Finite Group Databasebeta

← C23×#18275 #19165901

Subgroups — C23×#18275 #19165901

Order 5888.

938 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3561C211yesmaximal, characteristic
2C23×#3113C211yesmaximal, characteristic, complemented
3C23×#3108C211yesmaximal, characteristic, complemented
4C23×#2592C211yesmaximal, characteristic, complemented
5C23×#1612C212yesmaximal, complemented
6C23×#1608C212yesmaximal, complemented
7C23×#1606C212yesmaximal, complemented
8C23×#1602C212yesmaximal, complemented
9C23×#1509C211yesmaximal, characteristic, complemented
10C23×#1503C211yesmaximal, characteristic, complemented
11C23×#1469C211yesmaximal, characteristic, complemented
12C23×#563—21no—
13C23×#562—21no—
14C23×#556—21no—
15C23×#551—22no—
16D4×Q8×C23—21no—
17C23×#548—21no—
18C23×#546—21no—
19C23×#543C2212yes—
20C23×#542—21no—
21C23×#542—22no—
22C23×#538C2212yes—
23C23×#536—21no—
24C23×#533C2211yescharacteristic
25C23×#529C2211yescharacteristic
26C2×C23×#124C2211yescharacteristic
27C2×C4×Q8×C23C2211yescharacteristic
28C23×#483C2211yescharacteristic
29C23×#483C2212yes—
30C23×#480C2211yescharacteristic
31C23×#479—21no—
32C23×#475—21no—
33C23×#464—21no—
34C23×#462—21no—
35C23×#451C2211yescharacteristic
36C23×#451C2212yes—
37C23×#448C2211yescharacteristic
38C23×#441—22no—
39C4×Q16×C23—21no—
40C4×SD16×C23—21no—
41C23×#432C2211yescharacteristic
42C23×#431C2211yescharacteristic
43C23×#397C2212yes—
44C23×#393C2212yes—
45C23×#391—21no—
46C23×#390—21no—
47C23×#388—22no—
48C4×C4⋊C4×C23C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#328C2212yes—
52C23×#327C2212yes—
53C23×#323C2212yes—
54C23×#323C2212yes—
55C2×#41×C23—21no—
56C22×Q8×C23—21no—
57C22×Q8×C23D411yescharacteristic
58C22×Q8×C23C2311yescharacteristic
59C22×D4×C23D411yescharacteristic
60C2×Q16×C23—21no—
61C2×Q16×C23—22no—
62C2×SD16×C23—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—22no—
65C23×C4⋊Q8D411yescharacteristic
66C23×C4⋊Q8D411yescharacteristic
67C23×#126—22no—
68C23×#126—22no—
69C23×#126—22no—
70C23×#126—21no—
71C23×#125—21no—
72C23×#125D411yescharacteristic
73C23×#124—22no—
74C23×#124—21no—
75C23×#124—22no—
76C23×#124—21no—
77C23×#124—21no—
78C23×#124—21no—
79C23×#124—21no—
80C23×#124D411yescharacteristic
81C23×#123—22no—
82C23×#123—22no—
83C23×#123—22no—
84C23×#123—21no—
85C23×#123—22no—
86C23×C22⋊Q8—21no—
87C23×C22⋊Q8—22no—
88C23×C22⋊Q8—22no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—21no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8C2311yescharacteristic
96C23×C22⋊Q8C2312yes—
97C23×#121—21no—
98C23×#121—22no—
99C23×#121—22no—
100C23×#121C2311yescharacteristic
101C23×#120—22no—
102C4×Q8×C23—21no—
103C4×Q8×C23—21no—
104C4×Q8×C23—21no—
105C4×Q8×C23—21no—
106C4×Q8×C23—21no—
107C4×Q8×C23—21no—
108C4×Q8×C23D411yescharacteristic
109C4×Q8×C23D412yes—
110C4×D4×C23—22no—
111C4×D4×C23—21no—
112C4×D4×C23—22no—
113C4×D4×C23—21no—
114C4×D4×C23D411yescharacteristic
115C23×#117—22no—
116C23×#117—21no—
117C23×#117—21no—
118C2×C4⋊C4×C23—21no—
119C2×C4⋊C4×C23—21no—
120C2×C4⋊C4×C23C2311yescharacteristic
121C2×C22⋊C4×C23—22no—
122C2×C22⋊C4×C23—22no—
123C2×C22⋊C4×C23—22no—
124C2×C22⋊C4×C23D412yes—
125C2×C22⋊C4×C23C2312yes—
126C2×C42×C23—21no—
127C2×C42×C23—21no—
128C2×C42×C23C2311yescharacteristic
129C23×#107—21no—
130C23×#106—21no—
131C23×#106—22no—
132C23×C4⋊C8—21no—
133C23×C4⋊C8—21no—
