Finite Group Databasebeta

← C23×#18837 #19166463

Subgroups — C23×#18837 #19166463

Order 5888.

876 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3515C211yesmaximal, characteristic
2C23×#3238C211yesmaximal, characteristic, complemented
3C23×#3209C211yesmaximal, characteristic, complemented
4C23×#3079C211yesmaximal, characteristic, complemented
5C23×#1636C212yesmaximal, complemented
6C23×#1635C212yesmaximal, complemented
7C23×#1632C212yesmaximal, complemented
8C23×#1632C212yesmaximal, complemented
9C23×#1559C211yesmaximal, characteristic, complemented
10C23×#1495C211yesmaximal, characteristic, complemented
11C23×#1488C211yesmaximal, characteristic, complemented
12C23×#557C2211yescharacteristic
13C23×#554C2211yescharacteristic
14C23×#554C2211yescharacteristic
15C23×#553—21no—
16C23×#552—21no—
17C23×#552—22no—
18C23×#552—21no—
19C23×#547—21no—
20C23×#547C2211yescharacteristic
21C23×#541—21no—
22C23×#539—22no—
23C23×#533C2211yescharacteristic
24C2×C23×C4⋊Q8C2211yescharacteristic
25C23×#529—21no—
26C23×#529C2211yescharacteristic
27C2×C23×#125—21no—
28C23×#520—21no—
29C4×#41×C23C2211yescharacteristic
30C23×#501—21no—
31C23×#494C2212yes—
32C23×#489—21no—
33C23×#487C2212yes—
34C23×#483—21no—
35C23×#480—21no—
36C23×#477C2212yes—
37C23×#477C2212yes—
38C23×#476C2212yes—
39C23×#476C2212yes—
40C23×#475—21no—
41C23×#474—21no—
42C23×#462—22no—
43C23×#451—22no—
44C23×#432C2211yescharacteristic
45C23×#431C2211yescharacteristic
46C23×#428—21no—
47C23×#423C2211yescharacteristic
48C23×#417—21no—
49C23×#333C2212yes—
50C23×#332C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×D4⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C2×#41×C23—21no—
56C2×#41×C23D411yescharacteristic
57C22×Q8×C23—21no—
58C2×Q16×C23—24no—
59C2×M16×C23—21no—
60C23×C4⋊Q8—21no—
61C23×C4⋊Q8—21no—
62C23×C4⋊Q8—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8C2312yes—
65C23×C4⋊Q8C2311yescharacteristic
66C23×C4⋊Q8C2311yescharacteristic
67C23×#126—22no—
68C23×#126—22no—
69C23×#126—22no—
70C23×#126—21no—
71C23×#125—22no—
72C23×#125—22no—
73C23×#125—22no—
74C23×#125—21no—
75C23×#125—21no—
76C23×#125—22no—
77C23×#125—21no—
78C23×#125—21no—
79C23×#125D411yescharacteristic
80C23×#124D411yescharacteristic
81C23×#123—22no—
82C23×#123—22no—
83C23×#123—22no—
84C23×#123—21no—
85C23×#123—22no—
86C23×C22⋊Q8—21no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—22no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—21no—
91C23×C22⋊Q8—21no—
92C23×C22⋊Q8D411yescharacteristic
93C23×C22⋊Q8D412yes—
94C23×#121—22no—
95C23×#121—21no—
96C23×#121D411yescharacteristic
97C4×Q8×C23—21no—
98C4×Q8×C23—21no—
99C4×Q8×C23—21no—
100C4×Q8×C23D412yes—
101C4×Q8×C23D411yescharacteristic
102C4×Q8×C23C2311yescharacteristic
103C4×Q8×C23C2311yescharacteristic
104C4×D4×C23—22no—
105C4×D4×C23—22no—
106C4×D4×C23—21no—
107C4×D4×C23—21no—
108C4×D4×C23—21no—
109C4×D4×C23—21no—
110C4×D4×C23—21no—
111C4×D4×C23D411yescharacteristic
112C4×D4×C23C2311yescharacteristic
113C4×D4×C23C2311yescharacteristic
114C23×#117—22no—
115C23×#117—22no—
116C23×#117—21no—
117C23×#117—21no—
118C23×#117—21no—
119C23×#117D411yescharacteristic
120C2×C4⋊C4×C23—22no—
121C2×C4⋊C4×C23—22no—
122C2×C4⋊C4×C23—21no—
