Finite Group Databasebeta

← #10826856

Subgroups — #10826856

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
1C22×C3×#124C211yesmaximal, characteristic, complemented
2#87251C211yesmaximal, characteristic, complemented
3#84038C212yesmaximal, complemented
4C2×#5062C212yesmaximal, complemented
5#72133C211yesmaximal, characteristic, complemented
6#71730C214yesmaximal, complemented
7#71724C212yesmaximal, complemented
8#71532C212yesmaximal, complemented
9#23910—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C23×C3×D4C2211yescharacteristic
12C2×#1007C2211yescharacteristic, complemented
13C2×C3×#124—22no—
14C2×C3×#124—28no—
15C2×C3×#124C2212yes—
16C2×C3×#124C2212yes—
17C22×C3×C22⋊C4—22no—
18C22×C3×C42C2211yescharacteristic
19#5817—24no—
20#5816—22no—
21#5816C2212yescomplemented
22Q8×#67—24no—
23#5812C2211yescharacteristic, complemented
24C2×#932C2214yes—
25#5791—22no—
26#5791C2212yes—
27C2×#836C2212yescomplemented
28C2×#836C2212yescomplemented
29#5218—24no—
30#5217—24no—
31#5215C2214yes—
32#5207C2212yes—
33#5062—28no—
34C2×#805C2214yescomplemented
35#4523C2214yescomplemented
36#3486—31no—
37C22×#124D311yescharacteristic, complemented
38C2×#454—32no—
39#3003—32no—
40#2003—34no—
41#1896—32no—
42#1888—32no—
43#1784—31no—
44C25×C3—21no—
45C22×C3×Q8—24no—
46C22×C3×Q8—24no—
47C22×C3×Q8—42no—
48C22×C3×Q8D412yes—
49C22×C3×Q8C2311yescharacteristic
50C22×C3×Q8C2311yescharacteristic
51C22×C3×D4—24no—
52C22×C3×D4—21no—
53C22×C3×D4—24no—
54C22×C3×D4—42no—
55C22×C3×D4C2312yes—
56C23×C3×C4—21no—
57C23×C3×C4D412yes—
58C23×C3×C4C2311yescharacteristic
59#1007—24no—
60#1007—42no—
61C2×#239—22no—
62C2×D3×Q8—22no—
63C2×#235—22no—
64C2×#235C2312yes—
65C3×#124—216no—
66C3×#124—24no—
67C3×#124—24no—
68C3×#124—24no—
69C3×#124D418yescomplemented
70C2×C3×C22⋊C4—216no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C42—21no—
74C2×C3×C42—24no—
75C2×C3×C42C2312yes—
76#953—22no—
77#947—24no—
78#947—22no—
79#946—28no—
80#946—24no—
81Q8×Dic3—24no—
82#944—24no—
83#944—22no—
84C2×#214—24no—
85#932—28no—
86#932—44no—
87C2×#213—24no—
88#930—24no—
89#930D414yes—
90#929—24no—
91#929—22no—
92C4×#67—28no—
93C4×#67—24no—
94#926—22no—
95#926C2312yes—
96#924—24no—
97#924D414yes—
98C2×#208C2314yes—
99#836—24no—
100#836—42no—
101#832—28no—
102#830—24no—
103#805—28no—
104C2×2−1+4—31no—
105C23×Q8C2×D311yescharacteristic
106C23×D4C2×D311yescharacteristic
107C2×AGL(1,ℤ/8)—34no—
108D4×Q8—64no—
109#548—64no—
110#536—32no—
111#536—62no—
112C2×#124—22no—
113C2×#124—28no—
114C2×#124C2×D312yes—
115C2×#124C2×D312yes—
116#525—32no—
117#525—62no—
118#519—31no—
119C22×C22⋊C4—22no—
120C22×C42C2×D311yescharacteristic
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×C3—24no—
131C24×C3—22no—
132C24×C3—48no—
133C24×C3—42no—
134C24×C3—42no—
135C24×C3C2×D411yescharacteristic
136C2×C3×Q8—24no—
137C2×C3×Q8—24no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—42no—
144C2×C3×Q8—416no—
145C2×C3×Q8C2×D414yes—
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—22no—
149C2×C3×D4—24no—
