Finite Group Databasebeta

← C23×#18341 #19165967

Subgroups — C23×#18341 #19165967

Order 5888.

898 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×#3455C211yesmaximal, characteristic
2C23×#3236C211yesmaximal, characteristic, complemented
3C23×#3234C211yesmaximal, characteristic, complemented
4C23×#2611C211yesmaximal, characteristic, complemented
5C23×#1618C212yesmaximal, complemented
6C23×#1615C212yesmaximal, complemented
7C23×#1612C212yesmaximal, complemented
8C23×#1609C212yesmaximal, complemented
9C23×#1506C211yesmaximal, characteristic, complemented
10C23×#1505C211yesmaximal, 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—
24C2×C23×#125C2211yescharacteristic
25C23×#525—21no—
26C4×#41×C23—21no—
27C2×C23×#117C2211yescharacteristic
28C23×#501—21no—
29C23×#501—21no—
30C23×#491—21no—
31C23×#490—21no—
32C23×#484C2212yes—
33C23×#483C2212yes—
34C23×#481C2212yes—
35C23×#480C2212yes—
36C23×#479—22no—
37C23×#478—22no—
38C23×#432C2211yescharacteristic
39C23×#432C2211yescharacteristic
40C23×#400C2211yescharacteristic
41C23×#399C2211yescharacteristic
42C23×#398C2211yescharacteristic
43C23×#397C2211yescharacteristic
44C23×#389—21no—
45C23×#388—21no—
46C23×#387—21no—
47C23×#385—21no—
48C23×#376C2211yescharacteristic
49C23×#328C2212yes—
50C23×#328C2212yes—
51C23×#327C2212yes—
52C23×#327C2212yes—
53C23×#324C2212yes—
54C23×#323C2212yes—
55C2×#41×C23—21no—
56C23×C4×C23D411yescharacteristic
57C23×C4⋊Q8D411yescharacteristic
58C23×C4⋊Q8D411yescharacteristic
59C23×C4⋊Q8D411yescharacteristic
60C23×#127D411yescharacteristic
61C23×#126—22no—
62C23×#126—22no—
63C23×#126—22no—
64C23×#126—22no—
65C23×#125—22no—
66C23×#125—22no—
67C23×#125—21no—
68C23×#125—21no—
69C23×#125—21no—
70C23×#125—21no—
71C23×#125—21no—
72C23×#124—22no—
73C23×#124—22no—
74C23×#124—21no—
75C23×#123—22no—
76C23×#123—22no—
77C23×#123—22no—
78C23×#123—22no—
79C23×#123—22no—
80C23×C22⋊Q8—21no—
81C23×C22⋊Q8—21no—
82C23×C22⋊Q8—22no—
83C23×C22⋊Q8—22no—
84C23×C22⋊Q8—21no—
85C23×C22⋊Q8C2312yes—
86C23×#121—21no—
87C23×#121—21no—
88C23×#121—22no—
89C23×#121—22no—
90C23×#121—21no—
91C23×#121C2312yes—
92C23×#120—22no—
93C4×Q8×C23—21no—
94C4×Q8×C23—21no—
95C4×Q8×C23—21no—
96C4×Q8×C23D412yes—
97C4×D4×C23—21no—
98C4×D4×C23—21no—
99C4×D4×C23—22no—
100C4×D4×C23—22no—
101C4×D4×C23—22no—
102C4×D4×C23—22no—
103C4×D4×C23—21no—
104C4×D4×C23D412yes—
105C23×#117—22no—
106C23×#117—21no—
107C23×#117—21no—
108C23×#117—21no—
109C23×#117—21no—
110C23×#117—21no—
111C2×C4⋊C4×C23—21no—
112C2×C4⋊C4×C23—21no—
113C2×C4⋊C4×C23—21no—
114C2×C4⋊C4×C23—21no—
115C2×C4⋊C4×C23D411yescharacteristic
116C2×C4⋊C4×C23C2311yescharacteristic
117C2×C4⋊C4×C23C2311yescharacteristic
118C2×C4⋊C4×C23C2311yescharacteristic
119C2×C22⋊C4×C23—21no—
120C2×C22⋊C4×C23—21no—
121C2×C22⋊C4×C23D411yescharacteristic
122C2×C22⋊C4×C23C2311yescharacteristic
123C2×C42×C23—22no—
124C2×C42×C23—21no—
125C2×C42×C23D411yescharacteristic
126C2×C42×C23C2311yescharacteristic
127C23×#107—22no—
128C23×#107—22no—
129C23×#106—22no—
130C23×#106—22no—
131C23×C4⋊C8—21no—
