Finite Group Databasebeta

← #10794559

Subgroups — #10794559

Order 768.

1582 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#117C211yesmaximal, characteristic
2C2×#4839C212yesmaximal, complemented
3C2×#4726C211yesmaximal, characteristic
4C2×D3×C22⋊C8C211yesmaximal, characteristic
5C3×#1798C211yesmaximal, characteristic, complemented
6#71583C211yesmaximal, characteristic, complemented
7#70475C212yesmaximal, complemented
8#69909C211yesmaximal, characteristic, complemented
9#69909C211yesmaximal, characteristic, complemented
10#68662C212yesmaximal, complemented
11#68661C212yesmaximal, complemented
12#20628—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C22×ΓL(1,𝔽25)C2212yes—
17C22×D3×C8C2212yes—
18D3×#117—28no—
19C2×#892C412yes—
20C2×D3×C4⋊C4C412yes—
21C2×D3×C22⋊C4C412yes—
22C2×#877C412yes—
23C2×#872C2211yescharacteristic
24C2×#872C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C2×C42×D3C2211yescharacteristic
27C2×C3×C4⋊C8C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×(C2×D3)⋊C8C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×Dic3⋊C8C2212yescomplemented
35C2×Dic3⋊C8C2212yescomplemented
36C2×C4⋊(C3⋊C8)C2212yescomplemented
37#4839—28no—
38#4726—24no—
39D3×C22⋊C8—24no—
40C3×#336C2212yescomplemented
41#4550C2212yescomplemented
42#4550C2212yescomplemented
43#4524C2212yescomplemented
44C22×#117—31no—
45C2×#424—32no—
46C2×#407—31no—
47C22×C22⋊C8—31no—
48#1798—31no—
49#1798—31no—
50#1798—31no—
51#1798D311yescharacteristic, complemented
52#1797—32no—
53#1797—32no—
54#1729—32no—
55C24×D3C2×C411yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2×C411yescharacteristic
58C22×C4×D3—24no—
59C22×C4×D3—24no—
60C22×C4×D3C2×C412yes—
61C22×C4×D3C2×C412yes—
62C22×C4×D3C2×C411yescharacteristic
63C22×C4×D3C2×C411yescharacteristic
64C22×C4×D3D414yes—
65C22×C4×D3Q814yes—
66C22×C4×D3C2311yescharacteristic
67C22×C4×D3C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×#117—24no—
71C2×C3×C4⋊C4C2×C412yes—
72C2×C3×C22⋊C4C2×C412yes—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×#217C2×C412yes—
76C2×(C2×D3)⋊C4C2×C412yes—
77C2×(C2×D3)⋊C4C2×C412yes—
78#926—24no—
79C2×C4⋊Dic3C2×C412yes—
80C2×Dic3⋊C4C2×C412yes—
81C2×Dic3⋊C4C2×C412yes—
82C2×C4×Dic3C2×C412yes—
83C2×C4×Dic3C2×C412yes—
84C2×C4×Dic3C2311yescharacteristic
85C2×C4×Dic3C2311yescharacteristic
86C22×C3⋊C8C2312yes—
87C22×C3⋊C8C2312yes—
88C2×ΓL(1,𝔽25)—24no—
89C2×ΓL(1,𝔽25)—24no—
90C2×ΓL(1,𝔽25)—24no—
91C2×D3×C8—24no—
92C2×D3×C8—24no—
93C2×D3×C8—24no—
94#892—28no—
95D3×C4⋊C4—28no—
96D3×C22⋊C4—28no—
97#877—28no—
98#872—24no—
99#872D418yes—
100C42×D3—24no—
101C42×D3—24no—
102C3×C4⋊C8—24no—
103C3×C22⋊C8—22no—
104C3×C22⋊C8—22no—
105#830—22no—
106#830—22no—
107(C2×D3)⋊C8—24no—
108(C2×D3)⋊C8—24no—
109Dic3⋊C8—24no—
110Dic3⋊C8—24no—
111C4⋊(C3⋊C8)—24no—
112C24×C4—31no—
113C22×M16—32no—
114C23×C8—32no—
115C2×#117—31no—
116C2×#117—32no—
117C2×#117—32no—
118C2×#117—31no—
119C2×#117—31no—
120C2×#117—68no—
121C2×#117C2×D311yescharacteristic
122C22×C4⋊C4—32no—
123C22×C22⋊C4—32no—
124C22×C42—31no—
125C22×C42—31no—
126#424—68no—
127C2×C4⋊C8—32no—
128C2×C4⋊C8—32no—
129C2×C4⋊C8—32no—
130C2×C4⋊C8C2×D312yescomplemented
131#407—64no—
132C2×C22⋊C8—31no—
133C2×C22⋊C8—31no—
134C2×C22⋊C8—32no—
135C2×C22⋊C8—32no—
136C2×C22⋊C8—64no—
137C2×C22⋊C8C2×D311yescharacteristic
138C2×C22⋊C8C2×D311yescharacteristic
139#336—32no—
140#336—32no—
141#336—32no—
142#336C2×D312yescomplemented
143C24×C3C22×C411yescharacteristic
144C23×D3—22no—
145C23×D3—22no—
146C23×D3—24no—
147C23×D3—42no—
148C23×D3C4⋊C414yes—
149C23×D3C22×C411yescharacteristic
150C23×D3C22×C411yescharacteristic
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4C22×C412yes—
