Finite Group Databasebeta

← #61706

Subgroups — #61706

Order 256.

1011 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3505C211yesmaximal, characteristic, complemented
2C2×#547C211yesmaximal, characteristic, complemented
3#3345C214yesmaximal, complemented
4#3343C214yesmaximal, complemented
5#3308C214yesmaximal, complemented
6#3306C214yesmaximal, complemented
7#3238C212yesmaximal, complemented
8#3194C212yesmaximal, complemented
9#3187C211yesmaximal, characteristic, complemented
10#3186C211yesmaximal, characteristic, complemented
11#3114C212yesmaximal, complemented
12#3092C212yesmaximal, complemented
13C2×#483C211yesmaximal, characteristic, complemented
14C2×#480C211yesmaximal, characteristic, complemented
15#2973C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17#553—22no—
18#553C2212yescomplemented
19#552C2212yescomplemented
20#551—22no—
21#548C2212yescomplemented
22#547—22no—
23#547—22no—
24#547C2214yes—
25#547C2214yescomplemented
26#546C2212yescomplemented
27#539—21no—
28#538—21no—
29#534—21no—
30C2×C4⋊Q8C2211yescharacteristic, complemented
31#529C2212yes—
32#525C2212yes—
33C2×#123—21no—
34C2×#123—21no—
35C2×C22⋊Q8—21no—
36C2×C22⋊Q8C2211yescharacteristic
37C2×#121C2211yescharacteristic
38C2×#121C2211yescharacteristic
39C2×#121C2211yescharacteristic, complemented
40#518—21no—
41C2×C4×D4C2211yescharacteristic
42C2×C4×D4C2211yescharacteristic, complemented
43C22×C4⋊C4C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic, complemented
45#501C2214yescomplemented
46#489C2214yescomplemented
47#484C2214yescomplemented
48#483C2214yescomplemented
49#483C2212yescomplemented
50#483C2212yescomplemented
51#483C2212yescomplemented
52#482C2214yescomplemented
53#481C2212yescomplemented
54#481C2214yescomplemented
55#480C2212yes—
56#480C2212yes—
57#480C2212yes—
58#480C2214yescomplemented
59#480C2212yescomplemented
60#478C2214yescomplemented
61#477C2214yescomplemented
62#476C2214yescomplemented
63#469C2214yescomplemented
64#468C2214yescomplemented
65#466C2214yescomplemented
66#465C2214yescomplemented
67#463C2214yescomplemented
68#450C2212yescomplemented
69#449C2214yescomplemented
70#448C2212yescomplemented
71#447C2212yes—
72#447C2212yescomplemented
73#442C2214yescomplemented
74#440C2214yescomplemented
75#435C2214yescomplemented
76#435C2214yescomplemented
77#432C2212yescomplemented
78#428C2214yescomplemented
79C2×#107C2212yescomplemented
80C2×#106C2212yescomplemented
81#417C2214yescomplemented
82C2×#103C2212yescomplemented
83C2×#102C2212yescomplemented
84#407C2212yescomplemented
85C2×C22⋊C8C2211yescharacteristic
86C2×C22⋊C8C2211yescharacteristic
87C2×#41—24no—
88C2×#41—24no—
89C2×#41—48no—
90C2×#41C2312yes—
91C22×Q8—21no—
92C22×Q8C2311yescharacteristic
93C22×D4—21no—
94C22×D4—21no—
95C22×D4—21no—
96C22×D4D411yescharacteristic
97C22×D4C2311yescharacteristic
98C22×D4C2311yescharacteristic
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C2311yescharacteristic
103C23×C4C2311yescharacteristic
104C2×SD16C2314yes—
105C2×D8C2314yes—
106C2×M16C2314yes—
107C22×C8C2312yes—
108C22×C8C2312yes—
109C4⋊Q8—22no—
110C4⋊Q8C2314yes—
111#126—22no—
112#126—22no—
113#126—22no—
114#125—22no—
115#125C2312yes—
116#124—22no—
117#124C2312yes—
118#123—22no—
119#123—22no—
120#123—22no—
121#123—22no—
122#123—24no—
123#123—24no—
124#123—24no—
125#123—24no—
126#123—22no—
127C22⋊Q8—21no—
128C22⋊Q8—21no—
129C22⋊Q8—24no—
130C22⋊Q8—24no—
131C22⋊Q8—22no—
132C22⋊Q8C2312yes—
133C22⋊Q8C2312yes—
134C22⋊Q8C2312yes—
135#121—22no—
136#121—22no—
137#121—21no—
138#121—21no—
139#121—21no—
140#121—21no—
141#121—22no—
142#121—22no—
143#121C2312yes—
144#121C2312yes—
145#121C2312yes—
146#121C2312yes—
147#121C2314yes—
148#121C2312yes—
149#120—22no—
150#120—22no—
151C4×Q8C2312yes—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4D412yes—
159C4×D4D412yes—
160C4×D4C2314yes—
161C4×D4C2312yes—
162C4×D4C2312yes—
163C4×D4C2312yes—
164#117—22no—
165#117—22no—
166#117C2312yes—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4D411yescharacteristic
175C2×C4⋊C4C2311yescharacteristic
176C2×C4⋊C4C2311yescharacteristic
177C2×C4⋊C4C2312yes—
178C2×C4⋊C4C2312yes—
179C2×C4⋊C4C2311yescharacteristic
180C2×C4⋊C4C2314yes—
181C2×C4⋊C4C2311yescharacteristic
182C2×C4⋊C4C2311yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4D411yescharacteristic
190C2×C22⋊C4D411yescharacteristic
191C2×C42C2311yescharacteristic
