Finite Group Databasebeta

← C23×#18348 #19165974

Subgroups — C23×#18348 #19165974

Order 5888.

1016 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3492C211yesmaximal, characteristic, complemented
2C23×#3194C212yesmaximal, complemented
3C23×#2618C211yesmaximal, characteristic, complemented
4C23×#1611C211yesmaximal, characteristic, complemented
5C23×#1610C211yesmaximal, characteristic, complemented
6C23×#1609C212yesmaximal, complemented
7C23×#1599C211yesmaximal, characteristic, complemented
8C23×#1598C211yesmaximal, characteristic, complemented
9C23×#1597C212yesmaximal, complemented
10C23×#1513C212yesmaximal, complemented
11C23×#1463C211yesmaximal, characteristic, complemented
12C23×#554—22no—
13C23×#553—22no—
14C23×#552—22no—
15C23×#547—22no—
16C23×#544C2211yescharacteristic
17C23×#539—21no—
18C23×#539C2212yes—
19C23×#537C2211yescharacteristic
20C23×#525—21no—
21C23×#525—21no—
22C23×#525C2212yes—
23C2×C23×#123—21no—
24C2×C23×#121C2211yescharacteristic
25C23×#520—21no—
26C22×C4⋊C4×C23C2211yescharacteristic
27C23×#483C2212yes—
28C23×#482—22no—
29C23×#480C2212yes—
30C23×#469—22no—
31C23×#468—22no—
32C23×#450C2212yes—
33C23×#449—22no—
34C23×#447C2212yes—
35C23×#407C2212yes—
36C23×#398C2211yescharacteristic
37C23×#398C2211yescharacteristic
38C23×#396—21no—
39C23×#396—21no—
40C23×#394C2211yescharacteristic
41C23×#394C2211yescharacteristic
42C23×#386—21no—
43C23×#380C2211yescharacteristic
44C23×#328C2212yes—
45C23×#328C2212yes—
46C23×#327C2212yes—
47C23×#327C2212yes—
48C23×#324C2212yes—
49C23×#324C2211yescharacteristic
50C23×#324C2211yescharacteristic
51C2×#41×C23—21no—
52C2×#41×C23—21no—
53C2×#41×C23D412yes—
54C22×D4×C23—21no—
55C22×D4×C23C2311yescharacteristic
56C23×C4×C23D411yescharacteristic
57C23×C4×C23D411yescharacteristic
58C23×C4×C23C2311yescharacteristic
59C2×SD16×C23—22no—
60C2×D8×C23—22no—
61C2×M16×C23—22no—
62C23×C4⋊Q8—22no—
63C23×#126—22no—
64C23×#126—22no—
65C23×#125—22no—
66C23×#125—22no—
67C23×#125—22no—
68C23×#123—22no—
69C23×#123—22no—
70C23×#123—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×#123—44no—
74C23×C22⋊Q8—22no—
75C23×C22⋊Q8—21no—
76C23×C22⋊Q8—21no—
77C23×C22⋊Q8—21no—
78C23×C22⋊Q8—22no—
79C23×C22⋊Q8—21no—
80C23×C22⋊Q8—22no—
81C23×C22⋊Q8D412yes—
82C23×C22⋊Q8C2312yes—
83C23×#121—21no—
84C23×#121—21no—
85C23×#121—21no—
86C23×#121—22no—
87C23×#121—21no—
88C23×#121—22no—
89C23×#121—22no—
90C23×#121D412yes—
91C23×#121C2312yes—
92C23×#120—22no—
93C4×Q8×C23—22no—
94C4×D4×C23—22no—
95C4×D4×C23—22no—
96C4×D4×C23—22no—
97C4×D4×C23—22no—
98C4×D4×C23—22no—
99C4×D4×C23—22no—
100C4×D4×C23—22no—
101C23×#117—21no—
102C23×#117—22no—
103C23×#117—21no—
104C23×#117D412yes—
105C2×C4⋊C4×C23—22no—
106C2×C4⋊C4×C23—22no—
107C2×C4⋊C4×C23—22no—
108C2×C4⋊C4×C23—21no—
109C2×C4⋊C4×C23—21no—
110C2×C4⋊C4×C23—21no—
111C2×C4⋊C4×C23D411yescharacteristic
112C2×C4⋊C4×C23D411yescharacteristic
113C2×C4⋊C4×C23C2311yescharacteristic
114C2×C4⋊C4×C23C2311yescharacteristic
115C2×C4⋊C4×C23C2311yescharacteristic
116C2×C22⋊C4×C23—21no—
117C2×C22⋊C4×C23—21no—
118C2×C22⋊C4×C23—21no—
119C2×C22⋊C4×C23—21no—
120C2×C22⋊C4×C23—21no—
121C2×C22⋊C4×C23—21no—
122C23×#107—22no—
123C23×#106—22no—
124C23×#103—22no—
125C23×#102—22no—
