Finite Group Databasebeta

← #10831052

Subgroups — #10831052

Order 768.

1564 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#213C211yesmaximal, characteristic, complemented
2C3×#3003C211yesmaximal, characteristic, complemented
3C2×#5199C212yesmaximal, complemented
4#82927C211yesmaximal, characteristic, complemented
5#82916C211yesmaximal, characteristic, complemented
6#72129C211yesmaximal, characteristic, complemented
7#72095C212yesmaximal, complemented
8#70319C212yesmaximal, complemented
9#70316C212yesmaximal, complemented
10#69988C211yesmaximal, characteristic, complemented
11#69982C211yesmaximal, characteristic, complemented
12#24281—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C2×C3×AGL(1,ℤ/8)C2212yescomplemented
16C3×#525C2211yescharacteristic
17C2×#949C2212yescomplemented
18C2×#948C2211yescharacteristic, complemented
19C22×#213—28no—
20C22×#213—22no—
21C22×#213—22no—
22C22×#213C2212yes—
23C22×#213C2212yes—
24C23×C3⋊C8C2211yescharacteristic
25C3×#450—22no—
26C3×#447—22no—
27C3×#418C2212yescomplemented
28C3×#407C2211yescharacteristic
29#5234C2211yescharacteristic, complemented
30C2×#833C2212yescomplemented
31C2×#833C2212yescomplemented
32#5207C2211yescharacteristic
33#5207C2211yescharacteristic
34#5199—28no—
35C2×#823C2212yescomplemented
36C2×#823C2212yescomplemented
37C2×#822C2212yescomplemented
38#5046—22no—
39#5036—22no—
40#5011—22no—
41D4×C3⋊C8—22no—
42#5001C2212yescomplemented
43#4999C2211yescharacteristic, complemented
44#4997C2211yescharacteristic, complemented
45#4558C2212yescomplemented
46C23×D8—31no—
47C2×#472—32no—
48#3003—31no—
49#3003D311yescharacteristic, complemented
50#2929—31no—
51#2045—32no—
52#2018—32no—
53#1978—32no—
54#1810—31no—
55#1783—31no—
56#1783—31no—
57C25×C3—21no—
58C24×D3—21no—
59C2×C3×#41C2311yescharacteristic
60C22×C3×D4—24no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—42no—
65C22×C3×D4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C22×D12—22no—
68C22×D12—24no—
69C22×D12—22no—
70C22×D12—21no—
71C22×D12—42no—
72C22×D12C2312yes—
73C3×AGL(1,ℤ/8)—24no—
74C3×AGL(1,ℤ/8)—42no—
75C2×C3×SD16—22no—
76C2×C3×D8—22no—
77C2×C3×M16C2312yes—
78C3×#123—21no—
79C3×C22⋊Q8D412yes—
80C3×#121D412yes—
81C3×#120—21no—
82C3×C4×D4—22no—
83C3×#117C2311yescharacteristic
84#949—24no—
85#949—42no—
86#948—24no—
87C2×#215—22no—
88C2×#213—216no—
89C2×#213—216no—
90C2×#213—24no—
91C2×#213—24no—
92C2×#213—22no—
93C2×#213—24no—
94C2×#213—22no—
95C2×#213—22no—
96C2×#213—24no—
97C2×#213D418yescomplemented
98C2×#208—21no—
99C2×#208C2312yes—
100C2×#208C2311yescharacteristic
101C2×#208C2311yescharacteristic
102C22×C3⋊C8—24no—
103C22×C3⋊C8—21no—
104C22×C3⋊C8D412yes—
105C22×C3⋊C8D412yes—
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C3×#102—24no—
109C3×C22⋊C8—22no—
110#833—24no—
111#833—42no—
112#830—22no—
113#830—22no—
114#830—22no—
115#823—24no—
116#823—42no—
117#822—24no—
118#809—24no—
119C4⋊(C3⋊C8)—22no—
120#803—21no—
121C4×C3⋊C8—21no—
122C23×D4—31no—
123C23×D4C2×D311yescharacteristic
124C2×AGL(1,ℤ/8)—32no—
125C2×AGL(1,ℤ/8)C2×D312yescomplemented
126C22×D8—32no—
127C22×D8—32no—
128C22×D8—62no—
129C22×D8—62no—
130C22×D8—68no—
131C2×#131—31no—
132C23×C8—31no—
133#525C2×D311yescharacteristic
134#472—68no—
135#450—22no—
136#450—62no—
137#447—22no—
138#447—62no—
139#435—62no—
140C8×D4—62no—
141C2×#108—32no—
142#424—31no—
143#418—32no—
144#418C2×D312yescomplemented
145C2×#100—32no—
146C2×#100—32no—
147C2×#100—32no—
148C2×#100—32no—
149#408—31no—
150#407—31no—
151#407—31no—
152#407C2×D311yescharacteristic
153#405—31no—
154#341—32no—
155C24×C3—22no—
156C24×C3—44no—
157C24×C3—42no—
158C24×C3—84no—
159C24×C3#4111yescharacteristic
160C23×D3—22no—
161C23×D3—44no—
162C23×D3—42no—
163C23×D3—84no—
164C3×#41—22no—
165C3×#41—22no—
166C2×C3×Q8C2×D411yescharacteristic
167C2×C3×D4—28no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—48no—
176C2×C3×D4—44no—
177C2×C3×D4—44no—
178C2×C3×D4—42no—
179C2×C3×D4C2×D414yes—
180C2×C3×D4C2×D411yescharacteristic
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4#4111yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C2×D12—28no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—48no—
195C2×D12—44no—
196C2×D12—44no—
197C2×D12—42no—
198C2×D12C2×D414yes—
199C3×SD16—44no—
200C3×D8—44no—
201C3×M16—22no—
202C3×M16—22no—
203C2×C3×C8—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C2×D412yes—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—22no—
209C3×C42—21no—
210#215—44no—
211#213—216no—
212#213—28no—
213#213—28no—
214#213—44no—
215#208—22no—
216#208—22no—
217#208—22no—
218#208—22no—
219#208—42no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—24no—
222C2×C3⋊C8—24no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—42no—
227C2×C3⋊C8C2×D412yes—
228C2×C3⋊C8C2×D411yescharacteristic
229C2×C3⋊C8C2×D411yescharacteristic
230C2×C3⋊C8C2×D414yes—
231C25—21no—
232C25—61no—
233C2×#41C22×D311yescharacteristic
234C22×D4—24no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—32no—
239C22×D4—42no—
240C22×D4—64no—
241C22×D4—62no—
