Finite Group Databasebeta

← C23×#18573 #19166199

Subgroups — C23×#18573 #19166199

Order 5888.

934 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3522C211yesmaximal, characteristic
2C23×#3218C211yesmaximal, characteristic, complemented
3C23×#3141C211yesmaximal, characteristic, complemented
4C2×C23×#475C211yesmaximal, characteristic, complemented
5C23×#1645C212yesmaximal, complemented
6C23×#1643C212yesmaximal, complemented
7C23×#1642C212yesmaximal, complemented
8C23×#1640C212yesmaximal, complemented
9C23×#1551C211yesmaximal, characteristic, complemented
10C23×#1491C211yesmaximal, characteristic, complemented
11C23×#1483C211yesmaximal, characteristic, complemented
12C23×#559—21no—
13C23×#559C2211yescharacteristic
14Q82×C23C2211yescharacteristic
15C23×#554—21no—
16C23×#554C2211yescharacteristic
17C23×#550C2211yescharacteristic
18C23×#548—21no—
19C23×#547—22no—
20C23×#547—21no—
21C23×#542—22no—
22C23×#542—21no—
23C23×#532—21no—
24C23×#532C2211yescharacteristic
25C2×C23×C4⋊Q8C2211yescharacteristic
26C23×#529—21no—
27C23×#529C2211yescharacteristic
28C23×#520—21no—
29C2×C4×Q8×C23C2211yescharacteristic
30C23×#497—21no—
31C23×#491—21no—
32C23×#489—22no—
33C23×#487C2212yes—
34C23×#486C2212yes—
35C23×#475—21no—
36C23×#475—21no—
37C23×#475C2212yes—
38C23×#475C2212yes—
39C23×#462C2212yes—
40C23×#461C2212yes—
41C23×#451—21no—
42C23×#448—21no—
43C23×#432C2211yescharacteristic
44C2×C23×#106—21no—
45C2×C23×C4⋊C8C2211yescharacteristic
46C23×#417—22no—
47C2×C23×#103—21no—
48C4×M16×C23C2211yescharacteristic
49C23×#333C2212yes—
50C23×#331C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×Q8⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C2×#41×C23—22no—
56C2×#41×C23—21no—
57C2×#41×C23D412yes—
58C22×Q8×C23—21no—
59C22×Q8×C23D411yescharacteristic
60C2×Q16×C23—22no—
61C2×SD16×C23—22no—
62C2×M16×C23—21no—
63C22×C8×C23—21no—
64C23×C4⋊Q8—22no—
65C23×C4⋊Q8—21no—
66C23×C4⋊Q8—21no—
67C23×C4⋊Q8—21no—
68C23×C4⋊Q8—21no—
69C23×C4⋊Q8—21no—
70C23×C4⋊Q8C2312yes—
71C23×C4⋊Q8C2311yescharacteristic
72C23×#127—21no—
73C23×#127C2311yescharacteristic
74C23×#126—22no—
75C23×#126—21no—
76C23×#125—21no—
77C23×#125D411yescharacteristic
78C23×#124—22no—
79C23×#124—22no—
80C23×#124—21no—
81C23×#124—21no—
82C23×#124D411yescharacteristic
83C23×#124D412yes—
84C23×#123—22no—
85C23×#123—22no—
86C23×#123—22no—
87C23×#123—22no—
88C23×#123—21no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—22no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—21no—
94C23×C22⋊Q8D411yescharacteristic
95C23×#121—22no—
96C23×#121—22no—
97C23×#121—21no—
98C23×#121—21no—
99C23×#121—21no—
100C23×#121D411yescharacteristic
101C4×Q8×C23—21no—
102C4×Q8×C23—21no—
103C4×Q8×C23—21no—
104C4×Q8×C23—21no—
105C4×Q8×C23D411yescharacteristic
106C4×Q8×C23D411yescharacteristic
107C4×Q8×C23C2311yescharacteristic
108C4×Q8×C23C2311yescharacteristic
109C4×Q8×C23C2311yescharacteristic
110C4×D4×C23—22no—
111C4×D4×C23—22no—
112C4×D4×C23—22no—
113C4×D4×C23—21no—
114C4×D4×C23—21no—
115C4×D4×C23—21no—
116C4×D4×C23—21no—
117C4×D4×C23—21no—
118C4×D4×C23—21no—
119C4×D4×C23—21no—
120C4×D4×C23C2311yescharacteristic
121C23×#117—22no—
122C23×#117—21no—
123C2×C4⋊C4×C23—22no—
124C2×C4⋊C4×C23—21no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23—21no—
127C2×C4⋊C4×C23—21no—
128C2×C4⋊C4×C23D411yescharacteristic
