Finite Group Databasebeta

← #10846015

Subgroups — #10846015

Order 768.

1206 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
1C3×#2541C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic, complemented
3C2×#5039C212yesmaximal, complemented
4#82417C211yesmaximal, characteristic, complemented
5C2×#5012C211yesmaximal, characteristic, complemented
6#71764C211yesmaximal, characteristic, complemented
7#71684C212yesmaximal, complemented
8#71682C212yesmaximal, complemented
9#71680C212yesmaximal, complemented
10C4×#809C211yesmaximal, characteristic, complemented
11#69467C211yesmaximal, characteristic, complemented
12#25392—31nomaximal, Sylow
13C2×C3×#127—21no—
14C2×C3×#127C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×C3×#121C2212yes—
17C2×C3×C4×D4C2211yescharacteristic
18C2×#938—21no—
19C22×#213C2212yes—
20C2×#930C2212yes—
21C22×#208C2212yes—
22C2×C4×D12C2211yescharacteristic
23C3×#392—22no—
24C3×#390—21no—
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#832C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29#5212C2212yes—
30C2×#830C2212yes—
31#5072—24no—
32#5039—28no—
33#5012—24no—
34#4987—21no—
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#808C2212yes—
38#4961—22no—
39#4935C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44#4529C2212yes—
45C2×#469—32no—
46C2×#459—31no—
47C2×#436—31no—
48#2541D311yescharacteristic, complemented
49#2537—31no—
50#2034—32no—
51#1999—32no—
52#1957—31no—
53#1948—31no—
54#1895—32no—
55C4×#102—31no—
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4D411yescharacteristic
61C22×C3×D4C2312yes—
62C23×C3×C4C2312yes—
63C22×#67—22no—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C3×#127—22no—
67C3×#127—22no—
68C3×#127—44no—
69C3×#121—24no—
70C3×#121—24no—
71C3×#121—42no—
72C3×#121—42no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4D412yes—
81C2×C3×C42—21no—
82C2×C3×C42C2311yescharacteristic
83C2×#217—22no—
84#938—44no—
85C2×#213—24no—
86C2×#213—24no—
87C2×#213—42no—
88#930—24no—
89#930—42no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×#208—24no—
97C2×#208—24no—
98C2×#208—42no—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C4×D12—22no—
103C4×D12—22no—
104C3×#95—21no—
105#832—24no—
106#832—24no—
107#832—42no—
108C3⋊#95—22no—
109#830—24no—
110#809—22no—
111#809D414yescomplemented
112#808—24no—
113C4⋊(C3⋊C8)D414yescomplemented
114C4×C3⋊C8—22no—
115C22×D8—32no—
116C22×M16—32no—
117C2×#127—21no—
118C2×#127C2×D311yescharacteristic
119C2×#121—32no—
120C2×#121—61no—
121C2×#121C2×D312yes—
122C2×#121C2×D312yes—
123C2×C4×D4—31no—
124C2×C4×D4C2×D311yescharacteristic
125#469—68no—
126#459—64no—
127#436—64no—
128C2×#106—32no—
129C2×C4⋊C8—31no—
130C2×#102—31no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#102—32no—
135C2×C22⋊C8—32no—
136C2×C4×C8—31no—
137#394—62no—
138#392—22no—
139#390—21no—
140#384—31no—
141#380—32no—
142#380—61no—
143C4×C4⋊C4C2×D311yescharacteristic
144#340—31no—
145#340—32no—
146#340—31no—
147C24×C3—22no—
148C24×C3C2×D412yes—
149C23×D3—21no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—44no—
160C2×C3×D4—44no—
161C2×C3×D4—44no—
162C2×C3×D4—44no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—81no—
168C2×C3×D4—81no—
169C2×C3×D4—82no—
170C2×C3×D4—82no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C2×#67—44no—
184C2×#67—44no—
185C2×#67—82no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12C2×D412yes—
194C2×C4×D3—42no—
195C2×C4×D3—42no—
196C2×C4×D3—42no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4C2×D412yes—
201C3×C4⋊C4#4112yes—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—44no—
204C3×C22⋊C4—44no—
205C3×C42—42no—
206C3×C42C2×D412yes—
207C3×C42#4112yes—
208#217—44no—
209#213—48no—
210(C2×D3)⋊C4—44no—
211(C2×D3)⋊C4—42no—
212C4⋊Dic3—22no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—22no—
216C4⋊Dic3—42no—
217C4⋊Dic3—81no—
218#208—48no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8C2×D414yes—
227C2×C3⋊C8C2×D412yes—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—31no—
233C22×D4—62no—
234C22×D4#6711yescharacteristic
235C22×D4C22×D312yes—
236C23×C4—61no—
237C23×C4C22×D312yes—
238C2×D8—64no—
239C2×D8—64no—
240C2×D8—122no—
241C2×M16—64no—
242C2×M16—64no—
243C2×M16—122no—
244C22×C8—31no—
245C22×C8—32no—
246C22×C8—31no—
247#127—22no—
248#127—22no—
249#127—44no—
250#121—24no—
251#121—24no—
252#121—42no—
253#121—42no—
254#121—64no—
255#121—122no—
256#121—124no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—62no—
260C4×D4—62no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4#6712yes—
274C2×C42—21no—
275C2×C42C22×D311yescharacteristic
276#106—64no—
277C4⋊C8—34no—
278#102—34no—
279#102—64no—
280#102—64no—
281#102—62no—
282#102—122no—
283C22⋊C8—64no—
284C4×C8—62no—
285#95—21no—
286#95—62no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3C22×D411yescharacteristic