154C22×C3×C4C22×C412yes—
155C22×C3×C4C22×C411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×Q812yes—
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3C4⋊C414yes—
162C22×Dic3C22×C412yes—
163C22×Dic3C22×C412yes—
164C22×Dic3C22×C411yescharacteristic
165C22×Dic3C22×C411yescharacteristic
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—216no—
173C2×C4×D3—216no—
174C2×C4×D3—24no—
175C2×C4×D3—24no—
176C2×C4×D3—24no—
177C2×C4×D3—24no—
178C2×C4×D3—42no—
179C2×C4×D3—42no—
180C2×C4×D3C4⋊C414yes—
181C2×C4×D3C4⋊C414yes—
182C2×C4×D3C2×D412yes—
183C2×C4×D3C2×D412yes—
184C2×C4×D3C2×D414yes—
185C2×C4×D3C2×Q812yes—
186C2×C4×D3C2×Q812yes—
187C2×C4×D3#4114yes—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—24no—
195C3×C22⋊C4—24no—
196C3×C42—22no—
197C3×C42C2×D414yes—
198#217—24no—
199(C2×D3)⋊C4—24no—
200(C2×D3)⋊C4C22⋊C418yes—
201C4⋊Dic3—24no—
202Dic3⋊C4—24no—
203Dic3⋊C4C22⋊C418yes—
204C4×Dic3—24no—
205C4×Dic3—22no—
206C4×Dic3—24no—
207C4×Dic3C2×D414yes—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214ΓL(1,𝔽25)—48no—
215D3×C8—48no—
216C25—31no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—34no—
222C23×C4—34no—
223C23×C4—31no—
224C23×C4—32no—
225C23×C4—32no—
226C23×C4—64no—
227C23×C4—64no—
228C23×C4C22×D311yescharacteristic
229C2×M16—64no—
230C2×M16—64no—
231C2×M16—64no—
232C22×C8—32no—
233C22×C8—32no—
234C22×C8—64no—
235C22×C8—64no—
236C22×C8—64no—
237C22×C8C22×D312yes—
238C22×C8C22×D312yes—
239#117—24no—
240#117—38no—
241#117—64no—
242#117—64no—
243#117—68no—
244#117—68no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—68no—
249C2×C4⋊C4C4×D312yes—
250C2×C22⋊C4—32no—
251C2×C22⋊C4—32no—
252C2×C22⋊C4—32no—
253C2×C22⋊C4—68no—
254C2×C22⋊C4C4×D312yes—
255C2×C42—31no—
256C2×C42—32no—
257C2×C42—32no—
258C2×C42—31no—
259C2×C42—64no—
260C2×C42—64no—
261C2×C42C22×D311yescharacteristic
262C2×C42C22×D311yescharacteristic
263C4⋊C8—24no—
264C4⋊C8—64no—
265C4⋊C8—64no—
266C4⋊C8—64no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—64no—
270C22⋊C8—64no—
271C22⋊C8—62no—
272C22⋊C8—62no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3C2×C4⋊C412yes—
278C23×C3C23×C411yescharacteristic
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—28no—
285C22×D3—28no—
286C22×D3—44no—
287C22×D3—48no—
288C22×D3—48no—
289C22×D3C2×C22⋊C412yes—
290C22×D3C2×C4⋊C412yes—
291C22×D3C2×C4⋊C412yes—
292C22×D3#11712yes—
293C22×D3#13114yes—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4C2×C22⋊C414yes—
304C2×C3×C4C2×C4⋊C412yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D412yes—
307C2×C3×C4C22×Q811yescharacteristic
308C2×C3×C4C2×#4112yes—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—28no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3C2×C22⋊C414yes—
318C2×Dic3C2×C22⋊C412yes—
319C2×Dic3C2×C4⋊C412yes—
320C2×Dic3C2×C4⋊C412yes—
321C2×Dic3#11712yes—
322C4×D3—28no—
323C4×D3—24no—
324C4×D3—24no—
325C4×D3—28no—
326C4×D3—48no—
327C4×D3—48no—
328C4×D3#12118yes—
329C4×D3C22⋊Q818yes—
330C3×C8—42no—
331C3×C8—42no—
332C3⋊C8—42no—
333C3⋊C8—42no—
334C24—31no—
335C24—31no—
336C24—34no—
337C24—62no—
338C24—62no—
339C24—64no—
340C24—122no—
341C24C2×C4×D311yescharacteristic
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—31no—
345C22×C4—34no—
346C22×C4—32no—
347C22×C4—32no—
348C22×C4—34no—
349C22×C4—34no—
350C22×C4—31no—
351C22×C4—32no—
352C22×C4—34no—
353C22×C4—34no—
354C22×C4—32no—
355C22×C4—32no—
356C22×C4—32no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—616no—
360C22×C4—616no—
361C22×C4—64no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—64no—
365C22×C4—64no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4—122no—
372C22×C4—122no—
