Finite Group Databasebeta

← C23×#18579 #19166205

Subgroups — C23×#18579 #19166205

Order 5888.

932 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3520C211yesmaximal, characteristic
2C23×#3238C211yesmaximal, characteristic, complemented
3C23×#3139C211yesmaximal, characteristic, complemented
4C2×C23×#476C211yesmaximal, characteristic, complemented
5C23×#1638C212yesmaximal, complemented
6C23×#1637C212yesmaximal, complemented
7C23×#1635C212yesmaximal, complemented
8C23×#1635C212yesmaximal, complemented
9C23×#1552C211yesmaximal, characteristic, complemented
10C23×#1490C211yesmaximal, characteristic, complemented
11C23×#1483C211yesmaximal, characteristic, complemented
12C23×#556C2211yescharacteristic
13C23×#554—21no—
14C23×#552—21no—
15C23×#551—22no—
16D4×Q8×C23C2211yescharacteristic
17C23×#548—21no—
18C23×#547C2211yescharacteristic
19C23×#547C2211yescharacteristic
20C23×#546—21no—
21C23×#542—21no—
22C23×#539—22no—
23C23×#533C2211yescharacteristic
24C2×C23×C4⋊Q8C2211yescharacteristic
25C23×#529C2211yescharacteristic
26C23×#525—21no—
27C2×C23×C22⋊Q8—21no—
28C23×#520—21no—
29C2×C4×D4×C23C2211yescharacteristic
30C23×#501—21no—
31C23×#491—22no—
32C23×#489—21no—
33C23×#487C2212yescomplemented
34C23×#487C2212yescomplemented
35C23×#483—21no—
36C23×#480—21no—
37C23×#477C2212yescomplemented
38C23×#476—21no—
39C23×#476—21no—
40C23×#476C2212yes—
41C23×#476C2212yes—
42C23×#476C2212yescomplemented
43C23×#432C2211yescharacteristic
44C2×C23×#106—21no—
45C2×C23×C4⋊C8C2211yescharacteristic
46C23×#419—22no—
47C2×C23×#102—21no—
48C4×M16×C23C2211yescharacteristic
49C23×#332C2212yescomplemented
50C23×#332C2212yescomplemented
51C23×Q8⋊C8C2212yescomplemented
52C23×D4⋊C8C2212yes—
53C23×D4⋊C8C2212yes—
54C23×D4⋊C8C2212yescomplemented
55C2×#41×C23—21no—
56C2×#41×C23—21no—
57C22×Q8×C23—21no—
58C22×Q8×C23—21no—
59C22×D4×C23D411yescharacteristic
60C23×C4×C23—21no—
61C2×M16×C23—21no—
62C22×C8×C23—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8—21no—
66C23×C4⋊Q8—21no—
67C23×C4⋊Q8C2312yes—
68C23×C4⋊Q8C2311yescharacteristic
69C23×C4⋊Q8C2311yescharacteristic
70C23×#126—22no—
71C23×#126—22no—
72C23×#126—21no—
73C23×#125—22no—
74C23×#125D411yescharacteristic
75C23×#125D412yes—
76C23×#124—22no—
77C23×#124—21no—
78C23×#124D411yescharacteristic
79C23×#123—22no—
80C23×#123—22no—
81C23×#123—21no—
82C23×#123—22no—
83C23×#123—22no—
84C23×C22⋊Q8—22no—
85C23×C22⋊Q8—22no—
86C23×C22⋊Q8—22no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—21no—
90C23×C22⋊Q8—21no—
91C23×C22⋊Q8—21no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—21no—
94C23×C22⋊Q8D411yescharacteristic
95C23×#121—22no—
96C23×#121—21no—
97C23×#121—21no—
98C23×#121D411yescharacteristic
99C23×#120—22no—
100C4×Q8×C23—21no—
101C4×Q8×C23—21no—
102C4×Q8×C23—21no—
103C4×Q8×C23C2311yescharacteristic
