Finite Group Databasebeta

← #10948988

Subgroups — #10948988

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×#919C211yesmaximal, characteristic, complemented
2C2×C3×#510C211yesmaximal, characteristic, complemented
3#83914C211yesmaximal, characteristic, complemented
4C2×#5193C211yesmaximal, characteristic, complemented
5#72061C211yesmaximal, characteristic, complemented
6#71503C214yesmaximal, complemented
7#71484C214yesmaximal, complemented
8#71440C211yesmaximal, characteristic, complemented
9#71431C211yesmaximal, characteristic, complemented
10#33519—31nomaximal, Sylow
11C22×#239—21no—
12C2×C3×#135C2211yescharacteristic
13C22×C3×Q16C2211yescharacteristic
14#5828—21no—
15C22×#215—21no—
16C2×#919—24no—
17C2×#919—24no—
18C2×#919—22no—
19C2×#919C2214yes—
20C22×D24C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22C3×#510—24no—
23C2×C3×Q32C2212yescomplemented
24C2×C3×SD32C2212yescomplemented
25C2×C3×M32C2211yescharacteristic
26#5233—21no—
27#5193—24no—
28#5188C2212yescomplemented
29C2×#828C2212yescomplemented
30#5171—22no—
31#5157C2212yescomplemented
32C2×#826C2212yescomplemented
33#5106C2211yescharacteristic
34C2×#812C2211yescharacteristic
35#4564C2214yescomplemented
36#4509C2214yescomplemented
37(C2×D3)⋊C16C2214yescomplemented
38C22×#135—31no—
39C2×#510D311yescharacteristic, complemented
40C2×#509—31no—
41#3050—31no—
42#2189—34no—
43#2186—34no—
44#2141—31no—
45#2140—31no—
46#2112—31no—
47C2×C3×#41—21no—
48C22×C3×Q8—21no—
49C2×#239—28no—
50C2×#239—24no—
51C2×#239—44no—
52C2×#239—42no—
53C22×D12—21no—
54C22×D12—41no—
55C22×C4×D3—41no—
56C22×C4×D3D411yescharacteristic
57C3×#135—22no—
58C3×#135—22no—
59C2×C3×Q16—21no—
60C2×C3×Q16—22no—
61C2×C3×Q16D414yescomplemented
62C2×C3×Q16C2311yescharacteristic
63C2×C3×Q16C2312yes—
64C2×C3×SD16—21no—
65C2×C3×D8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67#946—22no—
68C2×#215—28no—
69C2×#215—24no—
70#938—22no—
71#937—41no—
72#935—41no—
73C4×#67—42no—
74#926—21no—
75C22×C3⋊C8D411yescharacteristic
76#919—216no—
77#919—28no—
78#919—28no—
79C2×D24—21no—
80C2×D24—22no—
81C2×D24D414yescomplemented
82C2×D24C2312yes—
83C2×D3×C8—22no—
84C2×D3×C8—22no—
85C2×D3×C8D412yes—
86C2×D3×C8C2312yes—
87C3×Q32—24no—
88C3×SD32—24no—
89C3×M32—22no—
90C2×C3×C16C2312yes—
91#835—22no—
92#832—22no—
93#828—24no—
94#826—24no—
95#818C2312yes—
96#817—21no—
97#812—22no—
98C2×C3⋊C16C2312yes—
99C22×#41—61no—
100C2×#135—34no—
101C2×#135—62no—
102C2×#135—64no—
103C2×#135—64no—
104C2×#135C2×D311yescharacteristic
105C22×Q16C2×D311yescharacteristic
106C22×SD16—61no—
107C22×D8—31no—
108C23×C8—31no—
109#525—61no—
110#510—24no—
111#509—64no—
112C2×Q32C2×D312yescomplemented
113C2×SD32—32no—
114C2×SD32C2×D312yescomplemented
115C2×D16—32no—
116C2×M32—31no—
117C2×M32C2×D311yescharacteristic
118#467—32no—
119#466—32no—
120#465—62no—
121#427—31no—
122#416—61no—
123#358—34no—
124#357—34no—
125C22⋊C16—34no—
126C23×D3—41no—
127C23×D3D811yescharacteristic
128C3×#41—42no—
129C3×#41—42no—
130C2×C3×Q8—24no—
131C2×C3×Q8—22no—
132C2×C3×Q8—21no—
133C2×C3×Q8—41no—
