Finite Group Databasebeta

← C23×#18201 #19165827

Subgroups — C23×#18201 #19165827

Order 5888.

960 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×#3492C211yesmaximal, characteristic
2C23×#3091C212yesmaximal, complemented
3C23×#2484C211yesmaximal, characteristic, complemented
4C23×#1617C211yesmaximal, characteristic, complemented
5C23×#1616C211yesmaximal, characteristic, complemented
6C23×#1615C212yesmaximal, complemented
7C23×#1611C211yesmaximal, characteristic, complemented
8C23×#1610C211yesmaximal, characteristic, complemented
9C23×#1609C212yesmaximal, complemented
10C23×#1516C212yesmaximal, complemented
11C2×C23×#324C211yesmaximal, 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—
23C23×#520—21no—
24C2×C23×#117C2211yescharacteristic
25C22×C4⋊C4×C23C2211yescharacteristic
26C23×#485—22no—
27C23×#484C2212yes—
28C23×#483C2212yes—
29C23×#482—22no—
30C23×#481C2212yes—
31C23×#480C2212yes—
32C23×#427—22no—
33C23×#416—22no—
34C2×C23×C22⋊C8C2212yes—
35C23×#400C2211yescharacteristic
36C23×#400C2211yescharacteristic
37C23×#398C2211yescharacteristic
38C23×#398C2211yescharacteristic
39C23×#387—21no—
40C23×#387—21no—
41C23×#385—21no—
42C23×#381—21no—
43C23×#381C2211yescharacteristic
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—
54C23×C4×C23D411yescharacteristic
55C23×C4×C23D411yescharacteristic
56C23×C4×C23C2311yescharacteristic
57C22×C8×C23—22no—
58C23×C4⋊Q8—22no—
59C23×#126—22no—
60C23×#126—22no—
61C23×#125—22no—
62C23×#125—22no—
63C23×#125—22no—
64C23×#123—22no—
65C23×#123—22no—
66C23×#123—22no—
67C23×#123—22no—
68C23×#123—22no—
69C23×C22⋊Q8—22no—
70C23×C22⋊Q8—21no—
71C23×C22⋊Q8—21no—
72C23×C22⋊Q8—21no—
73C23×C22⋊Q8—22no—
74C23×C22⋊Q8—21no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8D412yes—
77C23×C22⋊Q8C2312yes—
78C23×#121—21no—
79C23×#121—21no—
80C23×#121—21no—
81C23×#121—22no—
82C23×#121—21no—
83C23×#121D412yes—
84C23×#121C2312yes—
85C23×#120—22no—
86C4×Q8×C23—22no—
87C4×D4×C23—22no—
88C4×D4×C23—22no—
89C4×D4×C23—22no—
90C4×D4×C23—22no—
91C4×D4×C23—22no—
92C4×D4×C23—22no—
93C4×D4×C23—22no—
94C23×#117—21no—
95C23×#117—22no—
96C23×#117—21no—
97C23×#117—22no—
98C23×#117—22no—
99C23×#117D412yes—
100C2×C4⋊C4×C23—22no—
101C2×C4⋊C4×C23—22no—
102C2×C4⋊C4×C23—22no—
103C2×C4⋊C4×C23—21no—
104C2×C4⋊C4×C23—21no—
105C2×C4⋊C4×C23D411yescharacteristic
106C2×C4⋊C4×C23D411yescharacteristic
107C2×C4⋊C4×C23C2311yescharacteristic
108C2×C4⋊C4×C23C2311yescharacteristic
109C2×C4⋊C4×C23C2311yescharacteristic
110C2×C4⋊C4×C23C2311yescharacteristic
111C2×C22⋊C4×C23—21no—
112C2×C22⋊C4×C23—21no—
113C2×C42×C23—21no—
114C23×#107—22no—
115C23×#107—22no—
116C23×#106—22no—
117C23×#106—22no—
118C23×#103—22no—
119C23×#103—22no—
120C23×#102—22no—
121C23×#102—22no—
122C23×C22⋊C8—22no—
123C23×C22⋊C8C2312yes—
124C23×C22⋊C8C2312yes—
125C23×#95—21no—
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—44no—
137#41×C23—42no—
138#41×C23—82no—
139C2×Q8×C23—21no—
140C2×Q8×C23—21no—
141C2×Q8×C23C2×D412yes—
142C2×D4×C23—21no—
143C2×D4×C23—22no—
144C2×D4×C23—22no—
145C2×D4×C23—22no—
146C2×D4×C23—21no—
147C2×D4×C23C2×D412yes—
148C22×C4×C23—22no—
149C22×C4×C23—21no—
150C22×C4×C23—21no—
151C22×C4×C23—21no—
152C22×C4×C23—22no—
153C22×C4×C23—21no—
154C22×C4×C23—22no—
155C22×C4×C23—22no—
156C22×C4×C23—21no—
157C22×C4×C23—21no—
158C22×C4×C23—21no—
159C22×C4×C23—21no—
160C22×C4×C23—41no—
161C22×C4×C23—42no—
162C22×C4×C23—42no—
163C22×C4×C23C2×D411yescharacteristic
164C22×C4×C23C2×D412yes—
165C22×C4×C23C2×D411yescharacteristic
166C22×C4×C23C2×D411yescharacteristic
167C22×C4×C23C2×D411yescharacteristic
168C22×C4×C23C2×D411yescharacteristic
169C22×C4×C23C2×D412yes—
170C22×C4×C23C2411yescharacteristic
