Finite Group Databasebeta

← #10826887

Subgroups — #10826887

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
1C3×#3485C211yesmaximal, characteristic, complemented
2C2×#5830C211yesmaximal, characteristic, complemented
3C2×#5241C212yesmaximal, complemented
4#82927C212yesmaximal, complemented
5#72137C211yesmaximal, characteristic, complemented
6#71867C214yesmaximal, complemented
7#71712C212yesmaximal, complemented
8#71650C212yesmaximal, complemented
9#23913—31nomaximal, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic, complemented
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic, complemented
13C3×#547—24no—
14C3×#546—24no—
15C3×#536C2212yescomplemented
16C3×#534—22no—
17C3×#525C2212yes—
18C2×C3×#123—22no—
19C3×#518C2211yescharacteristic, complemented
20#5830—28no—
21C2×#949C2214yes—
22C2×#947C2212yescomplemented
23C2×#940C2212yes—
24C2×#937—22no—
25C2×#930—22no—
26C2×#926C2211yescharacteristic, complemented
27#5241—28no—
28C2×#837C2214yescomplemented
29C2×#836C2212yescomplemented
30C2×#833C2212yescomplemented
31#5207C2212yescomplemented
32#5046—24no—
33#5036—24no—
34#5001C2214yescomplemented
35#4532C2214yescomplemented
36#3485D311yescharacteristic, complemented
37C2×#535—31no—
38C2×#453—32no—
39#3003—32no—
40#2003—34no—
41#1897—32no—
42#1887—32no—
43#1785—31no—
44C24×D3—21no—
45C3×H(5,𝔽2)—24no—
46C3×H(5,𝔽2)—42no—
47C2×C3×#41—22no—
48C2×C3×#41—24no—
49C2×C3×#41—24no—
50C2×C3×#41—42no—
51C2×C3×#41C2312yes—
52C22×C3×Q8C2311yescharacteristic
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4D412yes—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4D412yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67—24no—
60C22×D12—21no—
61C22×D12—24no—
62C22×D12—24no—
63C22×D12—42no—
64C22×D12C2312yes—
65C3×C4⋊Q8—22no—
66C3×#124—22no—
67C3×#123—28no—
68C3×#123—24no—
69C3×#123—22no—
70C3×#123—24no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8D414yescomplemented
73C3×#121—28no—
74C3×#121D414yescomplemented
75C3×#120—24no—
76C3×#120—22no—
77C3×C4×D4—24no—
78C3×C4×D4—24no—
79C3×#117C2312yes—
80C2×C3×C4⋊C4—24no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—22no—
83C2×#217—21no—
84#949—28no—
85#949—44no—
86#947—24no—
87#947—24no—
88C2×#215—24no—
89#940—24no—
90#940D418yescomplemented
91#937—216no—
92C2×#213—24no—
93#930—216no—
94C2×(C2×D3)⋊C4—24no—
95#926—24no—
96C2×C4⋊Dic3—21no—
97C2×C4×Dic3C2312yes—
98C2×#208C2314yes—
99#837—28no—
100#836—42no—
101#833—24no—
102#830—24no—
103#809—28no—
104C2×H(5,𝔽2)C2×D311yescharacteristic, complemented
105C22×#41C2×D311yescharacteristic
106C23×D4—31no—
107C2×AGL(1,ℤ/8)—34no—
108#547—24no—
109#546—24no—
110#536C2×D312yescomplemented
111#535—68no—
112#534—22no—
113C2×#127—32no—
114C2×#124—32no—
115#525C2×D312yes—
116C2×#123—22no—
117C2×#121—62no—
118C2×#120—62no—
119#518C2×D311yescharacteristic, complemented
120C2×#117—31no—
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—22no—
131C24×C3C2×D411yescharacteristic
132C24×C3C2×D412yes—
133C23×D3—22no—
134C23×D3—24no—
135C23×D3—42no—
136C23×D3—42no—
137C23×D3—84no—
138C3×#41—24no—
139C3×#41—24no—
140C3×#41—416no—
141C3×#41—48no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8C2×D412yes—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—22no—
150C2×C3×D4—24no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—42no—
