Finite Group Databasebeta

← #61017

Subgroups — #61017

Order 256.

896 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3510C211yesmaximal, characteristic, complemented
2#3508C211yesmaximal, characteristic, complemented
3#3257C214yesmaximal, complemented
4#3240C212yesmaximal, complemented
5#3235C212yesmaximal, complemented
6#3217C212yesmaximal, complemented
7#3213C212yesmaximal, complemented
8#3200C212yesmaximal, complemented
9#3199C212yesmaximal, complemented
10#3180C212yesmaximal, complemented
11#3171C212yesmaximal, complemented
12#3126C212yesmaximal, complemented
13#3122C212yesmaximal, complemented
14#3118C214yesmaximal, complemented
15#2923C211yesmaximal, characteristic, complemented
16#559—22no—
17#559—22no—
18#556—22no—
19#551—22no—
20#551—22no—
21#550—22no—
22#546—22no—
23#546—22no—
24#542C2212yes—
25#542C2212yescomplemented
26#540C2212yes—
27#540C2212yescomplemented
28#536C2211yescharacteristic
29#535C2211yescharacteristic
30#535C2212yes—
31#533C2211yescharacteristic, complemented
32#533C2211yescharacteristic, complemented
33#532C2212yes—
34#532C2212yescomplemented
35C2×#127C2211yescharacteristic, complemented
36#529C2212yes—
37#529C2212yescomplemented
38C2×#124C2211yescharacteristic, complemented
39#519C2211yescharacteristic, complemented
40#518C2212yescomplemented
41#518C2211yescharacteristic, complemented
42C4×#41C2211yescharacteristic
43#501C2212yescomplemented
44#501C2212yescomplemented
45#496C2212yes—
46#496C2212yes—
47#490C2212yescomplemented
48#490C2212yescomplemented
49#489C2212yescomplemented
50#489C2212yescomplemented
51#485C2214yescomplemented
52#482C2214yes—
53#482C2214yescomplemented
54#482C2214yescomplemented
55#479C2214yescomplemented
56#478C2214yescomplemented
57#476C2214yescomplemented
58#474C2214yescomplemented
59#469C2214yescomplemented
60#468C2214yescomplemented
61#466C2214yescomplemented
62#465C2214yescomplemented
63#463C2214yescomplemented
64#463C2214yescomplemented
65#460C2214yescomplemented
66#459C2214yes—
67#450C2214yescomplemented
68#448C2214yescomplemented
69#447C2214yes—
70#447C2214yescomplemented
71#442C2214yescomplemented
72#440C2214yescomplemented
73C4×SD16C2214yescomplemented
74C4×D8C2214yescomplemented
75#433C2212yescomplemented
76#433C2212yescomplemented
77#432C2212yescomplemented
78#432C2212yescomplemented
79#424C2212yescomplemented
80#408C2212yes—
81C4×M16C2211yescharacteristic, complemented
82C2×#97C2211yescharacteristic, complemented
83C2×#41—22no—
84C2×#41—22no—
85C2×#41C2311yescharacteristic
86C22×Q8C2311yescharacteristic
87C22×D4—22no—
88C22×D4C2311yescharacteristic
89C22×D4C2312yes—
90C2×SD16C2314yes—
91C2×SD16C2314yes—
92C2×D8C2314yes—
93C2×D8C2314yes—
94C2×M16C2312yes—
95C22×C8C2312yes—
96C4⋊Q8—21no—
97C4⋊Q8C2312yes—
98C4⋊Q8C2312yes—
99#127—22no—
100#127—21no—
101#127—22no—
102#127C2312yes—
103#127C2312yes—
104#126—22no—
105#126—22no—
106#126—22no—
107#126—22no—
108#125C2312yes—
109#125C2312yes—
110#124—21no—
111#124—22no—
112#124—21no—
113#124—22no—
114#124—22no—
115#124—22no—
116#124—22no—
117#124—22no—
118#124C2312yes—
119#124C2312yes—
120#123—21no—
121#123—21no—
122#123—22no—
123#123—22no—
124#123—24no—
125#123—24no—
126#123—22no—
127#123—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8C2312yes—
133C22⋊Q8C2312yes—
134#121—22no—
135#121—22no—
136#121—24no—
137#121—24no—
138#121C2312yes—
139#121C2312yes—
140#121C2314yes—
141#121C2314yes—
142#120—22no—
143#120—22no—
144#120—21no—
145#120—21no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8D412yes—
150C4×Q8C2312yes—
151C4×Q8C2312yes—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—24no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4D412yes—
159C4×D4D412yes—
160C4×D4D412yes—
161C4×D4C2312yes—
162C4×D4C2312yes—
163C4×D4C2314yes—
164C4×D4C2314yes—
165#117—22no—
166#117—22no—
167#117—22no—
168#117C2312yes—
169#117C2311yescharacteristic
170#117C2311yescharacteristic
171#117C2311yescharacteristic
172#117C2311yescharacteristic
173#117C2311yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180#107C2314yes—
181#107C2314yes—
182#106C2314yes—
183#106C2314yes—
184C4⋊C8C2314yes—
185C4⋊C8C2314yes—
186#103C2314yes—
187#103C2314yes—
188#102C2314yes—
189#102C2314yes—
190#102C2314yes—