104C4×D4×C23—22no—
105C4×D4×C23—22no—
106C4×D4×C23—22no—
107C4×D4×C23—21no—
108C4×D4×C23—21no—
109C4×D4×C23—21no—
110C4×D4×C23—21no—
111C4×D4×C23D411yescharacteristic
112C4×D4×C23D411yescharacteristic
113C4×D4×C23C2311yescharacteristic
114C4×D4×C23C2311yescharacteristic
115C4×D4×C23C2311yescharacteristic
116C23×#117—22no—
117C23×#117—21no—
118C23×#117—21no—
119C23×#117D412yes—
120C2×C4⋊C4×C23—22no—
121C2×C4⋊C4×C23—21no—
122C2×C4⋊C4×C23—21no—
123C2×C4⋊C4×C23—21no—
124C2×C4⋊C4×C23—21no—
125C2×C4⋊C4×C23D411yescharacteristic
126C2×C22⋊C4×C23—22no—
127C2×C22⋊C4×C23—21no—
128C2×C42×C23C2311yescharacteristic
129C23×#107—22no—
130C23×#106—22no—
131C23×#106—22no—
132C23×#106—22no—
133C23×C4⋊C8—21no—
134C23×C4⋊C8C2312yes—
135C23×C4⋊C8C2312yes—
136C23×#103—22no—
137C23×#103—24no—
138C23×#103—24no—
139C23×#102—22no—
140C23×#102—22no—
141C23×#102—22no—
142C23×C22⋊C8—21no—
143C23×#97—21no—
144C23×#97—21no—
145C4×C8×C23C2312yes—
146C24×C23—21no—
147#41×C23—42no—
148#41×C23—42no—
149#41×C23—82no—
150C2×Q8×C23—22no—
151C2×Q8×C23—21no—
152C2×Q8×C23—21no—
153C2×Q8×C23—22no—
154C2×Q8×C23—44no—
155C2×Q8×C23—41no—
156C2×Q8×C23—82no—
157C2×Q8×C23C2×D411yescharacteristic
158C2×D4×C23—22no—
159C2×D4×C23—22no—
160C2×D4×C23—21no—
161C2×D4×C23—22no—
162C2×D4×C23—21no—
163C2×D4×C23—41no—
164C2×D4×C23C2×D411yescharacteristic
165C2×D4×C23C2×D411yescharacteristic
166C2×D4×C23C2×D411yescharacteristic
167C22×C4×C23—22no—
168C22×C4×C23—22no—
169C22×C4×C23—22no—
170C22×C4×C23—21no—
171C22×C4×C23—21no—
172C22×C4×C23—21no—
173C22×C4×C23—21no—
174C22×C4×C23—21no—
175C22×C4×C23—21no—
176C22×C4×C23—41no—
177C22×C4×C23—82no—
178C22×C4×C23C2×D411yescharacteristic
179C22×C4×C23C2×D411yescharacteristic
180C22×C4×C23C2×D411yescharacteristic
181M16×C23—42no—
182C2×C8×C23—22no—
183C2×C8×C23—22no—
184C2×C8×C23—22no—
185C2×C8×C23—42no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—22no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—21no—
198C4⋊C4×C23—21no—
199C4⋊C4×C23—22no—
200C4⋊C4×C23—22no—
201C4⋊C4×C23—41no—
202C4⋊C4×C23—41no—
203C4⋊C4×C23C2×D412yes—
204C4⋊C4×C23C2×D412yes—
205C4⋊C4×C23C2×D411yescharacteristic
206C4⋊C4×C23C2×D411yescharacteristic
207C4⋊C4×C23C2×D411yescharacteristic
208C4⋊C4×C23C2×D411yescharacteristic
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—22no—
212C22⋊C4×C23—22no—
213C22⋊C4×C23—22no—
214C22⋊C4×C23—21no—
215C22⋊C4×C23—22no—
216C22⋊C4×C23—21no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—21no—
219C22⋊C4×C23—21no—
220C22⋊C4×C23—21no—
221C22⋊C4×C23—21no—
222C22⋊C4×C23—21no—
223C22⋊C4×C23—21no—
224C42×C23—22no—
225C42×C23—22no—
226C42×C23—21no—
227C42×C23C2×D411yescharacteristic
228C42×C23C2×D411yescharacteristic
229C42×C23C2×D411yescharacteristic
230C42×C23C2411yescharacteristic
