Finite Group Databasebeta

← #10943899

Subgroups — #10943899

Order 768.

3688 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5964C211yesmaximal, characteristic, complemented
2C3×#2431C211yesmaximal, characteristic, complemented
3C2×#5077C212yesmaximal, complemented
4C2×#5062C212yesmaximal, complemented
5#82624C212yesmaximal, complemented
6#72137C211yesmaximal, characteristic, complemented
7#69692C212yesmaximal, complemented
8#69662C212yesmaximal, complemented
9#69454C212yesmaximal, complemented
10#32890—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12#5964—28no—
13#5964—28no—
14C2×#1010—22no—
15C2×#1009—22no—
16C2×D3×#41—24no—
17C2×#1007C2212yes—
18C2×#1004—22no—
19C2×#1004C2212yes—
20C22×#235—22no—
21C22×#235C2212yes—
22C2×C3×#127C2211yescharacteristic, complemented
23C2×C3×#124C2211yescharacteristic, complemented
24C2×C3×#121C2211yescharacteristic, complemented
25C2×#932C2214yescomplemented
26C3×#392—22no—
27C3×#391C2211yescharacteristic, complemented
28C3×#390C2211yescharacteristic, complemented
29C3×#386—22no—
30C3×C4×C22⋊C4C2211yescharacteristic, complemented
31C2×#836C2212yescomplemented
32C2×#833C2212yescomplemented
33#5207C2212yescomplemented
34#5077—28no—
35#5062—28no—
36#5040—22no—
37#5037—24no—
38#5036—22no—
39#4966C2214yescomplemented
40C2×#805C2214yescomplemented
41C2×#805C2214yescomplemented
42C2×#585—31no—
43C2×#454—32no—
44C2×#453—32no—
45#3006—32no—
46#2431D311yescharacteristic, complemented
47#1793—32no—
48#1792—32no—
49#1789—32no—
50#1785—31no—
51C2×C3×#41—24no—
52C2×C3×#41—24no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—22no—
57C22×C3×D4—21no—
58C22×C3×D4D412yes—
59C22×C3×D4C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4D412yes—
62C23×C3×C4C2311yescharacteristic
63C22×#67—24no—
64C22×#67—22no—
65#1010—232no—
66#1009—232no—
67D3×#41—232no—
68D3×#41—232no—
69#1007—24no—
70#1007—28no—
71#1007D418yescomplemented
72C2×#239—24no—
73C2×D3×Q8—24no—
74#1004—216no—
75#1004—216no—
76#1004—24no—
77#1004—28no—
78#1004D418yescomplemented
79C2×#237—28no—
80C2×#237—24no—
81C2×D3×D4—28no—
82C2×D3×D4—24no—
83C2×#235—24no—
84C2×#235—24no—
85C2×#235—216no—
86C2×#235—216no—
87C2×#235—22no—
88C2×#235—24no—
89C2×#235—22no—
90C2×#235—28no—
91C2×#235—28no—
92C2×#235D414yescomplemented
93C2×#235C2314yes—
94C22×D12—22no—
95C22×D12D412yescomplemented
96C22×C4×D3—24no—
97C22×C4×D3—22no—
98C22×Dic6—22no—
99C22×Dic6D412yescomplemented
100C3×#127—24no—
101C3×#124—24no—
102C3×#121—22no—
103C3×#121—22no—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×#214—24no—
112#932—28no—
113#932—28no—
114C2×#213—24no—
115C2×#208C2314yes—
116C3×#95—21no—
117C3×#95—22no—
118#836—24no—
119#833—24no—
120#830—22no—
121#830—22no—
122#810—24no—
123#809—24no—
124#805—28no—
125#805—28no—
126#585—68no—
127#585—68no—
128C2×2−1+4—32no—
129C2×2−1+4—62no—
130C2×H(5,𝔽2)—32no—
131C2×H(5,𝔽2)—62no—
132C2×H(5,𝔽2)—62no—
133C22×#41—32no—
134C22×#41—64no—
135C22×#41—62no—
136C22×#41C2×D311yescharacteristic
137C2×AGL(1,ℤ/8)—34no—
138C2×#127C2×D311yescharacteristic, complemented
139C2×#124C2×D311yescharacteristic, complemented
140C2×#121C2×D311yescharacteristic, complemented
141#454—68no—
142#453—68no—
143#449—64no—
144#448—62no—
145#447—62no—
146C2×#104—34no—
147C2×#104—34no—
148#417—34no—
149C2×#101—32no—
150C2×#100—32no—
151#407—32no—
152#392—22no—
153#391C2×D311yescharacteristic, complemented
154#390C2×D311yescharacteristic, complemented
155#386—22no—
156C4×C22⋊C4C2×D311yescharacteristic, complemented
157C24×C3—21no—
158C24×C3C2×D412yes—
159C24×C3C2×D411yescharacteristic
160C23×D3—24no—
161C23×D3—22no—
162C3×#41—216no—
163C3×#41—216no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8C2×D414yes—
167C2×C3×D4—24no—
