Finite Group Databasebeta

← #11069322

Subgroups — #11069322

Order 768.

2958 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×#5825C212yesmaximal, complemented
3C2×D4×Dic6C211yesmaximal, characteristic, complemented
4#84093C212yesmaximal, complemented
5#83184C211yesmaximal, characteristic, complemented
6#83179C211yesmaximal, characteristic, complemented
7#83176C214yesmaximal, complemented
8#82685C218yesmaximal, complemented
9#82677C212yesmaximal, complemented
10#82654C212yesmaximal, complemented
11#82475C211yesmaximal, characteristic, complemented
12C4×#944C211yesmaximal, characteristic, complemented
13#82362C214yesmaximal, complemented
14#82319C211yesmaximal, characteristic, complemented
15#39363—31nomaximal, Sylow
16C22×C3×#41C2212yes—
17C23×Dic6C2212yes—
18C3×#550—28no—
19C2×C3×#127C2212yes—
20C2×C3×#127C2211yescharacteristic
21C2×C3×#121C2212yes—
22C2×C3×#121C2214yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26#5825—216no—
27C2×#944C2212yes—
28C2×#944C2211yescharacteristic
29C2×#936C2212yes—
30C2×#936C2212yes—
31C2×#936C2212yes—
32C2×#935C2218yes—
33C2×D4×Dic3C2212yes—
34C2×#926C2212yes—
35C2×#924C2212yes—
36C2×#924C2218yes—
37D4×Dic6—28no—
38C2×#888C2212yes—
39C2×#878C2214yes—
40C2×#869C2211yescharacteristic
41C2×C4×Dic6C2211yescharacteristic
42#5229C2211yescharacteristic
43#5222C2218yes—
44#5222C2218yes—
45#5222C2214yes—
46#5221C2212yes—
47#5221C2212yes—
48#5221C2214yes—
49#5221C2214yes—
50#5221C2214yes—
51#5220C2218yes—
52#5210C2218yes—
53C4×#217C2212yes—
54#4984C2212yes—
55#4981C2214yes—
56#4976C2214yes—
57#4975C2211yescharacteristic
58#4973C2211yescharacteristic
59#4973C2212yes—
60#4951C2214yes—
61#4948C2218yes—
62#4947C2218yes—
63C4×C4⋊Dic3C2211yescharacteristic
64C4×Dic3⋊C4C2214yes—
65C42×Dic3C2211yescharacteristic
66#4928C2214yes—
67C2×#550D311yescharacteristic
68C2×D4×Q8—31no—
69C2×#536—32no—
70#2858—31no—
71#2681—38no—
72#2620—32no—
73#2548—31no—
74#2546—31no—
75#2452—34no—
76#2445—34no—
77#2436—32no—
78#2432—32no—
79C4×C4⋊Q8—31no—
80#2293—31no—
81C2×C3×#41—28no—
82C2×C3×#41—28no—
83C2×C3×#41—44no—
84C22×C3×Q8C2311yescharacteristic
85C22×C3×D4C2312yes—
86C22×C3×D4C2312yes—
87C22×C3×D4C2314yes—
88C22×C3×D4C2311yescharacteristic
89C22×C3×D4C2314yes—
90C22×C3×D4C2312yes—
91C23×C3×C4C2314yes—
92C23×C3×C4C2312yes—
93C23×Dic3C2314yes—
94C22×Dic6—28no—
95C22×Dic6—28no—
96C22×Dic6—44no—
97C22×Dic6C2312yes—
98C22×Dic6C2314yes—
99C22×Dic6C2311yescharacteristic
100C3×#127—28no—
101C3×#127—24no—
102C3×#121—28no—
103C3×#121—216no—
104C3×C4×Q8—24no—
105C3×C4×D4—28no—
106C3×C4×D4D418yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4C2314yes—
111C2×C3×C22⋊C4C2312yes—
112C2×C3×C42C2312yes—
113C2×C3×C42C2311yescharacteristic
114C2×#217C2318yes—
115C2×#217C2318yes—
116C2×#217C2314yes—
117C2×#217C2314yes—
118C2×#217C2318yes—
119C2×#217C2318yes—
120C2×#217C2314yes—
121#944—28no—
122#944D418yescomplemented
123#936—28no—
124#936—28no—
125#936—44no—
126#935—232no—
127D4×Dic3—28no—
128#926—28no—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2312yes—
131C2×C4⋊Dic3C2314yes—
