Finite Group Databasebeta

← C23×#18278 #19165904

Subgroups — C23×#18278 #19165904

Order 5888.

910 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3563C211yesmaximal, characteristic, complemented
2C23×#3110C211yesmaximal, characteristic, complemented
3C23×#3105C211yesmaximal, characteristic
4C23×#2562C211yesmaximal, characteristic, complemented
5C23×#1617C212yesmaximal, complemented
6C23×#1615C212yesmaximal, complemented
7C23×#1599C212yesmaximal, complemented
8C23×#1597C212yesmaximal
9C23×#1509C211yesmaximal, characteristic, complemented
10C23×#1503C211yesmaximal, characteristic, complemented
11C23×#1470C211yesmaximal, characteristic, complemented
12C23×#564—21no—
13C23×#563—21no—
14C23×#557—21no—
15C23×#554—21no—
16C23×#552—21no—
17C23×#552—21no—
18C23×#552—22no—
19C23×#544—21no—
20C23×#544C2212yes—
21C23×#541—21no—
22C23×#541—22no—
23C23×#539C2212yes—
24C23×#533C2211yescharacteristic
25C23×#529C2211yescharacteristic
26C2×C23×#125C2211yescharacteristic
27C2×C4×D4×C23C2211yescharacteristic
28C23×#484C2211yescharacteristic
29C23×#484C2212yes—
30C23×#481C2211yescharacteristic
31C23×#478—21no—
32C23×#474—21no—
33C23×#463—21no—
34C23×#461—21no—
35C23×#450C2211yescharacteristic
36C23×#450C2212yes—
37C23×#447C2211yescharacteristic
38C23×#440—22no—
39C4×SD16×C23—21no—
40C4×D8×C23—21no—
41C23×#432C2211yescharacteristic
42C23×#431C2211yescharacteristic
43C23×#400C2212yes—
44C23×#394C2212yes—
45C23×#388—22no—
46C23×#380—21no—
47C23×#380—21no—
48C4×C4⋊C4×C23C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#328C2212yes—
52C23×#327C2212yes—
53C23×#324C2212yes—
54C23×#324C2212yes—
55C22×D4×C23C2311yescharacteristic
56C23×C4×C23—21no—
57C2×SD16×C23—21no—
58C2×SD16×C23—22no—
59C2×D8×C23—21no—
60C23×C4⋊Q8—21no—
61C23×C4⋊Q8—22no—
62C23×C4⋊Q8—22no—
63C23×C4⋊Q8D411yescharacteristic
64C23×C4⋊Q8D411yescharacteristic
65C23×#126—22no—
66C23×#126—22no—
67C23×#126—22no—
68C23×#126—22no—
69C23×#126—21no—
70C23×#125—22no—
71C23×#125—21no—
72C23×#125—22no—
73C23×#125—22no—
74C23×#125—22no—
75C23×#125—22no—
76C23×#125—22no—
77C23×#125—22no—
78C23×#125—22no—
79C23×#125—21no—
80C23×#125—21no—
81C23×#125—21no—
82C23×#125—21no—
83C23×#125—21no—
84C23×#125D411yescharacteristic
85C23×#124D411yescharacteristic
86C23×#123—22no—
87C23×#123—21no—
88C23×#123—22no—
89C23×C22⋊Q8—21no—
90C23×C22⋊Q8—21no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—21no—
95C23×C22⋊Q8—22no—
96C23×C22⋊Q8C2311yescharacteristic
97C23×C22⋊Q8C2312yes—
98C23×#121C2311yescharacteristic
99C4×Q8×C23—21no—
100C4×Q8×C23—22no—
101C4×Q8×C23—21no—
102C4×Q8×C23—21no—
103C4×Q8×C23D411yescharacteristic
104C4×Q8×C23D412yes—
105C4×D4×C23—21no—
106C4×D4×C23—22no—
107C4×D4×C23—21no—
108C4×D4×C23—21no—
109C4×D4×C23—21no—
110C4×D4×C23—21no—
111C4×D4×C23D411yescharacteristic
112C23×#117—22no—
113C23×#117—21no—
114C23×#117—21no—
115C23×#117—22no—
116C23×#117—21no—
117C2×C4⋊C4×C23—21no—
118C2×C4⋊C4×C23—21no—
119C2×C4⋊C4×C23D411yescharacteristic
120C2×C4⋊C4×C23D412yes—
121C2×C4⋊C4×C23D411yescharacteristic
122C2×C4⋊C4×C23C2311yescharacteristic
123C2×C4⋊C4×C23C2312yes—
124C2×C22⋊C4×C23—22no—
125C2×C22⋊C4×C23—21no—
126C2×C22⋊C4×C23—22no—
