Finite Group Databasebeta

← #31228

Subgroups — #31228

Order 256.

399 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#3096C212yesmaximal, complemented
2#3045C212yesmaximal, complemented
3#2063C212yesmaximal, complemented
4#2051C212yesmaximal, complemented
5#2002C212yesmaximal, complemented
6#1974C211yesmaximal, characteristic, complemented
7#1957C211yesmaximal, characteristic, complemented
8#1945C211yesmaximal, characteristic, complemented
9#1859C212yesmaximal, complemented
10C2×AGL(1,ℤ/8)C2212yes—
11C2×#135C2211yescharacteristic
12#535C2212yes—
13C2×#127C2211yescharacteristic
14#520C2212yes—
15#482—22no—
16#482—22no—
17#481—22no—
18#480—22no—
19#464—22no—
20#463—22no—
21#460—22no—
22#459—22no—
23C2×#108C2211yescharacteristic
24C2×#108C2211yescharacteristic
25#428C2212yes—
26#427C2211yescharacteristic
27#424C2212yes—
28C2×#104C2212yes—
29#418C2212yes—
30#417C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×#100C2212yes—
34#408C2212yes—
35C2×C4×C8C2211yescharacteristic
36#384C2211yescharacteristic
37#343C2212yes—
38#339C2212yes—
39C2×#41C2312yes—
40C22×D4—41no—
41C22×D4C2312yes—
42AGL(1,ℤ/8)—42no—
43AGL(1,ℤ/8)—42no—
44#135—41no—
45#135—41no—
46C2×Q16D411yescharacteristic, complemented
47C2×SD16—22no—
48C2×SD16D412yescomplemented
49C2×D8—22no—
50C2×D8D411yescharacteristic, complemented
51C2×M16C2312yes—
52C2×M16C2312yes—
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55#127—22no—
56#127—22no—
57#127—41no—
58#124—22no—
59#121—22no—
60#121—22no—
61#120—42no—
62C4×Q8—22no—
63C4×D4—22no—
64C4×D4—42no—
65#117—22no—
66#117C2312yes—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4C2312yes—
70C2×C42C2311yescharacteristic
71#108—41no—
72#108—41no—
73#107—22no—
74#107D411yescharacteristic, complemented
75#107D411yescharacteristic, complemented
76#106—22no—
77#106D412yescomplemented
78C4⋊C8—22no—
79C4⋊C8—22no—
80#104—24no—
81#103—22no—
82#102—22no—
83#102—22no—
84#102—22no—
85#102—22no—
86#102—22no—
87#102—22no—
88#102—22no—
89#100—42no—
90C22⋊C8—22no—
91C22⋊C8—22no—
92C4×C8—41no—
93C24—22no—
94#41—22no—
95#41—22no—
96#41—22no—
97#41—22no—
98C2×Q8C2×D412yes—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—44no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—82no—
110C2×D4—82no—
111C2×D4C2×D412yes—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—42no—
116C22×C4C2411yescharacteristic
117Q16—41no—
118SD16—42no—
119SD16—44no—
120D8—41no—
121D8—44no—
122M16—22no—
123M16—22no—
124M16—22no—
125M16—22no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8C2×D411yescharacteristic
133C2×C8C2×D411yescharacteristic
134C2×C8C2×D411yescharacteristic
135C2×C8C2×D411yescharacteristic
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—42no—
139C4⋊C4—82no—
140C4⋊C4C2×D412yes—
141C4⋊C4C2×D412yes—
142C22⋊C4—22no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C42—22no—
147C42—21no—
148C42—42no—
149C42#4112yes—
150C23—22no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—82no—
155C23—82no—
156C23H(5,𝔽2)11yescharacteristic
157Q8—22no—
158Q8—22no—
159D4—22no—
160D4—22no—
161D4—42no—
162D4—42no—
163D4—42no—
164D4—42no—
165D4—42no—
166D4—42no—
167D4—88no—
168D4—162no—
169D4—162no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—82no—
180C2×C4—162no—
181C2×C4C22×D411yescharacteristic
182C2×C4C22×D411yescharacteristic
183C2×C4C2×#4111yescharacteristic
184C2×C4H(5,𝔽2)11yescharacteristic
185C2×C4H(5,𝔽2)11yescharacteristic
186C2×C42−1+411yescharacteristic
187C8—21no—
188C8—21no—
189C8—21no—
190C8—21no—
191C8—42no—
192C8—42no—
193C22—21no—
194C22—21no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—84no—
199C22—82no—
200C22—82no—
201C22—82no—
202C22—162no—
203C22#53811yescharacteristic
204C22D4211yescharacteristic
205C22#56211yescharacteristic
206C4—24no—
207C4—42no—
208C4—42no—
209C4—42no—
210C4—82no—
211C4#53411yescharacteristic
212C4#53711yescharacteristic
213C4#54711yescharacteristic
214C4#55011yescharacteristic
215C2—21no—
216C2—21no—
217C2—82no—
218C2—162no—
219C2#268211yescharacteristic
220C2#329111yescharacteristic
221C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3096C2semidirect
2. #3045C2semidirect
3. #2063C2semidirect
4. #2051C2semidirect
5. #2002C2semidirect
6. #1974C2semidirect
7. #1957C2semidirect
8. #1945C2semidirect
9. #1859C2semidirect
46. C2×Q16D4semidirect
48. C2×SD16D4semidirect
50. C2×D8D4semidirect
74. #107D4semidirect
75. #107D4semidirect
77. #106D4semidirect

Maximal subgroups

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

subgroupindex
#30962
#30452
#20632
#20512
#20022
#19742
#19572
#19452
#18592