Finite Group Databasebeta

← #60389

Subgroups — #60389

Order 256.

2385 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 2385 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(7,𝔽2)C211yesmaximal, characteristic, complemented
2#3586C212yesmaximal, complemented
3#3500C211yesmaximal, characteristic, complemented
4#3022C216yesmaximal, complemented
5#3019C216yesmaximal, complemented
6#3018C216yesmaximal, complemented
7C2×#455C216yesmaximal, complemented
8#2901C212yesmaximal, complemented
9#2888C211yesmaximal, characteristic, complemented
10#585—28no—
11#585—22no—
12#585C2216yes—
13#585C2212yes—
14C2×2−1+4C2211yescharacteristic, complemented
15C2×H(5,𝔽2)—28no—
16C2×H(5,𝔽2)—26no—
17C2×H(5,𝔽2)C2216yes—
18C2×H(5,𝔽2)C2211yescharacteristic
19#578—28no—
20#577—28no—
21#575C22112yescomplemented
22C2×#137C22112yescomplemented
23#568C2212yescomplemented
24#548—22no—
25#546—26no—
26#536C2213yescomplemented
27#535C2213yescomplemented
28#532C2213yescomplemented
29C2×#124C2213yescomplemented
30C4×#41C2211yescharacteristic, complemented
31#456C22112yescomplemented
32#455C22112yescomplemented
33#455C22124yescomplemented
34#454C2214yescomplemented
35#453C22112yescomplemented
36#421C22112yescomplemented
37C2×#104C22112yescomplemented
38#413C2216yescomplemented
39C2×#100C2216yescomplemented
402−1+4—28no—
412−1+4—212no—
422−1+4—22no—
432−1+4—28no—
442−1+4D418yescomplemented
452−1+4C2314yes—
46H(5,𝔽2)—28no—
47H(5,𝔽2)—248no—
48H(5,𝔽2)—224no—
49H(5,𝔽2)—28no—
50H(5,𝔽2)—212no—
51H(5,𝔽2)—212no—
52H(5,𝔽2)—26no—
53H(5,𝔽2)—212no—
54H(5,𝔽2)D418yescomplemented
55H(5,𝔽2)C23112yes—
56C2×#41—216no—
57C2×#41—224no—
58C2×#41—224no—
59C2×#41—28no—
60C2×#41—212no—
61C2×#41—23no—
62C2×#41—212no—
63C2×#41D412yes—
64C2×#41C23112yes—
65C2×#41C2313yes—
66C2×#41C2311yescharacteristic
67C2×#41C2313yescomplemented
68C22×Q8—21no—
69C22×Q8C2313yescomplemented
70C22×D4—224no—
71C22×D4—212no—
72C22×D4—212no—
73C22×D4D416yes—
74C22×D4C2313yes—
75#137—224no—
76#137—212no—
77#137C23124yescomplemented
78AGL(1,ℤ/8)—212no—
79AGL(1,ℤ/8)C2318yescomplemented
80#135—224no—
81C2×Q16—212no—
82C2×SD16—212no—
83#131—28no—
84C2×M16C23112yescomplemented
85C4⋊Q8C2313yescomplemented
86#127C2313yescomplemented
87#124—26no—
88#124C2313yescomplemented
89#124C2316yescomplemented
90#124C2311yescharacteristic, complemented
91#123—26no—
92C22⋊Q8—26no—
93#121—26no—
94#121—26no—
95#120—26no—
96C4×Q8—21no—
97C4×D4—23no—
98C4×D4—26no—
99#117C2313yescomplemented
100C2×C22⋊C4—26no—
101C2×C42C2313yescomplemented
102#104C2318yescomplemented
103#104C23124yescomplemented
104#101C2314yescomplemented
105#100C23112yescomplemented
106C24—212no—
107C24C2×D416yes—
108#41—248no—
109#41—248no—
110#41—224no—
111#41—224no—
112#41—28no—
113#41—28no—
114#41—224no—
115#41—224no—
116#41—28no—
117#41—28no—
118#41—212no—
119#41—212no—
120#41—212no—
121#41—212no—
122#41—412no—
123#41C2×D4124yes—
124#41C2×D418yes—
125#41C2418yes—
126C2×Q8—212no—
127C2×Q8—28no—
128C2×Q8—28no—
129C2×Q8—23no—
130C2×Q8—212no—
131C2×Q8—44no—
132C2×Q8C2×D4112yes—
133C2×Q8C2×D411yescharacteristic
134C2×Q8C2413yes—
