Finite Group Databasebeta

← #10826883

Subgroups — #10826883

Order 768.

1898 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic, complemented
2#87273C211yesmaximal, characteristic, complemented
3#84038C212yesmaximal, complemented
4C2×#5077C212yesmaximal, complemented
5#72137C211yesmaximal, characteristic, complemented
6#71763C212yesmaximal, complemented
7#71730C214yesmaximal, complemented
8#71591C212yesmaximal, complemented
9#23913—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×D4C2211yescharacteristic, complemented
12C2×#1004C2211yescharacteristic, complemented
13C3×#535—28no—
14C2×C3×#127C2212yes—
15C2×C3×#124C2212yescomplemented
16C2×C3×#121—22no—
17C2×C3×#120—22no—
18C2×C3×#117C2211yescharacteristic, complemented
19#5829—24no—
20#5824—24no—
21#5816C2212yescomplemented
22#5804—22no—
23#5798C2211yescharacteristic, complemented
24C2×#932C2214yes—
25#5791C2212yes—
26C2×#929—22no—
27C2×#836C2212yescomplemented
28#5218—24no—
29#5217—24no—
30C2×#833C2212yescomplemented
31#5215C2214yescomplemented
32#5207C2212yescomplemented
33#5077—28no—
34C2×#805C2214yescomplemented
35#4523C2214yescomplemented
36#3485—31no—
37C2×#535D311yescharacteristic, complemented
38C2×#453—32no—
39#3003—32no—
40#2003—34no—
41#1897—32no—
42#1887—32no—
43#1785—31no—
44C25×C3—21no—
45C2×C3×#41—24no—
46C2×C3×#41—24no—
47C2×C3×#41—42no—
48C22×C3×Q8C2311yescharacteristic
49C22×C3×D4—24no—
50C22×C3×D4—24no—
51C22×C3×D4—21no—
52C22×C3×D4—24no—
53C22×C3×D4—42no—
54C22×C3×D4D412yes—
55C22×C3×D4C2312yes—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4D412yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67—22no—
60#1004—24no—
61#1004—42no—
62C2×#237—22no—
63C2×D3×D4—22no—
64C2×#235C2312yes—
65C3×#127—24no—
66C3×#127D418yescomplemented
67C3×#124—24no—
68C3×#124—24no—
69C3×#121—216no—
70C3×#120—216no—
71C3×#117—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C42C2312yes—
76#953—22no—
77C2×#217—22no—
78#947—22no—
79#946—24no—
80#944—22no—
81#939—28no—
82#937—24no—
83#935—24no—
84D4×Dic3—24no—
85C2×#214—24no—
86#932—28no—
87#932—44no—
88C2×#213—24no—
89#930D414yescomplemented
90#929—28no—
91#929—22no—
92#929—24no—
93C4×#67—24no—
94C2×(C2×D3)⋊C4—24no—
95#926C2312yes—
96#924D414yescomplemented
97C2×Dic3⋊C4—24no—
98C2×#208C2314yes—
99#836—42no—
100#833—24no—
101#832—28no—
102#830—24no—
103#805—28no—
104C2×H(5,𝔽2)—31no—
105C22×#41C2×D311yescharacteristic
106C23×D4C2×D311yescharacteristic, complemented
107C2×AGL(1,ℤ/8)—34no—
108#547—64no—
109#546—64no—
110#536—32no—
111#535—28no—
112#534—62no—
113C2×#127C2×D312yes—
114C2×#124C2×D312yescomplemented
115#525—32no—
116C2×#123—62no—
117C2×#121—22no—
118C2×#120—22no—
119#518—31no—
120C2×#117C2×D311yescharacteristic, complemented
121#453—68no—
122#450—64no—
123#447—64no—
124C2×#104—34no—
125#418—34no—
126C2×#101—32no—
127C2×#100—32no—
128#407—32no—
129#339—34no—
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—42no—
133C24×C3—42no—
134C24×C3—84no—
135C24×C3C2×D411yescharacteristic
136C24×C3C2×D412yes—
137C23×D3—22no—
138C3×#41—416no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×D4—24no—
143C2×C3×D4—24no—
144C2×C3×D4—24no—
145C2×C3×D4—216no—
146C2×C3×D4—22no—
147C2×C3×D4—24no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—24no—
151C2×C3×D4—416no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—88no—
