Finite Group Databasebeta

← #10944122

Subgroups — #10944122

Order 768.

1588 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5521C211yesmaximal, characteristic, complemented
2C3×#2620C211yesmaximal, characteristic, complemented
3C2×#5057C212yesmaximal, complemented
4C2×#5046C211yesmaximal, characteristic, complemented
5C2×#5036C211yesmaximal, characteristic
6#71693C211yesmaximal, characteristic, complemented
7#71678C211yesmaximal, characteristic, complemented
8#71583C211yesmaximal, characteristic, complemented
9#69687C212yesmaximal, complemented
10#69657C212yesmaximal, complemented
11#69584C212yesmaximal, complemented
12#32917—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic, complemented
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×#121C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C22×#215C2212yes—
19C22×#213C2212yes—
20C2×#926C2211yescharacteristic
21#5521—28no—
22C2×#893—22no—
23C2×D3×C22⋊C4—22no—
24C2×C4×D12C2212yes—
25C2×C4×D12C2212yes—
26C3×#396—22no—
27C3×#394—22no—
28C3×#391C2211yescharacteristic, complemented
29C3×#390C2211yescharacteristic, complemented
30C3×#384C2211yescharacteristic, complemented
31C2×#835C2212yescomplemented
32C2×#832C2212yescomplemented
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5057—28no—
36#5046—24no—
37#5036—24no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×C4⋊(C3⋊C8)C2212yescomplemented
41#4529C2212yescomplemented
42#4529C2212yescomplemented
43#4524C2212yescomplemented
44C2×#518—31no—
45C2×#463—32no—
46C2×#450—31no—
47C2×#447—31no—
48#2620D311yescharacteristic, complemented
49#1879—32no—
50#1877—32no—
51#1875—31no—
52#1873—31no—
53#1802—32no—
54#1798—31no—
55C24×D3—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×D12—21no—
61C22×D12—22no—
62C22×D12—24no—
63C22×D12—24no—
64C22×D12D414yes—
65C22×D12C2312yes—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C3×#124—24no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×#121—22no—
72C3×#121—22no—
73C3×#117—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4D412yes—
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#217—21no—
87C2×#215—24no—
88C2×#215—24no—
89C2×#215—24no—
90C2×#213—24no—
91C2×#213—24no—
92C2×#213—24no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95#926—24no—
96C2×C4⋊Dic3C2312yes—
97C2×C4×Dic3—21no—
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2312yes—
100#893—216no—
101D3×C22⋊C4—216no—
102C4×D12—24no—
103C4×D12—24no—
104C4×D12—24no—
105C4×D12D418yescomplemented
106C3×#95—22no—
107#835—24no—
108#832—24no—
109#830—22no—
110#830—22no—
111#809—24no—
112#809—24no—
113C4⋊(C3⋊C8)—24no—
114C23×D4—31no—
115C22×SD16—32no—
116C22×D8—32no—
117C2×#124C2×D311yescharacteristic, complemented
118C2×C22⋊Q8C2×D311yescharacteristic
119C2×#121C2×D311yescharacteristic
120#518—68no—
121C2×C4×D4—32no—
122C2×C4×D4—32no—
123C2×C4×D4—62no—
124C2×#117—31no—
125C2×#117C2×D311yescharacteristic
126C22×C22⋊C4—62no—
127#463—68no—
128#450—64no—
129#447—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×C22⋊C8—31no—
136C2×C22⋊C8—31no—
137#396—22no—
138#394—22no—
139#391C2×D311yescharacteristic, complemented
140#390C2×D311yescharacteristic, complemented
141#384C2×D311yescharacteristic, complemented
142#340—32no—
143#340—32no—
144#336—32no—
145C24×C3—21no—
146C24×C3C2×D411yescharacteristic
147C23×D3—22no—
148C23×D3—24no—
149C23×D3—24no—
150C23×D3—24no—
151C23×D3—82no—
152C23×D3—82no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×D4—22no—