231#18579C2311yesmaximal, characteristic, direct product factor, Sylow
232C23×C23—21no—
233C23×C23—21no—
234C23×C23—21no—
235C23×C23—42no—
236C23×C23—42no—
237C23×C23—81no—
238C23×C23#12011yescharacteristic
239Q8×C23—22no—
240Q8×C23—44no—
241Q8×C23—41no—
242Q8×C23—41no—
243Q8×C23—42no—
244Q8×C23—82no—
245Q8×C23—82no—
246D4×C23—21no—
247D4×C23—21no—
248D4×C23—21no—
249D4×C23—22no—
250D4×C23—42no—
251D4×C23—44no—
252D4×C23—42no—
253D4×C23—82no—
254D4×C23#13712yes—
255C2×C4×C23—22no—
256C2×C4×C23—21no—
257C2×C4×C23—21no—
258C2×C4×C23—22no—
259C2×C4×C23—21no—
260C2×C4×C23—22no—
261C2×C4×C23—21no—
262C2×C4×C23—21no—
263C2×C4×C23—22no—
264C2×C4×C23—21no—
265C2×C4×C23—21no—
266C2×C4×C23—44no—
267C2×C4×C23—41no—
268C2×C4×C23—41no—
269C2×C4×C23—42no—
270C2×C4×C23—42no—
271C2×C4×C23—42no—
272C2×C4×C23—81no—
273C2×C4×C23—81no—
274C2×C4×C23—81no—
275C2×C4×C23—82no—
276C2×C4×C23—82no—
277C2×C4×C23—82no—
278C2×C4×C23#12011yescharacteristic
279C2×C4×C23#12011yescharacteristic
280C2×C4×C23#12011yescharacteristic
281C2×C4×C23C22×D411yescharacteristic
282C2×C4×C23C22×D411yescharacteristic
283C2×C4×C23C22×D411yescharacteristic
284C8×C23—42no—
285C8×C23—42no—
286C8×C23—42no—
287#3520C2×C2311yescharacteristic
288#3238C2×C2311yescharacteristic, complemented
289#3139C2×C2311yescharacteristic, complemented
290C2×#476C2×C2311yescharacteristic, complemented
291#1638C2×C2312yescomplemented
292#1637C2×C2312yescomplemented
293#1635C2×C2312yescomplemented
294#1635C2×C2312yescomplemented
295#1552C2×C2311yescharacteristic, complemented
296#1490C2×C2311yescharacteristic, complemented
297#1483C2×C2311yescharacteristic, complemented
298C22×C23—21no—
299C22×C23—21no—
300C22×C23—22no—
301C22×C23—41no—
302C22×C23—41no—
303C22×C23—41no—
304C22×C23—41no—
305C22×C23—41no—
306C22×C23—41no—
307C22×C23—41no—
308C22×C23—41no—
309C22×C23—84no—
310C22×C23#45412yes—
311C22×C23C2×#12011yescharacteristic
312C4×C23—21no—
313C4×C23—21no—
314C4×C23—21no—
315C4×C23—21no—
316C4×C23—41no—
317C4×C23—42no—
318C4×C23—41no—
319C4×C23—41no—
320C4×C23—42no—
321C4×C23—41no—
322C4×C23—41no—
323C4×C23—82no—
324C4×C23C2×#13711yescharacteristic
325C4×C23#57511yescharacteristic
326C4×C23#57712yes—
327#556C22×C2311yescharacteristic
328#554—21no—
329#552—21no—
330#551—22no—
331D4×Q8C22×C2311yescharacteristic
332#548—21no—
333#547C22×C2311yescharacteristic
334#547C22×C2311yescharacteristic
335#546—21no—
336#542—21no—
337#539—22no—
338#533C22×C2311yescharacteristic
339C2×C4⋊Q8C22×C2311yescharacteristic
340#529C22×C2311yescharacteristic
341#525—21no—
342C2×C22⋊Q8—21no—
343#520—21no—
344C2×C4×D4C22×C2311yescharacteristic
345#501—21no—
346#491—22no—
347#489—21no—
348#487C22×C2312yescomplemented
349#487C22×C2312yescomplemented
350#483—21no—
351#480—21no—
352#477C22×C2312yescomplemented
353#476—21no—
