Finite Group Databasebeta

← #10848740

Subgroups — #10848740

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic, complemented
3C2×#4712C211yesmaximal, characteristic, complemented
4#74256C211yesmaximal, characteristic, complemented
5C3×#1993C211yesmaximal, characteristic, complemented
6#69807C211yesmaximal, characteristic, complemented
7#69553C211yesmaximal, characteristic, complemented
8#68897C212yesmaximal, complemented
9#68896C212yesmaximal, complemented
10#68887C212yesmaximal, complemented
11#68886C212yesmaximal, complemented
12#25602—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#947—21no—
16C22×D24C2212yes—
17C22×#204C2212yes—
18#5725—24no—
19#5725—24no—
20C2×#897C2211yescharacteristic
21C2×#895—21no—
22C2×#895C2212yes—
23C2×#895C2211yescharacteristic
24C2×#893—21no—
25C2×#893C2212yes—
26C2×#875C2211yescharacteristic
27C2×#874C2211yescharacteristic
28C2×C42×D3C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30C3×#382C2211yescharacteristic
31C2×(C2×D3)⋊C8C2212yes—
32#5089—24no—
33#4988C2211yescharacteristic
34#4973C2211yescharacteristic
35C2×#810C2212yes—
36C2×#809C2212yes—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#803C2211yescharacteristic
40#4712—24no—
41#4670—21no—
42#4670—21no—
43#4665—21no—
44#4665—21no—
45C3×#340C2212yes—
46C3×#340C2212yes—
47#4550C2212yes—
48C2×#532—31no—
49C2×#496—31no—
50C2×#489—31no—
51#2850—31no—
52#2000—32no—
53#1999—32no—
54#1993D311yescharacteristic, complemented
55#1961—31no—
56#1879—32no—
57#1877—32no—
58#1749—31no—
59C22×C3×Q8—21no—
60C2×#239—28no—
61C2×#239—44no—
62C2×#239—44no—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12—21no—
66C22×D12—21no—
67C22×D12C2311yescharacteristic
68C22×D12C2312yes—
69C22×C4×D3—22no—
70C22×C4×D3D411yescharacteristic
71C22×C4×D3D411yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×Dic6C2311yescharacteristic
74C22×C3×C8C2312yes—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8D414yescomplemented
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C42C2311yescharacteristic
82#947—28no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3—21no—
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3D411yescharacteristic
94C2×C4×Dic3C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96C2×D24—24no—
97C2×D24—24no—
98C2×D24—42no—
99C2×#204—24no—
100C2×#204—24no—
101C2×#204—42no—
102#897—42no—
103#895—28no—
104#895—24no—
105#895—24no—
106#895—42no—
107#893—28no—
108#893—24no—
109#893—24no—
110#875—22no—
111#875—22no—
112#874—22no—
113#874D414yescomplemented
114C42×D3—22no—
115C42×D3—22no—
116C3×#97—22no—
117C3×#95—21no—
118C3×#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121(C2×D3)⋊C8—24no—
122#810—24no—
123#809—24no—
124#809—42no—
125#809—42no—
126#803—22no—
127C22×#41—61no—
128C22×SD16—32no—
129C22×D8—32no—
130#532—64no—
131#532—64no—
132C2×C4⋊Q8C2×D311yescharacteristic
133C2×#127—31no—
134C2×#124—31no—
135C2×#124—61no—
136C2×C22⋊Q8—31no—
137C2×#121—32no—
138C2×#121—31no—
139C2×#121—61no—
140C2×C4×D4—32no—
141C2×C4×D4—61no—
142C22×C42—31no—
143#496—64no—
144#489—64no—
145C2×#103—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102—32no—
149C2×C22⋊C8—32no—
150C2×#97—31no—
151C2×#97C2×D311yescharacteristic
152#391—31no—
153#390—31no—
154#388—61no—
155#388—61no—
156#386—61no—
157#386—61no—
158#382C2×D311yescharacteristic
159#340C2×D312yes—
160#340C2×D312yes—
161#336—32no—
162C23×D3—22no—
163C23×D3—21no—
164C23×D3C2×D411yescharacteristic
165C2×C3×Q8—24no—
166C2×C3×Q8—42no—