132C23×C4⋊C8—21no—
133C23×#103—22no—
134C23×#103—22no—
135C23×#102—22no—
136C23×#102—22no—
137C23×C22⋊C8C2312yes—
138C23×C22⋊C8C2312yes—
139C23×#97—21no—
140C23×#97—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×C23C2×D412yes—
154C2×D4×C23—22no—
155C2×D4×C23—22no—
156C2×D4×C23—22no—
157C2×D4×C23—21no—
158C2×D4×C23—21no—
159C2×D4×C23C2×D412yes—
160C22×C4×C23—21no—
161C22×C4×C23—22no—
162C22×C4×C23—22no—
163C22×C4×C23—22no—
164C22×C4×C23—21no—
165C22×C4×C23—21no—
166C22×C4×C23—21no—
167C22×C4×C23—21no—
168C22×C4×C23—21no—
169C22×C4×C23—41no—
170C22×C4×C23—41no—
171C22×C4×C23C2×D411yescharacteristic
172C22×C4×C23C2×D411yescharacteristic
173C22×C4×C23C2×D411yescharacteristic
174C22×C4×C23C2×D411yescharacteristic
175C22×C4×C23C2×D411yescharacteristic
176C22×C4×C23C2411yescharacteristic
177C2×C8×C23—22no—
178C2×C8×C23—22no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—21no—
184C4⋊C4×C23—21no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—21no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—42no—
198C4⋊C4×C23—42no—
199C4⋊C4×C23—41no—
200C4⋊C4×C23—41no—
201C4⋊C4×C23—41no—
202C4⋊C4×C23—41no—
203C4⋊C4×C23C2×D412yes—
204C4⋊C4×C23C2×D412yes—
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—21no—
213C22⋊C4×C23—21no—
214C22⋊C4×C23—22no—
215C22⋊C4×C23—21no—
216C22⋊C4×C23—21no—
217C22⋊C4×C23—22no—
218C22⋊C4×C23—42no—
219C22⋊C4×C23—42no—
220C42×C23—22no—
221C42×C23—22no—
222C42×C23—21no—
223C42×C23—22no—
224C42×C23—21no—
225C42×C23—42no—
226C42×C23C2×D411yescharacteristic
227C42×C23C2×D411yescharacteristic
228C42×C23C2×D411yescharacteristic
229C42×C23C2×D411yescharacteristic
230#18341C2311yesmaximal, characteristic, direct product factor, Sylow
231C23×C23—21no—
232C23×C23—22no—
233C23×C23—21no—
234C23×C23—21no—
235C23×C23—41no—
236C23×C23—41no—
237C23×C23C22×D411yescharacteristic
238Q8×C23—42no—
239Q8×C23—42no—
240Q8×C23—81no—
241D4×C23—42no—
242D4×C23—42no—
243D4×C23—42no—
244D4×C23—42no—
245D4×C23—81no—
246D4×C23—82no—
247D4×C23—82no—
248C2×C4×C23—21no—
249C2×C4×C23—21no—
250C2×C4×C23—22no—
251C2×C4×C23—21no—
252C2×C4×C23—21no—
253C2×C4×C23—22no—
254C2×C4×C23—22no—
255C2×C4×C23—22no—
256C2×C4×C23—21no—
257C2×C4×C23—21no—
258C2×C4×C23—21no—
259C2×C4×C23—41no—
260C2×C4×C23—41no—
261C2×C4×C23—42no—
262C2×C4×C23—42no—
263C2×C4×C23—44no—
264C2×C4×C23—41no—
265C2×C4×C23—41no—
266C2×C4×C23—41no—
267C2×C4×C23—41no—
268C2×C4×C23—41no—
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—82no—
278C2×C4×C23—82no—
279C2×C4×C23#12011yescharacteristic
280C2×C4×C23#12011yescharacteristic
281C2×C4×C23#12011yescharacteristic
282C2×C4×C23#12011yescharacteristic
283C2×C4×C23C22×D411yescharacteristic
284C2×C4×C23C22×D411yescharacteristic
285C8×C23—42no—
286C8×C23—42no—
287#3455C2×C2311yescharacteristic
288#3236C2×C2311yescharacteristic, complemented
289#3234C2×C2311yescharacteristic, complemented
290#2611C2×C2311yescharacteristic, complemented
291#1618C2×C2312yescomplemented
292#1615C2×C2312yescomplemented
293#1612C2×C2312yescomplemented