168C2×C3×D4—24no—
169C2×C3×D4—24no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—44no—
176C2×C3×D4—42no—
177C2×C3×D4—44no—
178C2×C3×D4—82no—
179C2×C3×D4—82no—
180C2×C3×D4C2×D414yes—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—24no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2411yescharacteristic
195C2×#67—28no—
196C2×#67—216no—
197C2×#67—28no—
198C2×#67—28no—
199C2×#67—216no—
200C2×#67—24no—
201C2×#67—24no—
202C22×Dic3—24no—
203C22×Dic3—22no—
204#239—216no—
205#239—216no—
206D3×Q8—216no—
207D3×Q8—216no—
208#237—264no—
209#237—216no—
210#237—216no—
211D3×D4—264no—
212D3×D4—216no—
213D3×D4—216no—
214#235—28no—
215#235—28no—
216#235—216no—
217#235—232no—
218#235—232no—
219#235—216no—
220#235—216no—
221#235—24no—
222#235—24no—
223#235—28no—
224#235—28no—
225#235C2×D418yes—
226#235C2×D418yes—
227C2×D12—24no—
228C2×D12—216no—
229C2×D12—22no—
230C2×D12—28no—
231C2×D12C2×D414yes—
232C2×C4×D3—28no—
233C2×C4×D3—232no—
234C2×C4×D3—216no—
235C2×C4×D3—24no—
236C2×C4×D3—24no—
237C2×Dic6—24no—
238C2×Dic6—216no—
239C2×Dic6—22no—
240C2×Dic6—28no—
241C2×Dic6C2×D414yes—
242C3×C4⋊C4—22no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—41no—
245C3×C22⋊C4—41no—
246C3×C22⋊C4—44no—
247C3×C22⋊C4—82no—
248C3×C42—22no—
249C3×C42—22no—
250#214—48no—
251#213—48no—
252#208—24no—
253#208—24no—
254C2×C3⋊C8—24no—
2552−1+4—38no—
2562−1+4—68no—
2572−1+4—632no—
2582−1+4—64no—
259H(5,𝔽2)—38no—
260H(5,𝔽2)—68no—
261H(5,𝔽2)—616no—
262H(5,𝔽2)—616no—
263H(5,𝔽2)—632no—
264H(5,𝔽2)—64no—
265C2×#41—24no—
266C2×#41—24no—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—34no—
270C2×#41—34no—
271C2×#41—68no—
272C2×#41—68no—
273C2×#41—64no—
274C2×#41—64no—
275C2×#41—616no—
276C2×#41—616no—
277C2×#41—64no—
278C2×#41—68no—
279C2×#41—64no—
280C2×#41—632no—
281C2×#41—632no—
282C2×#41—62no—
283C2×#41—62no—
284C2×#41—64no—
285C22×Q8—32no—
286C22×Q8—62no—
287C22×Q8—64no—
288C22×Q8C22×D311yescharacteristic
289C22×D4—22no—
290C22×D4—21no—
291C22×D4—32no—
292C22×D4—62no—
293C22×D4—64no—
294C22×D4—68no—
295C22×D4—62no—
296C22×D4—64no—
297C22×D4#6712yes—
298C22×D4C22×D311yescharacteristic
299C22×D4C22×D311yescharacteristic
300C23×C4—64no—
301C23×C4—62no—
302C23×C4#6712yes—
303C23×C4C22×D311yescharacteristic
304AGL(1,ℤ/8)—68no—
305AGL(1,ℤ/8)—68no—
306C2×SD16—64no—
307C2×D8—64no—
308C2×M16—34no—
309#127—24no—
310#124—24no—
311#121—22no—
312#121—22no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C42C22×D311yescharacteristic
319C2×C42C22×D311yescharacteristic
320#104—68no—
321#104—68no—
322#103—64no—
323#102—64no—
324#101—64no—
325#100—64no—
326C22⋊C8—62no—
327C22⋊C8—62no—
328#95—21no—
329#95—22no—
330C23×C3—22no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3#12012yes—
344C23×C3#12014yes—
345C23×C3C22×D411yescharacteristic
346C22×D3—24no—
347C22×D3—216no—
348C22×D3—28no—
349C22×D3—82no—
350C22×D3—82no—
351C22×D3—84no—
352C22×D3—84no—
353C3×Q8—28no—
354C3×D4—216no—
355C3×D4—28no—
356C3×D4—44no—
357C3×D4—88no—
358C3×D4—84no—
359C2×C3×C4—28no—
360C2×C3×C4—28no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4#12011yescharacteristic
376C2×C3×C4#12011yescharacteristic
377C2×C3×C4#12014yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D412yes—
381#67—232no—
382#67—216no—
383#67—28no—
384#67—28no—
385C2×Dic3—216no—
386C2×Dic3—28no—
387C2×Dic3—24no—
388D12—28no—
389D12—216no—
390D12—24no—
391D12#12018yes—
392C4×D3—216no—
393C4×D3—232no—
394C4×D3—28no—
395C4×D3—28no—
396Dic6—28no—
397Dic6—216no—
398Dic6—24no—
399Dic6#12018yes—
400C3⋊C8—44no—
401C24—21no—
402C24—64no—
403C24—62no—
404C24C2×#6712yes—
405C24C2×#6711yescharacteristic
406#41—216no—
407#41—216no—
408#41—38no—
409#41—38no—
410#41—68no—
411#41—68no—
412#41—632no—
413#41—68no—