373C22×C4C2×C4×D312yes—
374C22×C4C2×C4×D312yes—
375C22×C4C2×C4×D311yescharacteristic
376C22×C4D3×D412yes—
377C22×C4D3×Q812yes—
378C22×C4C23×D311yescharacteristic
379M16—128no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—128no—
393C4⋊C4—24no—
394C4⋊C4—38no—
395C4⋊C4—64no—
396C4⋊C4—64no—
397C22⋊C4—24no—
398C22⋊C4—38no—
399C22⋊C4—64no—
400C22⋊C4—64no—
401C42—22no—
402C42—34no—
403C42—64no—
404C42—64no—
405C42—62no—
406C42D3×D414yes—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3C22×C22⋊C411yescharacteristic
413C22×C3C22×C4⋊C411yescharacteristic
414C22×C3C2×#11711yescharacteristic
415C22×C3C2×#13112yes—
416C22×C3#56812yes—
417C2×D3—24no—
418C2×D3—24no—
419C2×D3—22no—
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—24no—
423C2×D3—24no—
424C2×D3—22no—
425C2×D3—416no—
426C2×D3#38014yes—
427C2×D3#42414yescomplemented
428C3×C4—24no—
429C3×C4—22no—
430C3×C4—42no—
431C3×C4C2×#12112yes—
432C3×C4C2×C22⋊Q812yes—
433Dic3—24no—
434Dic3—22no—
435Dic3—42no—
436Dic3#38014yes—
437C23—21no—
438C23—22no—
439C23—22no—
440C23—21no—
441C23—32no—
442C23—34no—
443C23—32no—
444C23—32no—
445C23—32no—
446C23—68no—
447C23—68no—
448C23—61no—
449C23—62no—
450C23—62no—
451C23—62no—
452C23—61no—
453C23—124no—
454C23—128no—
455C23—128no—
456C23D3×C4⋊C412yes—
457C23C22×C4×D311yescharacteristic
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—32no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—38no—
471C2×C4—38no—
472C2×C4—34no—
473C2×C4—32no—
474C2×C4—68no—
475C2×C4—68no—
476C2×C4—68no—
477C2×C4—61no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—61no—
481C2×C4—64no—
482C2×C4—64no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—128no—
487C2×C4—128no—
488C2×C4D3×C22⋊C414yes—
489C2×C4D3×C4⋊C412yes—
490C2×C4C2×D3×D411yescharacteristic
491C2×C4C2×D3×D412yes—
492C2×C4C2×D3×Q811yescharacteristic
493C2×C4D3×#4112yes—
494C8—42no—
495C8—42no—
496C8—122no—
497C8—122no—
498C2×C3—42no—
499C2×C3C2×#38011yescharacteristic
500C2×C3#288012yes—
501C2×C3C2×#42411yescharacteristic
502C2×C3#290511yescharacteristic
503C2×C3#292412yes—
504D3—24no—
505D3—24no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—34no—
510C22—34no—
511C22—42no—
512C22—42no—
513C22—64no—
514C22—64no—
515C22—62no—
516C22—64no—
517C22—64no—
518C22—62no—
519C22—62no—
520C22—62no—
521C22—1216no—
522C22C2×D3×C22⋊C411yescharacteristic
523C22C2×D3×C4⋊C411yescharacteristic
524C22D3×#11711yescharacteristic
525C22D3×#13112yes—
526C22#575012yes—
527C4—22no—
528C4—24no—
529C4—34no—
530C4—42no—
531C4—62no—
532C4—64no—
533C4—122no—
534C4D3×#12112yes—
535C4D3×C22⋊Q812yes—
536C3#2062811yescharacteristic, complemented, Sylow
537C2—42no—
538C2—64no—
539C2—64no—
540C2D3×#38011yescharacteristic
541C2#7866312yes—
542C2D3×#42411yescharacteristic
543C2#7908011yescharacteristic
544C2#7937412yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1798C2
prime radicalC24×D3C2×C4
cosocleC22×C3×C4C24
hypercenter#1798D3
hypocenterC3#20628
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. C2×#4839C2semidirect
5. C3×#1798C2semidirect
6. #71583C2semidirect
7. #70475C2semidirect
8. #69909C2semidirect
9. #69909C2semidirect
10. #68662C2semidirect
11. #68661C2semidirect
27. C2×C3×C4⋊C8C22semidirect
34. C2×Dic3⋊C8C22semidirect
35. C2×Dic3⋊C8C22semidirect
36. C2×C4⋊(C3⋊C8)C22semidirect
40. C3×#336C22semidirect
41. #4550C22semidirect
42. #4550C22semidirect
43. #4524C22semidirect
51. #1798D3semidirect
130. C2×C4⋊C8C2×D3semidirect
142. #336C2×D3semidirect
427. C2×D3#424semidirect
536. C3#20628semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1172
C2×#48392
C2×#47262
C2×D3×C22⋊C82
C3×#17982
#715832
#704752
#699092
#686622
#686612
#206283