123C2×C4⋊C4×C23—21no—
124C2×C4⋊C4×C23—21no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23—21no—
127C2×C42×C23—21no—
128C2×C42×C23C2311yescharacteristic
129C23×#107—22no—
130C23×#107—22no—
131C23×#106—22no—
132C23×#106—22no—
133C23×C4⋊C8—21no—
134C23×C4⋊C8—21no—
135C23×C4⋊C8C2312yes—
136C23×C4⋊C8C2312yes—
137C23×#103—22no—
138C23×#103—22no—
139C23×#103—24no—
140C23×#102—22no—
141C23×#102—22no—
142C23×C22⋊C8—21no—
143C23×C22⋊C8—21no—
144C23×#97—21no—
145C4×C8×C23C2312yes—
146#41×C23—22no—
147#41×C23—41no—
148#41×C23—42no—
149#41×C23—42no—
150C2×Q8×C23—21no—
151C2×Q8×C23—22no—
152C2×Q8×C23—41no—
153C2×Q8×C23—82no—
154C2×Q8×C23C2×D411yescharacteristic
155C2×Q8×C23C2×D411yescharacteristic
156C2×Q8×C23C2×D412yes—
157C2×D4×C23—22no—
158C2×D4×C23—21no—
159C2×D4×C23—21no—
160C2×D4×C23C2×D411yescharacteristic
161C2×D4×C23C2×D411yescharacteristic
162C22×C4×C23—22no—
163C22×C4×C23—22no—
164C22×C4×C23—22no—
165C22×C4×C23—21no—
166C22×C4×C23—21no—
167C22×C4×C23—21no—
168C22×C4×C23—21no—
169C22×C4×C23—41no—
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2×D411yescharacteristic
172C22×C4×C23C2×D411yescharacteristic
173Q16×C23—84no—
174M16×C23—42no—
175C2×C8×C23—22no—
176C2×C8×C23—22no—
177C2×C8×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×C23—21no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—22no—
199C4⋊C4×C23—41no—
200C4⋊C4×C23—41no—
201C4⋊C4×C23C2×D412yes—
202C4⋊C4×C23C2×D411yescharacteristic
203C4⋊C4×C23C2×D411yescharacteristic
204C4⋊C4×C23C2×D411yescharacteristic
205C4⋊C4×C23C2×D411yescharacteristic
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—21no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—21no—
213C22⋊C4×C23—21no—
214C22⋊C4×C23—21no—
215C22⋊C4×C23—21no—
216C22⋊C4×C23—21no—
217C22⋊C4×C23—21no—
218C42×C23—22no—
219C42×C23—21no—
220C42×C23—21no—
221C42×C23—22no—
222C42×C23—21no—
223C42×C23—21no—
224C42×C23C2×D411yescharacteristic
225C42×C23C2×D411yescharacteristic
226C42×C23C2×D411yescharacteristic
227C42×C23C2411yescharacteristic
228#18837C2311yesmaximal, characteristic, direct product factor, Sylow
229C23×C23—21no—
230C23×C23—21no—
231C23×C23#12011yescharacteristic
232Q8×C23—21no—
233Q8×C23—21no—
234Q8×C23—22no—
235Q8×C23—42no—
236Q8×C23—44no—
237Q8×C23—41no—
238Q8×C23—41no—
239Q8×C23—82no—
240D4×C23—21no—
241D4×C23—22no—
242D4×C23—42no—
243D4×C23—44no—
244D4×C23—81no—
245D4×C23#13712yes—
246C2×C4×C23—21no—
247C2×C4×C23—21no—
248C2×C4×C23—22no—
249C2×C4×C23—21no—
250C2×C4×C23—21no—
251C2×C4×C23—21no—
252C2×C4×C23—21no—
253C2×C4×C23—22no—
254C2×C4×C23—22no—
255C2×C4×C23—21no—
256C2×C4×C23—21no—
257C2×C4×C23—41no—
258C2×C4×C23—42no—
259C2×C4×C23—44no—
260C2×C4×C23—41no—
261C2×C4×C23—41no—
262C2×C4×C23—42no—
263C2×C4×C23—42no—
264C2×C4×C23—81no—
265C2×C4×C23—82no—
266C2×C4×C23—82no—
267C2×C4×C23—82no—
268C2×C4×C23#12011yescharacteristic
269C2×C4×C23#12011yescharacteristic
270C2×C4×C23#12011yescharacteristic
271C2×C4×C23C22×D411yescharacteristic
272C2×C4×C23C22×D411yescharacteristic
273C2×C4×C23C22×D411yescharacteristic
274C8×C23—42no—
275C8×C23—42no—
276C8×C23—42no—
277#3515C2×C2311yescharacteristic
278#3238C2×C2311yescharacteristic, complemented
