Finite Group Databasebeta

← #10822679

Subgroups — #10822679

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
2C3×#2280C211yesmaximal, characteristic, complemented
3C2×#5207C211yesmaximal, characteristic, complemented
4C2×#5001C211yesmaximal, characteristic, complemented
5C2×#5001C211yesmaximal, characteristic, complemented
6C2×#4995C211yesmaximal, characteristic, complemented
7#71581C211yesmaximal, characteristic, complemented
8#69524C214yesmaximal, complemented
9#69523C214yesmaximal, complemented
10#23480—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C23×#67—21no—
13C23×D12C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#930—22no—
17C2×#930—24no—
18C2×#930—24no—
19C2×#930C2214yes—
20C22×(C2×D3)⋊C4—21no—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#381—21no—
25C3×#380C2212yescomplemented
26C3×#380C2212yescomplemented
27C3×#379—21no—
28C3×C4×C22⋊C4—22no—
29#5207—24no—
30C2×#830C2214yescomplemented
31#5001—24no—
32#5001—24no—
33#4995—24no—
34C2×#809C2212yescomplemented
35C2×#809C2214yescomplemented
36C2×#809C2212yescomplemented
37C2×#808C2212yescomplemented
38C2×#807C2212yescomplemented
39#4529C2214yescomplemented
40C22×#121—31no—
41C2×#428—31no—
42C2×#418—31no—
43C2×#418—31no—
44C2×#407—31no—
45#2280D311yescharacteristic, complemented
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—22no—
70C22×D12—21no—
71C22×D12D414yes—
72C22×D12C2312yes—
73C3×#117—22no—
74C3×#117—22no—
75C3×#117—22no—
76C3×#117—22no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—21no—
85C2×C3×C42—21no—
86#930—216no—
87#930—28no—
88#930—28no—
89C2×(C2×D3)⋊C4—28no—
90C2×(C2×D3)⋊C4—24no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3D414yes—
94C2×C4⋊Dic3C2312yes—
95C2×#208—22no—
96C2×#208—22no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208D414yescomplemented
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2312yes—
103C3×#95—22no—
104C3×#95—22no—
105#830—28no—
106#809—24no—
107#809—24no—
108#809—28no—
109#808—24no—
110#807—24no—
111C23×D4—31no—
112C23×D4—61no—
113C24×C4C2×D311yescharacteristic
114C22×M16—31no—
115C22×M16—31no—
116C2×#121—34no—
117C2×#121—64no—
118C2×#121—64no—
119C2×#121—62no—
120C2×#117C2×D311yescharacteristic
121C2×#117C2×D311yescharacteristic
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—64no—
133C2×C22⋊C8—34no—
134#381—21no—
135#380C2×D312yescomplemented
136#380C2×D312yescomplemented
137#379—21no—
138C4×C22⋊C4—22no—
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—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—44no—
169C22×C3×C4—44no—
170C22×C3×C4—44no—
171C22×C3×C4—44no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4#4112yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C22×C3×C4C2411yescharacteristic
181C2×#67—216no—
182C2×#67—432no—
183C2×#67—48no—
184C2×#67—48no—
185C22×Dic3—24no—
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—44no—
197C2×D12—42no—
198C2×D12—42no—
199C2×D12—44no—
200C2×D12C2×D414yes—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C22⋊C4—44no—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—44no—
211C3×C42—42no—
212C3×C42—42no—
213(C2×D3)⋊C4—216no—
214C4⋊Dic3—24no—
215C4⋊Dic3—22no—
216C4⋊Dic3C2×D414yes—
217#208—28no—
218#208—44no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8C2×D414yescomplemented
225C25—61no—
226C25#6711yescharacteristic
227C22×D4—34no—
228C22×D4—32no—
229C22×D4—68no—
230C22×D4—64no—
231C22×D4—62no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4—62no—
235C22×D4—62no—
236C22×D4—61no—
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×C4#6711yescharacteristic
246C23×C4#6712yes—
247C23×C4C22×D312yes—
248C23×C4C22×D311yescharacteristic
249C2×M16—34no—
250C2×M16—62no—
251C2×M16—62no—
252C2×M16—62no—
253C2×M16—62no—
254C2×M16—62no—
255C22×C8—32no—
256C22×C8—32no—
257#121—68no—
258#121—68no—
259#121—616no—
260#117—22no—
261#117—22no—
262#117—22no—
263#117—22no—
264C2×C4⋊C4—34no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4C22×D312yes—
269C2×C4⋊C4C22×D311yescharacteristic
