Finite Group Databasebeta

← #60567

Subgroups — #60567

Order 256.

1037 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
1C2×#542C211yesmaximal, characteristic, complemented
2#3428C211yesmaximal, characteristic, complemented
3#3125C214yesmaximal, complemented
4#3124C214yesmaximal, complemented
5#3121C214yesmaximal, complemented
6#3118C214yesmaximal, complemented
7#3095C211yesmaximal, characteristic, complemented
8#3094C211yesmaximal, characteristic, complemented
9#3079C212yesmaximal, complemented
10#3040C212yesmaximal, complemented
11#3007C211yesmaximal, characteristic, complemented
12#3006C211yesmaximal, characteristic
13C4×#137C212yesmaximal, complemented
14C4×AGL(1,ℤ/8)C212yesmaximal, complemented
15#2921C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×#137C2212yes—
18C2×AGL(1,ℤ/8)C2212yes—
19#542—22no—
20#542—22no—
21#542C2214yes—
22#542C2214yescomplemented
23C2×C4⋊Q8C2211yescharacteristic, complemented
24C2×#126—21no—
25C2×#124—21no—
26C2×#124C2211yescharacteristic, complemented
27C2×#123—21no—
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8C2211yescharacteristic
30C2×#121C2211yescharacteristic
31#520—21no—
32#520—21no—
33#520C2212yescomplemented
34#520C2212yescomplemented
35#519C2212yescomplemented
36#518—21no—
37#518—21no—
38#518C2212yescomplemented
39C4×#41—22no—
40C4×#41C2212yes—
41C4×#41C2212yes—
42C2×C4×Q8C2211yescharacteristic, complemented
43C2×C4×D4C2211yescharacteristic, complemented
44C2×#117—22no—
45C2×#117C2211yescharacteristic
46C2×#117C2211yescharacteristic
47#485C2214yescomplemented
48#484C2212yescomplemented
49#483C2212yescomplemented
50#482C2214yes—
51#481C2212yescomplemented
52#480C2212yescomplemented
53#477C2214yescomplemented
54#476C2214yescomplemented
55#475C2214yescomplemented
56#474C2214yescomplemented
57#464C2214yescomplemented
58#464C2214yescomplemented
59#463C2214yescomplemented
60#463C2214yescomplemented
61#452C2214yescomplemented
62#451C2212yescomplemented
63#450C2212yescomplemented
64#449C2214yes—
65#448C2212yes—
66#447C2212yes—
67#442C2214yescomplemented
68#441C2214yescomplemented
69#440C2214yescomplemented
70#440C2214yescomplemented
71C4×Q16C2214yescomplemented
72C4×SD16C2214yescomplemented
73C4×SD16C2214yescomplemented
74C4×D8C2214yescomplemented
75#433C2214yescomplemented
76#433C2214yescomplemented
77#428C2212yescomplemented
78#428C2212yescomplemented
79#423C2212yescomplemented
80#419C2212yescomplemented
81#418C2212yescomplemented
82#417C2212yes—
83#417C2212yes—
84#407C2211yescharacteristic
85#407C2211yescharacteristic
86C4×M16C2212yescomplemented
87C2×#41—28no—
88C2×#41—21no—
89C2×#41—21no—
90C2×#41C2312yes—
91C2×#41C2312yes—
92C22×Q8—21no—
93C22×Q8C2311yescharacteristic
94C22×Q8C2311yescharacteristic
95C22×D4—21no—
96C22×D4—21no—
97C22×D4C2311yescharacteristic
98C22×D4C2311yescharacteristic
99C23×C4—21no—
100C23×C4—21no—
101C23×C4C2311yescharacteristic
102#137—28no—
103AGL(1,ℤ/8)—28no—
104C2×Q16C2314yes—
105C2×SD16C2314yes—
106C2×SD16C2314yes—
107C2×D8C2314yes—
108C2×M16C2312yes—
109C2×M16C2312yes—
110C4⋊Q8—22no—
111C4⋊Q8C2314yes—
112#126—24no—
113#126—24no—
114#124—24no—
115#124—24no—
116#124—22no—
117#124C2314yes—
118#123—24no—
119#123—24no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C22⋊Q8—24no—
123C22⋊Q8—24no—
124C22⋊Q8C2312yes—
125C22⋊Q8C2312yes—
126#121—21no—
127#121—21no—
128#121C2312yes—
129#121C2312yes—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8C2314yes—
134C4×Q8C2312yes—
135C4×Q8C2312yes—
136C4×Q8C2312yes—
137C4×Q8C2312yes—
138C4×D4—22no—
139C4×D4—22no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4C2314yes—
150C4×D4C2312yes—
151C4×D4C2312yes—
152C4×D4C2312yes—
153C4×D4C2312yes—
154#117—21no—
155#117—21no—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117D412yes—
163#117D412yes—
164#117C2312yes—
165#117C2312yes—
166#117C2312yes—
167#117C2312yes—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4D411yescharacteristic
174C2×C4⋊C4D411yescharacteristic
175C2×C4⋊C4C2311yescharacteristic
176C2×C4⋊C4C2311yescharacteristic
177C2×C4⋊C4C2311yescharacteristic
178C2×C4⋊C4C2311yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4D411yescharacteristic
188C2×C22⋊C4D411yescharacteristic
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C2311yescharacteristic
194C2×C42C2311yescharacteristic
195#107C2314yes—
196#107C2314yes—
197#106C2314yes—
198#106C2314yes—
199C4⋊C8C2314yes—
200C4⋊C8C2314yes—