134C2×C3×Q8—42no—
135C2×C3×D4—21no—
136C2×C3×D4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C2×#67—42no—
141C22×Dic3D811yescharacteristic
142#239—216no—
143#239—48no—
144#239—416no—
145C2×D12—24no—
146C2×D12—22no—
147C2×D12—48no—
148C2×D12—44no—
149C2×D12—41no—
150C2×D12—42no—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—48no—
154C2×C4×D3—44no—
155C2×C4×D3D812yes—
156C2×C4×D3C2×D412yes—
157C3×Q16—24no—
158C3×Q16—22no—
159C3×Q16—21no—
160C3×Q16—21no—
161C3×Q16C2×D414yes—
162C3×SD16—42no—
163C3×D8—21no—
164C3×D8—21no—
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8C2×D412yes—
167C2×C3×C8C2×D412yes—
168C2×C3×C8C2411yescharacteristic
169#217—41no—
170#217—42no—
171#215—216no—
172(C2×D3)⋊C4—42no—
173C4⋊Dic3—22no—
174Dic3⋊C4—42no—
175C4×Dic3—41no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8C2×D412yes—
178D24—24no—
179D24—22no—
180D24C2×D414yes—
181D3×C8—24no—
182D3×C8—22no—
183D3×C8—22no—
184C3×C16—22no—
185C3⋊C16—22no—
186C2×#41—21no—
187C2×#41—64no—
188C2×#41—68no—
189C2×#41—122no—
190C2×#41—124no—
191C22×Q8—21no—
192C22×D4—61no—
193C22×D4—121no—
194C23×C4—31no—
195C23×C4—121no—
196#135—22no—
197#135—22no—
198#135—68no—
199#135—616no—
200#135—68no—
201C2×Q16—21no—
202C2×Q16—22no—
203C2×Q16#6714yescomplemented
204C2×Q16C22×D311yescharacteristic
205C2×Q16C22×D312yes—
206C2×SD16—21no—
207C2×SD16—64no—
208C2×SD16—68no—
209C2×D8—34no—
210C2×D8—32no—
211C2×D8—61no—
212C2×D8—62no—
213C2×D8C22×D311yescharacteristic
214C22×C8—31no—
215C22×C8—32no—
216C22×C8—32no—
217C22×C8—62no—
218C22×C8—62no—
219C22×C8C22×D311yescharacteristic
220#123—121no—
221C22⋊Q8—62no—
222#121—62no—
223#120—121no—
224C4×D4—122no—
225#117—61no—
226Q32—24no—
227SD32—24no—
228SD32—64no—
229D16—64no—
230M32—22no—
231M32—62no—
232C2×C16—32no—
233C2×C16C22×D312yes—
234#107—32no—
235#106—61no—
236#103—62no—
237#102—62no—
238C23×C3—41no—
239C23×C3C2×D811yescharacteristic
240C22×D3—22no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—44no—
244C22×D3—84no—
245C22×D3C2×D812yes—
246C3×Q8—24no—
247C3×Q8—42no—
248C3×Q8—44no—
249C3×Q8—41no—
250C3×Q8—41no—
251C3×D4—41no—
252C3×D4—41no—
253C3×D4—82no—
254C2×C3×C4—42no—
255C2×C3×C4—44no—
256C2×C3×C4—41no—
257C2×C3×C4—82no—
258C2×C3×C4#12012yes—
259C2×C3×C4C2×D811yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)12yes—
261C2×C3×C4C22×D411yescharacteristic
262#67—84no—
263C2×Dic3—22no—
264C2×Dic3—42no—
265C2×Dic3C2×D812yes—
266D12—24no—
267D12—416no—
268D12—42no—
269D12—44no—
270C4×D3—22no—
271C4×D3—22no—
272C4×D3—24no—
273C4×D3—416no—
274C3×C8#12012yes—
275C3×C8C22×D411yescharacteristic
276C3×C8C22×D411yescharacteristic
277C3⋊C8—22no—
278C24—31no—
279C24—121no—
280#41—42no—
281#41—42no—
282#41—616no—
283#41—1216no—
284#41—128no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—64no—
293C2×D4—62no—
294C2×D4—124no—
295C2×D4—128no—
296C2×D4—122no—
297C2×D4—121no—
298C2×D4—122no—
299C22×C4—31no—
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—124no—
307C22×C4—128no—
308C22×C4D3×D411yescharacteristic
309Q16—24no—