135C2×Q8C2411yescharacteristic
136C2×D4—248no—
137C2×D4—248no—
138C2×D4—224no—
139C2×D4—248no—
140C2×D4—212no—
141C2×D4—224no—
142C2×D4—212no—
143C2×D4—224no—
144C2×D4—224no—
145C2×D4—224no—
146C2×D4—23no—
147C2×D4—26no—
148C2×D4—212no—
149C2×D4C2×D416yes—
150C2×D4C2×D416yes—
151C2×D4C2×D4112yes—
152C2×D4C2×D413yes—
153C2×D4C2411yescharacteristic
154C2×D4C2413yes—
155C22×C4—212no—
156C22×C4—224no—
157C22×C4—212no—
158C22×C4—23no—
159C22×C4—23no—
160C22×C4—23no—
161C22×C4C2×D416yes—
162C22×C4C2413yes—
163Q16—224no—
164SD16—28no—
165SD16—224no—
166D8—28no—
167M16—212no—
168M16C2418yes—
169C2×C8—212no—
170C4⋊C4—23no—
171C4⋊C4—26no—
172C22⋊C4—26no—
173C22⋊C4—212no—
174C22⋊C4—23no—
175C22⋊C4—22no—
176C42—23no—
177C42C2411yescharacteristic
178C42C2413yes—
179C23—224no—
180C23—212no—
181C23—212no—
182C23—21no—
183C23—86no—
184C23—1612no—
185C23C22×D416yes—
186C23C22×D413yes—
187Q8—28no—
188Q8—212no—
189Q8—22no—
190Q8—26no—
191Q8—46no—
192Q8#12014yes—
193Q8C22×D418yes—
194D4—248no—
195D4—28no—
196D4—224no—
197D4—212no—
198D4—28no—
199D4—212no—
200D4—212no—
201D4—26no—
202D4—22no—
203D4—46no—
204D4—86no—
205D4#120112yes—
206D4C22×D418yes—
207C2×C4—224no—
208C2×C4—216no—
209C2×C4—212no—
210C2×C4—224no—
211C2×C4—212no—
212C2×C4—28no—
213C2×C4—21no—
214C2×C4—23no—
215C2×C4—23no—
216C2×C4—46no—
217C2×C4—46no—
218C2×C4—83no—
219C2×C4C22×D412yes—
220C2×C4C22×D413yes—
221C2×C4C22×D4112yes—
222C2×C4C2511yescharacteristic
223C8—28no—
224C22—26no—
225C22—28no—
226C22—43no—
227C22—46no—
228C22—42no—
229C22—41no—
230C22—812no—
231C22—812no—
232C22—824no—
233C22C2×#12016yes—
234C22C23×D411yescharacteristic
235C4—28no—
236C4—22no—
237C4—41no—
238C4—43no—
239C4—43no—
240C4C2×#12016yes—
241C4C23×D412yes—
242C2—21no—
243C2—26no—
244C2—46no—
245C2—48no—
246C2—81no—
247C2C22×#12011yescharacteristic
subgroupG/H
center, socleC2C22×#120
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. H(7,𝔽2)C2semidirect
2. #3586C2semidirect
3. #3500C2semidirect
4. #3022C2semidirect
5. #3019C2semidirect
6. #3018C2semidirect
7. C2×#455C2semidirect
8. #2901C2semidirect
9. #2888C2semidirect
14. C2×2−1+4C22semidirect
21. #575C22semidirect
22. C2×#137C22semidirect
23. #568C22semidirect
26. #536C22semidirect
27. #535C22semidirect
28. #532C22semidirect
29. C2×#124C22semidirect
30. C4×#41C22semidirect
31. #456C22semidirect
32. #455C22semidirect
33. #455C22semidirect
34. #454C22semidirect
35. #453C22semidirect
36. #421C22semidirect
37. C2×#104C22semidirect
38. #413C22semidirect
39. C2×#100C22semidirect
44. 2−1+4D4semidirect
54. H(5,𝔽2)D4semidirect
67. C2×#41C23semidirect
69. C22×Q8C23semidirect
77. #137C23semidirect
79. AGL(1,ℤ/8)C23semidirect
84. C2×M16C23semidirect
85. C4⋊Q8C23semidirect
86. #127C23semidirect
88. #124C23semidirect
89. #124C23semidirect
90. #124C23semidirect
99. #117C23semidirect
101. C2×C42C23semidirect
102. #104C23semidirect
103. #104C23semidirect
104. #101C23semidirect
105. #100C23semidirect

Maximal subgroups

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

subgroupindex
H(7,𝔽2)2
#35862
#35002
#30222
#30192
#30182
C2×#4552
#29012
#28882