Finite Group Databasebeta

← #10948151

Subgroups — #10948151

Order 768.

1154 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
1C2×#5769C211yesmaximal, characteristic
2#82007C211yesmaximal, characteristic, complemented
3#79252C211yesmaximal, characteristic, complemented
4C3×#2148C211yesmaximal, characteristic, complemented
5#72081C211yesmaximal, characteristic, complemented
6C2×#4516C211yesmaximal, characteristic, complemented
7C2×#4495C211yesmaximal, characteristic, complemented
8#69266C214yesmaximal, complemented
9D3×#361C214yesmaximal, complemented
10#33392—31nomaximal, Sylow
11C2×#1009—21no—
12C2×C3×#135C2211yescharacteristic
13C2×#949—21no—
14#5769—22no—
15#5769—22no—
16#5769C414yes—
17#5769C414yes—
18C2×#919C2212yes—
19C2×#914—22no—
20C2×D3×D8C2212yes—
21#5745—21no—
22C22×D24C2211yescharacteristic
23C2×#901C2211yescharacteristic
24C2×C3×M32C2211yescharacteristic, complemented
25C2×C3×#108C2211yescharacteristic, complemented
26#5124—21no—
27C2×#819C2211yescharacteristic, complemented
28C2×#812C2211yescharacteristic, complemented
29#4907—22no—
30D3×M32C2214yescomplemented
31#4901—22no—
32#4893—22no—
33D3×#108C2214yescomplemented
34C3×#362C2212yescomplemented
35C3×#361C2212yescomplemented
36#4565C2212yescomplemented
37#4563C2212yescomplemented
38#4516C2214yescomplemented
39#4495C2214yescomplemented
40C2×#576—31no—
41#3407—31no—
42#2916—31no—
43#2148—34no—
44#2148—31no—
45#2148D311yescharacteristic, complemented
46C2×#361—31no—
47C2×#361—31no—
48C2×#361—34no—
49C2×C3×#41—21no—
50#1009—24no—
51#1009—24no—
52#1009—42no—
53#1009—42no—
54C2×#239—22no—
55C2×D3×D4—22no—
56C2×D3×D4—42no—
57C2×#235—41no—
58C2×#235D411yescharacteristic
59C22×D12—21no—
60C22×D12—41no—
61C3×#135—21no—
62C3×#135—21no—
63C3×#135C2×C412yes—
64C3×#135C2×C412yes—
65C2×C3×Q16C2311yescharacteristic
66C2×C3×SD16—21no—
67C2×C3×D8C2311yescharacteristic
68C2×C3×M16—21no—
69C22×C3×C8C2311yescharacteristic
70#949—24no—
71#949—24no—
72C2×#215—22no—
73C2×#213—22no—
74C2×#208—21no—
75C2×#208D411yescharacteristic
76#919—24no—
77#919C2×C418yes—
78#914—28no—
79#914—28no—
80D3×D8—24no—
81D3×D8C2×C418yes—
82#907—42no—
83D3×M16—24no—
84C2×D24—21no—
85C2×D24—21no—
86C2×D24—21no—
87C2×D24C2×C412yes—
88C2×D24C2×C412yes—
89C2×D24C2312yes—
90#901—42no—
91#901C2×C412yes—
92#901C2×C412yes—
93#901D412yes—
94#901D412yes—
95C2×ΓL(1,𝔽25)D412yes—
96C2×D3×C8C2312yes—
97C3×M32—21no—
98C3×M32C2312yes—
99C2×C3×C16—21no—
100C3×#108—21no—
101C3×#108C2312yes—
102#822—24no—
103#819—21no—
104#819C2312yes—
105#812—21no—
106#812C2312yes—
107C2×C3⋊C16—21no—
108#798—24no—
109D3×C16—24no—
110C2×H(5,𝔽2)—61no—
111#576—34no—
112#576—34no—
113#576—62no—
114#576—62no—
115C2×AGL(1,ℤ/8)—62no—
116C2×AGL(1,ℤ/8)—61no—
