Finite Group Databasebeta

← C23×#18157 #19165783

Subgroups — C23×#18157 #19165783

Order 5888.

926 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3455C211yesmaximal, characteristic
2C23×#3113C211yesmaximal, characteristic, complemented
3C23×#3112C211yesmaximal, characteristic, complemented
4C23×#2599C211yesmaximal, characteristic, complemented
5C23×#1612C212yesmaximal, complemented
6C23×#1609C212yesmaximal, complemented
7C23×#1606C212yesmaximal, complemented
8C23×#1603C212yesmaximal, complemented
9C23×#1515C211yesmaximal, characteristic, complemented
10C23×#1514C211yesmaximal, characteristic, complemented
11C23×#1464C211yesmaximal, characteristic, complemented
12C23×#542—21no—
13C23×#542—21no—
14C23×#541—21no—
15C23×#540—21no—
16C23×#539—21no—
17C23×#539C2212yes—
18C23×#538—21no—
19C23×#538C2212yes—
20C23×#533C2211yescharacteristic
21C23×#532C2211yescharacteristic
22C23×#529—22no—
23C23×#529—22no—
24C23×#525—21no—
25C4×#41×C23—21no—
26C2×C4×Q8×C23C2211yescharacteristic
27C2×C23×#117C2211yescharacteristic
28C23×#483C2212yes—
29C23×#480C2212yes—
30C23×#477—21no—
31C23×#475—21no—
32C23×#462—21no—
33C23×#460—21no—
34C23×#451C2212yes—
35C23×#448C2212yes—
36C23×#441—22no—
37C23×#440—22no—
38C23×#432C2211yescharacteristic
39C23×#432C2211yescharacteristic
40C23×#398C2211yescharacteristic
41C23×#397C2211yescharacteristic
42C23×#395C2211yescharacteristic
43C23×#393C2211yescharacteristic
44C23×#388—21no—
45C23×#387—21no—
46C23×#387—21no—
47C23×#385—21no—
48C23×#383C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#327C2212yes—
52C23×#327C2212yes—
53C23×#324C2212yes—
54C23×#323C2212yes—
55C2×#41×C23—21no—
56C22×Q8×C23C2311yescharacteristic
57C23×C4×C23D411yescharacteristic
58C2×Q16×C23—22no—
59C2×SD16×C23—22no—
60C23×C4⋊Q8D411yescharacteristic
61C23×C4⋊Q8D411yescharacteristic
62C23×C4⋊Q8D411yescharacteristic
63C23×#127D411yescharacteristic
64C23×#126—22no—
65C23×#126—22no—
66C23×#126—22no—
67C23×#126—22no—
68C23×#125—22no—
69C23×#125—22no—
70C23×#125—21no—
71C23×#124—22no—
72C23×#124—22no—
73C23×#124—21no—
74C23×#123—22no—
75C23×#123—22no—
76C23×#123—22no—
77C23×#123—22no—
78C23×#123—22no—
79C23×C22⋊Q8—21no—
80C23×C22⋊Q8—21no—
81C23×C22⋊Q8—22no—
82C23×C22⋊Q8—22no—
83C23×C22⋊Q8—21no—
84C23×C22⋊Q8C2312yes—
85C23×#121—21no—
86C23×#121—21no—
87C23×#121—22no—
88C23×#121—22no—
89C23×#121—21no—
90C23×#121C2312yes—
91C23×#120—22no—
92C4×Q8×C23—21no—
93C4×Q8×C23—21no—
94C4×Q8×C23—21no—
95C4×Q8×C23—21no—
96C4×Q8×C23—21no—
97C4×Q8×C23—21no—
98C4×Q8×C23—21no—
99C4×Q8×C23D412yes—
100C4×D4×C23—21no—
101C4×D4×C23—21no—
102C4×D4×C23—22no—
103C4×D4×C23—22no—
104C4×D4×C23—22no—
105C4×D4×C23—22no—
106C4×D4×C23—21no—
107C4×D4×C23D412yes—
108C23×#117—22no—
109C23×#117—21no—
110C23×#117—21no—
111C23×#117—21no—
112C23×#117—21no—
113C23×#117—21no—
114C2×C4⋊C4×C23—21no—
115C2×C4⋊C4×C23—21no—
116C2×C4⋊C4×C23—21no—
117C2×C4⋊C4×C23D411yescharacteristic
118C2×C4⋊C4×C23C2311yescharacteristic
119C2×C4⋊C4×C23C2311yescharacteristic
120C2×C22⋊C4×C23—21no—
121C2×C22⋊C4×C23—21no—
122C2×C22⋊C4×C23D411yescharacteristic
123C2×C22⋊C4×C23C2311yescharacteristic
124C2×C42×C23—22no—
125C2×C42×C23—21no—
126C2×C42×C23D411yescharacteristic
127C2×C42×C23C2311yescharacteristic
128C23×#107—22no—
129C23×#106—22no—
130C23×C4⋊C8—21no—