126C23×#102—22no—
127C23×#102—22no—
128C23×C22⋊C8—22no—
129C23×C22⋊C8C2312yes—
130C23×C22⋊C8C2312yes—
131C23×#95—21no—
132C23×#95—21no—
133C23×#95C2311yescharacteristic
134C23×#95C2311yescharacteristic
135C24×C23—21no—
136C24×C23C2×D411yescharacteristic
137#41×C23—42no—
138#41×C23—42no—
139#41×C23—82no—
140#41×C23—82no—
141C2×Q8×C23—21no—
142C2×Q8×C23—21no—
143C2×Q8×C23C2×D412yes—
144C2×D4×C23—21no—
145C2×D4×C23—22no—
146C2×D4×C23—22no—
147C2×D4×C23—22no—
148C2×D4×C23—21no—
149C2×D4×C23—22no—
150C2×D4×C23—42no—
151C2×D4×C23—42no—
152C2×D4×C23—41no—
153C2×D4×C23—41no—
154C2×D4×C23—81no—
155C2×D4×C23C2×D412yes—
156C22×C4×C23—22no—
157C22×C4×C23—21no—
158C22×C4×C23—21no—
159C22×C4×C23—21no—
160C22×C4×C23—22no—
161C22×C4×C23—21no—
162C22×C4×C23—22no—
163C22×C4×C23—22no—
164C22×C4×C23—21no—
165C22×C4×C23—21no—
166C22×C4×C23—21no—
167C22×C4×C23—41no—
168C22×C4×C23—81no—
169C22×C4×C23—81no—
170C22×C4×C23C2×D411yescharacteristic
171C22×C4×C23C2×D411yescharacteristic
172C22×C4×C23C2×D411yescharacteristic
173C22×C4×C23C2×D411yescharacteristic
174C22×C4×C23C2×D411yescharacteristic
175C22×C4×C23C2×D412yes—
176C22×C4×C23C2×D412yes—
177C22×C4×C23C2411yescharacteristic
178SD16×C23—82no—
179D8×C23—82no—
180M16×C23—82no—
181C2×C8×C23—22no—
182C2×C8×C23—22no—
183C4⋊C4×C23—21no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23—22no—
195C4⋊C4×C23—22no—
196C4⋊C4×C23—22no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—42no—
199C4⋊C4×C23—42no—
200C4⋊C4×C23—81no—
201C4⋊C4×C23C2×D412yes—
202C4⋊C4×C23C2×D412yes—
203C22⋊C4×C23—22no—
204C22⋊C4×C23—22no—
205C22⋊C4×C23—22no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—22no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—22no—
212C22⋊C4×C23—42no—
213C22⋊C4×C23—42no—
214C22⋊C4×C23—42no—
215C22⋊C4×C23—42no—
216C22⋊C4×C23—81no—
217C22⋊C4×C23—81no—
218C42×C23—22no—
219C42×C23—22no—
220C42×C23—22no—
221#18348C2311yesmaximal, characteristic, direct product factor, Sylow
222C23×C23—21no—
223C23×C23—21no—
224C23×C23—22no—
225C23×C23—41no—
226C23×C23—41no—
227C23×C23—41no—
228C23×C23—41no—
229C23×C23—41no—
230C23×C23—41no—
231C23×C23—41no—
232C23×C23—41no—
233C23×C23—41no—
234C23×C23#12011yescharacteristic
235C23×C23#12011yescharacteristic
236C23×C23C22×D411yescharacteristic
237Q8×C23—42no—
238Q8×C23—42no—
239Q8×C23—81no—
240D4×C23—42no—
241D4×C23—42no—
242D4×C23—42no—
243D4×C23—42no—
244D4×C23—44no—
245D4×C23—44no—
246D4×C23—42no—
247D4×C23—42no—
248D4×C23—81no—
249D4×C23—82no—
250D4×C23—82no—
251C2×C4×C23—21no—
252C2×C4×C23—21no—
253C2×C4×C23—22no—
254C2×C4×C23—22no—
255C2×C4×C23—21no—
256C2×C4×C23—22no—
257C2×C4×C23—21no—
258C2×C4×C23—21no—
259C2×C4×C23—21no—
260C2×C4×C23—21no—
261C2×C4×C23—41no—
262C2×C4×C23—42no—
263C2×C4×C23—42no—
264C2×C4×C23—41no—
265C2×C4×C23—42no—
266C2×C4×C23—42no—
267C2×C4×C23—44no—
268C2×C4×C23—42no—
269C2×C4×C23—41no—
270C2×C4×C23—41no—
271C2×C4×C23—41no—
272C2×C4×C23—41no—
273C2×C4×C23—41no—
274C2×C4×C23—41no—
275C2×C4×C23—41no—
276C2×C4×C23—41no—
277C2×C4×C23—41no—
278C2×C4×C23—41no—
279C2×C4×C23—41no—
280C2×C4×C23—82no—
281C2×C4×C23—81no—
282C2×C4×C23—82no—
283C2×C4×C23#12012yes—
284C2×C4×C23AGL(1,ℤ/8)12yes—
285C2×C4×C23C22×D412yes—
286C8×C23—42no—