154C2×C3×D4—48no—
155C2×C3×D4—44no—
156C2×C3×D4—44no—
157C2×C3×D4C2×D414yes—
158C2×C3×D4C2×D412yes—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—42no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D414yes—
170C22×C3×C4C2411yescharacteristic
171C2×#67—216no—
172C2×#67—88no—
173C22×Dic3—22no—
174C2×D12—24no—
175C2×D12—24no—
176C2×D12—24no—
177C2×D12—22no—
178C2×D12—416no—
179C2×D12—42no—
180C2×D12—42no—
181C2×D12C2×D414yes—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4C2×D414yes—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—28no—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4—24no—
190C3×C42—22no—
191#217—24no—
192#215—48no—
193#213—48no—
194(C2×D3)⋊C4—216no—
195C4⋊Dic3—24no—
196C4×Dic3—22no—
197C4×Dic3C2×D414yes—
198#208—24no—
199#208—24no—
200C2×C3⋊C8—24no—
201C25—61no—
202H(5,𝔽2)—24no—
203H(5,𝔽2)—42no—
204C2×#41—22no—
205C2×#41—24no—
206C2×#41—24no—
207C2×#41—42no—
208C2×#41C22×D312yes—
209C22×Q8C22×D311yescharacteristic
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—32no—
213C22×D4—64no—
214C22×D4—64no—
215C22×D4—61no—
216C22×D4—64no—
217C22×D4—122no—
218C22×D4#6712yes—
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—22no—
228#127—38no—
229#127—64no—
230#124—22no—
231#124—64no—
232#124—64no—
233#123—28no—
234#123—24no—
235#123—22no—
236#123—24no—
237C22⋊Q8—24no—
238C22⋊Q8#6714yescomplemented
239#121—28no—
240#121—616no—
241#121#6714yescomplemented
242#120—24no—
243#120—22no—
244#120—616no—
245C4×D4—24no—
246C4×D4—24no—
247#117—64no—
248#117C22×D312yes—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—61no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—64no—
255C2×C42—32no—
256#104—68no—
257#102—68no—
258#101—122no—
259#100—64no—
260C22⋊C8—64no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—21no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—42no—
267C23×C3—44no—
268C23×C3—44no—
269C23×C3—84no—
270C23×C3#12014yes—
271C23×C3#12012yes—
272C23×C3C22×D411yescharacteristic
273C22×D3—24no—
274C22×D3—48no—
275C22×D3—48no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—88no—
279C22×D3—88no—
280C22×D3—88no—
281C3×Q8—24no—
282C3×Q8—44no—
283C3×D4—28no—
284C3×D4—24no—
285C3×D4—48no—
286C3×D4—48no—
287C3×D4—42no—
288C3×D4—42no—
289C3×D4—84no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—28no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—48no—
299C2×C3×C4—44no—
300C2×C3×C4—44no—
301C2×C3×C4—84no—
302C2×C3×C4#12012yes—
303C2×C3×C4C22×D412yes—
304C2×C3×C4C22×D412yes—
305C2×C3×C4H(5,𝔽2)12yes—
306#67—816no—
307C2×Dic3—24no—
308C2×Dic3—44no—
309D12—28no—
310D12—48no—
311D12—44no—
312C3⋊C8—44no—
313C24—22no—
314C24—64no—
315C24—62no—
316C24—122no—
317C24—122no—
318C24—244no—
319C24C2×#6712yes—
320C24C2×#6711yescharacteristic
321#41—24no—
322#41—24no—
323#41—416no—
324#41—48no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8C2×#6712yes—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—34no—
337C2×D4—42no—
338C2×D4—48no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—64no—
342C2×D4—616no—
343C2×D4—62no—
344C2×D4—64no—
345C2×D4—64no—
346C2×D4—1216no—
347C2×D4—122no—
348C2×D4—122no—
349C2×D4—248no—
350C2×D4D3×D414yes—
351C2×D4C2×#6712yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—62no—
361C22×C4C2×#6712yes—
362C22×C4C2×#6711yescharacteristic
363C22×C4C2×#6714yes—