279#3209C2×C2311yescharacteristic, complemented
280#3079C2×C2311yescharacteristic, complemented
281#1636C2×C2312yescomplemented
282#1635C2×C2312yescomplemented
283#1632C2×C2312yescomplemented
284#1632C2×C2312yescomplemented
285#1559C2×C2311yescharacteristic, complemented
286#1495C2×C2311yescharacteristic, complemented
287#1488C2×C2311yescharacteristic, complemented
288C22×C23—21no—
289C22×C23—22no—
290C22×C23—21no—
291C22×C23—21no—
292C22×C23—41no—
293C22×C23—41no—
294C22×C23—41no—
295C22×C23—41no—
296C22×C23—41no—
297C22×C23C2×#12011yescharacteristic
298C4×C23—21no—
299C4×C23—21no—
300C4×C23—21no—
301C4×C23—21no—
302C4×C23—41no—
303C4×C23—41no—
304C4×C23—42no—
305C4×C23—41no—
306C4×C23—41no—
307C4×C23—41no—
308C4×C23—41no—
309C4×C23—42no—
310C4×C23—82no—
311C4×C23C2×#13711yescharacteristic
312C4×C23#57511yescharacteristic
313C4×C23#57812yes—
314#557C22×C2311yescharacteristic
315#554C22×C2311yescharacteristic
316#554C22×C2311yescharacteristic
317#553—21no—
318#552—21no—
319#552—22no—
320#552—21no—
321#547—21no—
322#547C22×C2311yescharacteristic
323#541—21no—
324#539—22no—
325#533C22×C2311yescharacteristic
326C2×C4⋊Q8C22×C2311yescharacteristic
327#529—21no—
328#529C22×C2311yescharacteristic
329C2×#125—21no—
330#520—21no—
331C4×#41C22×C2311yescharacteristic
332#501—21no—
333#494C22×C2312yes—
334#489—21no—
335#487C22×C2312yes—
336#483—21no—
337#480—21no—
338#477C22×C2312yes—
339#477C22×C2312yes—
340#476C22×C2312yes—
341#476C22×C2312yes—
342#475—21no—
343#474—21no—
344#462—22no—
345#451—22no—
346#432C22×C2311yescharacteristic
347#431C22×C2311yescharacteristic
348#428—21no—
349#423C22×C2311yescharacteristic
350#417—21no—
351#333C22×C2312yes—
352#332C22×C2312yes—
353Q8⋊C8C22×C2312yes—
354Q8⋊C8C22×C2312yes—
355D4⋊C8C22×C2312yes—
356D4⋊C8C22×C2312yes—
357C2×C23—41no—
358C2×C23—42no—
359C2×C23—81no—
360C2×C23#300711yescharacteristic
361C2×C23#301111yescharacteristic
362C2×C23#302111yescharacteristic
363C2×#41—21no—
364C2×#41D4×C2311yescharacteristic
365C22×Q8—21no—
366C2×Q16—24no—
367C2×M16—21no—
368C4⋊Q8—21no—
369C4⋊Q8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8C23×C2312yes—
373C4⋊Q8C23×C2311yescharacteristic
374C4⋊Q8C23×C2311yescharacteristic
375#126—22no—
376#126—22no—
377#126—22no—
378#126—21no—
379#125—22no—
380#125—22no—
381#125—22no—
382#125—21no—
383#125—21no—
384#125—22no—
385#125—21no—
386#125—21no—
387#125D4×C2311yescharacteristic
388#124D4×C2311yescharacteristic
389#123—22no—
390#123—22no—
391#123—22no—
392#123—21no—
393#123—22no—
394C22⋊Q8—21no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8D4×C2311yescharacteristic
401C22⋊Q8D4×C2312yes—
402#121—22no—
403#121—21no—
404#121D4×C2311yescharacteristic
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8D4×C2311yescharacteristic
409C4×Q8D4×C2312yes—
410C4×Q8C23×C2311yescharacteristic
411C4×Q8C23×C2311yescharacteristic
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4D4×C2311yescharacteristic
420C4×D4C23×C2311yescharacteristic
421C4×D4C23×C2311yescharacteristic
422#117—22no—
423#117—22no—
424#117—21no—
425#117—21no—
426#117—21no—
427#117D4×C2311yescharacteristic
428C2×C4⋊C4—22no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C42—21no—