287C8×C23—42no—
288#3492C2×C2311yescharacteristic, complemented
289#3194C2×C2312yescomplemented
290#2618C2×C2311yescharacteristic, complemented
291#1611C2×C2311yescharacteristic, complemented
292#1610C2×C2311yescharacteristic, complemented
293#1609C2×C2312yescomplemented
294#1599C2×C2311yescharacteristic, complemented
295#1598C2×C2311yescharacteristic, complemented
296#1597C2×C2312yescomplemented
297#1513C2×C2312yescomplemented
298#1463C2×C2311yescharacteristic, complemented
299C22×C23—21no—
300C22×C23—21no—
301C22×C23—21no—
302C22×C23—21no—
303C22×C23—21no—
304C22×C23—21no—
305C22×C23—21no—
306C22×C23—21no—
307C22×C23—42no—
308C22×C23—42no—
309C22×C23—42no—
310C22×C23—42no—
311C22×C23—42no—
312C22×C23—81no—
313C22×C23—81no—
314C22×C23—81no—
315C22×C23—81no—
316C22×C23—81no—
317C22×C23—81no—
318C22×C23—81no—
319C22×C23C2×#12011yescharacteristic
320C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
321C22×C23#57511yescharacteristic
322C4×C23—22no—
323C4×C23—22no—
324C4×C23—22no—
325C4×C23—41no—
326C4×C23—42no—
327C4×C23—42no—
328C4×C23—41no—
329C4×C23—41no—
330C4×C23—42no—
331C4×C23—81no—
332C4×C23—81no—
333C4×C23—81no—
334#554—22no—
335#553—22no—
336#552—22no—
337#547—22no—
338#544C22×C2311yescharacteristic
339#539—21no—
340#539C22×C2312yes—
341#537C22×C2311yescharacteristic
342#525—21no—
343#525—21no—
344#525C22×C2312yes—
345C2×#123—21no—
346C2×#121C22×C2311yescharacteristic
347#520—21no—
348C22×C4⋊C4C22×C2311yescharacteristic
349#483C22×C2312yes—
350#482—22no—
351#480C22×C2312yes—
352#469—22no—
353#468—22no—
354#450C22×C2312yes—
355#449—22no—
356#447C22×C2312yes—
357#407C22×C2312yes—
358#398C22×C2311yescharacteristic
359#398C22×C2311yescharacteristic
360#396—21no—
361#396—21no—
362#394C22×C2311yescharacteristic
363#394C22×C2311yescharacteristic
364#386—21no—
365#380C22×C2311yescharacteristic
366#328C22×C2312yes—
367#328C22×C2312yes—
368#327C22×C2312yes—
369#327C22×C2312yes—
370#324C22×C2312yes—
371#324C22×C2311yescharacteristic
372#324C22×C2311yescharacteristic
373C2×C23—21no—
374C2×C23—21no—
375C2×C23—41no—
376C2×C23—41no—
377C2×C23—82no—
378C2×C23—161no—
379C2×C23#300611yescharacteristic
380C2×C23#302011yescharacteristic
381C2×C23#302611yescharacteristic
382C2×#41—21no—
383C2×#41—21no—
384C2×#41D4×C2312yes—
385C22×D4—21no—
386C22×D4C23×C2311yescharacteristic
387C23×C4D4×C2311yescharacteristic
388C23×C4D4×C2311yescharacteristic
389C23×C4C23×C2311yescharacteristic
390C2×SD16—22no—
391C2×D8—22no—
392C2×M16—22no—
393C4⋊Q8—22no—
394#126—22no—
395#126—22no—
396#125—22no—
397#125—22no—
398#125—22no—
399#123—22no—
400#123—22no—
401#123—22no—
402#123—22no—
403#123—22no—
404#123—44no—
405C22⋊Q8—21no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—22no—
412C22⋊Q8D4×C2312yes—
413C22⋊Q8C23×C2312yes—
414#121—21no—
415#121—21no—
416#121—21no—
417#121—22no—
418#121—21no—
419#121—22no—
420#121—22no—
421#121D4×C2312yes—
422#121C23×C2312yes—
423#120—22no—
424C4×Q8—22no—
425C4×D4—22no—
426C4×D4—22no—
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—22no—
432#117—22no—
433#117—21no—
434#117—21no—
435#117D4×C2312yes—
436C2×C4⋊C4—22no—
437C2×C4⋊C4—22no—
438C2×C4⋊C4—22no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4D4×C2311yescharacteristic
