Finite Group Databasebeta

← C23×#18177 #19165803

Subgroups — C23×#18177 #19165803

Order 5888.

950 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3491C211yesmaximal, characteristic
2C23×#3117C212yesmaximal, complemented
3C23×#2396C211yesmaximal, characteristic, complemented
4C23×#1615C212yesmaximal, complemented
5C23×#1614C211yesmaximal, characteristic, complemented
6C23×#1613C211yesmaximal, characteristic
7C23×#1603C212yesmaximal, complemented
8C23×#1602C211yesmaximal, characteristic, complemented
9C23×#1601C211yesmaximal, characteristic
10C23×#1509C212yesmaximal, complemented
11C23×#1464C211yesmaximal, characteristic, complemented
12C23×#552—22no—
13C23×#552—22no—
14C23×#551—22no—
15C23×#551—22no—
16C23×#543C2211yescharacteristic
17C23×#539—21no—
18C23×#539C2212yes—
19C23×#534C2211yescharacteristic
20C23×#529—21no—
21C23×#529C2212yes—
22C2×C23×#123—21no—
23C23×#518—21no—
24C2×C4×Q8×C23C2211yescharacteristic, complemented
25C2×C23×#117C2211yescharacteristic
26C23×#484C2212yescomplemented
27C23×#481C2212yes—
28C23×#479—22no—
29C23×#464—22no—
30C23×#451C2212yescomplemented
31C23×#448C2212yes—
32C23×#441—22no—
33C23×#440—22no—
34C23×#432C2212yescomplemented
35C23×#400C2211yescharacteristic, complemented
36C23×#399C2211yescharacteristic
37C23×#398—21no—
38C23×#396—21no—
39C23×#395C2211yescharacteristic, complemented
40C23×#393C2211yescharacteristic
41C4×C4⋊C4×C23—21no—
42C4×C4⋊C4×C23C2211yescharacteristic, complemented
43C4×C22⋊C4×C23—21no—
44C23×#328C2212yescomplemented
45C23×#328C2212yescomplemented
46C23×#327C2212yes—
47C23×#327C2212yes—
48C23×#324C2212yescomplemented
49C23×#323C2211yescharacteristic
50C23×#323C2211yescharacteristic
51C22×Q8×C23C2311yescharacteristic
52C22×D4×C23—21no—
53C23×C4×C23D411yescharacteristic
54C2×Q16×C23—22no—
55C2×SD16×C23—22no—
56C23×#126—22no—
57C23×#126—22no—
58C23×#126—22no—
59C23×#126—22no—
60C23×#125—21no—
61C23×#125—22no—
62C23×#125—22no—
63C23×#125D412yes—
64C23×#124—21no—
65C23×#124D412yes—
66C23×#123—22no—
67C23×#123—22no—
68C23×#123—22no—
69C23×#123—22no—
70C23×#123—22no—
71C23×#123—22no—
72C23×#123—22no—
73C23×C22⋊Q8—22no—
74C23×C22⋊Q8—22no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8—21no—
77C23×C22⋊Q8—22no—
78C23×C22⋊Q8C2312yes—
79C23×#121—21no—
80C23×#121C2312yes—
81C23×#120—22no—
82C4×Q8×C23—21no—
83C4×Q8×C23—22no—
84C4×Q8×C23—22no—
85C4×Q8×C23D412yes—
86C4×D4×C23—21no—
87C4×D4×C23—22no—
88C4×D4×C23—22no—
89C4×D4×C23—22no—
90C4×D4×C23—22no—
91C4×D4×C23D412yes—
92C23×#117—22no—
93C23×#117—22no—
94C23×#117—22no—
95C23×#117—22no—
96C23×#117—22no—
97C2×C4⋊C4×C23—21no—
98C2×C4⋊C4×C23—21no—
99C2×C4⋊C4×C23—21no—
100C2×C4⋊C4×C23—21no—
101C2×C4⋊C4×C23—21no—
102C2×C4⋊C4×C23—21no—
103C2×C4⋊C4×C23D411yescharacteristic
104C2×C4⋊C4×C23C2311yescharacteristic
105C2×C4⋊C4×C23C2311yescharacteristic
106C2×C22⋊C4×C23—22no—
107C2×C22⋊C4×C23—21no—
108C2×C22⋊C4×C23—21no—
109C2×C22⋊C4×C23—21no—
110C2×C22⋊C4×C23—21no—
111C2×C22⋊C4×C23—21no—
112C2×C22⋊C4×C23D411yescharacteristic
113C2×C22⋊C4×C23C2311yescharacteristic
114C2×C42×C23—21no—
115C2×C42×C23—21no—
116C2×C42×C23D411yescharacteristic