117C2×#135—32no—
118C2×#135C2×D311yescharacteristic
119C22×D8—32no—
120C22×D8—31no—
121#568—61no—
122C2×#131—31no—
123#504—62no—
124#504—62no—
125C2×M32—31no—
126C2×M32—34no—
127C2×M32C2×D311yescharacteristic, complemented
128#430—62no—
129#430—61no—
130C2×#108—31no—
131C2×#108—34no—
132C2×#108C2×D311yescharacteristic, complemented
133#362—32no—
134#362C2×D312yescomplemented
135#361—32no—
136#361—34no—
137#361—34no—
138#361C2×D312yescomplemented
139C23×D3—41no—
140C23×D3—42no—
141C3×#41—22no—
142C3×#41—22no—
143C3×#41—41no—
144C3×#41—41no—
145C2×C3×Q8—21no—
146C2×C3×D4—21no—
147C2×C3×D4—41no—
148C22×C3×C4—41no—
149C22×C3×C4C2×D411yescharacteristic
150C2×#67—42no—
151C2×#67SD1612yes—
152#239—28no—
153#239—44no—
154D3×D4—28no—
155D3×D4—48no—
156D3×D4—48no—
157D3×D4—44no—
158#235—44no—
159#235—44no—
160#235C22⋊C412yes—
161#235C22⋊C412yes—
162#235D812yes—
163#235D812yes—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—42no—
168C2×D12—41no—
169C2×D12—41no—
170C2×D12—41no—
171C2×D12—44no—
172C2×D12—44no—
173C2×D12SD1611yescharacteristic
174C2×C4×D3—42no—
175C2×C4×D3C2×D412yes—
176C2×Dic6SD1611yescharacteristic
177C3×Q16—21no—
178C3×Q16C22×C412yes—
179C3×SD16—22no—
180C3×SD16—22no—
181C3×D8—21no—
182C3×D8C22×C412yes—
183C3×M16—22no—
184C3×M16—41no—
185C2×C3×C8—41no—
186C2×C3×C8C22×C411yescharacteristic
187C2×C3×C8C22×C411yescharacteristic
188C2×C3×C8C2×D411yescharacteristic
189C2×C3×C8C2×D411yescharacteristic
190C2×C3×C8C2×D411yescharacteristic
191C2×C3×C8C2411yescharacteristic
192#215—28no—
193#213—28no—
194#208—22no—
195#208—41no—
196#208C22⋊C412yes—
197#208C22⋊C412yes—
198C2×C3⋊C8—41no—
199C2×C3⋊C8C2×D412yes—
200D24—22no—
201D24—22no—
202D24—22no—
203D24C22×C414yes—
204ΓL(1,𝔽25)—44no—
205ΓL(1,𝔽25)C22⋊C414yes—
206ΓL(1,𝔽25)C22⋊C414yes—
207D3×C8—44no—
208D3×C8C22×C414yes—
209D3×C8C2×D414yes—
210C3×C16—22no—
211C3⋊C16—22no—
212H(5,𝔽2)—64no—
213H(5,𝔽2)—64no—
214H(5,𝔽2)—122no—
215H(5,𝔽2)—122no—
216C2×#41—21no—
217C2×#41—31no—
218C2×#41—62no—
219C2×#41—121no—
220C22×D4—62no—
221C22×D4—61no—
222C22×D4—121no—
223C22×D4—122no—
224AGL(1,ℤ/8)—68no—
225AGL(1,ℤ/8)—68no—
226AGL(1,ℤ/8)—64no—
227AGL(1,ℤ/8)—64no—
228#135—21no—
229#135—21no—
230#135—38no—
231#135—64no—
232#135C4×D312yes—
233#135C4×D312yes—
234C2×Q16C22×D311yescharacteristic
235C2×SD16—21no—
236C2×SD16—62no—
237C2×D8—38no—
238C2×D8—32no—
239C2×D8—32no—
240C2×D8—32no—
241C2×D8—64no—
242C2×D8—62no—
243C2×D8—61no—
244C2×D8—61no—
245C2×D8—61no—
246C2×D8C22×D311yescharacteristic
247#131—32no—
248#131—32no—
249#131—32no—
250#131—32no—
251#131—122no—
252#131—122no—
253C2×M16—21no—
254C2×M16—31no—
255C2×M16—32no—
256C2×M16—64no—
257C2×M16—61no—
258C22×C8—32no—
259C22×C8C22×D311yescharacteristic
260M32—21no—
