Finite Group Databasebeta

← #10948970

Subgroups — #10948970

Order 768.

1150 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#919C211yesmaximal, characteristic, complemented
2C2×#4919C212yesmaximal, complemented
3#80758C211yesmaximal, characteristic, complemented
4C3×#2188C211yesmaximal, characteristic, complemented
5#72063C211yesmaximal, characteristic, complemented
6#71503C212yesmaximal, complemented
7#71085C211yesmaximal, characteristic, complemented
8#69224C212yesmaximal, complemented
9#69220C212yesmaximal, complemented
10#69108C211yesmaximal, characteristic, complemented
11D3×C22⋊C16C211yesmaximal, characteristic, complemented
12#33516—31nomaximal, Sylow
13C22×#239—21no—
14C22×C3×Q16C2211yescharacteristic
15C22×#215—21no—
16C2×#919—28no—
17C2×#919—21no—
18C2×#919—21no—
19C2×#919C2212yes—
20C2×#919C2212yes—
21C2×#919C2211yescharacteristic, complemented
22C22×D24C2211yescharacteristic
23C22×D3×C8C2211yescharacteristic
24#5627—21no—
25C2×C3×Q32C2212yescomplemented
26C3×#467C2211yescharacteristic
27#5188C2211yescharacteristic, complemented
28C2×#828C2212yescomplemented
29C2×#828C2212yescomplemented
30#5115C2211yescharacteristic
31#5048—21no—
32#4919—28no—
33C2×D48C2212yescomplemented
34C2×D3×C16C2212yescomplemented
35#4883C2211yescharacteristic, complemented
36#4880C2211yescharacteristic, complemented
37#4807—21no—
38#4804—21no—
39C3×#358C2212yescomplemented
40C3×C22⋊C16C2211yescharacteristic
41#4564C2212yescomplemented
42#4549C2211yescharacteristic
43#4509C2212yescomplemented
44(C2×D3)⋊C16C2212yescomplemented
45#4491C2212yescomplemented
46C22×#135—31no—
47C2×#508—32no—
48#3050—31no—
49#2189—32no—
50#2188D311yescharacteristic, complemented
51#2187—31no—
52#2186—32no—
53#2185—31no—
54#2184—31no—
55#2138—32no—
56C2×C22⋊C16—31no—
57C22×C3×Q8—21no—
58C2×#239—28no—
59C2×#239—22no—
60C2×#239—22no—
61C2×#239—21no—
62C2×#239—44no—
63C2×#239—41no—
64C2×#239—41no—
65C22×D12—21no—
66C22×D12—41no—
67C22×C4×D3—41no—
68C22×C4×D3D411yescharacteristic
69C2×C3×Q16—24no—
70C2×C3×Q16—21no—
71C2×C3×Q16C2311yescharacteristic
72C2×C3×Q16C2312yes—
73C22×C3×C8C2311yescharacteristic
74C3×C22⋊Q8—21no—
75#946—21no—
76C2×#215—28no—
77C2×#215—22no—
78C2×#215—22no—
79C2×#215—21no—
80#930—21no—
81C22×C3⋊C8D411yescharacteristic
82#919—28no—
83#919—28no—
84#919—24no—
85#919—24no—
86#919—24no—
87#919—24no—
88#919D418yescomplemented
89C2×D24—21no—
90C2×D24—24no—
91C2×D24C2311yescharacteristic
92C2×D24C2312yes—
93C2×D3×C8—24no—
94C2×D3×C8D411yescharacteristic
95C2×D3×C8D411yescharacteristic
96C2×D3×C8C2311yescharacteristic
97C2×D3×C8C2311yescharacteristic
98#895—21no—
99#894—41no—
100#893—41no—
101#892—21no—
102#882—41no—
103#67⋊C4—41no—
104C3×Q32—24no—
105C2×C3×C16C2312yes—
106C3×#107C2311yescharacteristic
107C3×#103—21no—
108#835—21no—
109#828—24no—
110#828—24no—
111#821—21no—
112#818C2311yescharacteristic
113C2×C3⋊C16C2312yes—
114#809—21no—
115#808—21no—
116D48—24no—
117D3×C16—24no—
118C22×#41—61no—
119C2×#135—32no—
120C2×#135—31no—
121C2×#135—32no—
122C2×#135—61no—
123C2×#135—61no—
124C2×#135—68no—
125C22×Q16C2×D311yescharacteristic
126C22×SD16—61no—
127C22×D8—31no—
128C23×C8—31no—
129#525—61no—
130#508—68no—
131C2×Q32C2×D312yescomplemented
132C2×SD32—32no—
133C2×SD32—32no—
134C2×D16—32no—
135C22×C16—32no—
136#467—31no—
137#467C2×D311yescharacteristic
138#466—31no—
139#466—31no—
140#465—61no—
141#465—61no—
142#427—31no—
143#416—61no—
144#358—32no—
145#358C2×D312yescomplemented