294#1609C2×C2312yescomplemented
295#1506C2×C2311yescharacteristic, complemented
296#1505C2×C2311yescharacteristic, complemented
297#1464C2×C2311yescharacteristic, complemented
298C22×C23—21no—
299C22×C23—21no—
300C22×C23—42no—
301C22×C23—41no—
302C22×C23—42no—
303C22×C23—42no—
304C22×C23—41no—
305C22×C23—42no—
306C22×C23—41no—
307C22×C23—81no—
308C22×C23C2×#12011yescharacteristic
309C22×C23#57511yescharacteristic
310C22×C23#57511yescharacteristic
311C4×C23—21no—
312C4×C23—21no—
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—42no—
322C4×C23—41no—
323C4×C23—42no—
324C4×C23—81no—
325C4×C23—81no—
326C4×C23—81no—
327#542—21no—
328#542—21no—
329#541—21no—
330#540—21no—
331#539—21no—
332#539C22×C2312yes—
333#538—21no—
334#538C22×C2312yes—
335#533C22×C2311yescharacteristic
336#532C22×C2311yescharacteristic
337#529—22no—
338#529—22no—
339C2×#125C22×C2311yescharacteristic
340#525—21no—
341C4×#41—21no—
342C2×#117C22×C2311yescharacteristic
343#501—21no—
344#501—21no—
345#491—21no—
346#490—21no—
347#484C22×C2312yes—
348#483C22×C2312yes—
349#481C22×C2312yes—
350#480C22×C2312yes—
351#479—22no—
352#478—22no—
353#432C22×C2311yescharacteristic
354#432C22×C2311yescharacteristic
355#400C22×C2311yescharacteristic
356#399C22×C2311yescharacteristic
357#398C22×C2311yescharacteristic
358#397C22×C2311yescharacteristic
359#389—21no—
360#388—21no—
361#387—21no—
362#385—21no—
363#376C22×C2311yescharacteristic
364#328C22×C2312yes—
365#328C22×C2312yes—
366#327C22×C2312yes—
367#327C22×C2312yes—
368#324C22×C2312yes—
369#323C22×C2312yes—
370C2×C23—21no—
371C2×C23—21no—
372C2×C23—82no—
373C2×C23—81no—
374C2×C23#300511yescharacteristic
375C2×C23#301811yescharacteristic
376C2×C23#302811yescharacteristic
377C2×#41—21no—
378C23×C4D4×C2311yescharacteristic
379C4⋊Q8D4×C2311yescharacteristic
380C4⋊Q8D4×C2311yescharacteristic
381C4⋊Q8D4×C2311yescharacteristic
382#127D4×C2311yescharacteristic
383#126—22no—
384#126—22no—
385#126—22no—
386#126—22no—
387#125—22no—
388#125—22no—
389#125—21no—
390#125—21no—
391#125—21no—
392#125—21no—
393#125—21no—
394#124—22no—
395#124—22no—
396#124—21no—
397#123—22no—
398#123—22no—
399#123—22no—
400#123—22no—
401#123—22no—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406C22⋊Q8—21no—
407C22⋊Q8C23×C2312yes—
408#121—21no—
409#121—21no—
410#121—22no—
411#121—22no—
412#121—21no—
413#121C23×C2312yes—
414#120—22no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8D4×C2312yes—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424C4×D4—22no—
425C4×D4—21no—
426C4×D4D4×C2312yes—
427#117—22no—
428#117—21no—
429#117—21no—
430#117—21no—
431#117—21no—
432#117—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4D4×C2311yescharacteristic
438C2×C4⋊C4C23×C2311yescharacteristic
439C2×C4⋊C4C23×C2311yescharacteristic
440C2×C4⋊C4C23×C2311yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4D4×C2311yescharacteristic
444C2×C22⋊C4C23×C2311yescharacteristic
445C2×C42—22no—
446C2×C42—21no—
447C2×C42D4×C2311yescharacteristic
448C2×C42C23×C2311yescharacteristic
449#107—22no—