167C2×C3×Q8—42no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178#239—432no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—28no—
184C2×D12—24no—
185C2×D12—24no—
186C2×D12—22no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—42no—
192C2×D12—42no—
193C2×D12—42no—
194C2×D12—42no—
195C2×D12—44no—
196C2×D12—44no—
197C2×D12—42no—
198C2×D12—42no—
199C2×D12—42no—
200C2×D12C2×D414yes—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—28no—
205C2×C4×D3—22no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—41no—
211C2×C4×D3—41no—
212C2×C4×D3—44no—
213C2×C4×D3—44no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8#4114yescomplemented
221C3×C4⋊C4—24no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4C2×D414yes—
226C3×C42C2×D412yes—
227C3×C42C2×D412yes—
228(C2×D3)⋊C4—28no—
229(C2×D3)⋊C4—28no—
230(C2×D3)⋊C4—42no—
231(C2×D3)⋊C4—42no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236C4⋊Dic3—42no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244D24—48no—
245#204—48no—
246C2×#41—68no—
247C2×#41—124no—
248C2×#41—124no—
249C22×Q8—21no—
250C22×Q8—31no—
251C22×D4—32no—
252C22×D4—31no—
253C22×D4—62no—
254C22×D4—61no—
255C22×D4—61no—
256C22×D4—61no—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—31no—
260C23×C4—62no—
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C2×D8—64no—
265C2×D8—64no—
266C2×D8—122no—
267C22×C8—32no—
268C22×C8C22×D312yes—
269C4⋊Q8—22no—
270C4⋊Q8D1214yescomplemented
271#127—34no—
272#127—62no—
273#124—68no—
274#124—62no—
275#124—62no—
276C22⋊Q8—122no—
277#121—64no—
278#121—68no—
279#121—64no—
280#121—122no—
281C4×D4—64no—
282C4×D4—68no—
283C4×D4—64no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D312yes—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—62no—
301C2×C42—62no—
302C2×C42C22×D311yescharacteristic
303#103—64no—
304#102—64no—
305#102—122no—
306#102—122no—
307C22⋊C8—64no—
308#97—22no—
309#97—62no—
310#95—21no—
311#95—21no—
312#95—61no—
313#95—61no—
314C23×C3C22×D411yescharacteristic
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—24no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3AGL(1,ℤ/8)12yes—
336C3×Q8—48no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—24no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4#12012yes—
351C2×C3×C4#12012yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C2×#4112yes—
355C2×C3×C4H(5,𝔽2)12yes—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3AGL(1,ℤ/8)12yes—
368D12—24no—
369D12—24no—
370D12—48no—
371D12—48no—
372D12—416no—
373D12—42no—
374D12—42no—
375D12—42no—
376D12—42no—
377D12—84no—
378C4×D3—28no—
379C4×D3—416no—
380C4×D3—44no—
381C4×D3—42no—
382C4×D3—42no—
383Dic6—42no—
384Dic6—42no—
385C3×C8—24no—
386C3⋊C8—42no—
387C24—31no—
388C24—62no—
389C24—61no—
390#41—1232no—
391C2×Q8—24no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—62no—
397C2×Q8—62no—
398C2×D4—34no—
399C2×D4—68no—
400C2×D4—64no—
401C2×D4—64no—
402C2×D4—62no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—122no—
412C2×D4—122no—
413C2×D4—122no—
414C2×D4—122no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4—122no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—68no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—121no—
437C22×C4—121no—
438C22×C4—124no—
439C22×C4—124no—
440C22×C4D3×D411yescharacteristic
441C22×C4D3×D411yescharacteristic
442C22×C4C23×D311yescharacteristic
443SD16—128no—
444D8—128no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—62no—
450C2×C8#23914yescomplemented
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—122no—
456C4⋊C4C2×D1214yes—
457C22⋊C4—68no—
