Finite Group Databasebeta

← C23×#18349 #19165975

Subgroups — C23×#18349 #19165975

Order 5888.

954 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3492C211yesmaximal, characteristic
2C23×#3196C212yesmaximal, complemented
3C23×#2627C211yesmaximal, characteristic, complemented
4C23×#1617C211yesmaximal, characteristic, complemented
5C23×#1616C211yesmaximal, characteristic, complemented
6C23×#1615C212yesmaximal, complemented
7C23×#1605C211yesmaximal, characteristic, complemented
8C23×#1604C211yesmaximal, characteristic, complemented
9C23×#1603C212yesmaximal, 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×C22⋊Q8C2211yescharacteristic
24C23×#520—21no—
25C22×C4⋊C4×C23C2211yescharacteristic
26C23×#485—22no—
27C23×#484C2212yes—
28C23×#481C2212yes—
29C23×#470—22no—
30C23×#468—22no—
31C23×#452—22no—
32C23×#451C2212yes—
33C23×#448C2212yes—
34C23×#407C2212yes—
35C23×#400C2211yescharacteristic
36C23×#400C2211yescharacteristic
37C23×#398—21no—
38C23×#398—21no—
39C23×#395C2211yescharacteristic
40C23×#395C2211yescharacteristic
41C23×#385—21no—
42C23×#381—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×Q8×C23C2311yescharacteristic
55C23×C4×C23D411yescharacteristic
56C23×C4×C23D411yescharacteristic
57C23×C4×C23C2311yescharacteristic
58C2×Q16×C23—22no—
59C2×SD16×C23—22no—
60C2×M16×C23—22no—
61C23×C4⋊Q8—22no—
62C23×#126—22no—
63C23×#126—22no—
64C23×#125—22no—
65C23×#125—22no—
66C23×#125—22no—
67C23×#123—22no—
68C23×#123—22no—
69C23×#123—22no—
70C23×#123—22no—
71C23×#123—22no—
72C23×C22⋊Q8—22no—
73C23×C22⋊Q8—21no—
74C23×C22⋊Q8—21no—
75C23×C22⋊Q8—21no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—21no—
78C23×C22⋊Q8—22no—
79C23×C22⋊Q8—22no—
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×#121D412yes—
89C23×#121C2312yes—
90C23×#120—22no—
91C4×Q8×C23—22no—
92C4×D4×C23—22no—
93C4×D4×C23—22no—
94C4×D4×C23—22no—
95C4×D4×C23—22no—
96C4×D4×C23—22no—
97C4×D4×C23—22no—
98C4×D4×C23—22no—
99C23×#117—21no—
100C23×#117—22no—
101C23×#117—21no—
102C23×#117D412yes—
103C2×C4⋊C4×C23—22no—
104C2×C4⋊C4×C23—22no—
105C2×C4⋊C4×C23—22no—
106C2×C4⋊C4×C23—21no—
107C2×C4⋊C4×C23—21no—
108C2×C4⋊C4×C23—21no—
109C2×C4⋊C4×C23—21no—
110C2×C4⋊C4×C23D411yescharacteristic
111C2×C4⋊C4×C23D411yescharacteristic
112C2×C4⋊C4×C23C2311yescharacteristic
113C2×C4⋊C4×C23C2311yescharacteristic
114C2×C4⋊C4×C23C2311yescharacteristic
115C2×C22⋊C4×C23—21no—
116C2×C22⋊C4×C23—21no—
117C23×#107—22no—
118C23×#106—22no—
119C23×#103—22no—
120C23×#103—22no—
121C23×#103—22no—
122C23×#102—22no—
123C23×C22⋊C8—22no—
124C23×C22⋊C8C2312yes—
125C23×C22⋊C8C2312yes—
126C23×#95—21no—
127C23×#95—21no—
128C23×#95—21no—
129C23×#95—21no—
130C23×#95—21no—
131C23×#95—21no—
132C23×#95C2311yescharacteristic
133C23×#95C2311yescharacteristic
134C24×C23C2×D411yescharacteristic
135#41×C23—42no—
136#41×C23—42no—
137#41×C23—82no—
138#41×C23—82no—
139C2×Q8×C23—21no—
140C2×Q8×C23—21no—
141C2×Q8×C23—22no—
142C2×Q8×C23—81no—
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×C23C2×D412yes—
150C22×C4×C23—22no—
151C22×C4×C23—21no—
152C22×C4×C23—21no—
153C22×C4×C23—21no—
154C22×C4×C23—22no—
155C22×C4×C23—21no—
156C22×C4×C23—22no—
157C22×C4×C23—22no—
158C22×C4×C23—21no—
159C22×C4×C23—21no—
160C22×C4×C23—21no—
161C22×C4×C23—21no—
162C22×C4×C23—41no—
163C22×C4×C23—81no—
164C22×C4×C23—81no—
165C22×C4×C23C2×D411yescharacteristic
166C22×C4×C23C2×D411yescharacteristic
167C22×C4×C23C2×D411yescharacteristic
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D411yescharacteristic
170C22×C4×C23C2×D412yes—
171C22×C4×C23C2×D412yes—
172C22×C4×C23C2411yescharacteristic
173Q16×C23—82no—
174SD16×C23—82no—
175M16×C23—82no—
176C2×C8×C23—22no—
177C2×C8×C23—22no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—21no—
182C4⋊C4×C23—21no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—42no—