450#107—22no—
451#106—22no—
452#106—22no—
453C4⋊C8—21no—
454C4⋊C8—21no—
455#103—22no—
456#103—22no—
457#102—22no—
458#102—22no—
459C22⋊C8C23×C2312yes—
460C22⋊C8C23×C2312yes—
461#97—21no—
462#97—21no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95C23×C2311yescharacteristic
469#95C23×C2311yescharacteristic
470C23#1834111yescharacteristic, direct product factor, Sylow
471C24C2×D4×C2311yescharacteristic
472#41—42no—
473#41—42no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8C2×D4×C2312yes—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4C2×D4×C2312yes—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4C2×D4×C2311yescharacteristic
495C22×C4C2×D4×C2311yescharacteristic
496C22×C4C2×D4×C2311yescharacteristic
497C22×C4C2×D4×C2311yescharacteristic
498C22×C4C2×D4×C2311yescharacteristic
499C22×C4C24×C2311yescharacteristic
500C2×C8—22no—
501C2×C8—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4C2×D4×C2312yes—
527C4⋊C4C2×D4×C2312yes—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—22no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C42—22no—
544C42—22no—
545C42—21no—
546C42—22no—
547C42—21no—
548C42—42no—
549C42C2×D4×C2311yescharacteristic
550C42C2×D4×C2311yescharacteristic
551C42C2×D4×C2311yescharacteristic
552C42C2×D4×C2311yescharacteristic
553C23—21no—
554C23—21no—
555C23—22no—
556C23—21no—
557C23—41no—
558C23—41no—
559C23C22×D4×C2311yescharacteristic
560Q8—42no—
561Q8—42no—
562Q8—81no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—81no—
568D4—82no—
569D4—82no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4C23×#12011yescharacteristic
602C2×C4C23×#12011yescharacteristic
603C2×C4C23×#12011yescharacteristic
604C2×C4C23×#12011yescharacteristic
605C2×C4C22×D4×C2311yescharacteristic
606C2×C4C22×D4×C2311yescharacteristic
607C8—42no—
608C8—42no—
609C22—21no—
610C22—21no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—81no—
619C22C2×C23×#12011yescharacteristic
620C22C23×#57511yescharacteristic
621C22C23×#57511yescharacteristic
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—81no—
638C2—21no—
639C2—21no—
640C2—81no—
641C2—82no—
642C2C23×#300511yescharacteristic
643C2C23×#301811yescharacteristic
644C2C23×#302811yescharacteristic
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×#3236C2semidirect
3. C23×#3234C2semidirect
4. C23×#2611C2semidirect
5. C23×#1618C2semidirect
6. C23×#1615C2semidirect
7. C23×#1612C2semidirect
8. C23×#1609C2semidirect
9. C23×#1506C2semidirect
10. C23×#1505C2semidirect
11. C23×#1464C2semidirect
230. #18341C23direct (N × H)
288. #3236C2×C23semidirect
289. #3234C2×C23semidirect
290. #2611C2×C23semidirect
291. #1618C2×C23semidirect
292. #1615C2×C23semidirect
293. #1612C2×C23semidirect
294. #1609C2×C23semidirect
295. #1506C2×C23semidirect
296. #1505C2×C23semidirect
297. #1464C2×C23semidirect
470. C23#18341direct (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×#32362
C23×#32342
C23×#26112
C23×#16182
C23×#16152
C23×#16122
C23×#16092
C23×#15062
C23×#15052
C23×#14642
#1834123