458C22⋊C4—68no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C42—62no—
466C42—62no—
467C42—61no—
468C42—61no—
469C42C2×D1212yes—
470C42D3×D412yes—
471C22×C3C2×#12011yescharacteristic
472C22×C3#53211yescharacteristic
473C22×C3#53511yescharacteristic
474C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
475C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
476C22×C3#57511yescharacteristic
477C22×C3#57511yescharacteristic
478C2×D3—22no—
479C2×D3—22no—
480C2×D3—22no—
481C2×D3—41no—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—44no—
486C2×D3—44no—
487C2×D3—44no—
488C2×D3—82no—
489C2×D3—82no—
490C2×D3—82no—
491C2×D3—82no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—44no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4#54612yes—
505C3×C4#55012yes—
506Dic3—22no—
507Dic3—41no—
508Dic3—41no—
509Dic3—81no—
510C23—32no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—64no—
518C23—122no—
519C23—122no—
520C23—122no—
521C23—122no—
522C23—122no—
523C23—122no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23C2×D3×D411yescharacteristic
532Q8—48no—
533Q8—122no—
534Q8—122no—
535D4—64no—
536D4—64no—
537D4—128no—
538D4—128no—
539D4—1216no—
540D4—122no—
541D4—122no—
542D4—122no—
543D4—122no—
544D4—244no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—32no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—68no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—124no—
569C2×C4—1216no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4#88212yes—
576C2×C4#88212yes—
577C2×C4C22×D1211yescharacteristic
578C2×C4C2×D3×D411yescharacteristic
579C2×C4C2×#23912yes—
580C2×C4#100912yes—
581C8—24no—
582C8—122no—
583C2×C3#243111yescharacteristic
584C2×C3#300311yescharacteristic
585C2×C3#300511yescharacteristic
586C2×C3#317111yescharacteristic
587C2×C3#318011yescharacteristic
588C2×C3#321311yescharacteristic
589C2×C3#321411yescharacteristic
590D3—42no—
591D3—84no—
592D3—161no—
593C22—62no—
594C22—62no—
595C22—62no—
596C22—121no—
597C22—121no—
598C22—121no—
599C22—121no—
600C22—124no—
601C22—124no—
602C22—124no—
603C22—242no—
604C22—242no—
605C22—242no—
606C22—242no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22C2×#88211yescharacteristic
615C22#567411yescharacteristic
616C22#572511yescharacteristic
617C22D3×AGL(1,ℤ/8)11yescharacteristic
618C22D3×AGL(1,ℤ/8)11yescharacteristic
619C22#577711yescharacteristic
620C22#577711yescharacteristic
621C4—22no—
622C4—22no—
623C4—44no—
624C4—62no—
625C4—81no—
626C4—81no—
627C4—121no—
628C4—121no—
629C4—241no—
630C4#557712yes—
631C4#563012yes—
632C3#2560211yescharacteristic, complemented, Sylow
633C2—122no—
634C2—244no—
635C2—481no—
636C2#7662611yescharacteristic
637C2#7892611yescharacteristic
638C2#7895911yescharacteristic
639C2#8168411yescharacteristic
640C2#8168511yescharacteristic
641C2#8195811yescharacteristic
642C2#8196211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
hypercenter#1993D3
hypocenterC3#25602
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×#5725C2semidirect
2. C2×#5089C2semidirect
3. C2×#4712C2semidirect
4. #74256C2semidirect
5. C3×#1993C2semidirect
6. #69807C2semidirect
7. #69553C2semidirect
8. #68897C2semidirect
9. #68896C2semidirect
10. #68887C2semidirect
11. #68886C2semidirect
54. #1993D3semidirect
76. C3×C4⋊Q8D4semidirect
113. #874D4semidirect
220. C2×C3×C8#41semidirect
270. C4⋊Q8D12semidirect
450. C2×C8#239semidirect
632. C3#25602semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#50892
C2×#47122
#742562
C3×#19932
#698072
#695532
#688972
#688962
#688872
#688862
#256023