194C4⋊C4×C23—81no—
195C4⋊C4×C23—81no—
196C4⋊C4×C23—81no—
197C4⋊C4×C23C2×D412yes—
198C4⋊C4×C23C2×D412yes—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—22no—
204C22⋊C4×C23—22no—
205C22⋊C4×C23—22no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—42no—
209C22⋊C4×C23—41no—
210C22⋊C4×C23—41no—
211C42×C23—22no—
212C42×C23—22no—
213C42×C23—22no—
214#18349C2311yesmaximal, characteristic, direct product factor, Sylow
215C23×C23—21no—
216C23×C23—21no—
217C23×C23—22no—
218C23×C23—41no—
219C23×C23—41no—
220C23×C23#12011yescharacteristic
221C23×C23#12011yescharacteristic
222C23×C23C22×D411yescharacteristic
223Q8×C23—42no—
224Q8×C23—42no—
225Q8×C23—42no—
226Q8×C23—42no—
227Q8×C23—81no—
228D4×C23—42no—
229D4×C23—42no—
230D4×C23—44no—
231D4×C23—44no—
232D4×C23—42no—
233D4×C23—42no—
234D4×C23—81no—
235C2×C4×C23—21no—
236C2×C4×C23—21no—
237C2×C4×C23—22no—
238C2×C4×C23—22no—
239C2×C4×C23—21no—
240C2×C4×C23—22no—
241C2×C4×C23—21no—
242C2×C4×C23—21no—
243C2×C4×C23—21no—
244C2×C4×C23—21no—
245C2×C4×C23—41no—
246C2×C4×C23—42no—
247C2×C4×C23—42no—
248C2×C4×C23—41no—
249C2×C4×C23—42no—
250C2×C4×C23—42no—
251C2×C4×C23—44no—
252C2×C4×C23—42no—
253C2×C4×C23—41no—
254C2×C4×C23—41no—
255C2×C4×C23—41no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—41no—
260C2×C4×C23—41no—
261C2×C4×C23—42no—
262C2×C4×C23—41no—
263C2×C4×C23—41no—
264C2×C4×C23—41no—
265C2×C4×C23—41no—
266C2×C4×C23—82no—
267C2×C4×C23—81no—
268C2×C4×C23—82no—
269C2×C4×C23#12012yes—
270C2×C4×C23#13712yes—
271C2×C4×C23C22×D412yes—
272C8×C23—42no—
273C8×C23—42no—
274#3492C2×C2311yescharacteristic
275#3196C2×C2312yescomplemented
276#2627C2×C2311yescharacteristic, complemented
277#1617C2×C2311yescharacteristic, complemented
278#1616C2×C2311yescharacteristic, complemented
279#1615C2×C2312yescomplemented
280#1605C2×C2311yescharacteristic, complemented
281#1604C2×C2311yescharacteristic, complemented
282#1603C2×C2312yescomplemented
283#1513C2×C2312yescomplemented
284#1463C2×C2311yescharacteristic, complemented
285C22×C23—21no—
286C22×C23—21no—
287C22×C23—21no—
288C22×C23—21no—
289C22×C23—21no—
290C22×C23—21no—
291C22×C23—21no—
292C22×C23—21no—
293C22×C23—42no—
294C22×C23—42no—
295C22×C23—42no—
296C22×C23—42no—
297C22×C23—42no—
298C22×C23C2×#12011yescharacteristic
299C22×C23C2×#13711yescharacteristic
300C22×C23#57511yescharacteristic
301C4×C23—22no—
302C4×C23—22no—
303C4×C23—22no—
304C4×C23—41no—
305C4×C23—42no—
306C4×C23—42no—
307C4×C23—41no—
308C4×C23—41no—
309C4×C23—42no—
310C4×C23—81no—
311C4×C23—81no—
312C4×C23—81no—
313C4×C23—81no—
314#554—22no—
315#553—22no—
316#552—22no—
317#547—22no—
318#544C22×C2311yescharacteristic
319#539—21no—
320#539C22×C2312yes—
321#537C22×C2311yescharacteristic
322#525—21no—
323#525—21no—
324#525C22×C2312yes—
325C2×C22⋊Q8C22×C2311yescharacteristic
326#520—21no—
327C22×C4⋊C4C22×C2311yescharacteristic
328#485—22no—
329#484C22×C2312yes—
330#481C22×C2312yes—
331#470—22no—
332#468—22no—
333#452—22no—
334#451C22×C2312yes—
335#448C22×C2312yes—
336#407C22×C2312yes—
337#400C22×C2311yescharacteristic
338#400C22×C2311yescharacteristic
339#398—21no—
340#398—21no—
341#395C22×C2311yescharacteristic
342#395C22×C2311yescharacteristic
343#385—21no—
344#381—21no—
345#380C22×C2311yescharacteristic
346#328C22×C2312yes—
347#328C22×C2312yes—
348#327C22×C2312yes—
349#327C22×C2312yes—
350#324C22×C2312yes—
351#324C22×C2311yescharacteristic
352#324C22×C2311yescharacteristic
353C2×C23—21no—
354C2×C23—21no—
355C2×C23—41no—
356C2×C23—41no—
357C2×C23—82no—
358C2×C23#300711yescharacteristic
359C2×C23#302011yescharacteristic
360C2×C23#302611yescharacteristic
361C2×#41—21no—
362C2×#41—21no—
363C2×#41D4×C2312yes—
364C22×Q8C23×C2311yescharacteristic
365C23×C4D4×C2311yescharacteristic
366C23×C4D4×C2311yescharacteristic