134C23×#103—21no—
135C23×#103—22no—
136C23×#103—21no—
137C23×#103—21no—
138C23×#103—22no—
139C23×#102—21no—
140C23×C22⋊C8C2312yes—
141C23×C22⋊C8C2311yescharacteristic
142C23×C22⋊C8C2311yescharacteristic
143C23×#97—21no—
144C4×C8×C23—21no—
145C23×#95—22no—
146C23×#95—22no—
147C23×#95C2312yes—
148C24×C23C2×D412yes—
149#41×C23—82no—
150C2×Q8×C23—22no—
151C2×Q8×C23—21no—
152C2×Q8×C23—21no—
153C2×Q8×C23—21no—
154C2×Q8×C23—21no—
155C2×Q8×C23—21no—
156C2×Q8×C23—42no—
157C2×Q8×C23—42no—
158C2×Q8×C23—82no—
159C2×Q8×C23C2×D411yescharacteristic
160C2×Q8×C23C2×D412yes—
161C2×D4×C23—21no—
162C2×D4×C23—22no—
163C2×D4×C23—22no—
164C2×D4×C23—21no—
165C2×D4×C23—21no—
166C2×D4×C23—42no—
167C2×D4×C23C2×D411yescharacteristic
168C22×C4×C23—21no—
169C22×C4×C23—22no—
170C22×C4×C23—22no—
171C22×C4×C23—22no—
172C22×C4×C23—21no—
173C22×C4×C23—21no—
174C22×C4×C23—22no—
175C22×C4×C23C2×D411yescharacteristic
176C22×C4×C23C2×D411yescharacteristic
177C22×C4×C23C2×D412yes—
178C22×C4×C23C2411yescharacteristic
179Q16×C23—42no—
180Q16×C23—44no—
181SD16×C23—42no—
182C2×C8×C23—21no—
183C2×C8×C23—21no—
184C2×C8×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—21no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—22no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—21no—
199C4⋊C4×C23—21no—
200C4⋊C4×C23—42no—
201C4⋊C4×C23—41no—
202C4⋊C4×C23—41no—
203C4⋊C4×C23C2×D411yescharacteristic
204C4⋊C4×C23C2×D412yes—
205C4⋊C4×C23C2×D411yescharacteristic
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—22no—
213C22⋊C4×C23—22no—
214C22⋊C4×C23—22no—
215C22⋊C4×C23—21no—
216C22⋊C4×C23—21no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—22no—
219C22⋊C4×C23—22no—
220C22⋊C4×C23—21no—
221C22⋊C4×C23—44no—
222C22⋊C4×C23—82no—
223C42×C23—22no—
224C42×C23—22no—
225C42×C23—21no—
226C42×C23—41no—
227C42×C23—41no—
228C42×C23—81no—
229C42×C23C2×D411yescharacteristic
230C42×C23C2×D411yescharacteristic
231C42×C23C2×D411yescharacteristic
232C42×C23C2×D411yescharacteristic
233#18275C2311yesmaximal, characteristic, direct product factor, Sylow
234C23×C23—22no—
235C23×C23—21no—
236C23×C23—22no—
237C23×C23—22no—
238C23×C23—42no—
239C23×C23—42no—
240C23×C23C22×D411yescharacteristic
241Q8×C23—22no—
242Q8×C23—22no—
243Q8×C23—41no—
244Q8×C23—42no—
245Q8×C23—41no—
246Q8×C23—41no—
247Q8×C23—42no—
248Q8×C23—42no—
249Q8×C23—81no—
250Q8×C23—82no—
251D4×C23—41no—
252D4×C23—41no—
253D4×C23—41no—
254D4×C23—42no—
255D4×C23—81no—
256D4×C23—82no—
257D4×C23—82no—
258C2×C4×C23—21no—
259C2×C4×C23—21no—
260C2×C4×C23—22no—
261C2×C4×C23—22no—
262C2×C4×C23—21no—
263C2×C4×C23—22no—
264C2×C4×C23—21no—
265C2×C4×C23—22no—
266C2×C4×C23—22no—
267C2×C4×C23—21no—
268C2×C4×C23—41no—
269C2×C4×C23—41no—
270C2×C4×C23—41no—
271C2×C4×C23—42no—
272C2×C4×C23—42no—
273C2×C4×C23—42no—
274C2×C4×C23—42no—
275C2×C4×C23—41no—
276C2×C4×C23—42no—
277C2×C4×C23—41no—
278C2×C4×C23—41no—
279C2×C4×C23—82no—
280C2×C4×C23—82no—
281C2×C4×C23—82no—
282C2×C4×C23#12011yescharacteristic
283C2×C4×C23#12011yescharacteristic
284C2×C4×C23#12011yescharacteristic
285C2×C4×C23#12011yescharacteristic
286C2×C4×C23C22×D411yescharacteristic
287C2×C4×C23C22×D411yescharacteristic
288C8×C23—41no—
289C8×C23—41no—
290C8×C23—42no—
291#3561C2×C2311yescharacteristic
292#3113C2×C2311yescharacteristic, complemented
293#3108C2×C2311yescharacteristic, complemented
294#2592C2×C2311yescharacteristic, complemented
