Finite Group Databasebeta

← C23×#18455 #19166081

Subgroups — C23×#18455 #19166081

Order 5888.

894 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×#3538C211yesmaximal, characteristic
2C23×#3196C211yesmaximal, characteristic
3C23×#3189C211yesmaximal, characteristic, complemented
4C23×#2817C211yesmaximal, characteristic, complemented
5C23×#1617C212yesmaximal, complemented
6C23×#1615C212yesmaximal, complemented
7C23×#1605C212yesmaximal, complemented
8C23×#1603C212yesmaximal, complemented
9C23×#1516C211yesmaximal, characteristic
10C23×#1511C211yesmaximal, characteristic, complemented
11C23×#1472C211yesmaximal, characteristic, complemented
12C23×#563—22no—
13C23×#552—21no—
14C23×#552—21no—
15C23×#551—22no—
16D4×Q8×C23—21no—
17D4×Q8×C23—21no—
18C23×#544—21no—
19C23×#544C2212yes—
20C23×#541—21no—
21C23×#539—21no—
22C23×#539C2212yes—
23C23×#536—21no—
24C23×#525C2211yescharacteristic
25C2×C23×C22⋊Q8C2211yescharacteristic
26C2×C23×C22⋊Q8C2211yescharacteristic
27C2×C23×C22⋊Q8C2211yescharacteristic
28C23×#485—21no—
29C23×#484—21no—
30C23×#484C2212yes—
31C23×#484C2211yescharacteristic
32C23×#481C2211yescharacteristic
33C23×#470—21no—
34C23×#468—21no—
35C23×#467—22no—
36C23×#452—21no—
37C23×#451—21no—
38C23×#451C2212yes—
39C23×#451C2211yescharacteristic
40C23×#448C2211yescharacteristic
41C23×#407C2211yescharacteristic
42C2×C23×C22⋊C8C2211yescharacteristic
43C23×#400—21no—
44C23×#400C2212yes—
45C23×#397—22no—
46C23×#395—21no—
47C23×#395C2212yes—
48C23×#381C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#328C2212yes—
52C23×#327C2212yes—
53C23×#324C2212yes—
54C23×#324C2212yes—
55C2×#41×C23D411yescharacteristic
56C22×Q8×C23—21no—
57C22×Q8×C23D411yescharacteristic
58C22×Q8×C23D411yescharacteristic
59C22×Q8×C23C2311yescharacteristic
60C23×C4×C23C2311yescharacteristic
61C2×Q16×C23—22no—
62C2×Q16×C23—21no—
63C2×SD16×C23—21no—
64C2×M16×C23—21no—
65C22×C8×C23—21no—
66C23×C4⋊Q8—21no—
67C23×C4⋊Q8—22no—
68C23×C4⋊Q8—22no—
69C23×C4⋊Q8—22no—
70C23×#126—22no—
71C23×#126—22no—
72C23×#126—22no—
73C23×#126—22no—
74C23×#125—21no—
75C23×#125—22no—
76C23×#125—22no—
77C23×#125—22no—
78C23×#125—22no—
79C23×#124—22no—
80C23×#123—22no—
81C23×#123—22no—
82C23×#123—22no—
83C23×#123—22no—
84C23×#123—21no—
85C23×C22⋊Q8—22no—
86C23×C22⋊Q8—21no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—22no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—21no—
92C23×C22⋊Q8—21no—
93C23×C22⋊Q8—21no—
94C23×C22⋊Q8—21no—
95C23×C22⋊Q8—21no—
96C23×C22⋊Q8—21no—
97C23×C22⋊Q8—21no—
98C23×C22⋊Q8—21no—
99C23×C22⋊Q8—21no—
100C23×C22⋊Q8—21no—
101C23×C22⋊Q8D412yes—
102C23×C22⋊Q8D411yescharacteristic
103C23×C22⋊Q8C2312yes—
104C23×C22⋊Q8C2311yescharacteristic
105C23×#121D411yescharacteristic
106C23×#121C2311yescharacteristic
107C23×#120—21no—
108C4×Q8×C23—21no—
109C4×Q8×C23—21no—
110C4×D4×C23—22no—
111C4×D4×C23—22no—
112C4×D4×C23—21no—
113C4×D4×C23—21no—
114C23×#117—22no—
115C23×#117—21no—
116C23×#117—22no—
117C23×#117D411yescharacteristic
118C2×C4⋊C4×C23—21no—
119C2×C4⋊C4×C23—21no—
120C2×C4⋊C4×C23—22no—
121C2×C4⋊C4×C23D411yescharacteristic
122C2×C4⋊C4×C23D411yescharacteristic
123C2×C4⋊C4×C23C2311yescharacteristic
124C2×C4⋊C4×C23C2312yes—
125C2×C22⋊C4×C23—21no—