367C23×C4C23×C2311yescharacteristic
368C2×Q16—22no—
369C2×SD16—22no—
370C2×M16—22no—
371C4⋊Q8—22no—
372#126—22no—
373#126—22no—
374#125—22no—
375#125—22no—
376#125—22no—
377#123—22no—
378#123—22no—
379#123—22no—
380#123—22no—
381#123—22no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—21no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8D4×C2312yes—
392C22⋊Q8C23×C2312yes—
393#121—21no—
394#121—21no—
395#121—21no—
396#121—22no—
397#121—21no—
398#121D4×C2312yes—
399#121C23×C2312yes—
400#120—22no—
401C4×Q8—22no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409#117—22no—
410#117—21no—
411#117—21no—
412#117D4×C2312yes—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4D4×C2311yescharacteristic
421C2×C4⋊C4D4×C2311yescharacteristic
422C2×C4⋊C4C23×C2311yescharacteristic
423C2×C4⋊C4C23×C2311yescharacteristic
424C2×C4⋊C4C23×C2311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427#107—22no—
428#106—22no—
429#103—22no—
430#103—22no—
431#103—22no—
432#102—22no—
433C22⋊C8—22no—
434C22⋊C8C23×C2312yes—
435C22⋊C8C23×C2312yes—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95C23×C2311yescharacteristic
443#95C23×C2311yescharacteristic
444C23#1834911yescharacteristic, direct product factor, Sylow
445C24C2×D4×C2311yescharacteristic
446#41—42no—
447#41—42no—
448#41—82no—
449#41—82no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—81no—
454C2×Q8C2×D4×C2312yes—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4C2×D4×C2312yes—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—41no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4C2×D4×C2311yescharacteristic
477C22×C4C2×D4×C2311yescharacteristic
478C22×C4C2×D4×C2311yescharacteristic
479C22×C4C2×D4×C2311yescharacteristic
480C22×C4C2×D4×C2311yescharacteristic
481C22×C4C2×D4×C2312yes—
482C22×C4C2×D4×C2312yes—
483C22×C4C24×C2311yescharacteristic
484Q16—82no—
485SD16—82no—
486M16—82no—
487C2×C8—22no—
488C2×C8—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4C2×D4×C2312yes—
509C4⋊C4C2×D4×C2312yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—42no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C42—22no—
523C42—22no—
524C42—22no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23C23×#12011yescharacteristic
531C23C23×#12011yescharacteristic
532C23C22×D4×C2311yescharacteristic
533Q8—42no—
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—81no—
538D4—42no—
539D4—42no—
540D4—44no—
541D4—44no—
542D4—42no—
543D4—42no—
544D4—81no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4C23×#12012yes—
580C2×C4C23×#13712yes—
581C2×C4C22×D4×C2312yes—
582C8—42no—
583C8—42no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22C2×C23×#12011yescharacteristic
598C22C2×C23×#13711yescharacteristic
599C22C23×#57511yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C2—21no—
614C2—21no—
615C2—41no—
616C2—41no—
617C2—82no—
618C2C23×#300711yescharacteristic
619C2C23×#302011yescharacteristic
620C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3492C2
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
2. C23×#3196C2semidirect
3. C23×#2627C2semidirect
4. C23×#1617C2semidirect
5. C23×#1616C2semidirect
6. C23×#1615C2semidirect
7. C23×#1605C2semidirect
8. C23×#1604C2semidirect
9. C23×#1603C2semidirect
10. C23×#1513C2semidirect
11. C23×#1463C2semidirect
214. #18349C23direct (N × H)
275. #3196C2×C23semidirect
276. #2627C2×C23semidirect
277. #1617C2×C23semidirect
278. #1616C2×C23semidirect
279. #1615C2×C23semidirect
280. #1605C2×C23semidirect
281. #1604C2×C23semidirect
282. #1603C2×C23semidirect
283. #1513C2×C23semidirect
284. #1463C2×C23semidirect
444. C23#18349direct (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×#31962
C23×#26272
C23×#16172
C23×#16162
C23×#16152
C23×#16052
C23×#16042
C23×#16032
C23×#15132
C23×#14632
#1834923