150C2×C3×D4—416no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4C2×D414yes—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—24no—
157C22×C3×C4—24no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—42no—
161C22×C3×C4—44no—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C2×#67—24no—
168C2×#67—22no—
169#239—48no—
170D3×Q8—48no—
171#235—24no—
172#235—24no—
173#235—44no—
174#235—44no—
175C2×D12—22no—
176C2×D12C2×D412yes—
177C2×C4×D3—24no—
178C2×C4×D3—22no—
179C2×Dic6—22no—
180C2×Dic6C2×D412yes—
181C3×C22⋊C4—216no—
182C3×C22⋊C4—216no—
183C3×C42—24no—
184C3×C42—22no—
185C3×C42C2×D414yes—
186#217—24no—
187#217—22no—
188#214—48no—
189#213—48no—
190(C2×D3)⋊C4—28no—
191(C2×D3)⋊C4—24no—
192C4⋊Dic3—24no—
193C4⋊Dic3C2×D414yes—
194Dic3⋊C4—28no—
195Dic3⋊C4—24no—
196C4×Dic3—24no—
197C4×Dic3—22no—
198#208—24no—
199#208—24no—
200C2×C3⋊C8—24no—
201C25—21no—
2022−1+4—64no—
2032−1+4—122no—
204C2×#41—32no—
205C2×#41—62no—
206C2×#41—62no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—42no—
210C22×Q8—62no—
211C22×Q8#6712yes—
212C22×Q8C22×D311yescharacteristic
213C22×Q8C22×D311yescharacteristic
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—21no—
217C22×D4—42no—
218C22×D4C22×D312yes—
219C23×C4—21no—
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—
228C4⋊Q8—64no—
229#124—216no—
230#124—24no—
231#124—24no—
232#124—24no—
233#124—62no—
234#124—64no—
235#124#6718yescomplemented
236#123—62no—
237#123—64no—
238C22⋊Q8—34no—
239C22⋊Q8—64no—
240C22⋊Q8—64no—
241C22⋊Q8—68no—
242#121—34no—
243#121—64no—
244#120—62no—
245C4×Q8—64no—
246C4×D4—68no—
247C4×D4—64no—
248#117—32no—
249#117—62no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C42—21no—
254C2×C42—24no—
255C2×C42C22×D312yes—
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—24no—
265C23×C3—48no—
266C23×C3—416no—
267C23×C3—48no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—88no—
271C23×C3—84no—
272C23×C3—84no—
273C23×C3#12012yes—
274C23×C3C22×D411yescharacteristic
275C22×D3—22no—
276C3×Q8—28no—
277C3×Q8—48no—
278C3×Q8—44no—
279C3×D4—28no—
280C3×D4—48no—
281C3×D4—44no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—28no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—48no—
292C2×C3×C4—48no—
293C2×C3×C4—44no—
294C2×C3×C4#12012yes—
295C2×C3×C4#12014yes—
296C2×C3×C4C22×D412yes—
297C2×C3×C4C22×D412yes—
298C2×C3×C4H(5,𝔽2)12yes—
299#67—42no—
300#67—42no—
301#67—48no—
302C2×Dic3—24no—
303C2×Dic3—22no—
304D12—24no—
305D12—44no—
306C4×D3—48no—
307C4×D3—44no—
308Dic6—24no—
309Dic6—44no—
310C3⋊C8—44no—
311C24—24no—
312C24—22no—
313C24—42no—
314C24—48no—
315C24—42no—
316C24C2×#6711yescharacteristic
317#41—64no—
318#41—64no—
319#41—124no—
320#41—128no—
321#41—124no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—32no—
330C2×Q8—42no—
331C2×Q8—416no—
332C2×Q8—62no—
333C2×Q8—128no—
334C2×Q8C2×#6714yes—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—32no—
340C2×D4—416no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—62no—
344C2×D4—64no—
345C2×D4—62no—
346C2×D4C2×#6714yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—64no—