117C2×C42×C23C2311yescharacteristic
118C23×#107—22no—
119C23×#106—22no—
120C23×C4⋊C8—22no—
121C23×#103—22no—
122C23×#103—22no—
123C23×#103—22no—
124C23×#102—22no—
125C23×C22⋊C8C2312yes—
126C23×C22⋊C8C2312yes—
127C23×#97—22no—
128C23×#95—21no—
129C23×#95—21no—
130C23×#95—21no—
131C23×#95—21no—
132C23×#95C2311yescharacteristic
133C23×#95C2311yescharacteristic
134C24×C23—21no—
135C24×C23C2×D411yescharacteristic
136C2×Q8×C23—21no—
137C2×Q8×C23—22no—
138C2×Q8×C23—42no—
139C2×Q8×C23C2×D412yes—
140C2×D4×C23—22no—
141C2×D4×C23—22no—
142C2×D4×C23—21no—
143C2×D4×C23—42no—
144C2×D4×C23—42no—
145C2×D4×C23C2×D412yes—
146C22×C4×C23—21no—
147C22×C4×C23—22no—
148C22×C4×C23—21no—
149C22×C4×C23—21no—
150C22×C4×C23—21no—
151C22×C4×C23—21no—
152C22×C4×C23—22no—
153C22×C4×C23—21no—
154C22×C4×C23—21no—
155C22×C4×C23—22no—
156C22×C4×C23—21no—
157C22×C4×C23—21no—
158C22×C4×C23—41no—
159C22×C4×C23C2×D411yescharacteristic
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×D411yescharacteristic
162C22×C4×C23C2×D411yescharacteristic
163C22×C4×C23C2×D411yescharacteristic
164C22×C4×C23C2411yescharacteristic
165Q16×C23—44no—
166SD16×C23—44no—
167C2×C8×C23—22no—
168C2×C8×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—21no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—42no—
183C4⋊C4×C23—42no—
184C4⋊C4×C23—42no—
185C4⋊C4×C23—42no—
186C4⋊C4×C23C2×D412yes—
187C4⋊C4×C23C2×D412yes—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—22no—
194C22⋊C4×C23—22no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—22no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—42no—
201C22⋊C4×C23—42no—
202C42×C23—22no—
203C42×C23—21no—
204C42×C23—21no—
205C42×C23—22no—
206C42×C23—42no—
207C42×C23—41no—
208C42×C23—41no—
209C42×C23C2×D412yes—
210C42×C23C2×D412yes—
211#18177C2311yesmaximal, characteristic, direct product factor, Sylow
212C23×C23—21no—
213C23×C23—21no—
214C23×C23—22no—
215C23×C23—41no—
216C23×C23—41no—
217C23×C23—42no—
218C23×C23—81no—
219C23×C23—81no—
220C23×C23#13712yes—
221C23×C23C22×D411yescharacteristic
222Q8×C23—24no—
223Q8×C23—42no—
224Q8×C23—42no—
225Q8×C23—81no—
226D4×C23—42no—
227D4×C23—44no—
228D4×C23—42no—
229D4×C23—42no—
230D4×C23—81no—
231C2×C4×C23—21no—
232C2×C4×C23—21no—
233C2×C4×C23—22no—
234C2×C4×C23—22no—
235C2×C4×C23—21no—
236C2×C4×C23—21no—
237C2×C4×C23—22no—
238C2×C4×C23—22no—
239C2×C4×C23—22no—
240C2×C4×C23—21no—
241C2×C4×C23—21no—
242C2×C4×C23—21no—
243C2×C4×C23—41no—
244C2×C4×C23—41no—
245C2×C4×C23—42no—
246C2×C4×C23—42no—
247C2×C4×C23—41no—
248C2×C4×C23—42no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
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—82no—
261C2×C4×C23—81no—
262C2×C4×C23—82no—
263C2×C4×C23—81no—
264C2×C4×C23#12011yescharacteristic
265C2×C4×C23#12011yescharacteristic
266C2×C4×C23#12012yes—
267C2×C4×C23C22×D412yes—
268C8×C23—42no—
269C8×C23—42no—
270#3491C2×C2311yescharacteristic
271#3117C2×C2312yescomplemented
272#2396C2×C2311yescharacteristic, complemented
273#1615C2×C2312yescomplemented
274#1614C2×C2311yescharacteristic, complemented
275#1613C2×C2311yescharacteristic
276#1603C2×C2312yescomplemented
277#1602C2×C2311yescharacteristic, complemented