146#357—32no—
147#357—32no—
148C22⋊C16—31no—
149C22⋊C16—32no—
150C22⋊C16C2×D311yescharacteristic
151C23×D3—41no—
152C23×D3D811yescharacteristic
153C2×C3×Q8—24no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—42no—
157C2×C3×Q8—41no—
158C22×C3×C4—41no—
159C22×C3×C4C2×D411yescharacteristic
160C2×#67—41no—
161C22×Dic3D811yescharacteristic
162#239—28no—
163#239—28no—
164#239—48no—
165#239—44no—
166#239—48no—
167#239—44no—
168#239—42no—
169#239—42no—
170C2×D12—24no—
171C2×D12—22no—
172C2×D12—21no—
173C2×D12—42no—
174C2×D12—48no—
175C2×D12—42no—
176C2×D12—42no—
177C2×D12—41no—
178C2×D12—41no—
179C2×D12—41no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—48no—
183C2×C4×D3—42no—
184C2×C4×D3—42no—
185C2×C4×D3—41no—
186C2×C4×D3D811yescharacteristic
187C2×C4×D3D811yescharacteristic
188C2×C4×D3C2×D411yescharacteristic
189C2×C4×D3C2×D411yescharacteristic
190C3×Q16—24no—
191C3×Q16—22no—
192C3×Q16—22no—
193C3×Q16C2×D414yes—
194C2×C3×C8—22no—
195C2×C3×C8C2×D411yescharacteristic
196C2×C3×C8C2411yescharacteristic
197C3×C4⋊C4—21no—
198C3×C4⋊C4—41no—
199C3×C22⋊C4—41no—
200#215—28no—
201#215—28no—
202#215—42no—
203(C2×D3)⋊C4—41no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4—41no—
206(C2×D3)⋊C4—41no—
207C4⋊Dic3—21no—
208Dic3⋊C4—41no—
209C4×Dic3—41no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8C2×D411yescharacteristic
212C2×C3⋊C8C2×D411yescharacteristic
213D24—24no—
214D24—22no—
215D24—22no—
216D24C2×D414yes—
217D3×C8—22no—
218D3×C8—22no—
219D3×C8—22no—
220D3×C8C2×D414yes—
221C3×C16—22no—
222C3⋊C16—22no—
223C2×#41—62no—
224C2×#41—62no—
225C2×#41—61no—
226C2×#41—68no—
227C2×#41—121no—
228C2×#41—121no—
229C2×#41—124no—
230C22×Q8—21no—
231C22×D4—61no—
232C22×D4—121no—
233C23×C4—31no—
234C23×C4—121no—
235#135—38no—
236#135—64no—
237#135—64no—
238#135—64no—
239#135—68no—
240#135—68no—
241#135—64no—
242C2×Q16—24no—
243C2×Q16—21no—
244C2×Q16C22×D311yescharacteristic
245C2×Q16C22×D312yes—
246C2×SD16—62no—
247C2×SD16—62no—
248C2×SD16—61no—
249C2×SD16—68no—
250C2×D8—32no—
251C2×D8—31no—
252C2×D8—64no—
253C2×D8—61no—
254C22×C8—31no—
255C22×C8—31no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—64no—
260C22×C8C22×D311yescharacteristic
261#123—121no—
262C22⋊Q8—21no—
263C22⋊Q8—61no—
264#121—61no—
265#121—61no—
266#120—121no—
267C4×D4—121no—
268C4×D4—121no—
269#117—61no—
270Q32—24no—
271SD32—64no—
272SD32—64no—
273D16—64no—
274C2×C16—32no—
275C2×C16—64no—
276C2×C16C22×D312yes—
277#107—31no—
278#107C22×D311yescharacteristic
279#106—61no—
280#103—21no—
281#103—61no—
282#102—61no—
283#102—61no—
284C23×C3C2×D811yescharacteristic
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—41no—
288C22×D3—44no—
289C22×D3—42no—
290C22×D3—41no—
291C22×D3—82no—
292C22×D3—82no—
293C22×D3C2×D811yescharacteristic
294C22×D3C2×D811yescharacteristic
295C3×Q8—24no—
296C3×Q8—44no—
297C3×Q8—42no—
298C3×Q8—41no—
299C3×Q8—41no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—44no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4C2×D811yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308#67—82no—
309C2×Dic3—22no—
310C2×Dic3—41no—
311C2×Dic3C2×D811yescharacteristic
312C2×Dic3C2×D811yescharacteristic
313D12—24no—
314D12—44no—
315D12—48no—
316D12—48no—
317D12—41no—
318D12—42no—
319D12—41no—
320D12—82no—
321D12—82no—
322C4×D3—22no—
323C4×D3—21no—
324C4×D3—21no—
325C4×D3—22no—
326C4×D3—48no—
327C4×D3—48no—
328C4×D3—82no—
329C4×D3#12012yes—
330C4×D3AGL(1,ℤ/8)12yes—
