Finite Group Databasebeta

← #60951

Subgroups — #60951

Order 256.

1041 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3444C211yesmaximal, characteristic, complemented
2C2×#532C211yesmaximal, characteristic, complemented
3#3254C212yesmaximal, complemented
4#3253C212yesmaximal, complemented
5#3238C212yesmaximal, complemented
6#3235C212yesmaximal, complemented
7#3217C212yesmaximal, complemented
8#3214C212yesmaximal, complemented
9#3194C214yesmaximal, complemented
10#3171C214yesmaximal, complemented
11#3106C214yesmaximal, complemented
12#3101C212yesmaximal, complemented
13#3100C212yesmaximal, complemented
14#2920C211yesmaximal, characteristic, complemented
15C22×#41—21no—
16#533C2211yescharacteristic
17#533C2211yescharacteristic, complemented
18#533C2211yescharacteristic, complemented
19#532—21no—
20#532—21no—
21#532—21no—
22#532—21no—
23#532C2211yescharacteristic
24#532C2212yes—
25#532C2212yes—
26#532C2212yes—
27#532C2212yes—
28#532C2211yescharacteristic, complemented
29#532C2211yescharacteristic, complemented
30C2×C4⋊Q8C2211yescharacteristic
31C2×#127C2211yescharacteristic
32#529—21no—
33#529—21no—
34#529—21no—
35#529—21no—
36#529C2212yes—
37#529C2212yescomplemented
38C2×#124—21no—
39#525—21no—
40#525—21no—
41#525C2212yes—
42C2×#121—21no—
43C2×#121C2212yes—
44C4×#41—21no—
45C4×#41—21no—
46C4×#41C2212yescomplemented
47C2×C4×D4—21no—
48C2×C4×D4C2212yes—
49C22×C42C2211yescharacteristic
50#501C2214yescomplemented
51#496C2214yescomplemented
52#490C2214yescomplemented
53#489C2214yescomplemented
54#483C2214yes—
55#482C2218yescomplemented
56#480C2214yes—
57#477C2214yescomplemented
58#476C2214yescomplemented
59#476C2214yescomplemented
60#474C2214yescomplemented
61#469C2218yescomplemented
62#468C2218yescomplemented
63#461C2214yescomplemented
64#460C2214yescomplemented
65#459C2214yescomplemented
66#459C2214yescomplemented
67#450C2214yes—
68#449C2218yescomplemented
69#447C2214yes—
70C4×SD16C2218yescomplemented
71C4×D8C2218yescomplemented
72#432C2212yes—
73#432C2212yescomplemented
74#431C2212yes—
75#431C2212yescomplemented
76#423C2212yescomplemented
77#407C2212yes—
78C4×M16C2212yescomplemented
79C2×#41—21no—
80C2×#41—21no—
81C2×#41—22no—
82C2×#41—22no—
83C2×#41—22no—
84C2×#41—22no—
85C2×#41—48no—
86C2×#41C2312yes—
87C22×Q8—21no—
88C22×D4—22no—
89C22×D4—21no—
90C22×D4—21no—
91C22×D4C2312yes—
92C23×C4—22no—
93C23×C4D411yescharacteristic
94C23×C4D411yescharacteristic
95C23×C4C2311yescharacteristic
96C2×SD16C2318yes—
97C2×D8C2318yes—
98C2×M16C2314yes—
99C4⋊Q8—21no—
100C4⋊Q8—21no—
101C4⋊Q8C2311yescharacteristic
102C4⋊Q8C2312yes—
103C4⋊Q8C2312yes—
104C4⋊Q8C2311yescharacteristic
105C4⋊Q8C2311yescharacteristic
106#127—21no—
107#127—21no—
108#127C2312yes—
109#127C2312yes—
110#127C2311yescharacteristic
111#126—22no—
112#126—22no—
113#125—21no—
114#125—21no—
115#125C2312yes—
116#124—22no—
117#124—22no—
118#124—22no—
119#124—22no—
120#124—21no—
121#124—21no—
122#124C2312yes—
123#123—22no—
124#123—22no—
125#123—22no—
126C22⋊Q8—21no—
127C22⋊Q8—21no—
128C22⋊Q8—21no—
129C22⋊Q8—21no—
130C22⋊Q8C2312yes—
131C22⋊Q8C2312yes—
132#121—22no—
133#121—22no—
134#121—22no—
135#121—22no—
136#121—22no—
137#121—22no—
138#121—21no—
139#121—21no—
140#121—21no—
141#121—21no—
142#121C2314yes—
143#121C2314yes—
144#121C2312yes—
145#121C2312yes—
146#120—22no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8C2312yes—
152C4×Q8C2312yes—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4C2314yes—
170C4×D4C2314yes—
171C4×D4C2312yes—
172C4×D4C2312yes—
173#117—21no—
174#117—21no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117C2312yes—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C2312yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C42—22no—
184C2×C42D411yescharacteristic
185C2×C42D411yescharacteristic
186C2×C42D411yescharacteristic
187C2×C42D411yescharacteristic
188C2×C42D411yescharacteristic
189C2×C42D411yescharacteristic
190C2×C42C2311yescharacteristic
191C2×C42C2311yescharacteristic
192C2×C42C2311yescharacteristic
193C2×C42C2311yescharacteristic
194C2×C42C2311yescharacteristic
195C2×C42C2311yescharacteristic
196#107C2318yes—
197#106C2318yes—
198C4⋊C8C2314yes—
199C4⋊C8C2314yes—
200#103C2318yes—
201#102C2318yes—
202#102C2318yes—
203#102C2318yes—