414#41—68no—
415#41—632no—
416#41—616no—
417#41—616no—
418#41—616no—
419#41—664no—
420#41—616no—
421#41—616no—
422#41—616no—
423#41—616no—
424#41—64no—
425#41—64no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—34no—
429C2×Q8—68no—
430C2×Q8—64no—
431C2×Q8—616no—
432C2×Q8—616no—
433C2×Q8—616no—
434C2×Q8—62no—
435C2×Q8D3×D414yes—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—34no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—44no—
448C2×D4—68no—
449C2×D4—64no—
450C2×D4—616no—
451C2×D4—616no—
452C2×D4—664no—
453C2×D4—616no—
454C2×D4—68no—
455C2×D4—68no—
456C2×D4—68no—
457C2×D4—616no—
458C2×D4—64no—
459C2×D4—616no—
460C2×D4—64no—
461C2×D4—62no—
462C2×D4—82no—
463C2×D4—82no—
464C2×D4D3×D414yes—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—68no—
475C22×C4—632no—
476C22×C4—616no—
477C22×C4—64no—
478C22×C4—62no—
479C22×C4—64no—
480C22×C4—64no—
481C22×C4D3×D412yes—
482C22×C4D3×D412yes—
483C22×C4C2×#6711yescharacteristic
484C22×C4C2×#6712yes—
485C22×C4C23×D311yescharacteristic
486SD16—128no—
487D8—128no—
488M16—64no—
489M16—64no—
490C2×C8—64no—
491C4⋊C4—22no—
492C22⋊C4—42no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—44no—
496C22⋊C4—82no—
497C42—22no—
498C42—22no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—22no—
502C22×C3—24no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—44no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—81no—
509C22×C3—81no—
510C22×C3—82no—
511C22×C3—82no—
512C22×C3—82no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3C2×#12011yescharacteristic
517C22×C3C2×#12011yescharacteristic
518C22×C3C2×#12011yescharacteristic
519C2×D3—28no—
520C2×D3—44no—
521C2×D3—44no—
522C2×D3—88no—
523C2×D3—816no—
524C3×C4—21no—
525C3×C4—24no—
526C3×C4—21no—
527C3×C4—42no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4C2×#12014yes—
531Dic3—28no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—21no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—68no—
546C23—616no—
547C23—64no—
548C23—242no—
549C23—242no—
550C23—244no—
551C23—244no—
552C23#93712yes—
553C23#93714yes—
554C23C22×#6711yescharacteristic
555Q8—28no—
556Q8—38no—
557Q8—68no—
558Q8—616no—
559Q8—64no—
560D4—216no—
561D4—28no—
562D4—38no—
563D4—44no—
564D4—68no—
565D4—616no—
566D4—632no—
567D4—616no—
568D4—68no—
569D4—68no—
570D4—64no—
571D4—88no—
572D4—84no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—616no—
590C2×C4—632no—
591C2×C4—616no—
592C2×C4—68no—
593C2×C4—64no—
594C2×C4—68no—
595C2×C4—68no—
596C2×C4#93714yes—
597C2×C4#93711yescharacteristic
598C2×C4#93711yescharacteristic
599C2×C4C2×D3×D412yes—
600C2×C4C2×D3×D412yes—
601C2×C4C2×D3×D412yes—
602C8—124no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—42no—
606C2×C3—44no—
607C2×C3—161no—
608C2×C3#284611yescharacteristic
609C2×C3#301811yescharacteristic
610C2×C3#302111yescharacteristic
611D3—48no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—24no—
617C22—22no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—68no—
622C22—81no—
623C22—81no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—124no—
631C22—124no—
632C22—248no—
633C22—2416no—
634C22C2×#93711yescharacteristic
635C22C2×#93711yescharacteristic
636C22C2×#93711yescharacteristic
637C4—21no—
638C4—21no—
639C4—24no—
640C4—42no—
641C4—68no—
642C4—81no—
643C4—81no—
644C4D3×#12014yes—
645C3#3289011yescharacteristic, complemented, Sylow
646C2—21no—
647C2—21no—
648C2—42no—
649C2—44no—
650C2—128no—
651C2—161no—
652C2#8261211yescharacteristic
653C2#8332211yescharacteristic
654C2#8339511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2431C2
socleC22×C3C2×#120
hypercenter#2431D3
hypocenterC3#32890
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2431