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—68no—
276C2×C22⋊C4—64no—
277C2×C42—21no—
278C2×C42—21no—
279#107—64no—
280#106—64no—
281#102—68no—
282#102—64no—
283#102—64no—
284C22⋊C8—68no—
285#95—22no—
286#95—22no—
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—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4#12012yes—
332C2×C3×C4#12114yes—
333C2×C3×C4#12114yes—
334C2×C3×C4#12112yes—
335C2×C3×C4#12314yes—
336C2×C3×C4#12312yes—
337C2×C3×C4#12414yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344#67—432no—
345C2×Dic3—24no—
346C2×Dic3—48no—
347C2×Dic3—44no—
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×#6712yes—
367C24C2×#6711yescharacteristic
368C2×D4—34no—
369C2×D4—616no—
370C2×D4—68no—
371C2×D4—68no—
372C2×D4—64no—
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—124no—
383C2×D4—122no—
384C2×D4—122no—
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—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—64no—
405C22×C4—62no—
406C22×C4—124no—
407C22×C4#23512yes—
408C22×C4#23512yes—
409C22×C4D3×D412yes—
410C22×C4D3×D412yes—
411C22×C4D3×D412yes—
412C22×C4#23712yes—
413C22×C4C2×#6712yes—
414C22×C4C2×#6711yescharacteristic
415C22×C4C23×D311yescharacteristic
416M16—68no—
417M16—124no—
418C2×C8—34no—
419C2×C8—62no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—34no—
431C4⋊C4—64no—
432C4⋊C4—62no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—616no—
438C42—42no—
439C42—42no—
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—81no—
466C3×C4—81no—
467C3×C4—82no—
468C3×C4#39414yes—
469C3×C4C2×#12111yescharacteristic
470C3×C4C2×#12111yescharacteristic
471C3×C4C2×#12311yescharacteristic
472C3×C4C2×#12411yescharacteristic
473Dic3—44no—
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—122no—
492C23—122no—
493C23—124no—
494C23—124no—
495C23—248no—
496C23—248no—
497C23#92912yes—
498C23#93712yes—
499C23#93912yes—
500C23#94914yes—
501C23C22×#6711yescharacteristic
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—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—64no—
525C2×C4—128no—
526C2×C4—124no—
527C2×C4#88314yes—
528C2×C4#88414yes—
529C2×C4#88414yes—
530C2×C4#88514yes—
531C2×C4#92912yes—
532C2×C4#93712yes—
533C2×C4#93912yes—
534C2×C4C2×#23511yescharacteristic
535C2×C4C2×#23511yescharacteristic
536C2×C4C2×D3×D411yescharacteristic
537C2×C4C2×D3×D411yescharacteristic
538C2×C4C2×D3×D411yescharacteristic
539C2×C4C2×#23711yescharacteristic
540C8—64no—
541C8—122no—
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—24no—
552C22—22no—
553C22—22no—
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#495914yes—
569C22C2×#92911yescharacteristic
570C22C2×#93711yescharacteristic
571C22C2×#93911yescharacteristic
572C22C2×#94912yes—
573C22#582212yes—
574C4—42no—
575C4—82no—
576C4—81no—
577C4—81no—
578C4—124no—
579C4#495914yes—
580C4C2×#88311yescharacteristic
581C4C2×#88411yescharacteristic
582C4C2×#88411yescharacteristic
583C4C2×#88511yescharacteristic
584C3#2348011yescharacteristic, complemented, Sylow
585C2—24no—
586C2—44no—
587C2—244no—
588C2C2×#495911yescharacteristic
589C2#8221712yes—
590C2#8292412yes—
591C2#8292711yescharacteristic
592C2#8292811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2280C2
socleC23×C3C22×D4
prime radicalC22×#930C2
hypercenter#2280D3
hypocenterC3#23480
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C3×#2280C2semidirect
3. C2×#5207C2semidirect
4. C2×#5001C2semidirect
5. C2×#5001C2semidirect
6. C2×#4995C2semidirect
7. #71581C2semidirect
8. #69524C2semidirect
9. #69523C2semidirect
25. C3×#380C22semidirect
26. C3×#380C22semidirect
30. C2×#830C22semidirect
34. C2×#809C22semidirect
35. C2×#809C22semidirect
36. C2×#809C22semidirect
37. C2×#808C22semidirect
38. C2×#807C22semidirect
39. #4529C22semidirect
45. #2280D3semidirect
100. C2×#208D4semidirect
135. #380C2×D3semidirect
136. #380C2×D3semidirect
224. C2×C3⋊C8C2×D4semidirect
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
C3×#22802
C2×#52072
C2×#50012
C2×#49952
#715812
#695242
#695232
#234803