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2312yes—
134#924—28no—
135#924—232no—
136C2×Dic3⋊C4C2318yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×C4×Dic3C2312yes—
142C2×C4×Dic3C2311yescharacteristic
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2311yescharacteristic
147#888—28no—
148#878—216no—
149#869—24no—
150C4×Dic6—24no—
151C3⋊#95C2318yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2312yes—
155C3⋊#95C2314yes—
156C22×#41C2×D312yes—
157C23×Q8—32no—
158#550—28no—
159D4×Q8—68no—
160#536—616no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8—32no—
163C2×C4⋊Q8—32no—
164C2×C4⋊Q8—31no—
165C2×#127C2×D311yescharacteristic
166C2×#127C2×D312yes—
167C2×#124—32no—
168C2×#124—32no—
169C2×#124—32no—
170C2×#123—38no—
171C2×C22⋊Q8—32no—
172C2×C22⋊Q8—38no—
173C2×C22⋊Q8—34no—
174C2×#121C2×D312yes—
175C2×#121C2×D314yes—
176C2×C4×Q8—31no—
177C2×C4×Q8C2×D311yescharacteristic
178C2×C4×D4—32no—
179C2×C4×D4C2×D312yes—
180C2×C4×D4C2×D311yescharacteristic
181C2×#117—32no—
182#398—34no—
183#396—38no—
184#394—38no—
185#394—34no—
186#394—38no—
187#393—38no—
188#391—31no—
189#391—31no—
190#391—32no—
191#390—34no—
192#390—34no—
193#390—32no—
194#390—32no—
195#390—34no—
196#389—34no—
197#389—32no—
198#389—34no—
199#388—34no—
200#386—38no—
201#385—38no—
202#384—31no—
203C4×C4⋊C4—34no—
204C4×C4⋊C4—31no—
205C4×C22⋊C4—32no—
206C43—31no—
207C24×C3C2414yes—
208C24×C3C2412yes—
209C3×#41—432no—
210C2×C3×Q8—22no—
211C2×C3×Q8—22no—
212C2×C3×Q8C2×D414yes—
213C2×C3×D4—28no—
214C2×C3×D4—28no—
215C2×C3×D4—28no—
216C2×C3×D4—24no—
217C2×C3×D4—24no—
218C2×C3×D4—24no—
219C2×C3×D4—24no—
220C2×C3×D4—24no—
221C2×C3×D4—24no—
222C2×C3×D4—28no—
223C2×C3×D4—44no—
224C2×C3×D4—44no—
225C2×C3×D4—48no—
226C2×C3×D4C2×D414yes—
227C22×C3×C4—24no—
228C22×C3×C4—24no—
229C22×C3×C4—28no—
230C22×C3×C4—28no—
231C22×C3×C4—44no—
232C22×C3×C4C2×D418yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2412yes—
236C22×C3×C4C2412yes—
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2411yescharacteristic
239C22×Dic3—28no—
240C22×Dic3—28no—
241C22×Dic3—44no—
242C22×Dic3C2412yes—
243C22×Dic3C2414yes—
244C22×Dic3C2412yes—
245C22×Dic3C2414yes—
246C22×Dic3C2412yes—
247C22×Dic3C2412yes—
248C2×Dic6—22no—
249C2×Dic6—22no—
250C2×Dic6—24no—
251C2×Dic6—28no—
252C2×Dic6—24no—
253C2×Dic6—24no—
254C2×Dic6—28no—
255C2×Dic6—432no—
256C2×Dic6—44no—
257C2×Dic6C2×D414yes—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4C2×D414yes—
261C3×C22⋊C4—28no—
262C3×C22⋊C4C2×D418yes—
263C3×C42—24no—
264C3×C42C2×D414yes—
265#217—28no—
266#217—216no—
267#217—216no—
268#217—28no—
269#217—48no—
270#217—44no—
271#217—48no—
272C4⋊Dic3—22no—
273C4⋊Dic3—24no—
274C4⋊Dic3—24no—
275C4⋊Dic3—24no—
276C4⋊Dic3—28no—
277Dic3⋊C4—24no—
278Dic3⋊C4—28no—
279Dic3⋊C4—216no—
280Dic3⋊C4—44no—
281Dic3⋊C4C2×D4116yes—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3—22no—
285C4×Dic3—24no—
286C4×Dic3—42no—
287C4×Dic3C2×D414yes—
288C2×#41—28no—
289C2×#41—28no—
290C2×#41—44no—
291C22×Q8—31no—
292C22×Q8—32no—
293C22×Q8—34no—
294C22×Q8—68no—
295C22×Q8—68no—
296C22×Q8—124no—