242C22×D4—61no—
243C22×D4—62no—
244C22×D4—122no—
245C22×D4C22×D312yes—
246C23×C4C22×D311yescharacteristic
247AGL(1,ℤ/8)—24no—
248AGL(1,ℤ/8)—42no—
249AGL(1,ℤ/8)—64no—
250AGL(1,ℤ/8)—122no—
251C2×SD16—22no—
252C2×SD16—62no—
253C2×D8—22no—
254C2×D8—38no—
255C2×D8—62no—
256C2×D8—62no—
257C2×D8—64no—
258C2×D8—64no—
259C2×D8—64no—
260C2×D8—616no—
261C2×D8—616no—
262C2×D8—64no—
263C2×D8—62no—
264#131—64no—
265C2×M16—31no—
266C2×M16—32no—
267C2×M16—31no—
268C2×M16—61no—
269C2×M16C22×D312yes—
270C22×C8—31no—
271C22×C8—32no—
272C22×C8—32no—
273C22×C8—31no—
274C22×C8—64no—
275C22×C8—61no—
276#123—21no—
277C22⋊Q8#6712yes—
278#121#6712yes—
279#120—21no—
280C4×D4—22no—
281#117C22×D311yescharacteristic
282#108—64no—
283C4⋊C8—62no—
284#102—24no—
285#102—64no—
286#100—64no—
287#100—64no—
288#100—122no—
289#100—122no—
290C22⋊C8—22no—
291C22⋊C8—62no—
292C22⋊C8—62no—
293C22⋊C8—62no—
294#97—61no—
295C4×C8—61no—
296C23×C3—22no—
297C23×C3—24no—
298C23×C3—21no—
299C23×C3—24no—
300C23×C3—48no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—84no—
304C23×C3—816no—
305C23×C3—84no—
306C23×C3—84no—
307C23×C3#12112yes—
308C23×C3C2×#4111yescharacteristic
309C22×D3—24no—
310C22×D3—48no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—816no—
314C22×D3—84no—
315C22×D3—84no—
316C22×D3—84no—
317C3×Q8—22no—
318C3×D4—24no—
319C3×D4—22no—
320C3×D4—24no—
321C3×D4—48no—
322C3×D4—42no—
323C3×D4—44no—
324C3×D4—42no—
325C3×D4—42no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4#12112yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D412yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4H(5,𝔽2)12yes—
339D12—24no—
340D12—24no—
341D12—48no—
342D12—42no—
343D12—42no—
344C3×C8—42no—
345C3⋊C8—22no—
346C3⋊C8—41no—
347C3⋊C8—41no—
348C3⋊C8—42no—
349C3⋊C8#12012yes—
350C3⋊C8#12012yes—
351C24—22no—
352C24—44no—
353C24—42no—
354C24—62no—
355C24—84no—
356C24—124no—
357C24—122no—
358C24—244no—
359C24#23711yescharacteristic
360#41—22no—
361#41—22no—
362C2×Q8C2×#6711yescharacteristic
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—34no—
372C2×D4—48no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—68no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—128no—
383C2×D4—124no—
384C2×D4—124no—
385C2×D4—122no—
386C2×D4D3×D414yes—
387C2×D4C2×#6711yescharacteristic
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4D3×D412yes—
392C22×C4#23711yescharacteristic
393C22×C4C2×#6712yes—
394C22×C4C23×D311yescharacteristic
395SD16—44no—
396SD16—124no—
397D8—44no—
398D8—68no—
399D8—68no—
400D8—616no—
401D8—124no—
402M16—22no—
403M16—22no—
404M16—62no—
405M16—62no—
406M16—62no—
407M16—62no—
408M16—122no—
409C2×C8—22no—
410C2×C8—31no—
411C2×C8—31no—
412C2×C8—34no—
413C2×C8—32no—
414C2×C8—64no—
415C2×C8—64no—
416C2×C8—62no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—62no—
420C2×C8—122no—
421C4⋊C4—22no—
422C4⋊C4C2×#6712yes—
423C22⋊C4—22no—
424C22⋊C4—21no—
425C22⋊C4—22no—
426C42—21no—
427C22×C3—28no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—41no—
432C22×C3—44no—
433C22×C3—42no—
434C22×C3—84no—
435C22×C3—84no—
436C22×C3—816no—
437C22×C3—84no—
438C22×C3#47214yescomplemented
439C22×C3C2×#12111yescharacteristic
440C22×C3#54612yes—
441C2×D3—44no—
442C2×D3—84no—
443C2×D3—816no—
444C2×D3—84no—
445C2×D3—84no—
446C3×C4—22no—
447C3×C4—41no—
448C3×C4—42no—
449C3×C4C2×#12011yescharacteristic
450C3×C4#53511yescharacteristic
451C3×C4D4212yes—
452C23—24no—
453C23—22no—
454C23—21no—
455C23—24no—
456C23—48no—
457C23—42no—
458C23—42no—
459C23—64no—
460C23—84no—
461C23—816no—
462C23—84no—
463C23—84no—
464C23—128no—
465C23—122no—
466C23—122no—
467C23—244no—
468C23—2416no—
469C23—244no—
470C23—244no—
471C23#93912yes—
472C23C2×#23711yescharacteristic
473Q8—22no—
474D4—22no—
475D4—24no—
476D4—24no—
477D4—48no—
478D4—44no—
479D4—42no—
480D4—42no—
481D4—42no—
482D4—64no—
483D4—64no—
484D4—128no—
485D4—122no—
486D4—122no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4#93912yes—
496C2×C4C2×D3×D412yes—
497C2×C4C2×D3×D411yescharacteristic
498C2×C4#100412yes—
499C2×C4C22×#6711yescharacteristic
500C8—32no—
501C8—32no—
502C8—42no—
503C8—62no—
504C8—122no—
505C8—121no—
506C8—121no—
507C2×C3—22no—
508C2×C3—24no—
509C2×C3—81no—
510C2×C3—84no—
511C2×C3#240811yescharacteristic
512C2×C3C2×#47212yes—
513D3—84no—
514C22—28no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—41no—
519C22—44no—
520C22—42no—
521C22—84no—
522C22—816no—
523C22—84no—
524C22—84no—
525C22—124no—
526C22—244no—
527C22—2416no—
528C22—244no—
529C22—244no—
530C22#519914yescomplemented
531C22#559112yes—
532C22C2×#93911yescharacteristic
533C4—22no—
534C4—41no—
535C4—42no—
536C4D3×#12011yescharacteristic
537C4#559211yescharacteristic
538C4D4×#6712yes—
539C3#2428111yescharacteristic, complemented, Sylow
540C2—24no—
541C2—22no—
542C2—81no—
543C2—84no—
544C2—244no—
545C2#8258211yescharacteristic
546C2C2×#519912yes—
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3003C2
socleC22×C3C2×#121
hypercenter#3003D3
hypocenterC3#24281
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3003