354#476—21no—
355#476C22×C2312yes—
356#476C22×C2312yes—
357#476C22×C2312yescomplemented
358#432C22×C2311yescharacteristic
359C2×#106—21no—
360C2×C4⋊C8C22×C2311yescharacteristic
361#419—22no—
362C2×#102—21no—
363C4×M16C22×C2311yescharacteristic
364#332C22×C2312yescomplemented
365#332C22×C2312yescomplemented
366Q8⋊C8C22×C2312yescomplemented
367D4⋊C8C22×C2312yes—
368D4⋊C8C22×C2312yes—
369D4⋊C8C22×C2312yescomplemented
370C2×C23—22no—
371C2×C23—42no—
372C2×C23—81no—
373C2×C23—81no—
374C2×C23#300711yescharacteristic
375C2×C23#300911yescharacteristic
376C2×C23C2×#45411yescharacteristic
377C2×#41—21no—
378C2×#41—21no—
379C22×Q8—21no—
380C22×Q8—21no—
381C22×D4D4×C2311yescharacteristic
382C23×C4—21no—
383C2×M16—21no—
384C22×C8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8C23×C2312yes—
390C4⋊Q8C23×C2311yescharacteristic
391C4⋊Q8C23×C2311yescharacteristic
392#126—22no—
393#126—22no—
394#126—21no—
395#125—22no—
396#125D4×C2311yescharacteristic
397#125D4×C2312yes—
398#124—22no—
399#124—21no—
400#124D4×C2311yescharacteristic
401#123—22no—
402#123—22no—
403#123—21no—
404#123—22no—
405#123—22no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—21no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—22no—
415C22⋊Q8—21no—
416C22⋊Q8D4×C2311yescharacteristic
417#121—22no—
418#121—21no—
419#121—21no—
420#121D4×C2311yescharacteristic
421#120—22no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8C23×C2311yescharacteristic
426C4×D4—22no—
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4D4×C2311yescharacteristic
434C4×D4D4×C2311yescharacteristic
435C4×D4C23×C2311yescharacteristic
436C4×D4C23×C2311yescharacteristic
437C4×D4C23×C2311yescharacteristic
438#117—22no—
439#117—21no—
440#117—21no—
441#117D4×C2312yes—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4D4×C2311yescharacteristic
448C2×C22⋊C4—22no—
449C2×C22⋊C4—21no—
450C2×C42C23×C2311yescharacteristic
451#107—22no—
452#106—22no—
453#106—22no—
454#106—22no—
455C4⋊C8—21no—
456C4⋊C8C23×C2312yes—
457C4⋊C8C23×C2312yes—
458#103—22no—
459#103—24no—
460#103—24no—
461#102—22no—
462#102—22no—
463#102—22no—
464C22⋊C8—21no—
465#97—21no—
466#97—21no—
467C4×C8C23×C2312yes—
468C23#1857911yescharacteristic, direct product factor, Sylow
469C24—21no—
470#41—42no—
471#41—42no—
472#41—82no—
473C2×Q8—22no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—22no—
477C2×Q8—44no—
478C2×Q8—41no—
479C2×Q8—82no—
480C2×Q8C2×D4×C2311yescharacteristic
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—41no—
487C2×D4C2×D4×C2311yescharacteristic
488C2×D4C2×D4×C2311yescharacteristic
489C2×D4C2×D4×C2311yescharacteristic
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—41no—
500C22×C4—82no—
501C22×C4C2×D4×C2311yescharacteristic
502C22×C4C2×D4×C2311yescharacteristic
503C22×C4C2×D4×C2311yescharacteristic
