Finite Group Databasebeta

← #10943977

Subgroups — #10943977

Order 768.

2008 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D12C211yesmaximal, characteristic, complemented
2C3×#2856C211yesmaximal, characteristic, complemented
3C2×#5072C212yesmaximal, complemented
4C2×#5036C212yesmaximal, complemented
5#71677C212yesmaximal, complemented
6#71530C211yesmaximal, characteristic, complemented
7#69657C214yesmaximal, complemented
8#69584C212yesmaximal, complemented
9#32902—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C23×C4×D3—21no—
12C2×C3×#127C2211yescharacteristic, complemented
13C2×C3×#121—21no—
14C2×C3×#121—21no—
15C2×C3×#121C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×#213C2214yes—
18C22×(C2×D3)⋊C4—21no—
19C22×C4⋊Dic3C2211yescharacteristic
20C2×C4×D12—21no—
21C2×C4×D12—21no—
22C2×C4×D12—28no—
23C2×C4×D12C2212yes—
24C2×C4×D12C2212yes—
25C3×#390—21no—
26C3×#390—21no—
27C3×#390C2212yescomplemented
28C3×#384C2211yescharacteristic, complemented
29C2×#832C2214yescomplemented
30C2×#830C2212yes—
31#5072—28no—
32#5036—28no—
33C2×#809C2214yes—
34C2×C4⋊(C3⋊C8)C2212yescomplemented
35#4529C2214yescomplemented
36#4524C2212yescomplemented
37C22×C4×D4—31no—
38C2×#459—32no—
39C2×#447—32no—
40#2856D311yescharacteristic, complemented
41#1879—34no—
42#1874—32no—
43#1802—32no—
44#1797—31no—
45C24×D3—21no—
46C22×C3×D4—22no—
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4C2312yes—
50C23×C3×C4D411yescharacteristic
51C23×C3×C4D411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×Dic3—21no—
54C22×D12—21no—
55C22×D12—22no—
56C22×D12—24no—
57C22×D12—24no—
58C22×D12D414yescomplemented
59C22×D12C2312yes—
60C22×C4×D3—28no—
61C22×C4×D3—22no—
62C22×C4×D3—22no—
63C22×C4×D3—44no—
64C22×C4×D3—44no—
65C3×#127—24no—
66C3×#121—24no—
67C3×#121—24no—
68C3×#121—44no—
69C3×#121—44no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C42—24no—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×#213—28no—
82C2×#213—28no—
83C2×#213—28no—
84C2×(C2×D3)⋊C4—28no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3C2312yes—
90C22×C3⋊C8C2314yes—
91C4×D12—28no—
92C4×D12—28no—
93C4×D12—24no—
94C4×D12—24no—
95C4×D12—24no—
96C4×D12D418yescomplemented
97#832—28no—
98#830—24no—
99#809—28no—
100C4⋊(C3⋊C8)—24no—
101C23×D4—31no—
102C24×C4—61no—
103C22×D8—34no—
104C2×#127C2×D311yescharacteristic, complemented
105C2×#121—21no—
106C2×#121—21no—
107C2×#121C2×D312yes—
108C2×C4×D4—32no—
109C2×C4×D4—32no—
110C2×C4×D4—68no—
111C2×C4×D4—61no—
112C2×C4×D4—61no—
113C22×C4⋊C4—31no—
114C22×C22⋊C4—61no—
115C22×C42C2×D311yescharacteristic
116#459—68no—
117#447—68no—
118C2×C4⋊C8—32no—
119C2×#102—34no—
120C2×#102—34no—
121C2×C22⋊C8—32no—
122#390—21no—
123#390—21no—
124#390C2×D312yescomplemented
125#384C2×D311yescharacteristic, complemented
126#340—34no—
127#336—32no—
128C24×C3—22no—
129C24×C3C2×D411yescharacteristic
130C23×D3—24no—
131C23×D3—24no—
132C23×D3—24no—
133C23×D3—22no—
134C23×D3—42no—
135C23×D3—42no—
136C23×D3—42no—
137C23×D3—42no—
138C2×C3×D4—24no—
139C2×C3×D4—22no—
140C2×C3×D4—24no—
141C2×C3×D4—22no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—41no—
159C22×C3×C4D814yescomplemented
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4#4112yes—
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—24no—
169C22×Dic3—22no—
170C22×Dic3—42no—
171C22×Dic3—42no—
172C2×D12—24no—
173C2×D12—24no—
174C2×D12—216no—
175C2×D12—24no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—216no—
180C2×D12C2×D414yes—
181C2×D12C2×D414yes—
182C2×D12#4114yes—
183C2×C4×D3—28no—
184C2×C4×D3—28no—
185C2×C4×D3—44no—
186C2×C4×D3—432no—
187C2×C4×D3—44no—
188C2×C4×D3—44no—