261M32—32no—
262M32—64no—
263M32—61no—
264M32C22×D312yes—
265C2×C16—21no—
266C2×C16—64no—
267C2×C16—61no—
268#108—21no—
269#108—32no—
270#108—64no—
271#108—61no—
272#108C22×D312yes—
273C23×C3—41no—
274C23×C3C2×SD1611yescharacteristic
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—48no—
279C22×D3—164no—
280C22×D3—162no—
281C22×D3C2×SD1612yes—
282C3×Q8—22no—
283C3×Q8—41no—
284C3×D4—22no—
285C3×D4—42no—
286C3×D4—42no—
287C3×D4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4C2×C22⋊C411yescharacteristic
292C2×C3×C4C2×C22⋊C411yescharacteristic
293C2×C3×C4C2×D811yescharacteristic
294C2×C3×C4C2×D811yescharacteristic
295C2×C3×C4C2×SD1611yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297#67—48no—
298#67#10214yes—
299#67#10214yes—
300C2×Dic3C2×SD1612yes—
301D12—24no—
302D12—48no—
303D12—42no—
304D12—42no—
305D12—42no—
306D12#10212yes—
307D12#10212yes—
308C4×D3—48no—
309C4×D3C2×C22⋊C414yes—
310C4×D3C2×D814yes—
311Dic6#10212yes—
312Dic6#10212yes—
313C3×C8—42no—
314C3×C8C2×C22⋊C411yescharacteristic
315C3×C8C2×C22⋊C411yescharacteristic
316C3×C8C23×C411yescharacteristic
317C3×C8C22×D411yescharacteristic
318C3⋊C8—42no—
319C3⋊C8C2×C22⋊C414yes—
320C24—122no—
321C24—121no—
322#41—22no—
323#41—22no—
324#41—32no—
325#41—32no—
326#41—32no—
327#41—32no—
328#41—41no—
329#41—41no—
330#41—68no—
331#41—124no—
332#41—124no—
333#41—124no—
334C2×Q8—21no—
335C2×Q8—31no—
336C2×D4—21no—
337C2×D4—32no—
338C2×D4—31no—
339C2×D4—41no—
340C2×D4—68no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—124no—
345C2×D4—122no—
346C2×D4—124no—
347C2×D4—124no—
348C2×D4—122no—
349C2×D4—128no—
350C2×D4—128no—
351C2×D4—121no—
352C2×D4—121no—
353C2×D4—121no—
354C22×C4—32no—
355C22×C4—41no—
356C22×C4—122no—
357C22×C4D3×D411yescharacteristic
358Q16—21no—
359Q16C2×C4×D312yes—
360SD16—22no—
361SD16—22no—
362SD16—68no—
363D8—21no—
364D8—34no—
365D8—68no—
366D8—62no—
367D8—62no—
368D8—62no—
369D8C2×C4×D312yes—
370M16—22no—
371M16—32no—
372M16—32no—
373M16—34no—
374M16—34no—
375M16—41no—
376M16—62no—
377M16—124no—
378M16—121no—
379C2×C8—32no—
380C2×C8—34no—
381C2×C8—34no—
382C2×C8—41no—
383C2×C8—124no—
384C2×C8—121no—
385C2×C8C2×C4×D311yescharacteristic
386C2×C8C2×C4×D311yescharacteristic
387C2×C8D3×D411yescharacteristic
388C2×C8D3×D411yescharacteristic
389C2×C8D3×D411yescharacteristic
390C2×C8C23×D311yescharacteristic
391C16—22no—
392C16—62no—
393C22×C3—41no—
394C22×C3—42no—
395C22×C3—82no—
396C22×C3C2×#10211yescharacteristic
397C22×C3C2×#10211yescharacteristic
398C22×C3C22×SD1611yescharacteristic
399C2×D3—42no—
400C2×D3—44no—
401C2×D3—84no—
402C2×D3—168no—
403C2×D3—162no—
404C2×D3—162no—
405C2×D3C2×#10214yes—
406C3×C4—42no—
407C3×C4C2×#10211yescharacteristic
