Finite Group Databasebeta

← #10944125

Subgroups — #10944125

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
2#82677C211yesmaximal, characteristic, complemented
3C2×#5036C211yesmaximal, characteristic
4C2×#4787C212yesmaximal, complemented
5C2×#4731C211yesmaximal, characteristic, complemented
6C3×#1875C211yesmaximal, characteristic, complemented
7#71585C211yesmaximal, characteristic, complemented
8#70208C212yesmaximal, complemented
9#69909C211yesmaximal, characteristic, complemented
10#68896C212yesmaximal, complemented
11#68892C212yesmaximal, complemented
12#32917—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C2×C3×#121C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#936C2211yescharacteristic, complemented
17C22×#213C2212yes—
18C2×#926C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C22×#204C2212yes—
21#5521—28no—
22C2×#893C2212yes—
23C2×#893C2212yes—
24C2×D3×C22⋊C4—22no—
25C2×C4×D12—22no—
26C2×C3×#102C2212yescomplemented
27C2×C3×C22⋊C8C2211yescharacteristic
28#5222—22no—
29#5221C2211yescharacteristic, complemented
30C2×#830C2211yescharacteristic
31C2×#821C2212yes—
32C2×Dic3⋊C8C2212yescomplemented
33#5036—24no—
34#4975C2211yescharacteristic, complemented
35#4973C2211yescharacteristic, complemented
36C2×#810C2212yescomplemented
37C2×#809C2212yes—
38#4948—22no—
39#4787—28no—
40#4731—24no—
41C3×#340C2212yescomplemented
42#4550C2212yescomplemented
43#4529C2212yescomplemented
44C2×#518—31no—
45C2×#463—32no—
46C2×#450—31no—
47C2×#447—31no—
48#2620—31no—
49#1879—32no—
50#1877—32no—
51#1875D311yescharacteristic, complemented
52#1873—31no—
53#1802—32no—
54#1798—31no—
55C24×D3—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×D12—21no—
60C22×D12—22no—
61C22×D12—24no—
62C22×D12—24no—
63C22×D12D414yes—
64C22×D12C2312yes—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×Dic6C2311yescharacteristic
68C22×C3×C8C2312yes—
69C3×#121—22no—
70C3×#121—22no—
71C3×#117—24no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42—21no—
77C2×#217—21no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217D412yes—
81#936—24no—
82C2×#213—24no—
83C2×#213—24no—
84C2×#213—24no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87#926—24no—
88C2×C4⋊Dic3D412yes—
89C2×C4⋊Dic3C2311yescharacteristic
90#924—22no—
91#924—22no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3C2311yescharacteristic
95C2×C4×Dic3C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C2×#204—24no—
98C2×#204—24no—
99C2×#204—24no—
100#893—24no—
101#893—24no—
102#893—24no—
103#893D418yescomplemented
104D3×C22⋊C4—216no—
105C4×D12—216no—
106C3×#102—24no—
107C3×C22⋊C8—22no—
108C3⋊#95—22no—
109#830—22no—
110#821—24no—
111Dic3⋊C8—24no—
112#810—24no—
113#809—24no—
114C23×D4—31no—
115C22×SD16—32no—
116C22×D8—32no—
117C2×#124—31no—
118C2×C22⋊Q8—31no—
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×#102C2×D312yescomplemented
135C2×C22⋊C8—31no—
136C2×C22⋊C8C2×D311yescharacteristic
137#396—62no—
138#394—62no—
139#391—31no—
140#390—31no—
141#384—31no—
142#340—32no—
143#340C2×D312yescomplemented
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×D4—22no—
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—42no—
158C2×C3×D4—82no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4#4112yes—
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3C2×D412yes—
170C22×Dic3C2×D412yes—
171C2×D12—24no—
172C2×D12—24no—
173C2×D12—216no—
174C2×D12—24no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—216no—
179C2×D12C2×D414yes—
180C2×D12C2×D414yes—
181C2×D12#4114yes—
182C2×C4×D3—28no—
183C2×C4×D3—28no—
184C2×C4×D3—82no—
185C2×C4×D3—82no—
186C2×C4×D3—82no—
187C2×C4×D3—82no—
188C2×Dic6—22no—
189C2×Dic6—21no—
190C2×Dic6—22no—
191C2×Dic6—21no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4C2×D414yes—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—42no—
201C3×C42—24no—
202#217—24no—
203#217—42no—
204#217—44no—
205#217—82no—
206#213—48no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—28no—
209C4⋊Dic3—24no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212Dic3⋊C4—42no—
213Dic3⋊C4—82no—
214C4×Dic3—22no—
215C4×Dic3C2×D414yes—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219#204—48no—
220C25—61no—
221C22×Q8—31no—
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×C8C22×D312yes—
241#124—64no—
242C22⋊Q8—62no—
243C22⋊Q8—62no—