364C22×C4C23×D311yescharacteristic
365SD16—128no—
366D8—128no—
367M16—64no—
368M16—64no—
369C2×C8—64no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—64no—
373C4⋊C4C2×#6714yes—
374C22⋊C4—24no—
375C22⋊C4—28no—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—24no—
379C22⋊C4—616no—
380C22⋊C4—64no—
381C42—22no—
382C42—34no—
383C42—62no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—44no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—44no—
392C22×C3—42no—
393C22×C3—44no—
394C22×C3—82no—
395C22×C3—164no—
396C22×C3C2×#12011yescharacteristic
397C22×C3C2×#12011yescharacteristic
398C22×C3#53411yescharacteristic
399C2×D3—48no—
400C2×D3—44no—
401C2×D3—84no—
402C2×D3—816no—
403C2×D3—84no—
404C3×C4—22no—
405C3×C4—42no—
406C3×C4—42no—
407C3×C4—44no—
408C3×C4D4214yes—
409Dic3—44no—
410C23—21no—
411C23—21no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—64no—
419C23—84no—
420C23—128no—
421C23—128no—
422C23—122no—
423C23—122no—
424C23—248no—
425C23—248no—
426C23—248no—
427C23#93714yes—
428C23#95312yes—
429C23C22×#6711yescharacteristic
430Q8—24no—
431Q8—44no—
432D4—28no—
433D4—24no—
434D4—48no—
435D4—48no—
436D4—42no—
437D4—42no—
438D4—68no—
439D4—84no—
440D4—128no—
441D4—124no—
442D4—2416no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—28no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—48no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—64no—
455C2×C4—84no—
456C2×C4—124no—
457C2×C4#95312yes—
458C2×C4C2×D3×D412yes—
459C2×C4#100412yes—
460C2×C4C22×#6712yes—
461C8—124no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—42no—
465C2×C3—44no—
466C2×C3—82no—
467C2×C3#240311yescharacteristic
468C2×C3#302112yes—
469D3—84no—
470C22—22no—
471C22—22no—
472C22—24no—
473C22—24no—
474C22—44no—
475C22—42no—
476C22—42no—
477C22—44no—
478C22—42no—
479C22—44no—
480C22—82no—
481C22—128no—
482C22—124no—
483C22—164no—
484C22—244no—
485C22—2416no—
486C22—244no—
487C22C2×#93711yescharacteristic
488C22#580411yescharacteristic
489C22C2×#95311yescharacteristic
490C4—22no—
491C4—42no—
492C4—42no—
493C4—44no—
494C4—124no—
495C4D4×#6714yes—
496C3#2391311yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—42no—
500C2—44no—
501C2—82no—
502C2—244no—
503C2#8405211yescharacteristic
504C2#8412212yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3485C2
socleC22×C3C2×#120
hypercenter#3485D3
hypocenterC3#23913
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. C3×#3485C2semidirect
2. C2×#5830C2semidirect
3. C2×#5241C2semidirect
4. #82927C2semidirect
5. #72137C2semidirect
6. #71867C2semidirect
7. #71712C2semidirect
8. #71650C2semidirect
10. C2×C3×H(5,𝔽2)C22semidirect
12. C23×D12C22semidirect
15. C3×#536C22semidirect
19. C3×#518C22semidirect
22. C2×#947C22semidirect
26. C2×#926C22semidirect
28. C2×#837C22semidirect
29. C2×#836C22semidirect
30. C2×#833C22semidirect
31. #5207C22semidirect
34. #5001C22semidirect
35. #4532C22semidirect
36. #3485D3semidirect
72. C3×C22⋊Q8D4semidirect
74. C3×#121D4semidirect
90. #940D4semidirect
104. C2×H(5,𝔽2)C2×D3semidirect
110. #536C2×D3semidirect
119. #518C2×D3semidirect
238. C22⋊Q8#67semidirect
241. #121#67semidirect
496. C3#23913semidirect

Maximal subgroups

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

subgroupindex
C3×#34852
C2×#58302
C2×#52412
#829272
#721372
#718672
#717122
#716502
#239133