297C22×Q8C22×D311yescharacteristic
298C22×D4C22×D312yes—
299C22×D4C22×D312yes—
300C22×D4C22×D314yes—
301C22×D4C22×D311yescharacteristic
302C22×D4C22×D314yes—
303C22×D4C22×D312yes—
304C23×C4—34no—
305C23×C4C22×D314yes—
306C23×C4C22×D312yes—
307C4⋊Q8—38no—
308C4⋊Q8—68no—
309C4⋊Q8—68no—
310C4⋊Q8—64no—
311#127—28no—
312#127—24no—
313#124—68no—
314#124—68no—
315#124—124no—
316#123—632no—
317C22⋊Q8—632no—
318C22⋊Q8—68no—
319C22⋊Q8—616no—
320#121—28no—
321#121—216no—
322C4×Q8—24no—
323C4×Q8—64no—
324C4×D4—28no—
325C4×D4—68no—
326C4×D4#6718yes—
327#117—68no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—38no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—34no—
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D312yes—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—38no—
344C2×C22⋊C4—38no—
345C2×C22⋊C4—38no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—38no—
348C2×C22⋊C4C22×D314yes—
349C2×C22⋊C4C22×D312yes—
350C2×C42—31no—
351C2×C42—32no—
352C2×C42—32no—
353C2×C42—31no—
354C2×C42—32no—
355C2×C42—32no—
356C2×C42C22×D312yes—
357C2×C42C22×D311yescharacteristic
358#95—34no—
359#95—34no—
360#95—34no—
361#95—32no—
362#95—38no—
363C23×C3—22no—
364C23×C3—22no—
365C23×C3—24no—
366C23×C3—24no—
367C23×C3—24no—
368C23×C3—24no—
369C23×C3—24no—
370C23×C3—28no—
371C23×C3—28no—
372C23×C3C22×D414yes—
373C23×C3C2511yescharacteristic
374C3×Q8—28no—
375C3×D4—28no—
376C3×D4—416no—
377C3×D4—48no—
378C3×D4—48no—
379C3×D4—84no—
380C3×D4—88no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—216no—
387C2×C3×C4—22no—
388C2×C3×C4—24no—
389C2×C3×C4—416no—
390C2×C3×C4C22×D414yes—
391C2×C3×C4C22×D412yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4C22×D414yes—
394C2×C3×C4C22×D412yes—
395C2×C3×C4C22×D412yes—
396C2×C3×C4H(5,𝔽2)12yes—
397C2×C3×C42−1+414yes—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—24no—
406C2×Dic3—22no—
407C2×Dic3—24no—
408C2×Dic3—22no—
409C2×Dic3—24no—
410C2×Dic3—22no—
411C2×Dic3—24no—
412C2×Dic3—24no—
413C2×Dic3—416no—
414C2×Dic3C22×D414yes—
415C2×Dic3C22×D414yes—
416Dic6—28no—
417Dic6—416no—
418Dic6—48no—
419C24C23×D314yes—
420C24C23×D312yes—
421#41—432no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—34no—
425C2×Q8—68no—
426C2×Q8—64no—
427C2×Q8—62no—
428C2×Q8—64no—
429C2×Q8—62no—
430C2×Q8—68no—
431C2×Q8—64no—
432C2×Q8—124no—
433C2×Q8—1232no—
434C2×Q8D3×D414yes—
435C2×D4—28no—
436C2×D4—28no—
437C2×D4—28no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—48no—
448C2×D4C2×#6714yes—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—32no—
456C22×C4—34no—
457C22×C4—32no—
458C22×C4—34no—
459C22×C4—44no—
460C22×C4—68no—
461C22×C4—68no—
462C22×C4—124no—
463C22×C4C2×#6718yes—
464C22×C4C23×D312yes—
465C22×C4C23×D312yes—
466C22×C4C23×D311yescharacteristic
467C22×C4C23×D312yes—
468C22×C4C23×D311yescharacteristic
469C22×C4C23×D311yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—316no—
473C4⋊C4—68no—
474C4⋊C4—616no—
475C4⋊C4—64no—
476C4⋊C4—64no—
477C4⋊C4—64no—
478C4⋊C4—62no—
479C4⋊C4—68no—
480C4⋊C4—64no—
481C4⋊C4—124no—
482C4⋊C4C2×#6714yes—
483C22⋊C4—28no—