408C3×C4C2×#10211yescharacteristic
409C3×C4C22×C22⋊C411yescharacteristic
410C3×C4C22×D811yescharacteristic
411Dic3C2×#10214yes—
412C23—32no—
413C23—41no—
414C23—128no—
415C23—122no—
416C23—122no—
417C23—122no—
418C23—482no—
419C23—484no—
420C23D3×SD1611yescharacteristic
421Q8—22no—
422Q8—32no—
423Q8—32no—
424Q8—41no—
425D4—22no—
426D4—34no—
427D4—34no—
428D4—32no—
429D4—32no—
430D4—42no—
431D4—42no—
432D4—41no—
433D4—64no—
434D4—128no—
435D4—128no—
436D4—122no—
437D4—122no—
438D4—122no—
439C2×C4—32no—
440C2×C4—34no—
441C2×C4—34no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—128no—
446C2×C4D3×C22⋊C411yescharacteristic
447C2×C4D3×C22⋊C411yescharacteristic
448C2×C4D3×D811yescharacteristic
449C2×C4D3×D811yescharacteristic
450C2×C4D3×SD1611yescharacteristic
451C2×C4C2×D3×D411yescharacteristic
452C8—34no—
453C8—42no—
454C8—122no—
455C8D3×C22⋊C411yescharacteristic
456C8D3×C22⋊C411yescharacteristic
457C8C22×C4×D311yescharacteristic
458C8C2×D3×D411yescharacteristic
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—82no—
463C2×C3C22×#10211yescharacteristic
464D3—24no—
465D3—84no—
466C22—34no—
467C22—41no—
468C22—42no—
469C22—82no—
470C22—122no—
471C22—124no—
472C22—244no—
473C22—482no—
474C22—482no—
475C22—488no—
476C22D3×#10211yescharacteristic
477C22D3×#10211yescharacteristic
478C22C2×D3×SD1611yescharacteristic
479C4—34no—
480C4—42no—
481C4D3×#10211yescharacteristic
482C4D3×#10211yescharacteristic
483C4C2×D3×C22⋊C411yescharacteristic
484C4C2×D3×D811yescharacteristic
485C3#3339211yescharacteristic, complemented, Sylow
486C2—21no—
487C2—21no—
488C2—21no—
489C2—64no—
490C2—82no—
491C2—244no—
492C2C2×D3×#10211yescharacteristic
subgroupG/H
centerC2C2×D3×#102
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC3×#2148C2
socleC2×C3C22×#102
prime radicalC2×#5769C2
cosocleC2×C3×C8C24
hypercenter#2148D3
hypocenterC3#33392
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. #82007C2semidirect
3. #79252C2semidirect
4. C3×#2148C2semidirect
5. #72081C2semidirect
6. C2×#4516C2semidirect
7. C2×#4495C2semidirect
8. #69266C2semidirect
9. D3×#361C2semidirect
24. C2×C3×M32C22semidirect
25. C2×C3×#108C22semidirect
27. C2×#819C22semidirect
28. C2×#812C22semidirect
30. D3×M32C22semidirect
33. D3×#108C22semidirect
34. C3×#362C22semidirect
35. C3×#361C22semidirect
36. #4565C22semidirect
37. #4563C22semidirect
38. #4516C22semidirect
39. #4495C22semidirect
45. #2148D3semidirect
127. C2×M32C2×D3semidirect
132. C2×#108C2×D3semidirect
134. #362C2×D3semidirect
138. #361C2×D3semidirect
485. C3#33392semidirect

Maximal subgroups

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

subgroupindex
C2×#57692
#820072
#792522
C3×#21482
#720812
C2×#45162
C2×#44952
#692662
D3×#3612
#333923