129C2×C42×C23—21no—
130C2×C42×C23C2311yescharacteristic
131C23×#106—22no—
132C23×#106—22no—
133C23×C4⋊C8—21no—
134C23×C4⋊C8C2312yes—
135C23×C4⋊C8C2312yes—
136C23×#103—22no—
137C23×#103—22no—
138C23×#103—24no—
139C23×#103—22no—
140C23×#103—22no—
141C23×#102—24no—
142C23×C22⋊C8—21no—
143C23×#97—21no—
144C23×#97—21no—
145C4×C8×C23C2312yes—
146#41×C23—44no—
147#41×C23—42no—
148#41×C23—42no—
149#41×C23—84no—
150C2×Q8×C23—22no—
151C2×Q8×C23—21no—
152C2×Q8×C23—22no—
153C2×Q8×C23—22no—
154C2×Q8×C23—41no—
155C2×Q8×C23—41no—
156C2×Q8×C23—82no—
157C2×Q8×C23C2×D411yescharacteristic
158C2×Q8×C23C2×D411yescharacteristic
159C2×Q8×C23C2×D411yescharacteristic
160C2×Q8×C23C2×D412yes—
161C2×D4×C23—22no—
162C2×D4×C23—22no—
163C2×D4×C23—21no—
164C2×D4×C23—22no—
165C2×D4×C23—21no—
166C2×D4×C23—21no—
167C2×D4×C23C2×D411yescharacteristic
168C2×D4×C23C2×D412yes—
169C22×C4×C23—22no—
170C22×C4×C23—21no—
171C22×C4×C23—21no—
172C22×C4×C23—22no—
173C22×C4×C23—22no—
174C22×C4×C23—21no—
175C22×C4×C23—22no—
176C22×C4×C23—41no—
177C22×C4×C23C2×D411yescharacteristic
178C22×C4×C23C2×D411yescharacteristic
179C22×C4×C23C2×D411yescharacteristic
180Q16×C23—82no—
181SD16×C23—82no—
182M16×C23—42no—
183C2×C8×C23—22no—
184C2×C8×C23—22no—
185C2×C8×C23—22no—
186C2×C8×C23—42no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
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—21no—
198C4⋊C4×C23—21no—
199C4⋊C4×C23—21no—
200C4⋊C4×C23—21no—
201C4⋊C4×C23—21no—
202C4⋊C4×C23—22no—
203C4⋊C4×C23—22no—
204C4⋊C4×C23—41no—
205C4⋊C4×C23—41no—
206C4⋊C4×C23C2×D411yescharacteristic
207C4⋊C4×C23C2×D411yescharacteristic
208C4⋊C4×C23C2×D411yescharacteristic
209C4⋊C4×C23C2×D411yescharacteristic
210C22⋊C4×C23—22no—
211C22⋊C4×C23—22no—
212C22⋊C4×C23—22no—
213C22⋊C4×C23—22no—
214C22⋊C4×C23—22no—
215C22⋊C4×C23—21no—
216C22⋊C4×C23—22no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—21no—
219C22⋊C4×C23—22no—
220C22⋊C4×C23—21no—
221C22⋊C4×C23—21no—
222C42×C23—22no—
223C42×C23—21no—
224C42×C23—21no—
225C42×C23—21no—
226C42×C23—21no—
227C42×C23—21no—
228C42×C23C2×D411yescharacteristic
229C42×C23C2×D411yescharacteristic
230C42×C23C2×D411yescharacteristic
231C42×C23C2411yescharacteristic
232#18573C2311yesmaximal, characteristic, direct product factor, Sylow
233C23×C23—21no—
234C23×C23—22no—
235C23×C23#12011yescharacteristic
236Q8×C23—21no—
237Q8×C23—21no—
238Q8×C23—21no—
239Q8×C23—22no—
240Q8×C23—42no—
241Q8×C23—42no—
242Q8×C23—42no—
243Q8×C23—42no—
244Q8×C23—82no—
245Q8×C23—82no—
246Q8×C23#13712yes—
247D4×C23—22no—
248D4×C23—42no—
249D4×C23—44no—
250D4×C23—44no—
251D4×C23—42no—
252D4×C23—41no—
253D4×C23—41no—
254D4×C23—42no—
255D4×C23—82no—
256C2×C4×C23—21no—
257C2×C4×C23—22no—
258C2×C4×C23—21no—
259C2×C4×C23—21no—
260C2×C4×C23—22no—
261C2×C4×C23—21no—
262C2×C4×C23—21no—
263C2×C4×C23—21no—
264C2×C4×C23—21no—
265C2×C4×C23—22no—
266C2×C4×C23—21no—
267C2×C4×C23—21no—
268C2×C4×C23—42no—
269C2×C4×C23—44no—
270C2×C4×C23—42no—
271C2×C4×C23—41no—
272C2×C4×C23—41no—
273C2×C4×C23—42no—
274C2×C4×C23—82no—
275C2×C4×C23—82no—
276C2×C4×C23—82no—
277C2×C4×C23—82no—
278C2×C4×C23#12011yescharacteristic
279C2×C4×C23#12011yescharacteristic
280C2×C4×C23#12011yescharacteristic
281C2×C4×C23C22×D411yescharacteristic
282C2×C4×C23C22×D411yescharacteristic
283C2×C4×C23C22×D411yescharacteristic
284C8×C23—42no—