131C23×C4⋊C8—21no—
132C23×#103—22no—
133C23×#103—22no—
134C23×#103—22no—
135C23×#102—22no—
136C23×C22⋊C8C2312yes—
137C23×C22⋊C8C2312yes—
138C23×#97—21no—
139C23×#97—21no—
140C23×#95—21no—
141C23×#95—21no—
142C23×#95—21no—
143C23×#95—21no—
144C23×#95—21no—
145C23×#95—21no—
146C23×#95C2311yescharacteristic
147C23×#95C2311yescharacteristic
148C24×C23C2×D411yescharacteristic
149#41×C23—42no—
150#41×C23—42no—
151C2×Q8×C23—21no—
152C2×Q8×C23—21no—
153C2×Q8×C23—21no—
154C2×Q8×C23—21no—
155C2×Q8×C23—42no—
156C2×Q8×C23C2×D412yes—
157C2×D4×C23—22no—
158C2×D4×C23—22no—
159C2×D4×C23—22no—
160C2×D4×C23—21no—
161C2×D4×C23—21no—
162C2×D4×C23C2×D412yes—
163C22×C4×C23—21no—
164C22×C4×C23—22no—
165C22×C4×C23—22no—
166C22×C4×C23—22no—
167C22×C4×C23—21no—
168C22×C4×C23—21no—
169C22×C4×C23—21no—
170C22×C4×C23—21no—
171C22×C4×C23—21no—
172C22×C4×C23—41no—
173C22×C4×C23—41no—
174C22×C4×C23C2×D411yescharacteristic
175C22×C4×C23C2×D411yescharacteristic
176C22×C4×C23C2×D411yescharacteristic
177C22×C4×C23C2×D411yescharacteristic
178C22×C4×C23C2×D411yescharacteristic
179C22×C4×C23C2411yescharacteristic
180Q16×C23—44no—
181SD16×C23—44no—
182C2×C8×C23—22no—
183C2×C8×C23—22no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—21no—
189C4⋊C4×C23—21no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—21no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—22no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—22no—
198C4⋊C4×C23—21no—
199C4⋊C4×C23—21no—
200C4⋊C4×C23—42no—
201C4⋊C4×C23—42no—
202C4⋊C4×C23—41no—
203C4⋊C4×C23—41no—
204C4⋊C4×C23C2×D412yes—
205C4⋊C4×C23C2×D412yes—
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—22no—
213C22⋊C4×C23—21no—
214C22⋊C4×C23—21no—
215C22⋊C4×C23—22no—
216C22⋊C4×C23—21no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—22no—
219C22⋊C4×C23—42no—
220C22⋊C4×C23—42no—
221C42×C23—22no—
222C42×C23—22no—
223C42×C23—21no—
224C42×C23—22no—
225C42×C23—21no—
226C42×C23—41no—
227C42×C23—41no—
228C42×C23C2×D411yescharacteristic
229C42×C23C2×D411yescharacteristic
230C42×C23C2×D411yescharacteristic
231C42×C23C2×D411yescharacteristic
232#18157C2311yesmaximal, characteristic, direct product factor, Sylow
233C23×C23—21no—
234C23×C23—22no—
235C23×C23—21no—
236C23×C23—21no—
237C23×C23—41no—
238C23×C23—41no—
239C23×C23C22×D411yescharacteristic
240Q8×C23—22no—
241Q8×C23—22no—
242Q8×C23—42no—
243Q8×C23—41no—
244Q8×C23—41no—
245Q8×C23—42no—
246Q8×C23—81no—
247D4×C23—42no—
248D4×C23—42no—
249D4×C23—42no—
250D4×C23—42no—
251D4×C23—81no—
252D4×C23—82no—
253D4×C23—82no—
254C2×C4×C23—21no—
255C2×C4×C23—21no—
256C2×C4×C23—22no—
257C2×C4×C23—21no—
258C2×C4×C23—21no—
259C2×C4×C23—22no—
260C2×C4×C23—22no—
261C2×C4×C23—22no—
262C2×C4×C23—21no—
263C2×C4×C23—21no—
264C2×C4×C23—21no—
265C2×C4×C23—41no—
266C2×C4×C23—41no—
267C2×C4×C23—42no—
268C2×C4×C23—42no—
269C2×C4×C23—44no—
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—41no—
281C2×C4×C23—41no—
282C2×C4×C23—41no—
283C2×C4×C23—82no—
284C2×C4×C23—82no—
285C2×C4×C23#12011yescharacteristic
286C2×C4×C23#12011yescharacteristic
287C2×C4×C23#12011yescharacteristic
288C2×C4×C23#12011yescharacteristic
289C2×C4×C23C22×D411yescharacteristic
290C2×C4×C23C22×D411yescharacteristic
291C8×C23—42no—
292C8×C23—42no—
293#3455C2×C2311yescharacteristic