171C2×C8×C23—22no—
172C2×C8×C23—22no—
173C2×C8×C23—44no—
174C4⋊C4×C23—21no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
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—42no—
191C4⋊C4×C23—42no—
192C4⋊C4×C23C2×D412yes—
193C4⋊C4×C23C2×D412yes—
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—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—42no—
204C22⋊C4×C23—41no—
205C22⋊C4×C23—41no—
206C42×C23—22no—
207C42×C23—22no—
208C42×C23—22no—
209C42×C23—42no—
210#18201C2311yesmaximal, characteristic, direct product factor, Sylow
211C23×C23—21no—
212C23×C23—21no—
213C23×C23—22no—
214C23×C23—22no—
215C23×C23—41no—
216C23×C23#12011yescharacteristic
217C23×C23#12011yescharacteristic
218C23×C23C22×D411yescharacteristic
219Q8×C23—42no—
220Q8×C23—42no—
221Q8×C23—81no—
222D4×C23—42no—
223D4×C23—42no—
224D4×C23—44no—
225D4×C23—42no—
226D4×C23—42no—
227D4×C23—44no—
228D4×C23—81no—
229C2×C4×C23—21no—
230C2×C4×C23—21no—
231C2×C4×C23—24no—
232C2×C4×C23—22no—
233C2×C4×C23—22no—
234C2×C4×C23—21no—
235C2×C4×C23—22no—
236C2×C4×C23—21no—
237C2×C4×C23—21no—
238C2×C4×C23—21no—
239C2×C4×C23—21no—
240C2×C4×C23—42no—
241C2×C4×C23—41no—
242C2×C4×C23—41no—
243C2×C4×C23—42no—
244C2×C4×C23—42no—
245C2×C4×C23—44no—
246C2×C4×C23—42no—
247C2×C4×C23—41no—
248C2×C4×C23—41no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—42no—
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#12012yes—
264C2×C4×C23#13512yes—
265C2×C4×C23C22×D412yes—
266C8×C23—42no—
267C8×C23—42no—
268#3492C2×C2311yescharacteristic
269#3091C2×C2312yescomplemented
270#2484C2×C2311yescharacteristic, complemented
271#1617C2×C2311yescharacteristic, complemented
272#1616C2×C2311yescharacteristic, complemented
273#1615C2×C2312yescomplemented
274#1611C2×C2311yescharacteristic, complemented
275#1610C2×C2311yescharacteristic, complemented
276#1609C2×C2312yescomplemented
277#1516C2×C2312yescomplemented
278C2×#324C2×C2311yescharacteristic, complemented
279C22×C23—21no—
280C22×C23—21no—
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—21no—
284C22×C23—21no—
285C22×C23—21no—
286C22×C23—42no—
287C22×C23—42no—
288C22×C23—42no—
289C22×C23—42no—
290C22×C23—42no—
291C22×C23#45812yes—
292C22×C23C2×#12011yescharacteristic
293C22×C23C2×#13511yescharacteristic
294C22×C23#57511yescharacteristic
295C4×C23—22no—
296C4×C23—22no—
297C4×C23—22no—
298C4×C23—41no—
299C4×C23—42no—
300C4×C23—42no—
301C4×C23—41no—
302C4×C23—41no—
303C4×C23—42no—
304C4×C23—81no—
305C4×C23—81no—
306C4×C23—81no—
307C4×C23—81no—
308#554—22no—
309#553—22no—
310#552—22no—
311#547—22no—
312#544C22×C2311yescharacteristic
313#539—21no—
314#539C22×C2312yes—
315#537C22×C2311yescharacteristic
316#525—21no—
317#525—21no—
318#525C22×C2312yes—
319#520—21no—
320C2×#117C22×C2311yescharacteristic
321C22×C4⋊C4C22×C2311yescharacteristic
322#485—22no—
323#484C22×C2312yes—
324#483C22×C2312yes—
325#482—22no—
326#481C22×C2312yes—
327#480C22×C2312yes—
328#427—22no—
329#416—22no—
330C2×C22⋊C8C22×C2312yes—
331#400C22×C2311yescharacteristic
332#400C22×C2311yescharacteristic
333#398C22×C2311yescharacteristic
334#398C22×C2311yescharacteristic
335#387—21no—
336#387—21no—
337#385—21no—
338#381—21no—
339#381C22×C2311yescharacteristic
340#328C22×C2312yes—
341#328C22×C2312yes—
342#327C22×C2312yes—
343#327C22×C2312yes—
344#324C22×C2312yes—
345#324C22×C2311yescharacteristic
346#324C22×C2311yescharacteristic
347C2×C23—21no—
348C2×C23—21no—
349C2×C23—22no—
350C2×C23—41no—
351C2×C23—82no—
352C2×C23#300211yescharacteristic
353C2×C23#302011yescharacteristic
354C2×C23C2×#45811yescharacteristic
355C2×#41—21no—
356C2×#41—21no—
357C2×#41D4×C2312yes—
358C23×C4D4×C2311yescharacteristic
359C23×C4D4×C2311yescharacteristic