436C2×C42C23×C2311yescharacteristic
437#107—22no—
438#107—22no—
439#106—22no—
440#106—22no—
441C4⋊C8—21no—
442C4⋊C8—21no—
443C4⋊C8C23×C2312yes—
444C4⋊C8C23×C2312yes—
445#103—22no—
446#103—22no—
447#103—24no—
448#102—22no—
449#102—22no—
450C22⋊C8—21no—
451C22⋊C8—21no—
452#97—21no—
453C4×C8C23×C2312yes—
454C23#1883711yescharacteristic, direct product factor, Sylow
455#41—22no—
456#41—41no—
457#41—42no—
458#41—42no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—41no—
462C2×Q8—82no—
463C2×Q8C2×D4×C2311yescharacteristic
464C2×Q8C2×D4×C2311yescharacteristic
465C2×Q8C2×D4×C2312yes—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4C2×D4×C2311yescharacteristic
470C2×D4C2×D4×C2311yescharacteristic
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479C22×C4C2×D4×C2311yescharacteristic
480C22×C4C2×D4×C2311yescharacteristic
481C22×C4C2×D4×C2311yescharacteristic
482Q16—84no—
483M16—42no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4C2×D4×C2312yes—
511C4⋊C4C2×D4×C2311yescharacteristic
512C4⋊C4C2×D4×C2311yescharacteristic
513C4⋊C4C2×D4×C2311yescharacteristic
514C4⋊C4C2×D4×C2311yescharacteristic
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C42—22no—
528C42—21no—
529C42—21no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42C2×D4×C2311yescharacteristic
534C42C2×D4×C2311yescharacteristic
535C42C2×D4×C2311yescharacteristic
536C42C24×C2311yescharacteristic
537C23—21no—
538C23—21no—
539C23C23×#12011yescharacteristic
540Q8—21no—
541Q8—21no—
542Q8—22no—
543Q8—42no—
544Q8—44no—
545Q8—41no—
546Q8—41no—
547Q8—82no—
548D4—21no—
549D4—22no—
550D4—42no—
551D4—44no—
552D4—81no—
553D4C23×#13712yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4C23×#12011yescharacteristic
577C2×C4C23×#12011yescharacteristic
578C2×C4C23×#12011yescharacteristic
579C2×C4C22×D4×C2311yescharacteristic
580C2×C4C22×D4×C2311yescharacteristic
581C2×C4C22×D4×C2311yescharacteristic
582C8—42no—
583C8—42no—
584C8—42no—
585C22—21no—
586C22—22no—
587C22—21no—
588C22—21no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22C2×C23×#12011yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—82no—
608C4C2×C23×#13711yescharacteristic
609C4C23×#57511yescharacteristic
610C4C23×#57812yes—
611C2—41no—
612C2—42no—
613C2—81no—
614C2C23×#300711yescharacteristic
615C2C23×#301111yescharacteristic
616C2C23×#302111yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3515C2
LCSG ▷ C42 ▷ 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×#3238C2semidirect
3. C23×#3209C2semidirect
4. C23×#3079C2semidirect
5. C23×#1636C2semidirect
6. C23×#1635C2semidirect
7. C23×#1632C2semidirect
8. C23×#1632C2semidirect
9. C23×#1559C2semidirect
10. C23×#1495C2semidirect
11. C23×#1488C2semidirect
228. #18837C23direct (N × H)
278. #3238C2×C23semidirect
279. #3209C2×C23semidirect
280. #3079C2×C23semidirect
281. #1636C2×C23semidirect
282. #1635C2×C23semidirect
283. #1632C2×C23semidirect
284. #1632C2×C23semidirect
285. #1559C2×C23semidirect
286. #1495C2×C23semidirect
287. #1488C2×C23semidirect
454. C23#18837direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35152
C23×#32382
C23×#32092
C23×#30792
C23×#16362
C23×#16352
C23×#16322
C23×#15592
C23×#14952
C23×#14882
#1883723