192#107C2314yes—
193#107C2314yes—
194#107C2314yes—
195#106C2314yes—
196#106C2314yes—
197#106C2314yes—
198C4⋊C8C2314yes—
199#103C2314yes—
200#103C2314yes—
201#103C2314yes—
202#102C2314yes—
203#102C2314yes—
204#102C2314yes—
205#102C2314yes—
206#102C2314yes—
207C22⋊C8C2312yes—
208C22⋊C8C2312yes—
209C22⋊C8C2312yes—
210C22⋊C8C2312yes—
211C22⋊C8C2314yes—
212#97C2314yes—
213C24—21no—
214C24—21no—
215C24C2×D411yescharacteristic
216C24C2×D411yescharacteristic
217#41—44no—
218#41—416no—
219#41—416no—
220C2×Q8—24no—
221C2×Q8—21no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8C2412yes—
225C2×D4—24no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4C2×D412yes—
241C2×D4C2×D412yes—
242C2×D4C2412yes—
243C2×D4C2412yes—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4C2×D411yescharacteristic
262C22×C4C2×D411yescharacteristic
263C22×C4C2×D411yescharacteristic
264C22×C4C2×D411yescharacteristic
265C22×C4C2×D411yescharacteristic
266C22×C4C2×D411yescharacteristic
267C22×C4C2×D412yes—
268C22×C4C2411yescharacteristic
269C22×C4C2411yescharacteristic
270C22×C4C2411yescharacteristic
271C22×C4C2412yes—
272SD16—44no—
273D8—44no—
274M16—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8C2414yes—
278C2×C8C2414yes—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4C2×D412yes—
295C4⋊C4C2×D412yes—
296C4⋊C4C2412yes—
297C4⋊C4C2412yes—
298C4⋊C4C2412yes—
299C4⋊C4C2412yes—
300C4⋊C4C2412yes—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4C2×D412yes—
316C22⋊C4C2×D412yes—
317C22⋊C4C2×D412yes—
318C22⋊C4C2×D412yes—
319C42—22no—
320C42—22no—
321C42C2×D412yes—
322C42C2412yes—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23AGL(1,ℤ/8)14yes—
337C23C22×D412yes—
338C23C22×D411yescharacteristic
339C23C22×D411yescharacteristic
340C23C22×D411yescharacteristic
341Q8—24no—
342Q8—48no—
343Q8—81no—
344D4—24no—
345D4—24no—
346D4—24no—
347D4—24no—
348D4—42no—
349D4—48no—
350D4—44no—
351D4—44no—
352D4—44no—
353D4—44no—
354D4—81no—
355D4—82no—
356D4—82no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4C22×D412yes—
387C2×C4C22×D412yes—
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D412yes—
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D411yescharacteristic
392C2×C4C2511yescharacteristic
393C8—24no—
394C8—24no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—44no—
405C22—44no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—84no—
413C22—162no—
414C22C2×AGL(1,ℤ/8)12yes—
415C22C2×AGL(1,ℤ/8)12yes—
416C22C2×AGL(1,ℤ/8)12yes—
417C22C23×D411yescharacteristic
418C4—22no—
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4#58512yes—
431C2—22no—
432C2—22no—
433C2—22no—
434C2—42no—
435C2—82no—
436C2—82no—
437C2#348511yescharacteristic
438C2C22×AGL(1,ℤ/8)11yescharacteristic
439C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3505C2semidirect
2. C2×#547C2semidirect
3. #3345C2semidirect
4. #3343C2semidirect
5. #3308C2semidirect
6. #3306C2semidirect
7. #3238C2semidirect
8. #3194C2semidirect
9. #3187C2semidirect
10. #3186C2semidirect
11. #3114C2semidirect
12. #3092C2semidirect
13. C2×#483C2semidirect
14. C2×#480C2semidirect
15. #2973C2semidirect
18. #553C22semidirect
19. #552C22semidirect
21. #548C22semidirect
25. #547C22semidirect
26. #546C22semidirect
30. C2×C4⋊Q8C22semidirect
39. C2×#121C22semidirect
42. C2×C4×D4C22semidirect
44. C22×C4⋊C4C22semidirect
45. #501C22semidirect
46. #489C22semidirect
47. #484C22semidirect
48. #483C22semidirect
49. #483C22semidirect
50. #483C22semidirect
51. #483C22semidirect
52. #482C22semidirect
53. #481C22semidirect
54. #481C22semidirect
58. #480C22semidirect
59. #480C22semidirect
60. #478C22semidirect
61. #477C22semidirect
62. #476C22semidirect
63. #469C22semidirect
64. #468C22semidirect
65. #466C22semidirect
66. #465C22semidirect
67. #463C22semidirect
68. #450C22semidirect
69. #449C22semidirect
70. #448C22semidirect
72. #447C22semidirect
73. #442C22semidirect
74. #440C22semidirect
75. #435C22semidirect
76. #435C22semidirect
77. #432C22semidirect
78. #428C22semidirect
79. C2×#107C22semidirect
80. C2×#106C22semidirect
81. #417C22semidirect
82. C2×#103C22semidirect
83. C2×#102C22semidirect
84. #407C22semidirect

Maximal subgroups

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

subgroupindex
#35052
C2×#5472
#33452
#33432
#33082
#33062
#32382
#31942
#31872
#31862
#31142
#30922
C2×#4832
C2×#4802
#29732