244#121—22no—
245#121—22no—
246C4×D4—38no—
247C4×D4—616no—
248C4×D4—64no—
249C4×D4—64no—
250C4×D4—64no—
251#117—24no—
252#117—64no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C4⋊C4C22×D312yes—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—32no—
262C2×C22⋊C4—616no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—61no—
267C2×C22⋊C4—62no—
268C2×C42—21no—
269C2×C42—31no—
270C2×C42—31no—
271C4⋊C8—64no—
272#103—64no—
273#102—24no—
274#102—64no—
275#102—64no—
276C22⋊C8—22no—
277C22⋊C8—62no—
278#95—62no—
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×D4—42no—
303C3×D4—42no—
304C3×D4—84no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4#12112yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D412yes—
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4112yes—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3#12014yes—
330D12—216no—
331D12—24no—
332D12—24no—
333D12—24no—
334D12#12118yes—
335C4×D3—88no—
336C4×D3—88no—
337Dic6—42no—
338Dic6—42no—
339C3×C8—42no—
340C3⋊C8—42no—
341C24—21no—
342C24—64no—
343C24—64no—
344C24—62no—
345C24—64no—
346C24—242no—
347C24—242no—
348C24D3×D411yescharacteristic
349C2×Q8—61no—
350C2×Q8—62no—
351C2×Q8—61no—
352C2×Q8—62no—
353C2×D4—21no—
354C2×D4—22no—
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—22no—
373C22×C4—21no—
374C22×C4—32no—
375C22×C4—32no—
376C22×C4—68no—
377C22×C4—68no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—242no—
381C22×C4—242no—
382C22×C4—242no—
383C22×C4—242no—
384C22×C4#23512yes—
385C22×C4D3×D411yescharacteristic
386C22×C4D3×D412yes—
387C22×C4C23×D311yescharacteristic
388SD16—128no—
389D8—128no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—64no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—122no—
403C4⋊C4—242no—
404C4⋊C4D3×D414yes—
405C22⋊C4—24no—
406C22⋊C4—42no—
407C22⋊C4—68no—
408C22⋊C4—68no—
409C22⋊C4—64no—
410C22⋊C4—124no—
411C22⋊C4—122no—
412C22⋊C4—242no—
413C42—24no—
414C42—34no—
415C42—62no—
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—44no—
441C3×C4—81no—
442C3×C4C2×#12112yes—
443C3×C4#52512yes—
444Dic3—24no—
445Dic3—42no—
446Dic3—81no—
447Dic3—81no—
448C23—21no—
449C23—22no—
450C23—22no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—41no—
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#88412yes—
470C23C2×D3×D411yescharacteristic
471Q8—122no—
472Q8—122no—
473D4—38no—
474D4—42no—
475D4—42no—
476D4—616no—
477D4—64no—
478D4—64no—
479D4—64no—
480D4—84no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—34no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—248no—
502C2×C4—248no—
503C2×C4#88412yes—
504C2×C4C2×#23511yescharacteristic
505C2×C4C2×D3×D412yes—
506C2×C4C2×D3×D411yescharacteristic
507C2×C4D3×#4112yes—
508C8—42no—
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—81no—
525C22—81no—
526C22—81no—
527C22—82no—
528C22—82no—
529C22—124no—
530C22—124no—
531C22—124no—
532C22—124no—
533C22—124no—
534C22—124no—
535C22—2416no—
536C22C2×#88411yescharacteristic
537C22D3×#12011yescharacteristic
538C22#565611yescharacteristic
539C22#577012yes—
540C22#577412yes—
541C4—22no—
542C4—44no—
543C4—64no—
544C4—81no—
545C4—122no—
546C4—241no—
547C4—241no—
548C4#554712yes—
549C4D3×#12112yes—
550C3#3291711yescharacteristic, complemented, Sylow
551C2—42no—
552C2—128no—
553C2—161no—
554C2#7493511yescharacteristic
555C2#7886311yescharacteristic
556C2#7894411yescharacteristic
557C2#8008712yes—
558C2#8135212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1875C2
socleC23×C3C22×D4
prime radicalC2×#5036C2
hypercenter#1875D3
hypocenterC3#32917
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
1. C2×#5521C2semidirect
2. #82677C2semidirect
4. C2×#4787C2semidirect
5. C2×#4731C2semidirect
6. C3×#1875C2semidirect
7. #71585C2semidirect
8. #70208C2semidirect
9. #69909C2semidirect
10. #68896C2semidirect
11. #68892C2semidirect
16. C2×#936C22semidirect
26. C2×C3×#102C22semidirect
29. #5221C22semidirect
32. C2×Dic3⋊C8C22semidirect
34. #4975C22semidirect
35. #4973C22semidirect
36. C2×#810C22semidirect
41. C3×#340C22semidirect
42. #4550C22semidirect
43. #4529C22semidirect
51. #1875D3semidirect
103. #893D4semidirect
134. C2×#102C2×D3semidirect
143. #340C2×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
#826772
C2×#50362
C2×#47872
C2×#47312
C3×#18752
#715852
#702082
#699092
#688962
#688922
#329173