Complements

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

NH ≅ G/Ntype
1. C2×#5964C2semidirect
2. C3×#2431C2semidirect
3. C2×#5077C2semidirect
4. C2×#5062C2semidirect
5. #82624C2semidirect
6. #72137C2semidirect
7. #69692C2semidirect
8. #69662C2semidirect
9. #69454C2semidirect
22. C2×C3×#127C22semidirect
23. C2×C3×#124C22semidirect
24. C2×C3×#121C22semidirect
25. C2×#932C22semidirect
27. C3×#391C22semidirect
28. C3×#390C22semidirect
30. C3×C4×C22⋊C4C22semidirect
31. C2×#836C22semidirect
32. C2×#833C22semidirect
33. #5207C22semidirect
39. #4966C22semidirect
40. C2×#805C22semidirect
41. C2×#805C22semidirect
46. #2431D3semidirect
71. #1007D4semidirect
78. #1004D4semidirect
92. C2×#235D4semidirect
95. C22×D12D4semidirect
99. C22×Dic6D4semidirect
138. C2×#127C2×D3semidirect
139. C2×#124C2×D3semidirect
140. C2×#121C2×D3semidirect
153. #391C2×D3semidirect
154. #390C2×D3semidirect
156. C4×C22⋊C4C2×D3semidirect
645. C3#32890semidirect

Maximal subgroups

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

subgroupindex
C2×#59642
C3×#24312
C2×#50772
C2×#50622
#826242
#721372
#696922
#696622
#694542
#328903