484C22⋊C4—68no—
485C22⋊C4—616no—
486C22⋊C4—616no—
487C22⋊C4—68no—
488C22⋊C4—128no—
489C22⋊C4—124no—
490C22⋊C4—128no—
491C22⋊C4C2×#6718yes—
492C42—24no—
493C42—34no—
494C42—64no—
495C42—62no—
496C42—64no—
497C42—64no—
498C42—122no—
499C42C2×#6714yes—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—24no—
503C22×C3—44no—
504C22×C3—44no—
505C22×C3—44no—
506C22×C3—44no—
507C22×C3—48no—
508C22×C3—44no—
509C22×C3—48no—
510C22×C3C23×D411yescharacteristic
511C22×C3C23×D411yescharacteristic
512C22×C3C2×H(5,𝔽2)11yescharacteristic
513C22×C3C2×2−1+412yes—
514C22×C3#58512yes—
515C3×C4—24no—
516C3×C4—24no—
517C3×C4—22no—
518C3×C4—42no—
519C3×C4#53614yescomplemented
520C3×C4D4214yes—
521Dic3—24no—
522Dic3—24no—
523Dic3—44no—
524Dic3—44no—
525Dic3—42no—
526Dic3—42no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—24no—
536C23C22×#6714yes—
537C23C24×D311yescharacteristic
538Q8—28no—
539Q8—68no—
540Q8—1216no—
541Q8—128no—
542D4—28no—
543D4—416no—
544D4—48no—
545D4—48no—
546D4—84no—
547D4—88no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—216no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—34no—
557C2×C4—34no—
558C2×C4—416no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—64no—
569C2×C4—62no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—64no—
574C2×C4—1216no—
575C2×C4C2×D3×D414yes—
576C2×C4C2×D3×D412yes—
577C2×C4#100412yes—
578C2×C4#101014yes—
579C2×C4C22×#6712yes—
580C2×C4C22×#6714yes—
581C2×C4C22×#6712yes—
582C2×C4C22×#6712yes—
583C2×C3—44no—
584C2×C3—84no—
585C2×C3C2×#53611yescharacteristic
586C2×C3C2×D4211yescharacteristic
587C2×C3#349712yes—
588C2×C3#350011yescharacteristic
589C2×C3#355412yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—44no—
594C22—44no—
595C22—48no—
596C22—48no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22C22×D3×D411yescharacteristic
601C22C2×#100411yescharacteristic
602C22C2×#101012yes—
603C22#596612yes—
604C22C23×#6711yescharacteristic
605C4—24no—
606C4—24no—
607C4—22no—
608C4—42no—
609C4—64no—
610C4—64no—
611C4—124no—
612C4—122no—
613C4—124no—
614C4—122no—
615C4D4×#6714yes—
616C4#582514yescomplemented
617C3#3936311yescharacteristic, complemented, Sylow
618C2—44no—
619C2—84no—
620C2#8678412yes—
621C2#8678912yes—
622C2C2×D4×#6711yescharacteristic
623C2C2×#582511yescharacteristic
624C2#8728311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC2×C3×#550C2
hypercenterC2×#550D3
hypocenterC3#39363
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#550

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#5825C2semidirect
3. C2×D4×Dic6C2semidirect
4. #84093C2semidirect
5. #83184C2semidirect
6. #83179C2semidirect
7. #83176C2semidirect
8. #82685C2semidirect
9. #82677C2semidirect
10. #82654C2semidirect
11. #82475C2semidirect
12. C4×#944C2semidirect
13. #82362C2semidirect
14. #82319C2semidirect
122. #944D4semidirect
519. C3×C4#536semidirect
616. C4#5825semidirect
617. C3#39363semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×#58252
C2×D4×Dic62
#840932
#831842
#831792
#831762
#826852
#826772
#826542
#824752
C4×#9442
#823622
#823192
#393633