295#1612C2×C2312yescomplemented
296#1608C2×C2312yescomplemented
297#1606C2×C2312yescomplemented
298#1602C2×C2312yescomplemented
299#1509C2×C2311yescharacteristic, complemented
300#1503C2×C2311yescharacteristic, complemented
301#1469C2×C2311yescharacteristic, complemented
302C22×C23—21no—
303C22×C23—21no—
304C22×C23—41no—
305C22×C23—42no—
306C22×C23—41no—
307C22×C23—42no—
308C22×C23—42no—
309C22×C23—41no—
310C22×C23—82no—
311C22×C23—82no—
312C22×C23C2×#12011yescharacteristic
313C22×C23#57711yescharacteristic
314C22×C23#57811yescharacteristic
315C4×C23—21no—
316C4×C23—21no—
317C4×C23—21no—
318C4×C23—21no—
319C4×C23—21no—
320C4×C23—41no—
321C4×C23—41no—
322C4×C23—42no—
323C4×C23—42no—
324C4×C23—42no—
325C4×C23—41no—
326C4×C23—42no—
327C4×C23—82no—
328C4×C23—81no—
329C4×C23#45512yes—
330#563—21no—
331#562—21no—
332#556—21no—
333#551—22no—
334D4×Q8—21no—
335#548—21no—
336#546—21no—
337#543C22×C2312yes—
338#542—21no—
339#542—22no—
340#538C22×C2312yes—
341#536—21no—
342#533C22×C2311yescharacteristic
343#529C22×C2311yescharacteristic
344C2×#124C22×C2311yescharacteristic
345C2×C4×Q8C22×C2311yescharacteristic
346#483C22×C2311yescharacteristic
347#483C22×C2312yes—
348#480C22×C2311yescharacteristic
349#479—21no—
350#475—21no—
351#464—21no—
352#462—21no—
353#451C22×C2311yescharacteristic
354#451C22×C2312yes—
355#448C22×C2311yescharacteristic
356#441—22no—
357C4×Q16—21no—
358C4×SD16—21no—
359#432C22×C2311yescharacteristic
360#431C22×C2311yescharacteristic
361#397C22×C2312yes—
362#393C22×C2312yes—
363#391—21no—
364#390—21no—
365#388—22no—
366C4×C4⋊C4C22×C2311yescharacteristic
367#328C22×C2312yes—
368#328C22×C2312yes—
369#328C22×C2312yes—
370#327C22×C2312yes—
371#323C22×C2312yes—
372#323C22×C2312yes—
373C2×C23—21no—
374C2×C23—21no—
375C2×C23—81no—
376C2×C23—82no—
377C2×C23#301311yescharacteristic
378C2×C23C2×#45511yescharacteristic
379C2×C23#302511yescharacteristic
380C2×#41—21no—
381C22×Q8—21no—
382C22×Q8D4×C2311yescharacteristic
383C22×Q8C23×C2311yescharacteristic
384C22×D4D4×C2311yescharacteristic
385C2×Q16—21no—
386C2×Q16—22no—
387C2×SD16—21no—
388C4⋊Q8—21no—
389C4⋊Q8—22no—
390C4⋊Q8D4×C2311yescharacteristic
391C4⋊Q8D4×C2311yescharacteristic
392#126—22no—
393#126—22no—
394#126—22no—
395#126—21no—
396#125—21no—
397#125D4×C2311yescharacteristic
398#124—21no—
399#124—22no—
400#124—22no—
401#124—21no—
402#124—21no—
403#124—21no—
404#124—21no—
405#124D4×C2311yescharacteristic
406#123—22no—
407#123—22no—
408#123—22no—
409#123—21no—
410#123—22no—
411C22⋊Q8—22no—
412C22⋊Q8—21no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8—21no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8C23×C2311yescharacteristic
421C22⋊Q8C23×C2312yes—
422#121—21no—
423#121—22no—
424#121—22no—
425#121C23×C2311yescharacteristic
426#120—22no—
427C4×Q8—21no—
428C4×Q8—21no—
429C4×Q8—21no—
430C4×Q8—21no—
431C4×Q8—21no—
432C4×Q8—21no—
433C4×Q8D4×C2311yescharacteristic
434C4×Q8D4×C2312yes—
435C4×D4—21no—
436C4×D4—22no—
437C4×D4—21no—
438C4×D4—22no—
439C4×D4D4×C2311yescharacteristic
440#117—22no—
441#117—21no—
442#117—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4C23×C2311yescharacteristic
446C2×C22⋊C4—22no—
447C2×C22⋊C4—22no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4D4×C2312yes—
450C2×C22⋊C4C23×C2312yes—
451C2×C42—21no—
452C2×C42—21no—
453C2×C42C23×C2311yescharacteristic
