Finite Group Databasebeta

← #10826860

Subgroups — #10826860

Order 768.

1866 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3486C211yesmaximal, characteristic, complemented
2C22×#947C211yesmaximal, characteristic, complemented
3C2×#5245C212yesmaximal, complemented
4#82927C212yesmaximal, complemented
5#72133C211yesmaximal, characteristic, complemented
6#71867C214yesmaximal, complemented
7#71711C212yesmaximal, complemented
8#71661C212yesmaximal, complemented
9#23910—31nomaximal, Sylow
10C2×C3×2−1+4C2211yescharacteristic, complemented
11C23×C3×Q8C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C3×D4×Q8—24no—
14C3×#548—24no—
15C3×#536—22no—
16C3×#536C2212yescomplemented
17C3×#525—22no—
18C3×#525C2212yes—
19C3×#519C2211yescharacteristic, complemented
20C2×#949C2214yes—
21C2×#947—22no—
22C2×#947—28no—
23C2×#947C2212yes—
24C2×#947C2212yes—
25C22×(C2×D3)⋊C4—22no—
26C22×C4×Dic3C2211yescharacteristic
27#5245—28no—
28C2×#837C2214yescomplemented
29C2×#836C2212yescomplemented
30C2×#836C2212yescomplemented
31#5207C2212yes—
32#5046—24no—
33#5036—24no—
34#5001C2214yes—
35#4532C2214yescomplemented
36#3486D311yescharacteristic, complemented
37C22×#124—31no—
38C2×#454—32no—
39#3003—32no—
40#2003—34no—
41#1896—32no—
42#1888—32no—
43#1784—31no—
44C24×D3—21no—
45C3×2−1+4—24no—
46C3×2−1+4—42no—
47C2×C3×#41—22no—
48C2×C3×#41—22no—
49C2×C3×#41C2312yes—
50C22×C3×Q8—24no—
51C22×C3×Q8—22no—
52C22×C3×Q8—24no—
53C22×C3×Q8—42no—
54C22×C3×Q8D412yes—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C23×C3×C4D412yes—
58C23×C3×C4C2311yescharacteristic
59C23×Dic3—21no—
60C22×D12—21no—
61C22×D12—24no—
62C22×D12—24no—
63C22×D12—42no—
64C22×D12C2312yes—
65C3×C4⋊Q8—24no—
66C3×C4⋊Q8—22no—
67C3×#124—24no—
68C3×#124—22no—
69C3×#123—24no—
70C3×#123—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—28no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8D414yes—
75C3×#121—24no—
76C3×#121D414yes—
77C3×#120—22no—
78C3×C4×Q8—24no—
79C3×C4×D4—28no—
80C3×C4×D4—24no—
81C3×#117—22no—
82C3×#117C2312yes—
83#949—28no—
84#949—44no—
85#947—216no—
86#947—24no—
87#947—24no—
88#947—24no—
89#947D418yescomplemented
90C2×#215—24no—
91C2×#213—24no—
92C2×(C2×D3)⋊C4—216no—
93C2×(C2×D3)⋊C4—24no—
94C2×(C2×D3)⋊C4—24no—
95C2×C4×Dic3—21no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3C2312yes—
98C2×#208C2314yes—
99#837—28no—
100#836—24no—
101#836—42no—
102#830—24no—
103#809—28no—
104C2×2−1+4C2×D311yescharacteristic, complemented
105C23×Q8C2×D311yescharacteristic
106C23×D4—31no—
107C2×AGL(1,ℤ/8)—34no—
108D4×Q8—24no—
109#548—24no—
110#536—22no—
111#536C2×D312yescomplemented
112C2×#124—32no—
113C2×#124—32no—
114C2×#124—68no—
115C2×#124—62no—
116#525—22no—
117#525C2×D312yes—
118#519C2×D311yescharacteristic, complemented
119C22×C22⋊C4—62no—
120C22×C42—31no—
121#454—68no—
122#450—64no—
123#447—64no—
124C2×#104—34no—
125#418—34no—
126C2×#101—32no—
127C2×#101—32no—
128#407—32no—
129#339—34no—
130C24×C3C2×D411yescharacteristic
131C23×D3—24no—
132C23×D3—22no—
133C23×D3—42no—
134C23×D3—48no—
135C23×D3—42no—
136C3×#41—24no—
137C3×#41—24no—
138C3×#41—44no—
139C3×#41—48no—
140C3×#41—44no—