Complements

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

NH ≅ G/Ntype
1. C23×#213C2semidirect
2. C3×#3003C2semidirect
3. C2×#5199C2semidirect
4. #82927C2semidirect
5. #82916C2semidirect
6. #72129C2semidirect
7. #72095C2semidirect
8. #70319C2semidirect
9. #70316C2semidirect
10. #69988C2semidirect
11. #69982C2semidirect
15. C2×C3×AGL(1,ℤ/8)C22semidirect
17. C2×#949C22semidirect
18. C2×#948C22semidirect
27. C3×#418C22semidirect
29. #5234C22semidirect
30. C2×#833C22semidirect
31. C2×#833C22semidirect
35. C2×#823C22semidirect
36. C2×#823C22semidirect
37. C2×#822C22semidirect
42. #5001C22semidirect
43. #4999C22semidirect
44. #4997C22semidirect
45. #4558C22semidirect
49. #3003D3semidirect
97. C2×#213D4semidirect
125. C2×AGL(1,ℤ/8)C2×D3semidirect
144. #418C2×D3semidirect
438. C22×C3#472semidirect
530. C22#5199semidirect
539. C3#24281semidirect

Maximal subgroups

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

subgroupindex
C23×#2132
C3×#30032
C2×#51992
#829272
#829162
#721292
#720952
#703192
#703162
#699882
#699822
#242813