156C2×C3×D4C2×D414yes—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—42no—
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C2×#67—24no—
169C2×#67—22no—
170C2×#67—24no—
171C2×#67—44no—
172C2×#67—44no—
173C22×Dic3—22no—
174C22×Dic3—24no—
175#237—48no—
176D3×D4—48no—
177#235—24no—
178#235—24no—
179C2×D12C2×D412yes—
180C2×C4×D3—22no—
181C2×Dic6C2×D412yes—
182C3×C4⋊C4—24no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—216no—
185C3×C42—22no—
186C3×C42C2×D414yes—
187#217—24no—
188#217—24no—
189#217—22no—
190#214—48no—
191#213—48no—
192(C2×D3)⋊C4—28no—
193(C2×D3)⋊C4—24no—
194C4⋊Dic3C2×D414yes—
195Dic3⋊C4—28no—
196Dic3⋊C4—24no—
197C4×Dic3—22no—
198#208—24no—
199#208—24no—
200C2×C3⋊C8—24no—
201C25—21no—
202H(5,𝔽2)—64no—
203H(5,𝔽2)—122no—
204C2×#41—24no—
205C2×#41—24no—
206C2×#41—32no—
207C2×#41—42no—
208C2×#41—62no—
209C22×Q8C22×D311yescharacteristic
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—42no—
215C22×D4—62no—
216C22×D4—62no—
217C22×D4#6712yes—
218C22×D4C22×D312yes—
219C22×D4C22×D311yescharacteristic
220C23×C4#6712yes—
221C23×C4C22×D311yescharacteristic
222AGL(1,ℤ/8)—68no—
223AGL(1,ℤ/8)—124no—
224C2×SD16—64no—
225C2×D8—64no—
226C2×M16—34no—
227C4⋊Q8—62no—
228#127—24no—
229#127#6718yescomplemented
230#124—24no—
231#124—24no—
232#124—62no—
233#123—68no—
234#123—64no—
235#123—62no—
236#123—64no—
237C22⋊Q8—34no—
238C22⋊Q8—64no—
239#121—216no—
240#121—34no—
241#121—68no—
242#120—216no—
243#120—64no—
244#120—62no—
245C4×D4—64no—
246C4×D4—64no—
247#117—24no—
248#117—32no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—64no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—64no—
254C2×C22⋊C4—62no—
255C2×C42C22×D312yes—
256#104—68no—
257#102—68no—
258#101—122no—
259#100—64no—
260C22⋊C8—64no—
261C23×C3—22no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—22no—
265C23×C3—24no—
266C23×C3—42no—
267C23×C3—44no—
268C23×C3—48no—
269C23×C3—48no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—88no—
273C23×C3—88no—
274C23×C3—88no—
275C23×C3#12012yes—
276C23×C3#12014yes—
277C23×C3C22×D411yescharacteristic
278C22×D3—22no—
279C22×D3—44no—
280C22×D3—84no—
281C3×Q8—44no—
282C3×D4—28no—
283C3×D4—28no—
284C3×D4—48no—
285C3×D4—48no—
286C3×D4—44no—
287C3×D4—816no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—28no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—48no—
296C2×C3×C4—44no—
297C2×C3×C4#12012yes—
298C2×C3×C4C22×D412yes—
299C2×C3×C4C22×D412yes—
300C2×C3×C4H(5,𝔽2)12yes—
301#67—48no—
302#67—42no—
303#67—42no—
304#67—84no—
305C2×Dic3—24no—
306C2×Dic3—22no—
307C2×Dic3—44no—
308C2×Dic3—84no—
309D12—24no—
310C4×D3—44no—
311Dic6—24no—
312C3⋊C8—44no—
313C24—24no—
314C24—22no—
315C24—42no—
316C24—42no—
317C24—62no—
318C24—84no—
319C24C2×#6711yescharacteristic
320C24C2×#6712yes—
321#41—416no—
322#41—64no—
323#41—64no—
324#41—128no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—32no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—216no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—32no—
339C2×D4—416no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—64no—
344C2×D4—62no—
345C2×D4—64no—
346C2×D4—88no—
347C2×D4—128no—
348C2×D4—124no—
349C2×D4—124no—
350C2×D4C2×#6714yes—
351C2×D4C2×#6714yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—42no—
358C22×C4—62no—
359C22×C4—64no—
360C22×C4—62no—
