Finite Group Databasebeta

← #10948987

Subgroups — #10948987

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#904C211yesmaximal, characteristic, complemented
2C3×#3050C211yesmaximal, characteristic, complemented
3C2×#5195C211yesmaximal, characteristic, complemented
4C2×#5193C211yesmaximal, characteristic, complemented
5#71963C211yesmaximal, characteristic, complemented
6#71086C214yesmaximal, complemented
7#71076C214yesmaximal, complemented
8#70654C211yesmaximal, characteristic, complemented
9#70652C211yesmaximal, characteristic, complemented
10#33519—31nomaximal, Sylow
11C22×#235—21no—
12C2×C3×#135C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C3×#525—21no—
15C22×Dic12C2211yescharacteristic
16C2×#904—22no—
17C2×#904—24no—
18C2×#904—24no—
19C2×#904C2214yes—
20C22×D24C2211yescharacteristic
21C22×#204—21no—
22C3×#467C2212yescomplemented
23C3×#466C2212yescomplemented
24C3×#465—22no—
25C3×#427C2211yescharacteristic
26C3×#416—21no—
27#5195—24no—
28#5193—24no—
29C2×#829C2212yescomplemented
30C2×#828C2212yescomplemented
31C2×#827C2212yescomplemented
32C2×#826C2212yescomplemented
33C2×#812C2211yescharacteristic
34C2×#812C2211yescharacteristic
35#4549C2214yescomplemented
36#4492C2214yescomplemented
37#4491C2214yescomplemented
38C22×#135—31no—
39C2×#510—31no—
40C2×#509—31no—
41#3050D311yescharacteristic, complemented
42#2189—34no—
43#2186—34no—
44#2141—31no—
45#2140—31no—
46#2112—31no—
47C2×C3×#41—21no—
48C23×C3×C4D411yescharacteristic
49C22×#67—41no—
50C2×#235—28no—
51C2×#235—24no—
52C2×#235—44no—
53C2×#235—42no—
54C22×D12—21no—
55C22×C4×D3—41no—
56C22×Dic6—21no—
57C3×#135—22no—
58C3×#135—22no—
59C2×C3×Q16C2311yescharacteristic
60C2×C3×SD16—21no—
61C2×C3×D8C2311yescharacteristic
62C22×C3×C8—22no—
63C22×C3×C8—22no—
64C22×C3×C8D411yescharacteristic
65C22×C3×C8D412yes—
66C22×C3×C8C2312yes—
67C22×C3×C8C2311yescharacteristic
68C3×#123—41no—
69C3×C22⋊Q8—22no—
70C3×#121—22no—
71C3×#120—41no—
72C3×C4×D4—42no—
73C3×#117—21no—
74C2×Dic12—22no—
75C2×Dic12—21no—
76C2×Dic12D414yescomplemented
77C2×Dic12C2312yes—
78#904—216no—
79#904—28no—
80#904—28no—
81C2×D24—22no—
82C2×D24—21no—
83C2×D24D414yescomplemented
84C2×D24C2312yes—
85C2×#204—28no—
86C2×#204—24no—
87C3×#107C2312yes—
88C3×#106—21no—
89C3×#103—22no—
90C3×#102—22no—
91#829—24no—
92#828—24no—
93#827—24no—
94#826—24no—
95#812—22no—
96#812—22no—
97C2×C3⋊C16C2312yes—
98C2×C3⋊C16C2312yes—
99C22×#41—61no—
100C2×#135—34no—
101C2×#135—64no—
102C2×#135—64no—
103C2×#135—62no—
104C2×#135C2×D311yescharacteristic
105C22×Q16—31no—
106C22×SD16—61no—
107C22×D8—31no—
108C23×C8C2×D311yescharacteristic
109#525—21no—
110#510—64no—
111#509—64no—
112C2×Q32—32no—
113C2×SD32—32no—
114C2×SD32—32no—
115C2×D16—32no—
116C2×M32—31no—
117C2×M32—31no—
118#467C2×D312yescomplemented
119#466C2×D312yescomplemented
120#465—22no—
121#427C2×D311yescharacteristic
122#416—21no—
123#358—34no—
124#357—34no—
125C22⋊C16—34no—
126C24×C3D811yescharacteristic
127C23×D3—41no—
128C3×#41—42no—
129C3×#41—42no—
130C2×C3×Q8—21no—
131C2×C3×D4—21no—
132C2×C3×D4—42no—