141C2×C3×Q8—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8—24no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—42no—
150C2×C3×Q8—416no—
151C2×C3×Q8—48no—
152C2×C3×Q8C2×D414yes—
153C2×C3×Q8C2×D412yes—
154C2×C3×D4—22no—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4C2×D412yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—42no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—24no—
171C22×Dic3—22no—
172C22×Dic3—44no—
173C2×D12—24no—
174C2×D12—24no—
175C2×D12—24no—
176C2×D12—22no—
177C2×D12—416no—
178C2×D12—42no—
179C2×D12—42no—
180C2×D12C2×D414yes—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4C2×D414yes—
185C3×C22⋊C4—28no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—22no—
188C3×C22⋊C4—24no—
189C3×C42—24no—
190C3×C42—22no—
191#215—48no—
192#213—48no—
193(C2×D3)⋊C4—216no—
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—
2022−1+4—24no—
2032−1+4—42no—
204C2×#41—22no—
205C2×#41—22no—
206C2×#41C22×D312yes—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—42no—
211C22×Q8#6712yes—
212C22×Q8C22×D311yescharacteristic
213C22×Q8C22×D311yescharacteristic
214C22×D4—32no—
215C22×D4—64no—
216C22×D4—61no—
217C22×D4—64no—
218C22×D4—122no—
219C23×C4—61no—
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—24no—
228C4⋊Q8—22no—
229#124—24no—
230#124—22no—
231#124—38no—
232#124—64no—
233#124—64no—
234#124—616no—
235#124—64no—
236#123—24no—
237#123—22no—
238C22⋊Q8—24no—
239C22⋊Q8—28no—
240C22⋊Q8—24no—
241C22⋊Q8#6714yes—
242#121—24no—
243#121#6714yes—
244#120—22no—
245C4×Q8—24no—
246C4×D4—28no—
247C4×D4—24no—
248#117—22no—
249#117C22×D312yes—
250C2×C22⋊C4—616no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—64no—
253C2×C42—32no—
254C2×C42—64no—
255C2×C42—61no—
256#104—68no—
257#102—68no—
258#101—64no—
259#101—122no—
260C22⋊C8—64no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—24no—
264C23×C3—22no—
265C23×C3#12012yes—
266C23×C3C22×D411yescharacteristic
267C22×D3—24no—
268C22×D3—48no—
269C22×D3—416no—
270C22×D3—48no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—88no—
274C22×D3—84no—
275C22×D3—84no—
276C3×Q8—28no—
277C3×Q8—24no—
278C3×Q8—48no—
279C3×Q8—44no—
280C3×Q8—44no—
281C3×D4—24no—
282C3×D4—44no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4—48no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—22no—
290C2×C3×C4—28no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—48no—
297C2×C3×C4—48no—
298C2×C3×C4—44no—
299C2×C3×C4#12014yes—
300C2×C3×C4#12012yes—
301C2×C3×C4C22×D412yes—
302C2×C3×C4C22×D412yes—
303C2×C3×C4H(5,𝔽2)12yes—
304C2×Dic3—24no—
305C2×Dic3—48no—
306C2×Dic3—44no—
307D12—28no—
308D12—48no—
309D12—44no—
310C3⋊C8—44no—
311C24—64no—
312C24—62no—
313C24—122no—
314C24—128no—
315C24—122no—
316C24C2×#6711yescharacteristic
317#41—24no—
318#41—24no—
319#41—44no—
320#41—48no—
321#41—44no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—42no—
331C2×Q8—416no—
332C2×Q8—48no—
333C2×Q8D3×D414yes—
334C2×Q8C2×#6712yes—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—34no—
339C2×D4—64no—
340C2×D4—62no—
341C2×D4—64no—
342C2×D4—64no—
343C2×D4—1216no—
344C2×D4—122no—
345C2×D4—122no—
346C2×D4C2×#6712yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—64no—