443C2×C4⋊C4D4×C2311yescharacteristic
444C2×C4⋊C4C23×C2311yescharacteristic
445C2×C4⋊C4C23×C2311yescharacteristic
446C2×C4⋊C4C23×C2311yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453#107—22no—
454#106—22no—
455#103—22no—
456#102—22no—
457#102—22no—
458#102—22no—
459C22⋊C8—22no—
460C22⋊C8C23×C2312yes—
461C22⋊C8C23×C2312yes—
462#95—21no—
463#95—21no—
464#95C23×C2311yescharacteristic
465#95C23×C2311yescharacteristic
466C23#1834811yescharacteristic, direct product factor, Sylow
467C24—21no—
468C24C2×D4×C2311yescharacteristic
469#41—42no—
470#41—42no—
471#41—82no—
472#41—82no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8C2×D4×C2312yes—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—41no—
485C2×D4—41no—
486C2×D4—81no—
487C2×D4C2×D4×C2312yes—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—41no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4C2×D4×C2311yescharacteristic
503C22×C4C2×D4×C2311yescharacteristic
504C22×C4C2×D4×C2311yescharacteristic
505C22×C4C2×D4×C2311yescharacteristic
506C22×C4C2×D4×C2311yescharacteristic
507C22×C4C2×D4×C2312yes—
508C22×C4C2×D4×C2312yes—
509C22×C4C24×C2311yescharacteristic
510SD16—82no—
511D8—82no—
512M16—82no—
513C2×C8—22no—
514C2×C8—22no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—81no—
533C4⋊C4C2×D4×C2312yes—
534C4⋊C4C2×D4×C2312yes—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C42—22no—
551C42—22no—
552C42—22no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C23×#12011yescharacteristic
566C23C23×#12011yescharacteristic
567C23C22×D4×C2311yescharacteristic
568Q8—42no—
569Q8—42no—
570Q8—81no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—44no—
576D4—44no—
577D4—42no—
578D4—42no—
579D4—81no—
580D4—82no—
581D4—82no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—82no—
612C2×C4—81no—
613C2×C4—82no—
614C2×C4C23×#12012yes—
615C2×C4C23×AGL(1,ℤ/8)12yes—
616C2×C4C22×D4×C2312yes—
617C8—42no—
618C8—42no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22C2×C23×#12011yescharacteristic
640C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
641C22C23×#57511yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—81no—
652C4—81no—
653C4—81no—
654C2—21no—
655C2—21no—
656C2—41no—
657C2—41no—
658C2—82no—
659C2—161no—
660C2C23×#300611yescharacteristic
661C2C23×#302011yescharacteristic
662C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×#3492C2semidirect
2. C23×#3194C2semidirect
3. C23×#2618C2semidirect
4. C23×#1611C2semidirect
5. C23×#1610C2semidirect
6. C23×#1609C2semidirect
7. C23×#1599C2semidirect
8. C23×#1598C2semidirect
9. C23×#1597C2semidirect
10. C23×#1513C2semidirect
11. C23×#1463C2semidirect
221. #18348C23direct (N × H)
288. #3492C2×C23semidirect
289. #3194C2×C23semidirect
290. #2618C2×C23semidirect
291. #1611C2×C23semidirect
292. #1610C2×C23semidirect
293. #1609C2×C23semidirect
294. #1599C2×C23semidirect
295. #1598C2×C23semidirect
296. #1597C2×C23semidirect
297. #1513C2×C23semidirect
298. #1463C2×C23semidirect
466. C23#18348direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34922
C23×#31942
C23×#26182
C23×#16112
C23×#16102
C23×#16092
C23×#15992
C23×#15982
C23×#15972
C23×#15132
C23×#14632
#1834823