126C2×C22⋊C4×C23—21no—
127C2×C22⋊C4×C23—21no—
128C2×C22⋊C4×C23D412yes—
129C23×#107—22no—
130C23×#107—21no—
131C23×#106—21no—
132C23×#103—22no—
133C23×#103—21no—
134C23×#103—21no—
135C23×#103—22no—
136C23×#103—21no—
137C23×#102—21no—
138C23×C22⋊C8—21no—
139C23×C22⋊C8—21no—
140C23×C22⋊C8C2312yes—
141C23×C22⋊C8C2311yescharacteristic
142C23×C22⋊C8C2311yescharacteristic
143C23×#95—21no—
144C23×#95—22no—
145C23×#95—22no—
146C23×#95—22no—
147C23×#95C2312yes—
148C24×C23C2×D411yescharacteristic
149#41×C23—81no—
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—81no—
157C2×Q8×C23—81no—
158C2×Q8×C23—82no—
159C2×Q8×C23—81no—
160C2×Q8×C23C2×D412yes—
161C2×Q8×C23C2×D411yescharacteristic
162C2×D4×C23—21no—
163C2×D4×C23—21no—
164C2×D4×C23—21no—
165C2×D4×C23C2×D411yescharacteristic
166C22×C4×C23—21no—
167C22×C4×C23—22no—
168C22×C4×C23—21no—
169C22×C4×C23—21no—
170C22×C4×C23—21no—
171C22×C4×C23—21no—
172C22×C4×C23—22no—
173C22×C4×C23—81no—
174C22×C4×C23C2×D412yes—
175C22×C4×C23C2×D412yes—
176C22×C4×C23C2×D411yescharacteristic
177C22×C4×C23C2×D411yescharacteristic
178C22×C4×C23C2×D411yescharacteristic
179C22×C4×C23C2×D411yescharacteristic
180C22×C4×C23C2×D411yescharacteristic
181C22×C4×C23C2411yescharacteristic
182Q16×C23—82no—
183Q16×C23—81no—
184SD16×C23—81no—
185M16×C23—81no—
186C2×C8×C23—21no—
187C2×C8×C23—22no—
188C2×C8×C23—21no—
189C2×C8×C23—81no—
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—22no—
199C4⋊C4×C23—21no—
200C4⋊C4×C23—21no—
201C4⋊C4×C23—21no—
202C4⋊C4×C23—21no—
203C4⋊C4×C23—22no—
204C4⋊C4×C23—22no—
205C4⋊C4×C23—22no—
206C4⋊C4×C23—21no—
207C4⋊C4×C23—21no—
208C4⋊C4×C23—21no—
209C4⋊C4×C23—41no—
210C4⋊C4×C23—41no—
211C4⋊C4×C23—82no—
212C4⋊C4×C23C2×D411yescharacteristic
213C4⋊C4×C23C2×D412yes—
214C4⋊C4×C23C2×D411yescharacteristic
215C22⋊C4×C23—22no—
216C22⋊C4×C23—22no—
217C22⋊C4×C23—22no—
218C22⋊C4×C23—22no—
219C22⋊C4×C23—22no—
220C22⋊C4×C23—22no—
221C22⋊C4×C23—21no—
222C22⋊C4×C23—21no—
223C22⋊C4×C23—22no—
224C22⋊C4×C23—21no—
225C22⋊C4×C23—21no—
226C22⋊C4×C23—21no—
227C22⋊C4×C23—41no—
228C22⋊C4×C23—41no—
229C22⋊C4×C23—81no—
230C42×C23—22no—
231C42×C23—22no—
232C42×C23—22no—
233C42×C23—21no—
234#18455C2311yesmaximal, characteristic, direct product factor, Sylow
235C23×C23—22no—
236C23×C23—21no—
237C23×C23—41no—
238C23×C23—41no—
239C23×C23#12011yescharacteristic
240C23×C23#12011yescharacteristic
241C23×C23C22×D411yescharacteristic
242Q8×C23—42no—
243Q8×C23—41no—
244Q8×C23—42no—
245Q8×C23—42no—
246Q8×C23—41no—
247Q8×C23—41no—
248Q8×C23—42no—
249Q8×C23—81no—
250Q8×C23—82no—
251D4×C23—41no—
252D4×C23—42no—
253D4×C23—42no—
254D4×C23—81no—
255C2×C4×C23—21no—
256C2×C4×C23—21no—
257C2×C4×C23—22no—
258C2×C4×C23—22no—
259C2×C4×C23—21no—
260C2×C4×C23—22no—
261C2×C4×C23—22no—
262C2×C4×C23—24no—
263C2×C4×C23—41no—
264C2×C4×C23—41no—
265C2×C4×C23—41no—
266C2×C4×C23—41no—
267C2×C4×C23—41no—
268C2×C4×C23—41no—
269C2×C4×C23—42no—
270C2×C4×C23—42no—
271C2×C4×C23—42no—
272C2×C4×C23—42no—
273C2×C4×C23—42no—
274C2×C4×C23—42no—
275C2×C4×C23—42no—
276C2×C4×C23—41no—
277C2×C4×C23—41no—
278C2×C4×C23—42no—
279C2×C4×C23—81no—
280C2×C4×C23—82no—
281C2×C4×C23#12011yescharacteristic
282C2×C4×C23#12011yescharacteristic