294#3113C2×C2311yescharacteristic, complemented
295#3112C2×C2311yescharacteristic, complemented
296#2599C2×C2311yescharacteristic, complemented
297#1612C2×C2312yescomplemented
298#1609C2×C2312yescomplemented
299#1606C2×C2312yescomplemented
300#1603C2×C2312yescomplemented
301#1515C2×C2311yescharacteristic, complemented
302#1514C2×C2311yescharacteristic, complemented
303#1464C2×C2311yescharacteristic, complemented
304C22×C23—21no—
305C22×C23—21no—
306C22×C23—42no—
307C22×C23—41no—
308C22×C23—42no—
309C22×C23—42no—
310C22×C23—41no—
311C22×C23—42no—
312C22×C23—41no—
313C22×C23—81no—
314C22×C23C2×#12011yescharacteristic
315C22×C23#57511yescharacteristic
316C22×C23#57511yescharacteristic
317C4×C23—21no—
318C4×C23—21no—
319C4×C23—21no—
320C4×C23—21no—
321C4×C23—21no—
322C4×C23—21no—
323C4×C23—41no—
324C4×C23—42no—
325C4×C23—41no—
326C4×C23—42no—
327C4×C23—42no—
328C4×C23—41no—
329C4×C23—42no—
330C4×C23—81no—
331C4×C23—81no—
332C4×C23—81no—
333#542—21no—
334#542—21no—
335#541—21no—
336#540—21no—
337#539—21no—
338#539C22×C2312yes—
339#538—21no—
340#538C22×C2312yes—
341#533C22×C2311yescharacteristic
342#532C22×C2311yescharacteristic
343#529—22no—
344#529—22no—
345#525—21no—
346C4×#41—21no—
347C2×C4×Q8C22×C2311yescharacteristic
348C2×#117C22×C2311yescharacteristic
349#483C22×C2312yes—
350#480C22×C2312yes—
351#477—21no—
352#475—21no—
353#462—21no—
354#460—21no—
355#451C22×C2312yes—
356#448C22×C2312yes—
357#441—22no—
358#440—22no—
359#432C22×C2311yescharacteristic
360#432C22×C2311yescharacteristic
361#398C22×C2311yescharacteristic
362#397C22×C2311yescharacteristic
363#395C22×C2311yescharacteristic
364#393C22×C2311yescharacteristic
365#388—21no—
366#387—21no—
367#387—21no—
368#385—21no—
369#383C22×C2311yescharacteristic
370#328C22×C2312yes—
371#328C22×C2312yes—
372#327C22×C2312yes—
373#327C22×C2312yes—
374#324C22×C2312yes—
375#323C22×C2312yes—
376C2×C23—21no—
377C2×C23—21no—
378C2×C23—82no—
379C2×C23—81no—
380C2×C23#300511yescharacteristic
381C2×C23#302211yescharacteristic
382C2×C23#302511yescharacteristic
383C2×#41—21no—
384C22×Q8C23×C2311yescharacteristic
385C23×C4D4×C2311yescharacteristic
386C2×Q16—22no—
387C2×SD16—22no—
388C4⋊Q8D4×C2311yescharacteristic
389C4⋊Q8D4×C2311yescharacteristic
390C4⋊Q8D4×C2311yescharacteristic
391#127D4×C2311yescharacteristic
392#126—22no—
393#126—22no—
394#126—22no—
395#126—22no—
396#125—22no—
397#125—22no—
398#125—21no—
399#124—22no—
400#124—22no—
401#124—21no—
402#123—22no—
403#123—22no—
404#123—22no—
405#123—22no—
406#123—22no—
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8—21no—
412C22⋊Q8C23×C2312yes—
413#121—21no—
414#121—21no—
415#121—22no—
416#121—22no—
417#121—21no—
418#121C23×C2312yes—
419#120—22no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8—21no—
427C4×Q8D4×C2312yes—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—21no—
435C4×D4D4×C2312yes—
436#117—22no—
437#117—21no—
438#117—21no—
439#117—21no—
440#117—21no—
441#117—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4D4×C2311yescharacteristic
446C2×C4⋊C4C23×C2311yescharacteristic
447C2×C4⋊C4C23×C2311yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4D4×C2311yescharacteristic
451C2×C22⋊C4C23×C2311yescharacteristic
452C2×C42—22no—
453C2×C42—21no—
454C2×C42D4×C2311yescharacteristic