302C22×D3—22no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—41no—
309C3×D4—42no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—44no—
314C3×D4—84no—
315C3×D4—88no—
316C3×D4—88no—
317C3×D4—84no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4#12112yes—
334C2×C3×C4#12112yes—
335C2×C3×C4AGL(1,ℤ/8)12yes—
336C2×C3×C4AGL(1,ℤ/8)12yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4H(5,𝔽2)11yescharacteristic
342C2×C3×C4H(5,𝔽2)11yescharacteristic
343#67—88no—
344C2×Dic3—41no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—42no—
350D12—22no—
351D12—22no—
352D12—41no—
353D12—41no—
354C4×D3—84no—
355C3⋊C8—22no—
356C3⋊C8—24no—
357C3⋊C8—41no—
358C24—22no—
359C24—61no—
360C24C2×#6712yes—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—32no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4—124no—
389C2×D4—124no—
390C2×D4—242no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—61no—
400C22×C4—62no—
401C22×C4—122no—
402C22×C4—122no—
403C22×C4—122no—
404C22×C4C2×#6712yes—
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C23×D311yescharacteristic
408D8—128no—
409M16—128no—
410C2×C8—34no—
411C2×C8—32no—
412C2×C8—61no—
413C2×C8—62no—
414C2×C8—61no—
415C2×C8—62no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—61no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C4⋊C4—62no—
426C4⋊C4—122no—
427C4⋊C4—241no—
428C4⋊C4D3×D412yes—
429C4⋊C4#23712yes—
430C22⋊C4—24no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—124no—
434C22⋊C4—124no—
435C22⋊C4—122no—
436C42—42no—
437C42D3×D412yes—
438C42#23712yes—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3C2×#12111yescharacteristic
454C22×C3D4211yescharacteristic
455C22×C3#54611yescharacteristic
456C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
457C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
458C22×C3#57611yescharacteristic
459C22×C3#57711yescharacteristic
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—81no—
464C2×D3—81no—
465C2×D3—81no—
466C2×D3—81no—
467C3×C4—22no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4#46914yescomplemented
472C3×C4#53211yescharacteristic
473C3×C4#54011yescharacteristic
474C3×C4D4211yescharacteristic
475C3×C4#55011yescharacteristic
476Dic3—82no—
477Dic3—81no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—62no—
493C23—121no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23C22×#6711yescharacteristic
500D4—42no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—44no—
505D4—62no—
506D4—62no—
507D4—84no—
508D4—88no—
509D4—88no—
510D4—84no—
511D4—121no—
512D4—121no—
513D4—248no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—244no—
536C2×C4#93212yes—
537C2×C4#93912yes—
538C2×C4#93912yes—
539C2×C4#94912yes—
540C2×C4C2×D3×D411yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C2×C4C2×#23711yescharacteristic
543C2×C4C2×#23711yescharacteristic
544C2×C4#100411yescharacteristic
545C2×C4#100411yescharacteristic
546C8—62no—
547C8—64no—
548C8—121no—
549C2×C3—82no—
550C2×C3—162no—
551C2×C3#242811yescharacteristic
552C2×C3C2×#46911yescharacteristic
553C2×C3#306211yescharacteristic
554C2×C3#327211yescharacteristic
555C2×C3#327411yescharacteristic
556C2×C3#332511yescharacteristic
557C2×C3#333211yescharacteristic
558D3—82no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—122no—
574C22—122no—
575C22—122no—
576C22—241no—
577C22—241no—
578C22—241no—
579C22—241no—
580C22C2×#93211yescharacteristic
581C22D4×#6711yescharacteristic
582C22C2×#93911yescharacteristic
583C22#580611yescharacteristic
584C22C2×#94911yescharacteristic
585C22#583511yescharacteristic
586C22#583611yescharacteristic
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—242no—
592C4—241no—
593C4#503914yescomplemented
594C4#571611yescharacteristic
595C4#571711yescharacteristic
596C4#571811yescharacteristic
597C4#571911yescharacteristic
598C3#2539211yescharacteristic, complemented, Sylow
599C2—82no—
600C2—162no—
601C2—242no—
602C2C2×#503911yescharacteristic
603C2#8318611yescharacteristic
604C2#8337911yescharacteristic
605C2#8348711yescharacteristic
606C2#8348911yescharacteristic
607C2#8357411yescharacteristic
608C2#8357811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2541C2
socleC23×C3C22×D4
hypercenter#2541D3
hypocenterC3#25392
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C3×#2541C2semidirect
2. C2×#5072C2semidirect
3. C2×#5039C2semidirect
4. #82417C2semidirect
5. C2×#5012C2semidirect
6. #71764C2semidirect
7. #71684C2semidirect
8. #71682C2semidirect
9. #71680C2semidirect
10. C4×#809C2semidirect
11. #69467C2semidirect
48. #2541D3semidirect
111. #809D4semidirect
113. C4⋊(C3⋊C8)D4semidirect
471. C3×C4#469semidirect
593. C4#5039semidirect
598. C3#25392semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
C2×#50722
C2×#50392
#824172
C2×#50122
#717642
#716842
#716822
#716802
C4×#8092
#694672
#253923