283C2×C4×C23C22×D411yescharacteristic
284C2×C4×C23C22×D411yescharacteristic
285C8×C23—42no—
286C8×C23—41no—
287C8×C23—41no—
288#3538C2×C2311yescharacteristic
289#3196C2×C2311yescharacteristic
290#3189C2×C2311yescharacteristic, complemented
291#2817C2×C2311yescharacteristic, complemented
292#1617C2×C2312yescomplemented
293#1615C2×C2312yescomplemented
294#1605C2×C2312yescomplemented
295#1603C2×C2312yescomplemented
296#1516C2×C2311yescharacteristic
297#1511C2×C2311yescharacteristic, complemented
298#1472C2×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—41no—
308C22×C23—41no—
309C22×C23—41no—
310C22×C23—82no—
311C22×C23C2×#12011yescharacteristic
312C22×C23#57812yes—
313C4×C23—21no—
314C4×C23—21no—
315C4×C23—21no—
316C4×C23—21no—
317C4×C23—42no—
318C4×C23—41no—
319C4×C23—42no—
320C4×C23—41no—
321C4×C23—41no—
322C4×C23—42no—
323C4×C23—42no—
324C4×C23—82no—
325C4×C23—81no—
326C4×C23—81no—
327#563—22no—
328#552—21no—
329#552—21no—
330#551—22no—
331D4×Q8—21no—
332D4×Q8—21no—
333#544—21no—
334#544C22×C2312yes—
335#541—21no—
336#539—21no—
337#539C22×C2312yes—
338#536—21no—
339#525C22×C2311yescharacteristic
340C2×C22⋊Q8C22×C2311yescharacteristic
341C2×C22⋊Q8C22×C2311yescharacteristic
342C2×C22⋊Q8C22×C2311yescharacteristic
343#485—21no—
344#484—21no—
345#484C22×C2312yes—
346#484C22×C2311yescharacteristic
347#481C22×C2311yescharacteristic
348#470—21no—
349#468—21no—
350#467—22no—
351#452—21no—
352#451—21no—
353#451C22×C2312yes—
354#451C22×C2311yescharacteristic
355#448C22×C2311yescharacteristic
356#407C22×C2311yescharacteristic
357C2×C22⋊C8C22×C2311yescharacteristic
358#400—21no—
359#400C22×C2312yes—
360#397—22no—
361#395—21no—
362#395C22×C2312yes—
363#381C22×C2311yescharacteristic
364#328C22×C2312yes—
365#328C22×C2312yes—
366#328C22×C2312yes—
367#327C22×C2312yes—
368#324C22×C2312yes—
369#324C22×C2312yes—
370C2×C23—22no—
371C2×C23—41no—
372C2×C23—41no—
373C2×C23—81no—
374C2×C23#301111yescharacteristic
375C2×C23#302011yescharacteristic
376C2×C23#302611yescharacteristic
377C2×#41D4×C2311yescharacteristic
378C22×Q8—21no—
379C22×Q8D4×C2311yescharacteristic
380C22×Q8D4×C2311yescharacteristic
381C22×Q8C23×C2311yescharacteristic
382C23×C4C23×C2311yescharacteristic
383C2×Q16—22no—
384C2×Q16—21no—
385C2×SD16—21no—
386C2×M16—21no—
387C22×C8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—22no—
390C4⋊Q8—22no—
391C4⋊Q8—22no—
392#126—22no—
393#126—22no—
394#126—22no—
395#126—22no—
396#125—21no—
397#125—22no—
398#125—22no—
399#125—22no—
400#125—22no—
401#124—22no—
402#123—22no—
403#123—22no—
404#123—22no—
405#123—21no—
406#123—22no—
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418C22⋊Q8—22no—
419C22⋊Q8—21no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8D4×C2312yes—
424C22⋊Q8D4×C2311yescharacteristic
425C22⋊Q8C23×C2312yes—
426C22⋊Q8C23×C2311yescharacteristic
427#121D4×C2311yescharacteristic
428#121C23×C2311yescharacteristic
429#120—21no—
430C4×Q8—21no—
431C4×Q8—21no—
432C4×D4—22no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—22no—
436#117—21no—
437#117—22no—
438#117—22no—
439#117D4×C2311yescharacteristic
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—22no—
443C2×C4⋊C4D4×C2311yescharacteristic
444C2×C4⋊C4D4×C2311yescharacteristic