127C2×C42×C23—21no—
128C2×C42×C23C2311yescharacteristic
129C23×#107—21no—
130C23×#107—22no—
131C23×#106—21no—
132C23×C4⋊C8—21no—
133C23×C4⋊C8—21no—
134C23×#103—21no—
135C23×#103—22no—
136C23×#102—21no—
137C23×#102—21no—
138C23×#102—21no—
139C23×#102—22no—
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×C23—21no—
149C2×Q8×C23—22no—
150C2×Q8×C23—21no—
151C2×Q8×C23C2×D412yes—
152C2×Q8×C23C2×D411yescharacteristic
153C2×D4×C23—21no—
154C2×D4×C23—21no—
155C2×D4×C23—21no—
156C2×D4×C23—42no—
157C2×D4×C23C2×D411yescharacteristic
158C22×C4×C23—22no—
159C22×C4×C23—21no—
160C22×C4×C23—22no—
161C22×C4×C23—21no—
162C22×C4×C23—42no—
163C22×C4×C23—42no—
164C22×C4×C23—41no—
165C22×C4×C23—41no—
166C22×C4×C23C2×D412yes—
167C22×C4×C23C2×D411yescharacteristic
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D412yes—
170C22×C4×C23C2411yescharacteristic
171SD16×C23—42no—
172SD16×C23—44no—
173D8×C23—42no—
174C2×C8×C23—21no—
175C2×C8×C23—21no—
176C2×C8×C23—22no—
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—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—21no—
190C4⋊C4×C23—22no—
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—22no—
200C4⋊C4×C23—21no—
201C4⋊C4×C23—22no—
202C4⋊C4×C23—21no—
203C4⋊C4×C23—21no—
204C4⋊C4×C23—42no—
205C4⋊C4×C23C2×D411yescharacteristic
206C4⋊C4×C23C2×D411yescharacteristic
207C4⋊C4×C23C2×D412yes—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—22no—
212C22⋊C4×C23—21no—
213C22⋊C4×C23—21no—
214C22⋊C4×C23—21no—
215C22⋊C4×C23—21no—
216C22⋊C4×C23—21no—
217C22⋊C4×C23—44no—
218C22⋊C4×C23—41no—
219C22⋊C4×C23—41no—
220C22⋊C4×C23—82no—
221C42×C23—22no—
222C42×C23—22no—
223C42×C23—22no—
224C42×C23—22no—
225C42×C23—21no—
226C42×C23—81no—
227C42×C23C2×D411yescharacteristic
228C42×C23C2×D411yescharacteristic
229C42×C23C2×D411yescharacteristic
230C42×C23C2×D411yescharacteristic
231#18278C2311yesmaximal, characteristic, direct product factor, Sylow
232C23×C23—22no—
233C23×C23—21no—
234C23×C23—41no—
235C23×C23—41no—
236C23×C23—41no—
237C23×C23—41no—
238C23×C23—41no—
239C23×C23—41no—
240C23×C23C22×D411yescharacteristic
241Q8×C23—41no—
242Q8×C23—42no—
243Q8×C23—42no—
244Q8×C23—82no—
245D4×C23—22no—
246D4×C23—22no—
247D4×C23—41no—
248D4×C23—41no—
249D4×C23—41no—
250D4×C23—41no—
251D4×C23—41no—
252C2×C4×C23—21no—
253C2×C4×C23—21no—
254C2×C4×C23—22no—
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—22no—
262C2×C4×C23—22no—
263C2×C4×C23—22no—
264C2×C4×C23—41no—
265C2×C4×C23—42no—
266C2×C4×C23—42no—
267C2×C4×C23—42no—
268C2×C4×C23—42no—
269C2×C4×C23—41no—
270C2×C4×C23—41no—
271C2×C4×C23—41no—
272C2×C4×C23—42no—
273C2×C4×C23—82no—
274C2×C4×C23—82no—
275C2×C4×C23—82no—
276C2×C4×C23#12011yescharacteristic
277C2×C4×C23#12011yescharacteristic
278C2×C4×C23#12011yescharacteristic
279C2×C4×C23#12011yescharacteristic
280C2×C4×C23C22×D411yescharacteristic
281C2×C4×C23C22×D411yescharacteristic
282C8×C23—41no—
283C8×C23—41no—
284C8×C23—42no—
285#3563C2×C2311yescharacteristic, complemented
286#3110C2×C2311yescharacteristic, complemented
287#3105C2×C2311yescharacteristic
288#2562C2×C2311yescharacteristic, complemented
289#1617C2×C2312yescomplemented
290#1615C2×C2312yescomplemented
