Finite Group Databasebeta

← #10822680

Subgroups — #10822680

Order 768.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#930C211yesmaximal, characteristic
2C2×C3×#407C211yesmaximal, characteristic, complemented
3#84027C211yesmaximal, characteristic, complemented
4C2×#5131C211yesmaximal, characteristic, complemented
5C2×#5131C211yesmaximal, characteristic, complemented
6C2×#5121C211yesmaximal, characteristic, complemented
7#71985C211yesmaximal, characteristic, complemented
8#69935C214yesmaximal, complemented
9#69934C214yesmaximal, complemented
10#23480—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C23×#67—21no—
13C23×D12C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×#930—22no—
17C2×#930—24no—
18C2×#930—24no—
19C2×#930C2214yes—
20C22×(C2×D3)⋊C4—21no—
21C2×#926C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C3×#407—24no—
25C2×C3×C22⋊C8C2214yescomplemented
26#5247—21no—
27#5212C2212yescomplemented
28#5212C2212yescomplemented
29#5211—21no—
30C4×#217—22no—
31#5131—24no—
32#5131—24no—
33#5121—24no—
34C2×#821C2212yescomplemented
35C2×#821C2214yescomplemented
36C2×#821C2212yescomplemented
37C2×#818C2212yescomplemented
38C2×#817C2212yescomplemented
39#4551C2214yescomplemented
40C22×#121—31no—
41C2×#428—31no—
42C2×#418—31no—
43C2×#418—31no—
44C2×#407D311yescharacteristic, complemented
45#2280—31no—
46#2034—34no—
47#2033—34no—
48#1736—31no—
49C25×C3D411yescharacteristic
50C24×D3—21no—
51C23×C3×C4—22no—
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—22no—
57C23×C3×C4D412yes—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—28no—
62C22×#67—24no—
63C22×#67—48no—
64C23×Dic3—21no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12—22no—
68C22×D12—22no—
69C22×D12—21no—
70C22×D12—22no—
71C22×D12D414yes—
72C22×D12C2312yes—
73C2×C3×M16—22no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16—22no—
77C2×C3×M16—22no—
78C2×C3×M16D414yescomplemented
79C22×C3×C8C2312yes—
80C22×C3×C8C2312yes—
81C2×#217—22no—
82C2×#217—22no—
83C2×#217—21no—
84C2×#217—21no—
85#930—216no—
86#930—28no—
87#930—28no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—24no—
90#926—22no—
91#926—22no—
92#926—22no—
93#926—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3D414yes—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2312yes—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C3×C22⋊C8—28no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106#821—24no—
107#821—24no—
108#821—28no—
109#818—24no—
110#817—24no—
111C23×D4—31no—
112C23×D4—61no—
113C24×C4C2×D311yescharacteristic
114C22×M16C2×D311yescharacteristic
115C22×M16C2×D311yescharacteristic
116C2×#121—34no—
117C2×#121—64no—
118C2×#121—64no—
119C2×#121—62no—
120C2×#117—31no—
121C2×#117—31no—
122C22×C4⋊C4—31no—
123C22×C22⋊C4—61no—
124#428—64no—
125C2×#107—32no—
126C2×#106—32no—
127#418—64no—
128#418—64no—
129C2×#102—32no—
130C2×#102—34no—
131C2×#102—32no—
132#407—24no—
133C2×C22⋊C8C2×D314yescomplemented
134#381—61no—
135#380—32no—
136#380—32no—
137#379—61no—
138C4×C22⋊C4—62no—
139#340—34no—
140C24×C3—22no—
141C24×C3—24no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—42no—
145C24×C3C2×D412yes—
146C24×C3C2×D411yescharacteristic
147C23×D3—24no—
148C23×D3—22no—
149C23×D3—44no—
150C23×D3—44no—
151C23×D3—42no—
152C23×D3—42no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—24no—
157C22×C3×C4—24no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—44no—
166C22×C3×C4—44no—
167C22×C3×C4—44no—
168C22×C3×C4—44no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×C3×C4#4112yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—216no—
179C2×#67—432no—
180C2×#67—48no—