355C22×C4—62no—
356C22×C4—124no—
357C22×C4C2×#6712yes—
358C22×C4C2×#6712yes—
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6714yes—
361C22×C4C23×D311yescharacteristic
362SD16—128no—
363D8—128no—
364M16—64no—
365M16—64no—
366C2×C8—64no—
367C4⋊C4—24no—
368C4⋊C4—28no—
369C4⋊C4—24no—
370C4⋊C4C2×#6714yes—
371C22⋊C4—28no—
372C22⋊C4—24no—
373C22⋊C4—22no—
374C22⋊C4—24no—
375C22⋊C4—616no—
376C22⋊C4—616no—
377C42—24no—
378C42—22no—
379C42—34no—
380C42—64no—
381C42—62no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—42no—
388C22×C3—44no—
389C22×C3#45414yescomplemented
390C22×C3C2×#12011yescharacteristic
391C22×C3C2×#12011yescharacteristic
392C22×C3#53411yescharacteristic
393C2×D3—48no—
394C2×D3—44no—
395C2×D3—84no—
396C2×D3—816no—
397C2×D3—84no—
398C3×C4—22no—
399C3×C4—24no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—44no—
403C3×C4D4214yes—
404Dic3—44no—
405C23—24no—
406C23—21no—
407C23—21no—
408C23—22no—
409C23—64no—
410C23—128no—
411C23—1216no—
412C23—128no—
413C23—122no—
414C23—122no—
415C23—248no—
416C23—244no—
417C23—244no—
418C23#95312yes—
419C23C22×#6711yescharacteristic
420Q8—28no—
421Q8—24no—
422Q8—48no—
423Q8—44no—
424Q8—44no—
425D4—24no—
426D4—44no—
427D4—48no—
428D4—42no—
429D4—42no—
430D4—68no—
431D4—128no—
432D4—124no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—64no—
447C2×C4—128no—
448C2×C4—124no—
449C2×C4#93714yes—
450C2×C4#95312yes—
451C2×C4C2×D3×D412yes—
452C2×C4#100412yes—
453C2×C4C22×#6712yes—
454C8—124no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—24no—
458C2×C3—82no—
459C2×C3#240311yescharacteristic
460C2×C3C2×#45412yes—
461D3—84no—
462C22—24no—
463C22—24no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—42no—
468C22—44no—
469C22—128no—
470C22—124no—
471C22—2416no—
472C22—244no—
473C22—244no—
474C22#524514yescomplemented
475C22C2×#93711yescharacteristic
476C22#580411yescharacteristic
477C22C2×#95311yescharacteristic
478C4—22no—
479C4—24no—
480C4—42no—
481C4—42no—
482C4—44no—
483C4—124no—
484C4D4×#6714yes—
485C3#2391011yescharacteristic, complemented, Sylow
486C2—21no—
487C2—24no—
488C2—21no—
489C2—82no—
490C2—244no—
491C2#8405211yescharacteristic
492C2C2×#524512yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3486C2
socleC22×C3C2×#120
hypercenter#3486D3
hypocenterC3#23910
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3486

Complements

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

NH ≅ G/Ntype
1. C3×#3486C2semidirect
2. C22×#947C2semidirect
3. C2×#5245C2semidirect
4. #82927C2semidirect
5. #72133C2semidirect
6. #71867C2semidirect
7. #71711C2semidirect
8. #71661C2semidirect
10. C2×C3×2−1+4C22semidirect
16. C3×#536C22semidirect
19. C3×#519C22semidirect
28. C2×#837C22semidirect
29. C2×#836C22semidirect
30. C2×#836C22semidirect
35. #4532C22semidirect
36. #3486D3semidirect
89. #947D4semidirect
104. C2×2−1+4C2×D3semidirect
111. #536C2×D3semidirect
118. #519C2×D3semidirect
389. C22×C3#454semidirect
474. C22#5245semidirect
485. C3#23910semidirect

Maximal subgroups

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

subgroupindex
C3×#34862
C22×#9472
C2×#52452
#829272
#721332
#718672
#717112
#716612
#239103