278#1601C2×C2311yescharacteristic
279#1509C2×C2312yescomplemented
280#1464C2×C2311yescharacteristic, complemented
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—22no—
284C22×C23—42no—
285C22×C23—41no—
286C22×C23—41no—
287C22×C23—42no—
288C22×C23—42no—
289C22×C23—42no—
290C22×C23—81no—
291C22×C23—162no—
292C22×C23C2×#12011yescharacteristic
293C22×C23C2×#13711yescharacteristic
294C22×C23#57511yescharacteristic
295C4×C23—21no—
296C4×C23—21no—
297C4×C23—22no—
298C4×C23—22no—
299C4×C23—42no—
300C4×C23—41no—
301C4×C23—42no—
302C4×C23—41no—
303C4×C23—42no—
304C4×C23—42no—
305C4×C23—41no—
306C4×C23—42no—
307C4×C23—81no—
308C4×C23—81no—
309#552—22no—
310#552—22no—
311#551—22no—
312#551—22no—
313#543C22×C2311yescharacteristic
314#539—21no—
315#539C22×C2312yes—
316#534C22×C2311yescharacteristic
317#529—21no—
318#529C22×C2312yes—
319C2×#123—21no—
320#518—21no—
321C2×C4×Q8C22×C2311yescharacteristic, complemented
322C2×#117C22×C2311yescharacteristic
323#484C22×C2312yescomplemented
324#481C22×C2312yes—
325#479—22no—
326#464—22no—
327#451C22×C2312yescomplemented
328#448C22×C2312yes—
329#441—22no—
330#440—22no—
331#432C22×C2312yescomplemented
332#400C22×C2311yescharacteristic, complemented
333#399C22×C2311yescharacteristic
334#398—21no—
335#396—21no—
336#395C22×C2311yescharacteristic, complemented
337#393C22×C2311yescharacteristic
338C4×C4⋊C4—21no—
339C4×C4⋊C4C22×C2311yescharacteristic, complemented
340C4×C22⋊C4—21no—
341#328C22×C2312yescomplemented
342#328C22×C2312yescomplemented
343#327C22×C2312yes—
344#327C22×C2312yes—
345#324C22×C2312yescomplemented
346#323C22×C2311yescharacteristic
347#323C22×C2311yescharacteristic
348C2×C23—21no—
349C2×C23—21no—
350C2×C23—42no—
351C2×C23—82no—
352C2×C23#300711yescharacteristic
353C2×C23#301811yescharacteristic
354C2×C23#302511yescharacteristic
355C22×Q8C23×C2311yescharacteristic
356C22×D4—21no—
357C23×C4D4×C2311yescharacteristic
358C2×Q16—22no—
359C2×SD16—22no—
360#126—22no—
361#126—22no—
362#126—22no—
363#126—22no—
364#125—21no—
365#125—22no—
366#125—22no—
367#125D4×C2312yes—
368#124—21no—
369#124D4×C2312yes—
370#123—22no—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375#123—22no—
376#123—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—21no—
381C22⋊Q8—22no—
382C22⋊Q8C23×C2312yes—
383#121—21no—
384#121C23×C2312yes—
385#120—22no—
386C4×Q8—21no—
387C4×Q8—22no—
388C4×Q8—22no—
389C4×Q8D4×C2312yes—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—21no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4D4×C2312yes—
396#117—22no—
397#117—22no—
398#117—22no—
399#117—22no—
400#117—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4D4×C2311yescharacteristic
408C2×C4⋊C4C23×C2311yescharacteristic
409C2×C4⋊C4C23×C2311yescharacteristic
410C2×C22⋊C4—22no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4D4×C2311yescharacteristic
417C2×C22⋊C4C23×C2311yescharacteristic
418C2×C42—21no—
419C2×C42—21no—
420C2×C42D4×C2311yescharacteristic
421C2×C42C23×C2311yescharacteristic
422#107—22no—
423#106—22no—
424C4⋊C8—22no—
425#103—22no—
426#103—22no—
427#103—22no—
428#102—22no—
429C22⋊C8C23×C2312yes—