201#103C2314yes—
202#103C2314yes—
203#103C2314yes—
204#103C2314yes—
205#102C2314yes—
206#102C2314yes—
207#102C2314yes—
208#102C2314yes—
209C22⋊C8C2312yes—
210C22⋊C8C2312yes—
211C22⋊C8C2312yes—
212C22⋊C8C2312yes—
213#97C2314yes—
214C4×C8C2314yes—
215C24—21no—
216C24—21no—
217C24C2×D411yescharacteristic
218#41—28no—
219#41—28no—
220#41—24no—
221#41—24no—
222#41—24no—
223#41#4118yes—
224C2×Q8—24no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—82no—
230C2×Q8C2412yes—
231C2×Q8C2412yes—
232C2×D4—28no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—24no—
241C2×D4—82no—
242C2×D4C2412yes—
243C2×D4C2412yes—
244C22×C4—28no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—82no—
260C22×C4C2×D411yescharacteristic
261C22×C4C2×D411yescharacteristic
262C22×C4C2×D411yescharacteristic
263C22×C4C2×D411yescharacteristic
264C22×C4C2×D411yescharacteristic
265C22×C4C2×D411yescharacteristic
266C22×C4C2×D411yescharacteristic
267C22×C4C2411yescharacteristic
268C22×C4C2411yescharacteristic
269C22×C4C2411yescharacteristic
270Q16—28no—
271SD16—28no—
272SD16—28no—
273D8—28no—
274M16—24no—
275M16—24no—
276C2×C8C2414yes—
277C2×C8C2414yes—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4C2×D412yes—
291C4⋊C4C2×D412yes—
292C4⋊C4C2×D412yes—
293C4⋊C4C2×D412yes—
294C4⋊C4C2412yes—
295C4⋊C4C2412yes—
296C4⋊C4C2412yes—
297C4⋊C4C2412yes—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4C2×D412yes—
312C22⋊C4C2×D412yes—
313C22⋊C4C2×D412yes—
314C22⋊C4C2×D412yes—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—24no—
322C42C2×D412yes—
323C42C2×D412yes—
324C42C2412yes—
325C42C2412yes—
326C23—24no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23C22×D411yescharacteristic
338C23C22×D411yescharacteristic
339C23C22×D411yescharacteristic
340Q8—24no—
341Q8—22no—
342Q8—24no—
343Q8—42no—
344Q8—84no—
345Q8C2×#4114yes—
346D4—28no—
347D4—24no—
348D4—28no—
349D4—22no—
350D4—24no—
351D4—42no—
352D4—82no—
353D4—82no—
354D4C2×#4114yes—
355C2×C4—24no—
356C2×C4—28no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—28no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—84no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4C22×D412yes—
384C2×C4C22×D411yescharacteristic
385C2×C4C22×D412yes—
386C2×C4C22×D412yes—
387C2×C4C22×D412yes—
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D411yescharacteristic
390C2×C4C2×#4114yes—
391C2×C4C2511yescharacteristic
392C8—24no—
393C8—24no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—24no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—82no—
409C22—84no—
410C22—84no—
411C22#52512yes—
412C22C23×D411yescharacteristic
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—21no—
418C4—24no—
419C4—21no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4#52512yes—
426C4C22×#4112yes—
427C2—22no—
428C2—41no—
429C2—41no—
430C2—44no—
431C2—82no—
432C2C2×#52511yescharacteristic
433C2#358511yescharacteristic
434C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3006C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#542C2semidirect
2. #3428C2semidirect
3. #3125C2semidirect
4. #3124C2semidirect
5. #3121C2semidirect
6. #3118C2semidirect
7. #3095C2semidirect
8. #3094C2semidirect
9. #3079C2semidirect
10. #3040C2semidirect
11. #3007C2semidirect
13. C4×#137C2semidirect
14. C4×AGL(1,ℤ/8)C2semidirect
15. #2921C2semidirect
22. #542C22semidirect
23. C2×C4⋊Q8C22semidirect
26. C2×#124C22semidirect
33. #520C22semidirect
34. #520C22semidirect
35. #519C22semidirect
38. #518C22semidirect
42. C2×C4×Q8C22semidirect
43. C2×C4×D4C22semidirect
47. #485C22semidirect
48. #484C22semidirect
49. #483C22semidirect
51. #481C22semidirect
52. #480C22semidirect
53. #477C22semidirect
54. #476C22semidirect
55. #475C22semidirect
56. #474C22semidirect
57. #464C22semidirect
58. #464C22semidirect
59. #463C22semidirect
60. #463C22semidirect
61. #452C22semidirect
62. #451C22semidirect
63. #450C22semidirect
67. #442C22semidirect
68. #441C22semidirect
69. #440C22semidirect
70. #440C22semidirect
71. C4×Q16C22semidirect
72. C4×SD16C22semidirect
73. C4×SD16C22semidirect
74. C4×D8C22semidirect
75. #433C22semidirect
76. #433C22semidirect
77. #428C22semidirect
78. #428C22semidirect
79. #423C22semidirect
80. #419C22semidirect
81. #418C22semidirect
86. C4×M16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
#34282
#31252
#31242
#31212
#31182
#30952
#30942
#30792
#30402
#30072
#30062
C4×#1372
C4×AGL(1,ℤ/8)2
#29212