158C2×C3×D4—21no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—42no—
162C2×C3×D4—82no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4#4112yes—
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—22no—
175C2×D12—24no—
176C2×D12—24no—
177C2×D12—216no—
178C2×D12—24no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—216no—
183C2×D12C2×D414yes—
184C2×D12C2×D414yes—
185C2×D12#4114yes—
186C2×C4×D3—28no—
187C2×C4×D3—28no—
188C2×C4×D3—82no—
189C2×C4×D3—82no—
190C2×C4×D3—82no—
191C2×C4×D3—82no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—82no—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—42no—
203C3×C22⋊C4—82no—
204C3×C42—22no—
205C3×C42C2×D414yes—
206#217—24no—
207#215—48no—
208#213—48no—
209(C2×D3)⋊C4—28no—
210(C2×D3)⋊C4—28no—
211C4⋊Dic3—22no—
212C4⋊Dic3C2×D414yes—
213C4×Dic3—24no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C25—61no—
221C22×Q8C22×D311yescharacteristic
222C22×D4—21no—
223C22×D4—34no—
224C22×D4—32no—
225C22×D4—64no—
226C22×D4—64no—
227C22×D4—62no—
228C22×D4—61no—
229C22×D4C22×D311yescharacteristic
230C23×C4—62no—
231C23×C4—62no—
232C23×C4C22×D311yescharacteristic
233C2×SD16—64no—
234C2×SD16—64no—
235C2×SD16—64no—
236C2×D8—64no—
237C2×D8—64no—
238C2×D8—64no—
239C22×C8—32no—
240C22×C8—32no—
241#124—24no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244#121—22no—
245#121—22no—
246C4×D4—38no—
247C4×D4—64no—
248C4×D4—64no—
249C4×D4—616no—
250C4×D4—64no—
251#117—24no—
252#117—64no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4#6712yes—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—616no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4#6712yes—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270C2×C42C22×D311yescharacteristic
271C4⋊C8—64no—
272#103—64no—
273#102—64no—
274#102—64no—
275#102—64no—
276C22⋊C8—62no—
277C22⋊C8—62no—
278#95—22no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3#12112yes—
289C23×C3C22×D411yescharacteristic
290C22×D3—24no—
291C22×D3—24no—
292C22×D3—24no—
293C22×D3—216no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—84no—
299C22×D3—84no—
300C22×D3—88no—
301C22×D3—88no—
302C3×Q8—42no—
303C3×Q8—42no—
304C3×D4—42no—
305C3×D4—42no—
306C3×D4—84no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4#12014yes—
324C2×C3×C4#12112yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D412yes—
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4112yes—
329C2×Dic3—24no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332D12—216no—
333D12—24no—
334D12—24no—
335D12—24no—
336D12#12118yes—
337C4×D3—88no—
338C4×D3—88no—
339C3⋊C8—42no—
340C3⋊C8—42no—
341C24—21no—
342C24—64no—
343C24—64no—
344C24—62no—
345C24—64no—
346C24—242no—
347C24—242no—
348C24C2×#6711yescharacteristic
349C2×Q8—22no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—22no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—34no—
358C2×D4—34no—
359C2×D4—34no—
360C2×D4—42no—
361C2×D4—616no—
362C2×D4—64no—
363C2×D4—64no—
364C2×D4—616no—
365C2×D4—64no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—82no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—68no—
376C22×C4—62no—
377C22×C4—68no—
378C22×C4—242no—
379C22×C4—242no—
380C22×C4—242no—
381C22×C4—242no—
382C22×C4D3×D412yes—
383C22×C4#23712yes—
384C22×C4C2×#6712yes—
385C22×C4C2×#6712yes—
386C22×C4C2×#6711yescharacteristic
387C22×C4C23×D311yescharacteristic
388SD16—128no—
389D8—128no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—24no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—34no—