189C2×C4×D3—44no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C3×C22⋊C4—44no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42C2×D414yes—
201#213—416no—
202(C2×D3)⋊C4—28no—
203(C2×D3)⋊C4—28no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3C2×D414yes—
207C2×C3⋊C8—24no—
208C2×C3⋊C8—24no—
209C2×C3⋊C8—24no—
210C25—61no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—34no—
215C22×D4—32no—
216C22×D4—64no—
217C22×D4—64no—
218C22×D4—62no—
219C22×D4—61no—
220C22×D4C22×D312yes—
221C23×C4—68no—
222C23×C4—61no—
223C23×C4—62no—
224C23×C4—62no—
225C23×C4—124no—
226C23×C4—124no—
227C23×C4#6711yescharacteristic
228C23×C4#6711yescharacteristic
229C23×C4C22×D311yescharacteristic
230C2×D8—68no—
231C2×D8—68no—
232C2×D8—68no—
233C22×C8—34no—
234#127—24no—
235#121—24no—
236#121—24no—
237#121—44no—
238#121—44no—
239C4×D4—38no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—68no—
243C4×D4—68no—
244C4×D4—64no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4C22×D312yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—68no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C42—24no—
258C2×C42#6711yescharacteristic
259C2×C42#6711yescharacteristic
260C2×C42C22×D311yescharacteristic
261C2×C42C22×D311yescharacteristic
262C4⋊C8—64no—
263#102—68no—
264#102—68no—
265C22⋊C8—64no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3#12112yes—
278C23×C3C2×D812yes—
279C23×C3AGL(1,ℤ/8)12yes—
280C23×C3C22×D411yescharacteristic
281C22×D3—216no—
282C22×D3—24no—
283C22×D3—24no—
284C22×D3—24no—
285C22×D3—48no—
286C22×D3—48no—
287C22×D3—48no—
288C22×D3—48no—
289C22×D3—48no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—48no—
295C3×D4—44no—
296C3×D4—44no—
297C3×D4—84no—
298C3×D4—88no—
299C3×D4—84no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—28no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—44no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12012yes—
318C2×C3×C4#12112yes—
319C2×C3×C4C2×D812yes—
320C2×C3×C4C2×D812yes—
321C2×C3×C4AGL(1,ℤ/8)12yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D412yes—
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4112yes—
326C2×Dic3—24no—
327C2×Dic3—48no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330D12—216no—
331D12—24no—
332D12—24no—
333D12—24no—
334D12#12118yescomplemented
335C4×D3—416no—
336C4×D3—416no—
337C3⋊C8—44no—
338C24—22no—
339C24—64no—
340C24—64no—
341C24—64no—
342C24—62no—
343C24—122no—
344C24—122no—
345C24—122no—
346C24—122no—
347C24C2×#6711yescharacteristic
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—34no—
353C2×D4—34no—
354C2×D4—34no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—616no—
365C2×D4—64no—
366C2×D4—64no—
367C2×D4—616no—
368C2×D4—64no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—41no—
380C22×C4—68no—
381C22×C4—64no—
382C22×C4—62no—
383C22×C4—68no—
384C22×C4—124no—
385C22×C4—1232no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—124no—
389C22×C4—124no—
390C22×C4—124no—
391C22×C4#21314yescomplemented
392C22×C4D3×D412yes—
393C22×C4#23712yes—
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398C22×C4C2×#6711yescharacteristic
399C22×C4C23×D311yescharacteristic
400D8—1216no—
401C2×C8—64no—
402C2×C8—64no—
403C2×C8—64no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—34no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—64no—
410C4⋊C4—62no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—68no—
415C22⋊C4—68no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42D3×D414yes—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3#45914yescomplemented
434C22×C3C2×#12011yescharacteristic
435C22×C3C2×#12111yescharacteristic