454#107—21no—
455#106—21no—
456#106—22no—
457C4⋊C8—21no—
458C4⋊C8—21no—
459#103—21no—
460#103—22no—
461#103—21no—
462#103—21no—
463#103—22no—
464#102—21no—
465C22⋊C8C23×C2312yes—
466C22⋊C8C23×C2311yescharacteristic
467C22⋊C8C23×C2311yescharacteristic
468#97—21no—
469C4×C8—21no—
470#95—22no—
471#95—22no—
472#95C23×C2312yes—
473C23#1827511yescharacteristic, direct product factor, Sylow
474C24C2×D4×C2312yes—
475#41—82no—
476C2×Q8—22no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8—82no—
485C2×Q8C2×D4×C2311yescharacteristic
486C2×Q8C2×D4×C2312yes—
487C2×D4—21no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—42no—
493C2×D4C2×D4×C2311yescharacteristic
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4C2×D4×C2311yescharacteristic
502C22×C4C2×D4×C2311yescharacteristic
503C22×C4C2×D4×C2312yes—
504C22×C4C24×C2311yescharacteristic
505Q16—42no—
506Q16—44no—
507SD16—42no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4C2×D4×C2311yescharacteristic
530C4⋊C4C2×D4×C2312yes—
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—22no—
540C22⋊C4—22no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—44no—
548C22⋊C4—82no—
549C42—22no—
550C42—22no—
551C42—21no—
552C42—41no—
553C42—41no—
554C42—81no—
555C42C2×D4×C2311yescharacteristic
556C42C2×D4×C2311yescharacteristic
557C42C2×D4×C2311yescharacteristic
558C42C2×D4×C2311yescharacteristic
559C23—22no—
560C23—21no—
561C23—22no—
562C23—22no—
563C23—42no—
564C23—42no—
565C23C22×D4×C2311yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—41no—
569Q8—42no—
570Q8—41no—
571Q8—41no—
572Q8—42no—
573Q8—42no—
574Q8—81no—
575Q8—82no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—42no—
580D4—81no—
581D4—82no—
582D4—82no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4C23×#12011yescharacteristic
608C2×C4C23×#12011yescharacteristic
609C2×C4C23×#12011yescharacteristic
610C2×C4C23×#12011yescharacteristic
611C2×C4C22×D4×C2311yescharacteristic
612C2×C4C22×D4×C2311yescharacteristic
613C8—41no—
614C8—41no—
615C8—42no—
616C22—21no—
617C22—21no—
618C22—41no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—41no—
624C22—82no—
625C22—82no—
626C22C2×C23×#12011yescharacteristic
627C22C23×#57711yescharacteristic
628C22C23×#57811yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—41no—
640C4—42no—
641C4—82no—
642C4—81no—
643C4C23×#45512yes—
644C2—21no—
645C2—21no—
646C2—82no—
647C2—81no—
648C2C23×#301311yescharacteristic
649C2C2×C23×#45511yescharacteristic
650C2C23×#302511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3561C2
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×#3113C2semidirect
3. C23×#3108C2semidirect
4. C23×#2592C2semidirect
5. C23×#1612C2semidirect
6. C23×#1608C2semidirect
7. C23×#1606C2semidirect
8. C23×#1602C2semidirect
9. C23×#1509C2semidirect
10. C23×#1503C2semidirect
11. C23×#1469C2semidirect
233. #18275C23direct (N × H)
292. #3113C2×C23semidirect
293. #3108C2×C23semidirect
294. #2592C2×C23semidirect
295. #1612C2×C23semidirect
296. #1608C2×C23semidirect
297. #1606C2×C23semidirect
298. #1602C2×C23semidirect
299. #1509C2×C23semidirect
300. #1503C2×C23semidirect
301. #1469C2×C23semidirect
473. C23#18275direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35612
C23×#31132
C23×#31082
C23×#25922
C23×#16122
C23×#16082
C23×#16062
C23×#16022
C23×#15092
C23×#15032
C23×#14692
#1827523