445C2×C4⋊C4C23×C2311yescharacteristic
446C2×C4⋊C4C23×C2312yes—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4D4×C2312yes—
451#107—22no—
452#107—21no—
453#106—21no—
454#103—22no—
455#103—21no—
456#103—21no—
457#103—22no—
458#103—21no—
459#102—21no—
460C22⋊C8—21no—
461C22⋊C8—21no—
462C22⋊C8C23×C2312yes—
463C22⋊C8C23×C2311yescharacteristic
464C22⋊C8C23×C2311yescharacteristic
465#95—21no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95C23×C2312yes—
470C23#1845511yescharacteristic, direct product factor, Sylow
471C24C2×D4×C2311yescharacteristic
472#41—81no—
473C2×Q8—22no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—81no—
480C2×Q8—81no—
481C2×Q8—82no—
482C2×Q8—81no—
483C2×Q8C2×D4×C2312yes—
484C2×Q8C2×D4×C2311yescharacteristic
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×D4×C2311yescharacteristic
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—81no—
497C22×C4C2×D4×C2312yes—
498C22×C4C2×D4×C2312yes—
499C22×C4C2×D4×C2311yescharacteristic
500C22×C4C2×D4×C2311yescharacteristic
501C22×C4C2×D4×C2311yescharacteristic
502C22×C4C2×D4×C2311yescharacteristic
503C22×C4C2×D4×C2311yescharacteristic
504C22×C4C24×C2311yescharacteristic
505Q16—82no—
506Q16—81no—
507SD16—81no—
508M16—81no—
509C2×C8—21no—
510C2×C8—22no—
511C2×C8—21no—
512C2×C8—81no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—82no—
535C4⋊C4C2×D4×C2311yescharacteristic
536C4⋊C4C2×D4×C2312yes—
537C4⋊C4C2×D4×C2311yescharacteristic
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—81no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—21no—
557C23—22no—
558C23—21no—
559C23—41no—
560C23—41no—
561C23C23×#12011yescharacteristic
562C23C23×#12011yescharacteristic
563C23C22×D4×C2311yescharacteristic
564Q8—42no—
565Q8—41no—
566Q8—42no—
567Q8—42no—
568Q8—41no—
569Q8—41no—
570Q8—42no—
571Q8—81no—
572Q8—82no—
573D4—41no—
574D4—42no—
575D4—42no—
576D4—81no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4C23×#12011yescharacteristic
604C2×C4C23×#12011yescharacteristic
605C2×C4C22×D4×C2311yescharacteristic
606C2×C4C22×D4×C2311yescharacteristic
607C8—42no—
608C8—41no—
609C8—41no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—82no—
622C22C2×C23×#12011yescharacteristic
623C22C23×#57812yes—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—81no—
637C4—81no—
638C2—22no—
639C2—41no—
640C2—41no—
641C2—81no—
642C2C23×#301111yescharacteristic
643C2C23×#302011yescharacteristic
644C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#525C22
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
3. C23×#3189C2semidirect
4. C23×#2817C2semidirect
5. C23×#1617C2semidirect
6. C23×#1615C2semidirect
7. C23×#1605C2semidirect
8. C23×#1603C2semidirect
10. C23×#1511C2semidirect
11. C23×#1472C2semidirect
234. #18455C23direct (N × H)
290. #3189C2×C23semidirect
291. #2817C2×C23semidirect
292. #1617C2×C23semidirect
293. #1615C2×C23semidirect
294. #1605C2×C23semidirect
295. #1603C2×C23semidirect
297. #1511C2×C23semidirect
298. #1472C2×C23semidirect
470. C23#18455direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35382
C23×#31962
C23×#31892
C23×#28172
C23×#16172
C23×#16152
C23×#16052
C23×#16032
C23×#15162
C23×#15112
C23×#14722
#1845523