356C22×C4—62no—
357C22×C4D3×D414yes—
358C22×C4C2×#6712yes—
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6712yes—
361C22×C4C23×D311yescharacteristic
362SD16—128no—
363D8—128no—
364M16—64no—
365M16—64no—
366C2×C8—64no—
367C4⋊C4—34no—
368C4⋊C4—64no—
369C4⋊C4—68no—
370C4⋊C4—64no—
371C22⋊C4—216no—
372C22⋊C4—216no—
373C22⋊C4—68no—
374C22⋊C4—64no—
375C22⋊C4—62no—
376C22⋊C4—64no—
377C42—24no—
378C42—22no—
379C42—64no—
380C42—62no—
381C42C2×#6714yes—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—48no—
388C22×C3—44no—
389C22×C3—84no—
390C22×C3—816no—
391C22×C3—84no—
392C22×C3#45414yescomplemented
393C22×C3C2×#12011yescharacteristic
394C22×C3C2×#12011yescharacteristic
395C22×C3#53411yescharacteristic
396C2×D3—42no—
397C2×D3—44no—
398C3×C4—24no—
399C3×C4—22no—
400C3×C4—44no—
401C3×C4—42no—
402C3×C4D4214yes—
403Dic3—42no—
404Dic3—44no—
405C23—21no—
406C23—24no—
407C23—21no—
408C23—24no—
409C23—48no—
410C23—416no—
411C23—48no—
412C23—42no—
413C23—42no—
414C23—62no—
415C23—88no—
416C23—84no—
417C23—84no—
418C23#93712yes—
419C23C22×#6711yescharacteristic
420Q8—28no—
421Q8—48no—
422Q8—44no—
423Q8—64no—
424Q8—124no—
425D4—28no—
426D4—48no—
427D4—44no—
428D4—64no—
429D4—124no—
430D4—122no—
431D4—128no—
432D4—122no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—28no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—48no—
443C2×C4—48no—
444C2×C4—44no—
445C2×C4—64no—
446C2×C4—62no—
447C2×C4—128no—
448C2×C4—124no—
449C2×C4#93712yes—
450C2×C4#95314yes—
451C2×C4C2×D3×D412yes—
452C2×C4#100412yes—
453C2×C4C22×#6712yes—
454C8—124no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—24no—
458C2×C3—84no—
459C2×C3#240311yescharacteristic
460C2×C3C2×#45412yes—
461D3—82no—
462C22—24no—
463C22—24no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—48no—
468C22—44no—
469C22—84no—
470C22—816no—
471C22—84no—
472C22—124no—
473C22—122no—
474C22#506214yescomplemented
475C22C2×#93711yescharacteristic
476C22#580411yescharacteristic
477C22C2×#95311yescharacteristic
478C4—22no—
479C4—24no—
480C4—44no—
481C4—42no—
482C4—124no—
483C4—122no—
484C4D4×#6714yes—
485C3#2391011yescharacteristic, complemented, Sylow
486C2—21no—
487C2—21no—
488C2—24no—
489C2—84no—
490C2—242no—
491C2C2×#506212yes—
492C2#8405211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#124C2
socleC22×C3C2×#120
hypercenterC22×#124D3
hypocenterC3#23910
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
1. C22×C3×#124C2semidirect
2. #87251C2semidirect
3. #84038C2semidirect
4. C2×#5062C2semidirect
5. #72133C2semidirect
6. #71730C2semidirect
7. #71724C2semidirect
8. #71532C2semidirect
12. C2×#1007C22semidirect
21. #5816C22semidirect
23. #5812C22semidirect
27. C2×#836C22semidirect
28. C2×#836C22semidirect
34. C2×#805C22semidirect
35. #4523C22semidirect
37. C22×#124D3semidirect
69. C3×#124D4semidirect
235. #124#67semidirect
392. C22×C3#454semidirect
474. C22#5062semidirect
485. C3#23910semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1242
#872512
#840382
C2×#50622
#721332
#717302
#717242
#715322
#239103