360C23×C4C23×C2311yescharacteristic
361C22×C8—22no—
362C4⋊Q8—22no—
363#126—22no—
364#126—22no—
365#125—22no—
366#125—22no—
367#125—22no—
368#123—22no—
369#123—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8D4×C2312yes—
381C22⋊Q8C23×C2312yes—
382#121—21no—
383#121—21no—
384#121—21no—
385#121—22no—
386#121—21no—
387#121D4×C2312yes—
388#121C23×C2312yes—
389#120—22no—
390C4×Q8—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398#117—22no—
399#117—21no—
400#117—21no—
401#117—22no—
402#117—22no—
403#117D4×C2312yes—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4D4×C2311yescharacteristic
410C2×C4⋊C4D4×C2311yescharacteristic
411C2×C4⋊C4C23×C2311yescharacteristic
412C2×C4⋊C4C23×C2311yescharacteristic
413C2×C4⋊C4C23×C2311yescharacteristic
414C2×C4⋊C4C23×C2311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C42—21no—
418#107—22no—
419#107—22no—
420#106—22no—
421#106—22no—
422#103—22no—
423#103—22no—
424#102—22no—
425#102—22no—
426C22⋊C8—22no—
427C22⋊C8C23×C2312yes—
428C22⋊C8C23×C2312yes—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95C23×C2311yescharacteristic
437#95C23×C2311yescharacteristic
438C23#1820111yescharacteristic, direct product factor, Sylow
439C24C2×D4×C2311yescharacteristic
440#41—42no—
441#41—44no—
442#41—42no—
443#41—82no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8C2×D4×C2312yes—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4C2×D4×C2312yes—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4C2×D4×C2311yescharacteristic
469C22×C4C2×D4×C2312yes—
470C22×C4C2×D4×C2311yescharacteristic
471C22×C4C2×D4×C2311yescharacteristic
472C22×C4C2×D4×C2311yescharacteristic
473C22×C4C2×D4×C2311yescharacteristic
474C22×C4C2×D4×C2312yes—
475C22×C4C24×C2311yescharacteristic
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—44no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4C2×D4×C2312yes—
498C4⋊C4C2×D4×C2312yes—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—42no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—42no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23C23×#12011yescharacteristic
521C23C23×#12011yescharacteristic
522C23C22×D4×C2311yescharacteristic
523Q8—42no—
524Q8—42no—
525Q8—81no—
526D4—42no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—42no—
531D4—44no—
532D4—81no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
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×C4C23×#12012yes—
568C2×C4C23×#13512yes—
569C2×C4C22×D4×C2312yes—
570C8—42no—
571C8—42no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22C23×#45812yes—
585C22C2×C23×#12011yescharacteristic
586C22C2×C23×#13511yescharacteristic
587C22C23×#57511yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C2—21no—
602C2—21no—
603C2—22no—
604C2—41no—
605C2—82no—
606C2C23×#300211yescharacteristic
607C2C23×#302011yescharacteristic
608C2C2×C23×#45811yescharacteristic
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×#3091C2semidirect
3. C23×#2484C2semidirect
4. C23×#1617C2semidirect
5. C23×#1616C2semidirect
6. C23×#1615C2semidirect
7. C23×#1611C2semidirect
8. C23×#1610C2semidirect
9. C23×#1609C2semidirect
10. C23×#1516C2semidirect
11. C2×C23×#324C2semidirect
210. #18201C23direct (N × H)
269. #3091C2×C23semidirect
270. #2484C2×C23semidirect
271. #1617C2×C23semidirect
272. #1616C2×C23semidirect
273. #1615C2×C23semidirect
274. #1611C2×C23semidirect
275. #1610C2×C23semidirect
276. #1609C2×C23semidirect
277. #1516C2×C23semidirect
278. C2×#324C2×C23semidirect
438. C23#18201direct (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×#30912
C23×#24842
C23×#16172
C23×#16162
C23×#16152
C23×#16112
C23×#16102
C23×#16092
C23×#15162
C2×C23×#3242
#1820123