285C8×C23—42no—
286C8×C23—42no—
287#3522C2×C2311yescharacteristic
288#3218C2×C2311yescharacteristic, complemented
289#3141C2×C2311yescharacteristic, complemented
290C2×#475C2×C2311yescharacteristic, complemented
291#1645C2×C2312yescomplemented
292#1643C2×C2312yescomplemented
293#1642C2×C2312yescomplemented
294#1640C2×C2312yescomplemented
295#1551C2×C2311yescharacteristic, complemented
296#1491C2×C2311yescharacteristic, complemented
297#1483C2×C2311yescharacteristic, complemented
298C22×C23—21no—
299C22×C23—21no—
300C22×C23—42no—
301C22×C23—41no—
302C22×C23—41no—
303C22×C23—42no—
304C22×C23—42no—
305C22×C23—41no—
306C22×C23#45512yes—
307C22×C23C2×#12011yescharacteristic
308C4×C23—21no—
309C4×C23—22no—
310C4×C23—21no—
311C4×C23—21no—
312C4×C23—21no—
313C4×C23—41no—
314C4×C23—41no—
315C4×C23—41no—
316C4×C23—41no—
317C4×C23—41no—
318C4×C23—42no—
319C4×C23—41no—
320C4×C23—82no—
321C4×C23C2×#13711yescharacteristic
322C4×C23#57511yescharacteristic
323C4×C23#57712yes—
324#559—21no—
325#559C22×C2311yescharacteristic
326Q82C22×C2311yescharacteristic
327#554—21no—
328#554C22×C2311yescharacteristic
329#550C22×C2311yescharacteristic
330#548—21no—
331#547—22no—
332#547—21no—
333#542—22no—
334#542—21no—
335#532—21no—
336#532C22×C2311yescharacteristic
337C2×C4⋊Q8C22×C2311yescharacteristic
338#529—21no—
339#529C22×C2311yescharacteristic
340#520—21no—
341C2×C4×Q8C22×C2311yescharacteristic
342#497—21no—
343#491—21no—
344#489—22no—
345#487C22×C2312yes—
346#486C22×C2312yes—
347#475—21no—
348#475—21no—
349#475C22×C2312yes—
350#475C22×C2312yes—
351#462C22×C2312yes—
352#461C22×C2312yes—
353#451—21no—
354#448—21no—
355#432C22×C2311yescharacteristic
356C2×#106—21no—
357C2×C4⋊C8C22×C2311yescharacteristic
358#417—22no—
359C2×#103—21no—
360C4×M16C22×C2311yescharacteristic
361#333C22×C2312yes—
362#331C22×C2312yes—
363Q8⋊C8C22×C2312yes—
364Q8⋊C8C22×C2312yes—
365Q8⋊C8C22×C2312yes—
366D4⋊C8C22×C2312yes—
367C2×C23—22no—
368C2×C23—81no—
369C2×C23—82no—
370C2×C23#300711yescharacteristic
371C2×C23#301011yescharacteristic
372C2×C23C2×#45511yescharacteristic
373C2×#41—22no—
374C2×#41—21no—
375C2×#41D4×C2312yes—
376C22×Q8—21no—
377C22×Q8D4×C2311yescharacteristic
378C2×Q16—22no—
379C2×SD16—22no—
380C2×M16—21no—
381C22×C8—21no—
382C4⋊Q8—22no—
383C4⋊Q8—21no—
384C4⋊Q8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8C23×C2312yes—
389C4⋊Q8C23×C2311yescharacteristic
390#127—21no—
391#127C23×C2311yescharacteristic
392#126—22no—
393#126—21no—
394#125—21no—
395#125D4×C2311yescharacteristic
396#124—22no—
397#124—22no—
398#124—21no—
399#124—21no—
400#124D4×C2311yescharacteristic
401#124D4×C2312yes—
402#123—22no—
403#123—22no—
404#123—22no—
405#123—22no—
406#123—21no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8—21no—
412C22⋊Q8D4×C2311yescharacteristic
413#121—22no—
414#121—22no—
415#121—21no—
416#121—21no—
417#121—21no—
418#121D4×C2311yescharacteristic
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8D4×C2311yescharacteristic
424C4×Q8D4×C2311yescharacteristic
425C4×Q8C23×C2311yescharacteristic
426C4×Q8C23×C2311yescharacteristic
427C4×Q8C23×C2311yescharacteristic
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4C23×C2311yescharacteristic
439#117—22no—
440#117—21no—
441C2×C4⋊C4—22no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4D4×C2311yescharacteristic