133C2×C3×D4—41no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—41no—
137C22×C3×C4D812yes—
138C22×C3×C4D811yescharacteristic
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C2×#67—48no—
142C2×#67—44no—
143C22×Dic3—41no—
144#235—216no—
145#235—48no—
146#235—416no—
147C2×D12—24no—
148C2×D12—22no—
149C2×D12—41no—
150C2×D12—42no—
151C2×C4×D3—48no—
152C2×C4×D3—44no—
153C2×Dic6—24no—
154C2×Dic6—22no—
155C2×Dic6—41no—
156C2×Dic6—42no—
157C3×Q16—21no—
158C3×Q16—21no—
159C3×SD16—42no—
160C3×D8—21no—
161C3×D8—21no—
162C2×C3×C8—22no—
163C2×C3×C8—24no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8C2×D412yes—
167C2×C3×C8C2×D412yes—
168C2×C3×C8C2×D412yes—
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8C2411yescharacteristic
171C3×C4⋊C4—22no—
172C3×C4⋊C4—42no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—42no—
175C3×C22⋊C4—42no—
176C3×C42—41no—
177Dic12—24no—
178Dic12—22no—
179Dic12C2×D414yes—
180D24—24no—
181D24—22no—
182D24C2×D414yes—
183#204—216no—
184C3⋊C16—22no—
185C3⋊C16—22no—
186C2×#41—21no—
187C2×#41—68no—
188C2×#41—64no—
189C2×#41—124no—
190C2×#41—122no—
191C22×Q8—61no—
192C22×D4—61no—
193C22×D4—121no—
194C23×C4—121no—
195C23×C4#6711yescharacteristic
196#135—22no—
197#135—22no—
198#135—68no—
199#135—68no—
200#135—616no—
201C2×Q16—34no—
202C2×Q16—32no—
203C2×Q16—62no—
204C2×Q16—61no—
205C2×Q16C22×D311yescharacteristic
206C2×SD16—21no—
207C2×SD16—68no—
208C2×SD16—64no—
209C2×D8—34no—
210C2×D8—32no—
211C2×D8—62no—
212C2×D8—61no—
213C2×D8C22×D311yescharacteristic
214C22×C8—22no—
215C22×C8—22no—
216C22×C8#6711yescharacteristic
217C22×C8#6712yes—
218C22×C8C22×D312yes—
219C22×C8C22×D311yescharacteristic
220#123—41no—
221C22⋊Q8—22no—
222#121—22no—
223#120—41no—
224C4×D4—42no—
225#117—21no—
226Q32—64no—
227SD32—64no—
228SD32—64no—
229D16—64no—
230M32—62no—
231M32—62no—
232C2×C16—32no—
233C2×C16—32no—
234#107C22×D312yes—
235#106—21no—
236#103—22no—
237#102—22no—
238C23×C3—22no—
239C23×C3—42no—
240C23×C3—41no—
241C23×C3C2×D812yes—
242C23×C3C2×D811yescharacteristic
243C22×D3—42no—
244C22×D3—44no—
245C22×D3—84no—
246C3×Q8—41no—
247C3×Q8—41no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—84no—
251C3×D4—82no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—82no—
259C2×C3×C4#12012yes—
260C2×C3×C4C2×D812yes—
261C2×C3×C4C2×D811yescharacteristic
262C2×C3×C4AGL(1,ℤ/8)12yes—
263C2×C3×C4C22×D411yescharacteristic
264#67—416no—
265C2×Dic3—42no—
266C2×Dic3—44no—
267D12—24no—
268D12—42no—
269D12—44no—
270C4×D3—416no—
271Dic6—24no—
272Dic6—42no—
273Dic6—44no—
274C3×C8—22no—
275C3×C8#12012yes—
276C3×C8C22×D411yescharacteristic
277C3×C8C22×D411yescharacteristic
278C24—121no—
279C24#21311yescharacteristic
280#41—42no—
281#41—42no—
282#41—616no—
283#41—1216no—
284#41—128no—
285C2×Q8—21no—
286C2×Q8—64no—
287C2×Q8—62no—
288C2×Q8—121no—
289C2×Q8—122no—
290C2×D4—21no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—64no—
294C2×D4—62no—
295C2×D4—128no—
296C2×D4—124no—
297C2×D4—121no—
298C2×D4—122no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—128no—