361C22×C4D3×D414yes—
362C22×C4C2×#6712yes—
363C22×C4C2×#6711yescharacteristic
364C22×C4C23×D311yescharacteristic
365SD16—128no—
366D8—128no—
367M16—64no—
368M16—64no—
369C2×C8—64no—
370C4⋊C4—24no—
371C4⋊C4—34no—
372C4⋊C4—68no—
373C4⋊C4—64no—
374C22⋊C4—24no—
375C22⋊C4—216no—
376C22⋊C4—68no—
377C22⋊C4—64no—
378C22⋊C4—64no—
379C22⋊C4—62no—
380C22⋊C4—64no—
381C42—22no—
382C42—62no—
383C42C2×#6714yes—
384C22×C3—22no—
385C22×C3—24no—
386C22×C3—22no—
387C22×C3—24no—
388C22×C3—44no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—48no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—82no—
395C22×C3—84no—
396C22×C3—816no—
397C22×C3—84no—
398C22×C3C2×#12011yescharacteristic
399C22×C3C2×#12011yescharacteristic
400C22×C3#53411yescharacteristic
401C2×D3—42no—
402C2×D3—44no—
403C2×D3—164no—
404C3×C4—22no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4D4214yes—
408Dic3—42no—
409Dic3—44no—
410C23—22no—
411C23—21no—
412C23—21no—
413C23—22no—
414C23—24no—
415C23—42no—
416C23—44no—
417C23—48no—
418C23—48no—
419C23—42no—
420C23—42no—
421C23—62no—
422C23—88no—
423C23—88no—
424C23—88no—
425C23—124no—
426C23—244no—
427C23#93712yes—
428C23#95314yes—
429C23C22×#6711yescharacteristic
430Q8—44no—
431Q8—64no—
432D4—28no—
433D4—28no—
434D4—48no—
435D4—48no—
436D4—44no—
437D4—64no—
438D4—816no—
439D4—128no—
440D4—122no—
441D4—122no—
442D4—244no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—28no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—48no—
451C2×C4—44no—
452C2×C4—64no—
453C2×C4—62no—
454C2×C4—124no—
455C2×C4—124no—
456C2×C4—244no—
457C2×C4#93712yes—
458C2×C4C2×D3×D412yes—
459C2×C4#100412yes—
460C2×C4C22×#6712yes—
461C8—124no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—44no—
465C2×C3—42no—
466C2×C3—84no—
467C2×C3#240311yescharacteristic
468C2×C3#302112yes—
469D3—82no—
470C22—22no—
471C22—22no—
472C22—24no—
473C22—24no—
474C22—44no—
475C22—42no—
476C22—42no—
477C22—48no—
478C22—44no—
479C22—44no—
480C22—82no—
481C22—84no—
482C22—816no—
483C22—84no—
484C22—122no—
485C22—124no—
486C22—484no—
487C22C2×#93711yescharacteristic
488C22#580411yescharacteristic
489C22C2×#95311yescharacteristic
490C4—22no—
491C4—44no—
492C4—42no—
493C4—122no—
494C4—124no—
495C4D4×#6714yes—
496C3#2391311yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—42no—
500C2—44no—
501C2—84no—
502C2—242no—
503C2#8339512yes—
504C2#8405211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#535C2
socleC22×C3C2×#120
hypercenterC2×#535D3
hypocenterC3#23913
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. #87273C2semidirect
3. #84038C2semidirect
4. C2×#5077C2semidirect
5. #72137C2semidirect
6. #71763C2semidirect
7. #71730C2semidirect
8. #71591C2semidirect
11. C23×C3×D4C22semidirect
12. C2×#1004C22semidirect
15. C2×C3×#124C22semidirect
18. C2×C3×#117C22semidirect
21. #5816C22semidirect
23. #5798C22semidirect
27. C2×#836C22semidirect
30. C2×#833C22semidirect
31. #5215C22semidirect
32. #5207C22semidirect
34. C2×#805C22semidirect
35. #4523C22semidirect
37. C2×#535D3semidirect
66. C3×#127D4semidirect
89. #930D4semidirect
96. #924D4semidirect
106. C23×D4C2×D3semidirect
114. C2×#124C2×D3semidirect
120. C2×#117C2×D3semidirect
229. #127#67semidirect
496. C3#23913semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
#872732
#840382
C2×#50772
#721372
#717632
#717302
#715912
#239133