181C2×#67—48no—
182C22×Dic3—24no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—44no—
188C2×D12—28no—
189C2×D12—28no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—24no—
195C2×D12—22no—
196C2×D12—42no—
197C2×D12—44no—
198C2×D12—42no—
199C2×D12—44no—
200C2×D12C2×D414yes—
201C3×M16—28no—
202C3×M16—44no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8C2×D414yescomplemented
209#217—44no—
210#217—42no—
211#217—42no—
212#217—44no—
213(C2×D3)⋊C4—216no—
214C4⋊Dic3—24no—
215C4⋊Dic3—22no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3C2×D414yes—
223C4×Dic3—42no—
224C4×Dic3—42no—
225C25—61no—
226C25D1211yescharacteristic
227C22×D4—34no—
228C22×D4—32no—
229C22×D4—68no—
230C22×D4—64no—
231C22×D4—61no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4—62no—
235C22×D4—62no—
236C22×D4—62no—
237C22×D4—128no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—61no—
245C23×C4D1211yescharacteristic
246C23×C4D1212yes—
247C23×C4C22×D312yes—
248C23×C4C22×D311yescharacteristic
249C2×M16—22no—
250C2×M16—22no—
251C2×M16—22no—
252C2×M16—22no—
253C2×M16—22no—
254C2×M16#6714yescomplemented
255C22×C8C22×D312yes—
256C22×C8C22×D312yes—
257#121—68no—
258#121—68no—
259#121—616no—
260#117—62no—
261#117—62no—
262#117—62no—
263#117—62no—
264C2×C4⋊C4—34no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—62no—
271C2×C22⋊C4—68no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C42—61no—
278C2×C42—61no—
279#107—64no—
280#106—64no—
281#102—64no—
282#102—64no—
283#102—68no—
284C22⋊C8—28no—
285#95—62no—
286#95—62no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—28no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—48no—
297C23×C3—48no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3#12012yes—
301C23×C3#12112yes—
302C23×C3#12312yes—
303C23×C3AGL(1,ℤ/8)14yes—
304C23×C3C22×D411yescharacteristic
305C22×D3—24no—
306C22×D3—48no—
307C22×D3—48no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—44no—
311C22×D3—44no—
312C22×D3—48no—
313C22×D3—88no—
314C22×D3—88no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—44no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—48no—
324C2×C3×C4#12012yes—
325C2×C3×C4#12112yes—
326C2×C3×C4#12114yes—
327C2×C3×C4#12114yes—
328C2×C3×C4#12314yes—
329C2×C3×C4#12312yes—
330C2×C3×C4#12414yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337#67—432no—
338C2×Dic3—24no—
339C2×Dic3—48no—
340C2×Dic3—44no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348D12—24no—
349D12—24no—
350D12—42no—
351D12—42no—
352D12—48no—
353C3×C8—24no—
354C3×C8—42no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—42no—
360C24—64no—
361C24—62no—
362C24—124no—
363C24—124no—
364C24—122no—
365C24—122no—
366C24C2×D1212yes—
367C24C2×D1211yescharacteristic
368C2×D4—34no—
369C2×D4—616no—
370C2×D4—64no—
371C2×D4—68no—
372C2×D4—68no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—1232no—
379C2×D4—128no—
380C2×D4—128no—
381C2×D4—124no—
382C2×D4—122no—
383C2×D4—122no—
384C2×D4—124no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—64no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—124no—
407C22×C4C2×D1212yes—
408C22×C4C2×D1211yescharacteristic
409C22×C4#23512yes—
410C22×C4D3×D412yes—
411C22×C4D3×D412yes—
412C22×C4#23712yes—
413C22×C4#23712yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416M16—28no—
417M16—44no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8C2×#6714yescomplemented
424C4⋊C4—34no—
425C4⋊C4—64no—
426C4⋊C4—62no—
427C4⋊C4—62no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—62no—
433C22⋊C4—616no—
434C22⋊C4—124no—
435C22⋊C4—122no—
436C22⋊C4—124no—
437C22⋊C4—122no—