303C22×C4—121no—
304C22×C4—124no—
305C22×C4#21312yes—
306C22×C4#21311yescharacteristic
307C22×C4C2×#6712yes—
308C22×C4C2×#6711yescharacteristic
309Q16—21no—
310Q16—21no—
311Q16—34no—
312Q16—64no—
313Q16—62no—
314SD16—42no—
315SD16—616no—
316D8—21no—
317D8—21no—
318D8—34no—
319D8—64no—
320D8—62no—
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8D3×D412yes—
326C2×C8D3×D412yes—
327C2×C8C2×#6712yes—
328C2×C8C2×#6711yescharacteristic
329C2×C8C23×D311yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C42—41no—
336C16—62no—
337C16—62no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—44no—
343C22×C3—82no—
344C22×C3—81no—
345C22×C3#44712yes—
346C22×C3C22×D811yescharacteristic
347C2×D3—44no—
348C2×D3—88no—
349C2×D3—84no—
350C3×C4—22no—
351C3×C4—82no—
352C3×C4—81no—
353C3×C4#44712yes—
354C3×C4C2×#12011yescharacteristic
355C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
356Dic3—44no—
357C23—22no—
358C23—42no—
359C23—41no—
360C23—122no—
361C23—124no—
362C23—244no—
363C23C2×#21312yes—
364C23C2×#21311yescharacteristic
365Q8—41no—
366Q8—41no—
367Q8—64no—
368Q8—124no—
369Q8—122no—
370D4—41no—
371D4—41no—
372D4—64no—
373D4—84no—
374D4—82no—
375D4—1216no—
376D4—124no—
377D4—122no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—82no—
385C2×C4—1216no—
386C2×C4—122no—
387C2×C4—124no—
388C2×C4C2×#21312yes—
389C2×C4C2×#21311yescharacteristic
390C2×C4#93712yes—
391C2×C4#94912yes—
392C2×C4C22×#6711yescharacteristic
393C8—22no—
394C8#93712yes—
395C8C2×D3×D411yescharacteristic
396C8C2×D3×D411yescharacteristic
397C2×C3—22no—
398C2×C3—42no—
399C2×C3—161no—
400C2×C3C2×#44711yescharacteristic
401C2×C3#341412yes—
402D3—84no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—44no—
408C22—82no—
409C22—81no—
410C22—124no—
411C22—248no—
412C22—244no—
413C22#503612yes—
414C22C22×#21311yescharacteristic
415C4—22no—
416C4—82no—
417C4—81no—
418C4—124no—
419C4#503612yes—
420C4C2×#93711yescharacteristic
421C4C2×#94911yescharacteristic
422C3#3351911yescharacteristic, complemented, Sylow
423C2—22no—
424C2—42no—
425C2—161no—
426C2—244no—
427C2C2×#503611yescharacteristic
428C2#8389412yes—
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3050C2
socleC22×C3C22×D8
hypercenter#3050D3
hypocenterC3#33519
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C22×#904C2semidirect
2. C3×#3050C2semidirect
3. C2×#5195C2semidirect
4. C2×#5193C2semidirect
5. #71963C2semidirect
6. #71086C2semidirect
7. #71076C2semidirect
8. #70654C2semidirect
9. #70652C2semidirect
22. C3×#467C22semidirect
23. C3×#466C22semidirect
29. C2×#829C22semidirect
30. C2×#828C22semidirect
31. C2×#827C22semidirect
32. C2×#826C22semidirect
35. #4549C22semidirect
36. #4492C22semidirect
37. #4491C22semidirect
41. #3050D3semidirect
76. C2×Dic12D4semidirect
83. C2×D24D4semidirect
118. #467C2×D3semidirect
119. #466C2×D3semidirect
422. C3#33519semidirect

Maximal subgroups

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

subgroupindex
C22×#9042
C3×#30502
C2×#51952
C2×#51932
#719632
#710862
#710762
#706542
#706522
#335193