504M16—42no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—42no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4C2×D4×C2312yes—
527C4⋊C4C2×D4×C2312yes—
528C4⋊C4C2×D4×C2311yescharacteristic
529C4⋊C4C2×D4×C2311yescharacteristic
530C4⋊C4C2×D4×C2311yescharacteristic
531C4⋊C4C2×D4×C2311yescharacteristic
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42C2×D4×C2311yescharacteristic
551C42C2×D4×C2311yescharacteristic
552C42C2×D4×C2311yescharacteristic
553C42C24×C2311yescharacteristic
554C23—21no—
555C23—21no—
556C23—21no—
557C23—42no—
558C23—42no—
559C23—81no—
560C23C23×#12011yescharacteristic
561Q8—22no—
562Q8—44no—
563Q8—41no—
564Q8—41no—
565Q8—42no—
566Q8—82no—
567Q8—82no—
568D4—21no—
569D4—21no—
570D4—21no—
571D4—22no—
572D4—42no—
573D4—44no—
574D4—42no—
575D4—82no—
576D4C23×#13712yes—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4C23×#12011yescharacteristic
601C2×C4C23×#12011yescharacteristic
602C2×C4C23×#12011yescharacteristic
603C2×C4C22×D4×C2311yescharacteristic
604C2×C4C22×D4×C2311yescharacteristic
605C2×C4C22×D4×C2311yescharacteristic
606C8—42no—
607C8—42no—
608C8—42no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—84no—
621C22C23×#45412yes—
622C22C2×C23×#12011yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—82no—
635C4C2×C23×#13711yescharacteristic
636C4C23×#57511yescharacteristic
637C4C23×#57712yes—
638C2—22no—
639C2—42no—
640C2—81no—
641C2—81no—
642C2C23×#300711yescharacteristic
643C2C23×#300911yescharacteristic
644C2C2×C23×#45411yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3520C2
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×#3139C2semidirect
4. C2×C23×#476C2semidirect
5. C23×#1638C2semidirect
6. C23×#1637C2semidirect
7. C23×#1635C2semidirect
8. C23×#1635C2semidirect
9. C23×#1552C2semidirect
10. C23×#1490C2semidirect
11. C23×#1483C2semidirect
33. C23×#487C22semidirect
34. C23×#487C22semidirect
37. C23×#477C22semidirect
42. C23×#476C22semidirect
49. C23×#332C22semidirect
50. C23×#332C22semidirect
51. C23×Q8⋊C8C22semidirect
54. C23×D4⋊C8C22semidirect
231. #18579C23direct (N × H)
288. #3238C2×C23semidirect
289. #3139C2×C23semidirect
290. C2×#476C2×C23semidirect
291. #1638C2×C23semidirect
292. #1637C2×C23semidirect
293. #1635C2×C23semidirect
294. #1635C2×C23semidirect
295. #1552C2×C23semidirect
296. #1490C2×C23semidirect
297. #1483C2×C23semidirect
348. #487C22×C23semidirect
349. #487C22×C23semidirect
352. #477C22×C23semidirect
357. #476C22×C23semidirect
364. #332C22×C23semidirect
365. #332C22×C23semidirect
366. Q8⋊C8C22×C23semidirect
369. D4⋊C8C22×C23semidirect
468. C23#18579direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35202
C23×#32382
C23×#31392
C2×C23×#4762
C23×#16382
C23×#16372
C23×#16352
C23×#15522
C23×#14902
C23×#14832
#1857923