438C42—122no—
439C42—122no—
440C22×C3—24no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—24no—
444C22×C3—24no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3—416no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3#39414yes—
453C22×C3C2×#12011yescharacteristic
454C22×C3C2×#12111yescharacteristic
455C22×C3C2×#12311yescharacteristic
456C22×C3C2×AGL(1,ℤ/8)12yes—
457C22×C3#57512yes—
458C2×D3—44no—
459C2×D3—44no—
460C2×D3—44no—
461C2×D3—816no—
462C2×D3—84no—
463C2×D3—84no—
464C3×C4—42no—
465C3×C4#39414yes—
466C3×C4C2×#12111yescharacteristic
467C3×C4C2×#12111yescharacteristic
468C3×C4C2×#12311yescharacteristic
469C3×C4C2×#12411yescharacteristic
470Dic3—44no—
471Dic3—81no—
472Dic3—81no—
473Dic3—82no—
474C23—22no—
475C23—22no—
476C23—28no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—48no—
484C23—48no—
485C23—44no—
486C23—44no—
487C23—64no—
488C23—128no—
489C23—128no—
490C23—128no—
491C23—124no—
492C23—122no—
493C23—124no—
494C23—122no—
495C23—248no—
496C23—248no—
497C23#88212yes—
498C23#88612yes—
499C23#90814yes—
500C23#93012yes—
501C23C22×D1211yescharacteristic
502D4—64no—
503D4—64no—
504D4—1232no—
505D4—128no—
506D4—122no—
507D4—122no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—64no—
518C2×C4—128no—
519C2×C4—124no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4#88212yes—
528C2×C4#88314yes—
529C2×C4#88514yes—
530C2×C4#88612yes—
531C2×C4#93012yes—
532C2×C4#93814yes—
533C2×C4#93914yes—
534C2×C4C2×#23511yescharacteristic
535C2×C4C2×D3×D411yescharacteristic
536C2×C4C2×D3×D411yescharacteristic
537C2×C4C2×#23711yescharacteristic
538C2×C4C2×#23711yescharacteristic
539C2×C4C22×#6711yescharacteristic
540C8—24no—
541C8—42no—
542C2×C3—24no—
543C2×C3—44no—
544C2×C3C2×#39411yescharacteristic
545C2×C3#300311yescharacteristic
546C2×C3#300511yescharacteristic
547C2×C3#305512yes—
548C2×C3#309412yes—
549D3—84no—
550C22—24no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—416no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—124no—
563C22—124no—
564C22—124no—
565C22—2416no—
566C22—244no—
567C22—244no—
568C22#496114yes—
569C22C2×#88211yescharacteristic
570C22C2×#88611yescharacteristic
571C22C2×#90812yes—
572C22#574812yes—
573C22C2×#93011yescharacteristic
574C4—42no—
575C4—124no—
576C4—241no—
577C4—241no—
578C4—242no—
579C4#496114yes—
580C4C2×#88311yescharacteristic
581C4C2×#88511yescharacteristic
582C4C2×#93811yescharacteristic
583C4C2×#93911yescharacteristic
584C3#2348011yescharacteristic, complemented, Sylow
585C2—24no—
586C2—44no—
587C2—244no—
588C2#7862412yes—
589C2#7863711yescharacteristic
590C2#7863811yescharacteristic
591C2C2×#496111yescharacteristic
592C2#8377212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#407C2
socleC23×C3C22×D4
prime radicalC22×#930C2
hypercenterC2×#407D3
hypocenterC3#23480
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×C3×#407C2semidirect
3. #84027C2semidirect
4. C2×#5131C2semidirect
5. C2×#5131C2semidirect
6. C2×#5121C2semidirect
7. #71985C2semidirect
8. #69935C2semidirect
9. #69934C2semidirect
25. C2×C3×C22⋊C8C22semidirect
27. #5212C22semidirect
28. #5212C22semidirect
34. C2×#821C22semidirect
35. C2×#821C22semidirect
36. C2×#821C22semidirect
37. C2×#818C22semidirect
38. C2×#817C22semidirect
39. #4551C22semidirect
44. C2×#407D3semidirect
78. C2×C3×M16D4semidirect
133. C2×C22⋊C8C2×D3semidirect
208. C2×C3×C8C2×D4semidirect
254. C2×M16#67semidirect
423. C2×C8C2×#67semidirect
584. C3#23480semidirect

Maximal subgroups

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

subgroupindex
C22×#9302
C2×C3×#4072
#840272
C2×#51312
C2×#51212
#719852
#699352
#699342
#234803