291#1599C2×C2312yescomplemented
292#1597C2×C2312yes—
293#1509C2×C2311yescharacteristic, complemented
294#1503C2×C2311yescharacteristic, complemented
295#1470C2×C2311yescharacteristic, complemented
296C22×C23—21no—
297C22×C23—21no—
298C22×C23—42no—
299C22×C23—42no—
300C22×C23—41no—
301C22×C23—41no—
302C22×C23—41no—
303C22×C23—42no—
304C22×C23—81no—
305C22×C23—81no—
306C22×C23—81no—
307C22×C23—81no—
308C22×C23C2×#12011yescharacteristic
309C22×C23#57711yescharacteristic
310C22×C23#57811yescharacteristic
311C4×C23—21no—
312C4×C23—21no—
313C4×C23—21no—
314C4×C23—21no—
315C4×C23—21no—
316C4×C23—42no—
317C4×C23—41no—
318C4×C23—41no—
319C4×C23—42no—
320C4×C23—42no—
321C4×C23—42no—
322C4×C23—41no—
323C4×C23—82no—
324C4×C23—81no—
325C4×C23#45412yes—
326#564—21no—
327#563—21no—
328#557—21no—
329#554—21no—
330#552—21no—
331#552—21no—
332#552—22no—
333#544—21no—
334#544C22×C2312yes—
335#541—21no—
336#541—22no—
337#539C22×C2312yes—
338#533C22×C2311yescharacteristic
339#529C22×C2311yescharacteristic
340C2×#125C22×C2311yescharacteristic
341C2×C4×D4C22×C2311yescharacteristic
342#484C22×C2311yescharacteristic
343#484C22×C2312yes—
344#481C22×C2311yescharacteristic
345#478—21no—
346#474—21no—
347#463—21no—
348#461—21no—
349#450C22×C2311yescharacteristic
350#450C22×C2312yes—
351#447C22×C2311yescharacteristic
352#440—22no—
353C4×SD16—21no—
354C4×D8—21no—
355#432C22×C2311yescharacteristic
356#431C22×C2311yescharacteristic
357#400C22×C2312yes—
358#394C22×C2312yes—
359#388—22no—
360#380—21no—
361#380—21no—
362C4×C4⋊C4C22×C2311yescharacteristic
363#328C22×C2312yes—
364#328C22×C2312yes—
365#328C22×C2312yes—
366#327C22×C2312yes—
367#324C22×C2312yes—
368#324C22×C2312yes—
369C2×C23—21no—
370C2×C23—21no—
371C2×C23—81no—
372C2×C23—82no—
373C2×C23#301311yescharacteristic
374C2×C23C2×#45411yescharacteristic
375C2×C23#302511yescharacteristic
376C22×D4C23×C2311yescharacteristic
377C23×C4—21no—
378C2×SD16—21no—
379C2×SD16—22no—
380C2×D8—21no—
381C4⋊Q8—21no—
382C4⋊Q8—22no—
383C4⋊Q8—22no—
384C4⋊Q8D4×C2311yescharacteristic
385C4⋊Q8D4×C2311yescharacteristic
386#126—22no—
387#126—22no—
388#126—22no—
389#126—22no—
390#126—21no—
391#125—22no—
392#125—22no—
393#125—22no—
394#125—22no—
395#125—21no—
396#125—22no—
397#125—22no—
398#125—22no—
399#125—22no—
400#125—21no—
401#125—21no—
402#125—21no—
403#125—21no—
404#125—21no—
405#125D4×C2311yescharacteristic
406#124D4×C2311yescharacteristic
407#123—22no—
408#123—21no—
409#123—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—21no—
416C22⋊Q8—22no—
417C22⋊Q8C23×C2311yescharacteristic
418C22⋊Q8C23×C2312yes—
419#121C23×C2311yescharacteristic
420C4×Q8—21no—
421C4×Q8—22no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8D4×C2311yescharacteristic
425C4×Q8D4×C2312yes—
426C4×D4—21no—
427C4×D4—22no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4D4×C2311yescharacteristic
433#117—22no—
434#117—21no—
435#117—21no—
436#117—22no—
437#117—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4D4×C2311yescharacteristic
441C2×C4⋊C4D4×C2312yes—
442C2×C4⋊C4D4×C2311yescharacteristic
443C2×C4⋊C4C23×C2311yescharacteristic
444C2×C4⋊C4C23×C2312yes—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—22no—