455C2×C42C23×C2311yescharacteristic
456#107—22no—
457#106—22no—
458C4⋊C8—21no—
459C4⋊C8—21no—
460#103—22no—
461#103—22no—
462#103—22no—
463#102—22no—
464C22⋊C8C23×C2312yes—
465C22⋊C8C23×C2312yes—
466#97—21no—
467#97—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95C23×C2311yescharacteristic
475#95C23×C2311yescharacteristic
476C23#1815711yescharacteristic, direct product factor, Sylow
477C24C2×D4×C2311yescharacteristic
478#41—42no—
479#41—42no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—42no—
485C2×Q8C2×D4×C2312yes—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4C2×D4×C2312yes—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4C2×D4×C2311yescharacteristic
504C22×C4C2×D4×C2311yescharacteristic
505C22×C4C2×D4×C2311yescharacteristic
506C22×C4C2×D4×C2311yescharacteristic
507C22×C4C2×D4×C2311yescharacteristic
508C22×C4C24×C2311yescharacteristic
509Q16—44no—
510SD16—44no—
511C2×C8—22no—
512C2×C8—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—41no—
532C4⋊C4—41no—
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—21no—
543C22⋊C4—21no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C42—22no—
551C42—22no—
552C42—21no—
553C42—22no—
554C42—21no—
555C42—41no—
556C42—41no—
557C42C2×D4×C2311yescharacteristic
558C42C2×D4×C2311yescharacteristic
559C42C2×D4×C2311yescharacteristic
560C42C2×D4×C2311yescharacteristic
561C23—21no—
562C23—21no—
563C23—22no—
564C23—21no—
565C23—41no—
566C23—41no—
567C23C22×D4×C2311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—42no—
571Q8—41no—
572Q8—41no—
573Q8—42no—
574Q8—81no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—81no—
580D4—82no—
581D4—82no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
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—82no—
613C2×C4C23×#12011yescharacteristic
614C2×C4C23×#12011yescharacteristic
615C2×C4C23×#12011yescharacteristic
616C2×C4C23×#12011yescharacteristic
617C2×C4C22×D4×C2311yescharacteristic
618C2×C4C22×D4×C2311yescharacteristic
619C8—42no—
620C8—42no—
621C22—21no—
622C22—21no—
623C22—42no—
624C22—41no—
625C22—42no—
626C22—42no—
627C22—41no—
628C22—42no—
629C22—41no—
630C22—81no—
631C22C2×C23×#12011yescharacteristic
632C22C23×#57511yescharacteristic
633C22C23×#57511yescharacteristic
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—81no—
650C2—21no—
651C2—21no—
652C2—81no—
653C2—82no—
654C2C23×#300511yescharacteristic
655C2C23×#302211yescharacteristic
656C2C23×#302511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3455C2
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×#3112C2semidirect
4. C23×#2599C2semidirect
5. C23×#1612C2semidirect
6. C23×#1609C2semidirect
7. C23×#1606C2semidirect
8. C23×#1603C2semidirect
9. C23×#1515C2semidirect
10. C23×#1514C2semidirect
11. C23×#1464C2semidirect
232. #18157C23direct (N × H)
294. #3113C2×C23semidirect
295. #3112C2×C23semidirect
296. #2599C2×C23semidirect
297. #1612C2×C23semidirect
298. #1609C2×C23semidirect
299. #1606C2×C23semidirect
300. #1603C2×C23semidirect
301. #1515C2×C23semidirect
302. #1514C2×C23semidirect
303. #1464C2×C23semidirect
476. C23#18157direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34552
C23×#31132
C23×#31122
C23×#25992
C23×#16122
C23×#16092
C23×#16062
C23×#16032
C23×#15152
C23×#15142
C23×#14642
#1815723