331C3×C8—21no—
332C3×C8C22×D412yes—
333C3⋊C8—21no—
334C3⋊C8#12012yes—
335C24—31no—
336C24—121no—
337#41—68no—
338#41—68no—
339#41—124no—
340#41—122no—
341#41—122no—
342#41—128no—
343#41—128no—
344#41—124no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8—42no—
349C2×Q8—41no—
350C2×D4—64no—
351C2×D4—62no—
352C2×D4—61no—
353C2×D4—122no—
354C2×D4—122no—
355C2×D4—122no—
356C2×D4—128no—
357C2×D4—121no—
358C2×D4—121no—
359C2×D4—121no—
360C2×D4—121no—
361C22×C4—31no—
362C22×C4—31no—
363C22×C4—31no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—41no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—122no—
370C22×C4—122no—
371C22×C4—121no—
372C22×C4—128no—
373C22×C4D3×D411yescharacteristic
374Q16—24no—
375Q16—22no—
376Q16—22no—
377Q16D3×D414yes—
378SD16—68no—
379SD16—68no—
380SD16—122no—
381D8—34no—
382D8—64no—
383D8—62no—
384D8—62no—
385C2×C8—22no—
386C2×C8—31no—
387C2×C8—34no—
388C2×C8—31no—
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8D3×D411yescharacteristic
394C2×C8C23×D311yescharacteristic
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—61no—
398C4⋊C4—121no—
399C22⋊C4—41no—
400C22⋊C4—121no—
401C22⋊C4—121no—
402C22⋊C4—121no—
403C22⋊C4—121no—
404C42—121no—
405C16—22no—
406C16—62no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3#50812yescomplemented
410C22×C3C22×D811yescharacteristic
411C2×D3—21no—
412C2×D3—21no—
413C2×D3—21no—
414C2×D3—21no—
415C2×D3—21no—
416C2×D3—44no—
417C2×D3—44no—
418C2×D3—88no—
419C2×D3—82no—
420C2×D3—82no—
421C2×D3—81no—
422C2×D3—81no—
423C2×D3—81no—
424C2×D3#44712yes—
425C3×C4—21no—
426C3×C4—44no—
427C3×C4—81no—
428C3×C4—81no—
429C3×C4C2×#12011yescharacteristic
430C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
431Dic3—21no—
432Dic3—81no—
433Dic3#44712yes—
434C23—31no—
435C23—31no—
436C23—62no—
437C23—62no—
438C23—121no—
439C23—124no—
440C23—122no—
441C23—121no—
442C23—242no—
443C23—242no—
444C23D3×D811yescharacteristic
445Q8—24no—
446Q8—44no—
447Q8—42no—
448Q8—41no—
449Q8—41no—
450D4—64no—
451D4—124no—
452D4—128no—
453D4—128no—
454D4—122no—
455D4—121no—
456D4—121no—
457D4—242no—
458D4—242no—
459D4—242no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—31no—
463C2×C4—32no—
464C2×C4—32no—
465C2×C4—31no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—128no—
476C2×C4—128no—
477C2×C4—121no—
478C2×C4—242no—
479C2×C4D3×D811yescharacteristic
480C2×C4C2×D3×D411yescharacteristic
481C8—21no—
482C8—32no—
483C8—61no—
484C8C2×D3×D412yes—
485C2×C3—22no—
486C2×C3C2×#44711yescharacteristic
487C2×C3C2×#50811yescharacteristic
488C2×C3#341411yescharacteristic
489D3—22no—
490D3—41no—
491D3—84no—
492D3—161no—
493C22—21no—
494C22—21no—
495C22—32no—
496C22—61no—
497C22—61no—
498C22—61no—
499C22—61no—
500C22—61no—
501C22—124no—
502C22—124no—
503C22—248no—
504C22—242no—
505C22—242no—
506C22—241no—
507C22—241no—
508C22—241no—
509C22#491912yescomplemented
510C22C2×D3×D811yescharacteristic
511C4—21no—
512C4—32no—
513C4—44no—
514C4—61no—
515C4—81no—
516C4—81no—
517C4—241no—
518C4D3×#12011yescharacteristic
519C4D3×AGL(1,ℤ/8)11yescharacteristic
520C3#3351611yescharacteristic, complemented, Sylow
521C2—22no—
522C2—62no—
523C2—121no—
524C2—244no—
525C2—481no—
526C2D3×#44711yescharacteristic
527C2C2×#491911yescharacteristic
528C2#8203911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2188C2
socleC22×C3C22×D8
hypercenter#2188D3
hypocenterC3#33516
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2188