447C2×C42—21no—
448C2×C42C23×C2311yescharacteristic
449#106—22no—
450#106—22no—
451C4⋊C8—21no—
452C4⋊C8C23×C2312yes—
453C4⋊C8C23×C2312yes—
454#103—22no—
455#103—22no—
456#103—24no—
457#103—22no—
458#103—22no—
459#102—24no—
460C22⋊C8—21no—
461#97—21no—
462#97—21no—
463C4×C8C23×C2312yes—
464C23#1857311yescharacteristic, direct product factor, Sylow
465#41—44no—
466#41—42no—
467#41—42no—
468#41—84no—
469C2×Q8—22no—
470C2×Q8—21no—
471C2×Q8—22no—
472C2×Q8—22no—
473C2×Q8—41no—
474C2×Q8—41no—
475C2×Q8—82no—
476C2×Q8C2×D4×C2311yescharacteristic
477C2×Q8C2×D4×C2311yescharacteristic
478C2×Q8C2×D4×C2311yescharacteristic
479C2×Q8C2×D4×C2312yes—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×D4×C2311yescharacteristic
487C2×D4C2×D4×C2312yes—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—41no—
496C22×C4C2×D4×C2311yescharacteristic
497C22×C4C2×D4×C2311yescharacteristic
498C22×C4C2×D4×C2311yescharacteristic
499Q16—82no—
500SD16—82no—
501M16—42no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—42no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4C2×D4×C2311yescharacteristic
526C4⋊C4C2×D4×C2311yescharacteristic
527C4⋊C4C2×D4×C2311yescharacteristic
528C4⋊C4C2×D4×C2311yescharacteristic
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42C2×D4×C2311yescharacteristic
548C42C2×D4×C2311yescharacteristic
549C42C2×D4×C2311yescharacteristic
550C42C24×C2311yescharacteristic
551C23—21no—
552C23—22no—
553C23C23×#12011yescharacteristic
554Q8—21no—
555Q8—21no—
556Q8—21no—
557Q8—22no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—82no—
563Q8—82no—
564Q8C23×#13712yes—
565D4—22no—
566D4—42no—
567D4—44no—
568D4—44no—
569D4—42no—
570D4—41no—
571D4—41no—
572D4—42no—
573D4—82no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4C23×#12011yescharacteristic
597C2×C4C23×#12011yescharacteristic
598C2×C4C23×#12011yescharacteristic
599C2×C4C22×D4×C2311yescharacteristic
600C2×C4C22×D4×C2311yescharacteristic
601C2×C4C22×D4×C2311yescharacteristic
602C8—42no—
603C8—42no—
604C8—42no—
605C22—21no—
606C22—21no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22C23×#45512yes—
614C22C2×C23×#12011yescharacteristic
615C4—21no—
616C4—22no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—82no—
628C4C2×C23×#13711yescharacteristic
629C4C23×#57511yescharacteristic
630C4C23×#57712yes—
631C2—22no—
632C2—81no—
633C2—82no—
634C2C23×#300711yescharacteristic
635C2C23×#301011yescharacteristic
636C2C2×C23×#45511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3522C2
LCSG ▷ C42 ▷ 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×#3218C2semidirect
3. C23×#3141C2semidirect
4. C2×C23×#475C2semidirect
5. C23×#1645C2semidirect
6. C23×#1643C2semidirect
7. C23×#1642C2semidirect
8. C23×#1640C2semidirect
9. C23×#1551C2semidirect
10. C23×#1491C2semidirect
11. C23×#1483C2semidirect
232. #18573C23direct (N × H)
288. #3218C2×C23semidirect
289. #3141C2×C23semidirect
290. C2×#475C2×C23semidirect
291. #1645C2×C23semidirect
292. #1643C2×C23semidirect
293. #1642C2×C23semidirect
294. #1640C2×C23semidirect
295. #1551C2×C23semidirect
296. #1491C2×C23semidirect
297. #1483C2×C23semidirect
464. C23#18573direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35222
C23×#32182
C23×#31412
C2×C23×#4752
C23×#16452
C23×#16432
C23×#16422
C23×#16402
C23×#15512
C23×#14912
C23×#14832
#1857323