204C22⋊C8C2314yes—
205C22⋊C8C2314yes—
206#97C2314yes—
207C4×C8C2314yes—
208C24—22no—
209C24C2×D411yescharacteristic
210#41—44no—
211#41—44no—
212#41—48no—
213#41—48no—
214#41—48no—
215#41—48no—
216#41—44no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—44no—
222C2×Q8C2412yes—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—48no—
239C2×D4C2412yes—
240C2×D4C2414yes—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—48no—
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2×D411yescharacteristic
260C22×C4C2×D411yescharacteristic
261C22×C4C2×D411yescharacteristic
262C22×C4C2×D411yescharacteristic
263C22×C4C2×D411yescharacteristic
264C22×C4C2×D411yescharacteristic
265C22×C4C2×D411yescharacteristic
266C22×C4C2411yescharacteristic
267C22×C4C2411yescharacteristic
268C22×C4C2411yescharacteristic
269SD16—48no—
270D8—48no—
271M16—44no—
272C2×C8C2418yes—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4C2412yes—
287C4⋊C4C2412yes—
288C4⋊C4C2414yes—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42C2×D411yescharacteristic
306C42C2×D411yescharacteristic
307C42C2×D411yescharacteristic
308C42C2×D411yescharacteristic
309C42C2×D411yescharacteristic
310C42C2×D411yescharacteristic
311C42C2×D411yescharacteristic
312C42C2×D411yescharacteristic
313C42C2×D411yescharacteristic
314C42C2×D411yescharacteristic
315C42C2×D411yescharacteristic
316C42C2×D411yescharacteristic
317C42C2411yescharacteristic
318C42C2411yescharacteristic
319C42C2411yescharacteristic
320C42C2411yescharacteristic
321C23—22no—
322C23—22no—
323C23—24no—
324C23—41no—
325C23—41no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23C22×D411yescharacteristic
330C23C22×D411yescharacteristic
331C23C22×D411yescharacteristic
332Q8—24no—
333Q8—44no—
334Q8—44no—
335Q8—42no—
336Q8—42no—
337D4—24no—
338D4—24no—
339D4—24no—
340D4—44no—
341D4—44no—
342D4—44no—
343D4—44no—
344D4—42no—
345D4—48no—
346D4—48no—
347D4—44no—
348D4—44no—
349D4—44no—
350D4—42no—
351D4—44no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4AGL(1,ℤ/8)14yes—
380C2×C4C22×D411yescharacteristic
381C2×C4C22×D411yescharacteristic
382C2×C4C22×D411yescharacteristic
383C2×C4C22×D411yescharacteristic
384C2×C4C22×D411yescharacteristic
385C2×C4C22×D411yescharacteristic
386C2×C4C22×D411yescharacteristic
387C2×C4C22×D411yescharacteristic
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D411yescharacteristic
390C2×C4C22×D411yescharacteristic
391C2×C4C2511yescharacteristic
392C8—28no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—44no—
402C22—42no—
403C22—44no—
404C22—44no—
405C22—42no—
406C22—42no—
407C22—44no—
408C22—88no—
409C22C2×AGL(1,ℤ/8)12yes—
410C22C23×D411yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—44no—
421C4—42no—
422C4—42no—
423C4C2×AGL(1,ℤ/8)12yes—
424C4C2×AGL(1,ℤ/8)12yes—
425C4#57512yes—
426C4#58512yes—
427C2—22no—
428C2—41no—
429C2—41no—
430C2—82no—
431C2—84no—
432C2#344911yescharacteristic
433C2C22×AGL(1,ℤ/8)11yescharacteristic
434C2C2×#57511yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutator, Frattini, cosocleC2×C4C25
socleC22C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3444C2semidirect
2. C2×#532C2semidirect
3. #3254C2semidirect
4. #3253C2semidirect
5. #3238C2semidirect
6. #3235C2semidirect
7. #3217C2semidirect
8. #3214C2semidirect
9. #3194C2semidirect
10. #3171C2semidirect
11. #3106C2semidirect
12. #3101C2semidirect
13. #3100C2semidirect
14. #2920C2semidirect
17. #533C22semidirect
18. #533C22semidirect
28. #532C22semidirect
29. #532C22semidirect
37. #529C22semidirect
46. C4×#41C22semidirect
50. #501C22semidirect
51. #496C22semidirect
52. #490C22semidirect
53. #489C22semidirect
55. #482C22semidirect
57. #477C22semidirect
58. #476C22semidirect
59. #476C22semidirect
60. #474C22semidirect
61. #469C22semidirect
62. #468C22semidirect
63. #461C22semidirect
64. #460C22semidirect
65. #459C22semidirect
66. #459C22semidirect
68. #449C22semidirect
70. C4×SD16C22semidirect
71. C4×D8C22semidirect
73. #432C22semidirect
75. #431C22semidirect
76. #423C22semidirect
78. C4×M16C22semidirect

Maximal subgroups

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

subgroupindex
#34442
C2×#5322
#32542
#32532
#32382
#32352
#32172
#32142
#31942
#31712
#31062
#31012
#31002
#29202