430C22⋊C8C23×C2312yes—
431#97—22no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95C23×C2311yescharacteristic
437#95C23×C2311yescharacteristic
438C23#1817711yescharacteristic, direct product factor, Sylow
439C24—21no—
440C24C2×D4×C2311yescharacteristic
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—42no—
444C2×Q8C2×D4×C2312yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4C2×D4×C2312yes—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—41no—
464C22×C4C2×D4×C2311yescharacteristic
465C22×C4C2×D4×C2311yescharacteristic
466C22×C4C2×D4×C2311yescharacteristic
467C22×C4C2×D4×C2311yescharacteristic
468C22×C4C2×D4×C2311yescharacteristic
469C22×C4C24×C2311yescharacteristic
470Q16—44no—
471SD16—44no—
472C2×C8—22no—
473C2×C8—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4C2×D4×C2312yes—
492C4⋊C4C2×D4×C2312yes—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—22no—
511C42—42no—
512C42—41no—
513C42—41no—
514C42C2×D4×C2312yes—
515C42C2×D4×C2312yes—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—81no—
523C23—81no—
524C23C23×#13712yes—
525C23C22×D4×C2311yescharacteristic
526Q8—24no—
527Q8—42no—
528Q8—42no—
529Q8—81no—
530D4—42no—
531D4—44no—
532D4—42no—
533D4—42no—
534D4—81no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4C23×#12011yescharacteristic
569C2×C4C23×#12011yescharacteristic
570C2×C4C23×#12012yes—
571C2×C4C22×D4×C2312yes—
572C8—42no—
573C8—42no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—162no—
585C22C2×C23×#12011yescharacteristic
586C22C2×C23×#13711yescharacteristic
587C22C23×#57511yescharacteristic
588C4—21no—
589C4—21no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—81no—
601C4—81no—
602C2—21no—
603C2—21no—
604C2—42no—
605C2—82no—
606C2C23×#300711yescharacteristic
607C2C23×#301811yescharacteristic
608C2C23×#302511yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#534C22
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×#3117C2semidirect
3. C23×#2396C2semidirect
4. C23×#1615C2semidirect
5. C23×#1614C2semidirect
7. C23×#1603C2semidirect
8. C23×#1602C2semidirect
10. C23×#1509C2semidirect
11. C23×#1464C2semidirect
24. C2×C4×Q8×C23C22semidirect
26. C23×#484C22semidirect
30. C23×#451C22semidirect
34. C23×#432C22semidirect
35. C23×#400C22semidirect
39. C23×#395C22semidirect
42. C4×C4⋊C4×C23C22semidirect
44. C23×#328C22semidirect
45. C23×#328C22semidirect
48. C23×#324C22semidirect
211. #18177C23direct (N × H)
271. #3117C2×C23semidirect
272. #2396C2×C23semidirect
273. #1615C2×C23semidirect
274. #1614C2×C23semidirect
276. #1603C2×C23semidirect
277. #1602C2×C23semidirect
279. #1509C2×C23semidirect
280. #1464C2×C23semidirect
321. C2×C4×Q8C22×C23semidirect
323. #484C22×C23semidirect
327. #451C22×C23semidirect
331. #432C22×C23semidirect
332. #400C22×C23semidirect
336. #395C22×C23semidirect
339. C4×C4⋊C4C22×C23semidirect
341. #328C22×C23semidirect
342. #328C22×C23semidirect
345. #324C22×C23semidirect
438. C23#18177direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34912
C23×#31172
C23×#23962
C23×#16152
C23×#16142
C23×#16132
C23×#16032
C23×#16022
C23×#16012
C23×#15092
C23×#14642
#1817723