×D411yescharacteristic
165C22×C4×C23C2411yescharacteristic
166SD16×C23—44no—
167D8×C23—44no—
168C2×C8×C23—22no—
169C2×C8×C23—22no—
170C4⋊C4×C23—21no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—21no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23C2×D412yes—
187C4⋊C4×C23C2×D412yes—
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—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—42no—
198C22⋊C4×C23—42no—
199C22⋊C4×C23—42no—
200C22⋊C4×C23—42no—
201C22⋊C4×C23—42no—
202C42×C23—22no—
203C42×C23—22no—
204C42×C23—21no—
205C42×C23—21no—
206C42×C23—21no—
207C42×C23—22no—
208C42×C23—21no—
209C42×C23—42no—
210C42×C23C2×D412yes—
211C42×C23C2×D412yes—
212#18130C2311yesmaximal, characteristic, direct product factor, Sylow
213C23×C23—22no—
214C23×C23—22no—
215C23×C23—41no—
216C23×C23—41no—
217C23×C23—41no—
218C23×C23—41no—
219C23×C23—41no—
220C23×C23—41no—
221C23×C23C22×D411yescharacteristic
222Q8×C23—44no—
223Q8×C23—42no—
224Q8×C23—42no—
225Q8×C23—81no—
226D4×C23—24no—
227D4×C23—42no—
228D4×C23—42no—
229D4×C23—42no—
230D4×C23—42no—
231D4×C23—42no—
232D4×C23—81no—
233C2×C4×C23—21no—
234C2×C4×C23—21no—
235C2×C4×C23—21no—
236C2×C4×C23—21no—
237C2×C4×C23—22no—
238C2×C4×C23—21no—
239C2×C4×C23—22no—
240C2×C4×C23—22no—
241C2×C4×C23—21no—
242C2×C4×C23—21no—
243C2×C4×C23—22no—
244C2×C4×C23—21no—
245C2×C4×C23—21no—
246C2×C4×C23—41no—
247C2×C4×C23—44no—
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—41no—
255C2×C4×C23—41no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—41no—
260C2×C4×C23—82no—
261C2×C4×C23—82no—
262C2×C4×C23—82no—
263C2×C4×C23#12011yescharacteristic
264C2×C4×C23#12012yes—
265C2×C4×C23#12011yescharacteristic
266C2×C4×C23#13512yes—
267C2×C4×C23C22×D412yes—
268C8×C23—42no—
269C8×C23—42no—
270#3461C2×C2311yescharacteristic, complemented
271#3105C2×C2312yescomplemented
272#2398C2×C2311yescharacteristic, complemented
273#1617C2×C2311yescharacteristic, complemented
274#1616C2×C2311yescharacteristic, complemented
275#1615C2×C2312yescomplemented
276#1599C2×C2311yescharacteristic, complemented
277#1598C2×C2311yescharacteristic
278#1597C2×C2312yescomplemented
279#1500C2×C2312yescomplemented
280#1461C2×C2311yescharacteristic, complemented
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—42no—
284C22×C23—42no—
285C22×C23—42no—
286C22×C23—42no—
287C22×C23—42no—
288C22×C23—42no—
289C22×C23—81no—
290C22×C23—81no—
291C22×C23—81no—
292C22×C23—81no—
293C22×C23C2×#12011yescharacteristic
294C22×C23C2×#13511yescharacteristic
295C22×C23#57511yescharacteristic
296C4×C23—22no—
297C4×C23—22no—
298C4×C23—21no—
299C4×C23—22no—
300C4×C23—41no—
301C4×C23—42no—
302C4×C23—42no—
303C4×C23—41no—
304C4×C23—41no—
305C4×C23—42no—
306C4×C23—81no—
307C4×C23—81no—
308C4×C23—81no—
309C4×C23#45412yes—
310#544C22×C2311yescharacteristic
311#542—22no—
312#541—22no—
313#539—21no—
314#539C22×C2312yes—
315#537C22×C2311yescharacteristic
316#536—21no—
317#533—22no—
318#529—22no—
319#529—21no—
320#529C22×C2312yes—
321C4×#41—21no—
322C2×C4×Q8C22×C2311yescharacteristic
323C2×C4×D4C22×C2311yescharacteristic
324#484C22×C2312yes—
325#481C22×C2312yes—
326#478—22no—
327#463—22no—
328#450C22×C2312yes—
329#447C22×C2312yes—
330C4×SD16—22no—
331C4×D8—22no—
332#431C22×C2312yes—
333#400C22×C2311yescharacteristic
334#400C22×C2311yescharacteristic
335#397—21no—
336#394C22×C2311yescharacteristic
337#394C22×C2311yescharacteristic
338#393—21no—
339C4×C22⋊C4—21no—
340C4×C22⋊C4—21no—
341#376C22×C2311yescharacteristic
342#328C22×C2312yes—
343#328C22×C2312yes—
344#327C22×C2312yes—
345#327C22×C2312yes—
346#324C22×C2312yes—
347#324C22×C2311yescharacteristic
348#324C22×C2311yescharacteristic
349C2×C23—21no—
350C2×C23—21no—
351C2×C23—82no—
352C2×C23—82no—
353C2×C23#300211yescharacteristic
354C2×C23C2×#45411yescharacteristic
355C2×C23#302511yescharacteristic
356C2×#41—21no—
357C22×Q8D4×C2311yescharacteristic
358C22×D4C23×C2311yescharacteristic