Complements

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

NH ≅ G/Ntype
1. C22×#919C2semidirect
2. C2×#4919C2semidirect
3. #80758C2semidirect
4. C3×#2188C2semidirect
5. #72063C2semidirect
6. #71503C2semidirect
7. #71085C2semidirect
8. #69224C2semidirect
9. #69220C2semidirect
10. #69108C2semidirect
11. D3×C22⋊C16C2semidirect
21. C2×#919C22semidirect
25. C2×C3×Q32C22semidirect
27. #5188C22semidirect
28. C2×#828C22semidirect
29. C2×#828C22semidirect
33. C2×D48C22semidirect
34. C2×D3×C16C22semidirect
35. #4883C22semidirect
36. #4880C22semidirect
39. C3×#358C22semidirect
41. #4564C22semidirect
43. #4509C22semidirect
44. (C2×D3)⋊C16C22semidirect
45. #4491C22semidirect
50. #2188D3semidirect
88. #919D4semidirect
131. C2×Q32C2×D3semidirect
145. #358C2×D3semidirect
409. C22×C3#508semidirect
509. C22#4919semidirect
520. C3#33516semidirect

Maximal subgroups

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

subgroupindex
C22×#9192
C2×#49192
#807582
C3×#21882
#720632
#715032
#710852
#692242
#692202
#691082
D3×C22⋊C162
#335163