447C2×C22⋊C4—22no—
448C2×C42—21no—
449C2×C42C23×C2311yescharacteristic
450#107—21no—
451#107—22no—
452#106—21no—
453C4⋊C8—21no—
454C4⋊C8—21no—
455#103—21no—
456#103—22no—
457#102—21no—
458#102—21no—
459#102—21no—
460#102—22no—
461C22⋊C8C23×C2312yes—
462C22⋊C8C23×C2311yescharacteristic
463C22⋊C8C23×C2311yescharacteristic
464#97—21no—
465C4×C8—21no—
466#95—22no—
467#95—22no—
468#95C23×C2312yes—
469C23#1827811yescharacteristic, direct product factor, Sylow
470C24—21no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8C2×D4×C2312yes—
474C2×Q8C2×D4×C2311yescharacteristic
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—42no—
479C2×D4C2×D4×C2311yescharacteristic
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4C2×D4×C2312yes—
489C22×C4C2×D4×C2311yescharacteristic
490C22×C4C2×D4×C2311yescharacteristic
491C22×C4C2×D4×C2312yes—
492C22×C4C24×C2311yescharacteristic
493SD16—42no—
494SD16—44no—
495D8—42no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4C2×D4×C2311yescharacteristic
528C4⋊C4C2×D4×C2311yescharacteristic
529C4⋊C4C2×D4×C2312yes—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—44no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—82no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—21no—
548C42—81no—
549C42C2×D4×C2311yescharacteristic
550C42C2×D4×C2311yescharacteristic
551C42C2×D4×C2311yescharacteristic
552C42C2×D4×C2311yescharacteristic
553C23—22no—
554C23—21no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C22×D4×C2311yescharacteristic
562Q8—41no—
563Q8—42no—
564Q8—42no—
565Q8—82no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—41no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4C23×#12011yescharacteristic
598C2×C4C23×#12011yescharacteristic
599C2×C4C23×#12011yescharacteristic
600C2×C4C23×#12011yescharacteristic
601C2×C4C22×D4×C2311yescharacteristic
602C2×C4C22×D4×C2311yescharacteristic
603C8—41no—
604C8—41no—
605C8—42no—
606C22—21no—
607C22—21no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22C2×C23×#12011yescharacteristic
619C22C23×#57711yescharacteristic
620C22C23×#57811yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—82no—
634C4—81no—
635C4C23×#45412yes—
636C2—21no—
637C2—21no—
638C2—82no—
639C2—81no—
640C2C23×#301311yescharacteristic
641C2C2×C23×#45411yescharacteristic
642C2C23×#302511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#447C22
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
1. C23×#3563C2semidirect
2. C23×#3110C2semidirect
4. C23×#2562C2semidirect
5. C23×#1617C2semidirect
6. C23×#1615C2semidirect
7. C23×#1599C2semidirect
9. C23×#1509C2semidirect
10. C23×#1503C2semidirect
11. C23×#1470C2semidirect
231. #18278C23direct (N × H)
285. #3563C2×C23semidirect
286. #3110C2×C23semidirect
288. #2562C2×C23semidirect
289. #1617C2×C23semidirect
290. #1615C2×C23semidirect
291. #1599C2×C23semidirect
293. #1509C2×C23semidirect
294. #1503C2×C23semidirect
295. #1470C2×C23semidirect
469. C23#18278direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35632
C23×#31102
C23×#31052
C23×#25622
C23×#16172
C23×#16152
C23×#15992
C23×#15972
C23×#15092
C23×#15032
C23×#14702
#1827823