402C4⋊C4—42no—
403C4⋊C4—62no—
404C4⋊C4—82no—
405C22⋊C4—24no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—42no—
409C22⋊C4—68no—
410C22⋊C4—64no—
411C22⋊C4—68no—
412C22⋊C4—82no—
413C42—22no—
414C42—64no—
415C42D3×D414yes—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—81no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3C2×#12011yescharacteristic
427C22×C3C2×#12111yescharacteristic
428C22×C3#52511yescharacteristic
429C22×C3#57612yes—
430C22×C3#57712yes—
431C2×D3—28no—
432C2×D3—44no—
433C2×D3—44no—
434C2×D3—44no—
435C2×D3—44no—
436C2×D3—44no—
437C2×D3—44no—
438C2×D3—816no—
439C3×C4—22no—
440C3×C4—24no—
441C3×C4—42no—
442C3×C4—81no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4C2×#12112yes—
446C3×C4#52512yes—
447Dic3—44no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—616no—
458C23—64no—
459C23—64no—
460C23—64no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23—122no—
465C23—244no—
466C23—244no—
467C23—248no—
468C23—248no—
469C23#93812yes—
470C23C22×#6711yescharacteristic
471Q8—42no—
472Q8—42no—
473D4—38no—
474D4—42no—
475D4—42no—
476D4—616no—
477D4—64no—
478D4—64no—
479D4—64no—
480D4—84no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—64no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—248no—
501C2×C4—248no—
502C2×C4#93714yes—
503C2×C4#93812yes—
504C2×C4C2×D3×D411yescharacteristic
505C2×C4C2×D3×D412yes—
506C2×C4C2×#23711yescharacteristic
507C2×C4D3×#4112yes—
508C8—122no—
509C8—122no—
510C2×C3—42no—
511C2×C3—161no—
512C2×C3#284711yescharacteristic
513C2×C3#301212yes—
514C2×C3#304111yescharacteristic
515C2×C3#304211yescharacteristic
516C2×C3#311812yes—
517D3—48no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—68no—
524C22—82no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—82no—
529C22—124no—
530C22—124no—
531C22—124no—
532C22—124no—
533C22—124no—
534C22—124no—
535C22—2416no—
536C22C2×#93711yescharacteristic
537C22C2×#93811yescharacteristic
538C22#582811yescharacteristic
539C22#583512yes—
540C22#583712yes—
541C4—22no—
542C4—24no—
543C4—42no—
544C4—81no—
545C4—81no—
546C4—81no—
547C4—124no—
548C4D3×#12112yes—
549C4#560612yes—
550C3#3291711yescharacteristic, complemented, Sylow
551C2—42no—
552C2—128no—
553C2—161no—
554C2#8272311yescharacteristic
555C2#8338411yescharacteristic
556C2#8340511yescharacteristic
557C2#8342412yes—
558C2#8346012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2620C2
socleC23×C3C22×D4
prime radicalC2×#5036C2
hypercenter#2620D3
hypocenterC3#32917
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2620

Complements

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

NH ≅ G/Ntype
1. C2×#5521C2semidirect
2. C3×#2620C2semidirect
3. C2×#5057C2semidirect
4. C2×#5046C2semidirect
6. #71693C2semidirect
7. #71678C2semidirect
8. #71583C2semidirect
9. #69687C2semidirect
10. #69657C2semidirect
11. #69584C2semidirect
14. C2×C3×#124C22semidirect
28. C3×#391C22semidirect
29. C3×#390C22semidirect
30. C3×#384C22semidirect
31. C2×#835C22semidirect
32. C2×#832C22semidirect
40. C2×C4⋊(C3⋊C8)C22semidirect
41. #4529C22semidirect
42. #4529C22semidirect
43. #4524C22semidirect
48. #2620D3semidirect
105. C4×D12D4semidirect
117. C2×#124C2×D3semidirect
139. #391C2×D3semidirect
140. #390C2×D3semidirect
141. #384C2×D3semidirect
550. C3#32917semidirect

Maximal subgroups

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

subgroupindex
C2×#55212
C3×#26202
C2×#50572
C2×#50462
C2×#50362
#716932
#716782
#715832
#696872
#696572
#695842
#329173