359C23×C4—21no—
360C2×SD16—22no—
361C2×D8—22no—
362C4⋊Q8—22no—
363C4⋊Q8—22no—
364#126—22no—
365#126—22no—
366#126—22no—
367#126—22no—
368#125—21no—
369#125—22no—
370#125—22no—
371#125—22no—
372#125D4×C2312yes—
373#124—22no—
374#124—21no—
375#124D4×C2312yes—
376#123—22no—
377#123—22no—
378#123—22no—
379#123—22no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8C23×C2312yes—
387#121—22no—
388#121—21no—
389#121C23×C2312yes—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8D4×C2312yes—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4D4×C2312yes—
404#117—22no—
405#117—22no—
406#117—21no—
407#117—22no—
408C2×C4⋊C4D4×C2311yescharacteristic
409C2×C4⋊C4C23×C2311yescharacteristic
410C2×C4⋊C4C23×C2311yescharacteristic
411C2×C4⋊C4C23×C2311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C42—22no—
418C2×C42—21no—
419C2×C42D4×C2311yescharacteristic
420C2×C42D4×C2311yescharacteristic
421C2×C42C23×C2311yescharacteristic
422#107—22no—
423#106—22no—
424C4⋊C8—22no—
425#103—22no—
426#102—22no—
427#102—22no—
428#102—22no—
429C22⋊C8C23×C2312yes—
430C22⋊C8C23×C2312yes—
431C4×C8—22no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95C23×C2311yescharacteristic
438#95C23×C2311yescharacteristic
439C23#1813011yescharacteristic, direct product factor, Sylow
440C24—21no—
441#41—42no—
442#41—42no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—41no—
448C2×Q8C2×D4×C2312yes—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4C2×D4×C2312yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4C2×D4×C2311yescharacteristic
466C22×C4C2×D4×C2311yescharacteristic
467C22×C4C2×D4×C2311yescharacteristic
468C22×C4C2×D4×C2311yescharacteristic
469C22×C4C2×D4×C2311yescharacteristic
470C22×C4C2×D4×C2311yescharacteristic
471C22×C4C24×C2311yescharacteristic
472SD16—44no—
473D8—44no—
474C2×C8—22no—
475C2×C8—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4C2×D4×C2312yes—
493C4⋊C4C2×D4×C2312yes—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C42—22no—
509C42—22no—
510C42—21no—
511C42—22no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42C2×D4×C2312yes—
517C42C2×D4×C2312yes—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C22×D4×C2311yescharacteristic
527Q8—44no—
528Q8—42no—
529Q8—42no—
530Q8—81no—
531D4—24no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—81no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4C23×#12011yescharacteristic
569C2×C4C23×#12012yes—
570C2×C4C23×#12011yescharacteristic
571C2×C4C23×#13512yes—
572C2×C4C22×D4×C2312yes—
573C8—42no—
574C8—42no—
575C22—21no—
576C22—21no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×C23×#12011yescharacteristic
588C22C2×C23×#13511yescharacteristic
589C22C23×#57511yescharacteristic
590C4—22no—
591C4—22no—
592C4—21no—
593C4—22no—
594C4—41no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4C23×#45412yes—
604C2—21no—
605C2—21no—
606C2—82no—
607C2—82no—
608C2C23×#300211yescharacteristic
609C2C2×C23×#45411yescharacteristic
610C2C23×#302511yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutator, Frattini, cosocleC22×C4C24×C23
socleC22×C23C2×#120
prime radicalC23×#1598C2
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
1. C23×#3461C2semidirect
2. C23×#3105C2semidirect
3. C23×#2398C2semidirect
4. C23×#1617C2semidirect
5. C23×#1616C2semidirect
6. C23×#1615C2semidirect
7. C23×#1599C2semidirect
9. C23×#1597C2semidirect
10. C23×#1500C2semidirect
11. C23×#1461C2semidirect
212. #18130C23direct (N × H)
270. #3461C2×C23semidirect
271. #3105C2×C23semidirect
272. #2398C2×C23semidirect
273. #1617C2×C23semidirect
274. #1616C2×C23semidirect
275. #1615C2×C23semidirect
276. #1599C2×C23semidirect
278. #1597C2×C23semidirect
279. #1500C2×C23semidirect
280. #1461C2×C23semidirect
439. C23#18130direct (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×#31052
C23×#23982
C23×#16172
C23×#16162
C23×#16152
C23×#15992
C23×#15982
C23×#15972
C23×#15002
C23×#14612
#1813023