191#102C2314yes—
192#102C2314yes—
193#102C2314yes—
194C22⋊C8C2314yes—
195C22⋊C8C2314yes—
196#97C2312yes—
197#97C2312yes—
198#97C2312yes—
199C4×C8C2312yes—
200C24—21no—
201C24—22no—
202#41—42no—
203#41—84no—
204#41—84no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—42no—
209C2×Q8C2×D411yescharacteristic
210C2×Q8C2412yes—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—21no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—24no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—84no—
224C2×D4C2×D411yescharacteristic
225C2×D4C2×D411yescharacteristic
226C2×D4C2×D411yescharacteristic
227C2×D4C2412yes—
228C2×D4C2414yes—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4C2×D412yes—
235C22×C4C2×D412yes—
236C22×C4C2×D412yes—
237C22×C4C2411yescharacteristic
238C22×C4C2411yescharacteristic
239C22×C4C2411yescharacteristic
240SD16—44no—
241SD16—44no—
242D8—44no—
243D8—44no—
244M16—42no—
245C2×C8—42no—
246C2×C8C2414yes—
247C2×C8C2414yes—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4C2×D412yes—
258C4⋊C4C2×D412yes—
259C4⋊C4C2×D412yes—
260C4⋊C4C2412yes—
261C4⋊C4C2412yes—
262C4⋊C4C2414yes—
263C22⋊C4—22no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4C2×D412yes—
281C22⋊C4C2×D412yes—
282C22⋊C4C2×D412yes—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—21no—
287C42—21no—
288C42C2×D412yes—
289C42C2×D412yes—
290C42C2×D412yes—
291C42C2411yescharacteristic
292C42C2411yescharacteristic
293C42C2411yescharacteristic
294C42C2411yescharacteristic
295C23—22no—
296C23—24no—
297C23—42no—
298C23—44no—
299C23—84no—
300C23—82no—
301C23C22×D411yescharacteristic
302C23C22×D411yescharacteristic
303C23C22×D411yescharacteristic
304Q8—24no—
305Q8—41no—
306Q8—42no—
307Q8—44no—
308Q8—82no—
309D4—24no—
310D4—24no—
311D4—24no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—44no—
316D4—42no—
317D4—44no—
318D4—44no—
319D4—84no—
320D4—82no—
321D4—82no—
322D4—84no—
323D4—84no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4C22×D411yescharacteristic
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D412yes—
342C2×C4C22×D412yes—
343C2×C4C22×D412yes—
344C2×C4C22×D411yescharacteristic
345C2×C4C22×D412yes—
346C2×C4C2511yescharacteristic
347C8—24no—
348C8—24no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22—44no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—44no—
361C22—162no—
362C22—164no—
363C22C23×D411yescharacteristic
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—44no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4#58512yes—
376C2—41no—
377C2—41no—
378C2—41no—
379C2—84no—
380C2—82no—
381C2#344911yescharacteristic
382C2#358512yes—
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3510C2semidirect
2. #3508C2semidirect
3. #3257C2semidirect
4. #3240C2semidirect
5. #3235C2semidirect
6. #3217C2semidirect
7. #3213C2semidirect
8. #3200C2semidirect
9. #3199C2semidirect
10. #3180C2semidirect
11. #3171C2semidirect
12. #3126C2semidirect
13. #3122C2semidirect
14. #3118C2semidirect
15. #2923C2semidirect
25. #542C22semidirect
27. #540C22semidirect
31. #533C22semidirect
32. #533C22semidirect
34. #532C22semidirect
35. C2×#127C22semidirect
37. #529C22semidirect
38. C2×#124C22semidirect
39. #519C22semidirect
40. #518C22semidirect
41. #518C22semidirect
43. #501C22semidirect
44. #501C22semidirect
47. #490C22semidirect
48. #490C22semidirect
49. #489C22semidirect
50. #489C22semidirect
51. #485C22semidirect
53. #482C22semidirect
54. #482C22semidirect
55. #479C22semidirect
56. #478C22semidirect
57. #476C22semidirect
58. #474C22semidirect
59. #469C22semidirect
60. #468C22semidirect
61. #466C22semidirect
62. #465C22semidirect
63. #463C22semidirect
64. #463C22semidirect
65. #460C22semidirect
67. #450C22semidirect
68. #448C22semidirect
70. #447C22semidirect
71. #442C22semidirect
72. #440C22semidirect
73. C4×SD16C22semidirect
74. C4×D8C22semidirect
75. #433C22semidirect
76. #433C22semidirect
77. #432C22semidirect
78. #432C22semidirect
79. #424C22semidirect
81. C4×M16C22semidirect
82. C2×#97C22semidirect

Maximal subgroups

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

subgroupindex
#35102
#35082
#32572
#32402
#32352
#32172
#32132
#32002
#31992
#31802
#31712
#31262
#31222
#31182
#29232