436C22×C3#52511yescharacteristic
437C22×C3C22×D811yescharacteristic
438C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
439C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
440C22×C3#57511yescharacteristic
441C2×D3—28no—
442C2×D3—432no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—44no—
447C2×D3—44no—
448C2×D3—44no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—41no—
453C3×C4—82no—
454C3×C4#44714yescomplemented
455C3×C4C2×#12112yes—
456C3×C4#52512yes—
457Dic3—44no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—616no—
470C23—64no—
471C23—64no—
472C23—64no—
473C23—128no—
474C23—128no—
475C23—128no—
476C23—128no—
477C23—128no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23—128no—
483C23C2×#21312yes—
484C23#93212yes—
485C23#93812yes—
486C23C22×#6711yescharacteristic
487D4—38no—
488D4—44no—
489D4—44no—
490D4—616no—
491D4—64no—
492D4—64no—
493D4—64no—
494D4—84no—
495D4—88no—
496D4—84no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—64no—
514C2×C4—1216no—
515C2×C4—128no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—1216no—
519C2×C4C2×#21312yes—
520C2×C4C2×#21312yes—
521C2×C4#93712yes—
522C2×C4#93712yes—
523C2×C4#93812yes—
524C2×C4#94912yes—
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×D3×D412yes—
527C2×C4C2×#23711yescharacteristic
528C2×C4D3×#4112yes—
529C8—124no—
530C2×C3—24no—
531C2×C3—162no—
532C2×C3#284711yescharacteristic
533C2×C3C2×#44711yescharacteristic
534C2×C3#300611yescharacteristic
535C2×C3C2×#45911yescharacteristic
536C2×C3#303611yescharacteristic
537C2×C3#310011yescharacteristic
538C2×C3#310911yescharacteristic
539D3—48no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—68no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—1232no—
555C22—124no—
556C22—124no—
557C22—124no—
558C22—124no—
559C22—124no—
560C22—124no—
561C22#507214yescomplemented
562C22C22×#21311yescharacteristic
563C22C2×#93211yescharacteristic
564C22C2×#93711yescharacteristic
565C22C2×#93811yescharacteristic
566C22C2×#94911yescharacteristic
567C22#582211yescharacteristic
568C22#582811yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—41no—
573C4—82no—
574C4—124no—
575C4#503614yescomplemented
576C4D3×#12112yes—
577C4#560612yes—
578C3#3290211yescharacteristic, complemented, Sylow
579C2—24no—
580C2—128no—
581C2—162no—
582C2C2×#503611yescharacteristic
583C2#8262411yescharacteristic
584C2#8272311yescharacteristic
585C2C2×#507211yescharacteristic
586C2#8329711yescharacteristic
587C2#8329811yescharacteristic
588C2#8330011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2856C2
socleC23×C3C22×D4
hypercenter#2856D3
hypocenterC3#32902
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2856

Complements

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

NH ≅ G/Ntype
1. C22×C4×D12C2semidirect
2. C3×#2856C2semidirect
3. C2×#5072C2semidirect
4. C2×#5036C2semidirect
5. #71677C2semidirect
6. #71530C2semidirect
7. #69657C2semidirect
8. #69584C2semidirect
12. C2×C3×#127C22semidirect
27. C3×#390C22semidirect
28. C3×#384C22semidirect
29. C2×#832C22semidirect
34. C2×C4⋊(C3⋊C8)C22semidirect
35. #4529C22semidirect
36. #4524C22semidirect
40. #2856D3semidirect
58. C22×D12D4semidirect
96. C4×D12D4semidirect
104. C2×#127C2×D3semidirect
124. #390C2×D3semidirect
125. #384C2×D3semidirect
159. C22×C3×C4D8semidirect
334. D12#121semidirect
391. C22×C4#213semidirect
433. C22×C3#459semidirect
454. C3×C4#447semidirect
561. C22#5072semidirect
575. C4#5036semidirect
578. C3#32902semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D122
C3×#28562
C2×#50722
C2×#50362
#716772
#715302
#696572
#695842
#329023