310Q16—22no—
311Q16—21no—
312Q16—21no—
313Q16C2×#6714yes—
314SD16—42no—
315SD16—616no—
316D8—21no—
317D8—21no—
318D8—34no—
319D8—64no—
320D8—62no—
321C2×C8—32no—
322C2×C8—62no—
323C2×C8—64no—
324C2×C8—62no—
325C2×C8—62no—
326C2×C8D3×D411yescharacteristic
327C2×C8D3×D412yes—
328C2×C8C2×#6712yes—
329C2×C8C23×D311yescharacteristic
330C4⋊C4—62no—
331C4⋊C4—122no—
332C22⋊C4—122no—
333C22⋊C4—122no—
334C22⋊C4—121no—
335C42—121no—
336C16—22no—
337C16—62no—
338C22×C3—22no—
339C22×C3—82no—
340C22×C3—81no—
341C22×C3#44712yes—
342C22×C3C22×D811yescharacteristic
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—44no—
347C2×D3—44no—
348C2×D3—88no—
349C2×D3—84no—
350C3×C4—44no—
351C3×C4—81no—
352C3×C4#44712yes—
353C3×C4C2×#12011yescharacteristic
354C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
355Dic3—22no—
356Dic3—82no—
357C23—32no—
358C23—41no—
359C23—62no—
360C23—122no—
361C23—124no—
362C23—122no—
363C23—244no—
364C23D3×D811yescharacteristic
365Q8—24no—
366Q8—44no—
367Q8—42no—
368Q8—41no—
369Q8—41no—
370D4—41no—
371D4—41no—
372D4—64no—
373D4—82no—
374D4—1216no—
375D4—124no—
376D4—122no—
377D4—244no—
378C2×C4—32no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—62no—
383C2×C4—64no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—82no—
387C2×C4—1216no—
388C2×C4—122no—
389C2×C4D3×D811yescharacteristic
390C2×C4#91112yes—
391C2×C4#93712yes—
392C2×C4C2×D3×D411yescharacteristic
393C8—62no—
394C8#93712yes—
395C8C2×D3×D411yescharacteristic
396C8C22×#6711yescharacteristic
397C2×C3—22no—
398C2×C3—161no—
399C2×C3C2×#44711yescharacteristic
400C2×C3#341412yes—
401D3—42no—
402D3—84no—
403C22—22no—
404C22—62no—
405C22—62no—
406C22—62no—
407C22—82no—
408C22—81no—
409C22—124no—
410C22—124no—
411C22—248no—
412C22—244no—
413C22#515612yes—
414C22C2×D3×D811yescharacteristic
415C4—44no—
416C4—62no—
417C4—81no—
418C4—242no—
419C4#515612yes—
420C4C2×#91111yescharacteristic
421C4C2×#93711yescharacteristic
422C3#3351911yescharacteristic, complemented, Sylow
423C2—22no—
424C2—122no—
425C2—161no—
426C2—244no—
427C2#8203912yes—
428C2C2×#515611yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#510C2
socleC22×C3C22×D8
hypercenterC2×#510D3
hypocenterC3#33519
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
1. C22×#919C2semidirect
2. C2×C3×#510C2semidirect
3. #83914C2semidirect
4. C2×#5193C2semidirect
5. #72061C2semidirect
6. #71503C2semidirect
7. #71484C2semidirect
8. #71440C2semidirect
9. #71431C2semidirect
23. C2×C3×Q32C22semidirect
24. C2×C3×SD32C22semidirect
28. #5188C22semidirect
29. C2×#828C22semidirect
31. #5157C22semidirect
32. C2×#826C22semidirect
35. #4564C22semidirect
36. #4509C22semidirect
37. (C2×D3)⋊C16C22semidirect
39. C2×#510D3semidirect
61. C2×C3×Q16D4semidirect
81. C2×D24D4semidirect
112. C2×Q32C2×D3semidirect
114. C2×SD32C2×D3semidirect
203. C2×Q16#67semidirect
422. C3#33519semidirect

Maximal subgroups

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

subgroupindex
C22×#9192
C2×C3×#5102
#839142
C2×#51932
#720612
#715032
#714842
#714402
#714312
#335193