Finite Group Databasebeta

← #20306

Subgroups — #20306

Order 256.

295 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 99 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#3367C212yesmaximal
2#3328C212yesmaximal, complemented
3#1710C211yesmaximal, characteristic, complemented
4#1700C212yesmaximal, complemented
5#1699C212yesmaximal, complemented
6#1650C212yesmaximal, complemented
7#1649C212yesmaximal, complemented
8#1591C211yesmaximal, characteristic, complemented
9#1589C211yesmaximal, characteristic
10#556C2212yes—
11D4×Q8C2212yes—
12#500C2211yescharacteristic
13#500C2211yescharacteristic
14#489—22no—
15#487C2212yes—
16#487C2212yes—
17#485—22no—
18#477—22no—
19#475C2212yes—
20#474C2212yes—
21#470—22no—
22#463—22no—
23#462C2212yes—
24#461C2212yes—
25#451—22no—
26#446C2212yes—
27#440—22no—
28C4×Q16C2211yescharacteristic
29C4×D8C2211yescharacteristic
30#436C2212yes—
31#419—22no—
32#335C2211yescharacteristic
33#335C2211yescharacteristic
34#332C2212yes—
35#332C2212yes—
36Q8⋊C8C2212yes—
37D4⋊C8C2212yes—
38C82C2211yescharacteristic
39C22×Q8—22no—
40C2×Q16—22no—
41C2×Q16D411yescharacteristic
42C2×SD16—22no—
43C2×D8D411yescharacteristic
44C2×M16—22no—
45C4⋊Q8—22no—
46C4⋊Q8C2312yes—
47C4⋊Q8C2312yes—
48#126—42no—
49#124—22no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52C22⋊Q8—42no—
53C4×Q8—22no—
54C4×Q8C2312yes—
55C4×D4—22no—
56C4×D4C2312yes—
57#117—22no—
58#107—22no—
59#107—41no—
60#107—41no—
61#107D411yescharacteristic
62#107D411yescharacteristic
63#106—22no—
64C4⋊C8—22no—
65C4⋊C8C2312yes—
66C4⋊C8C2312yes—
67#103—22no—
68#103—22no—
69#103—22no—
70#103—22no—
71#103—22no—
72#103—22no—
73#103—22no—
74#103D412yes—
75#102—22no—
76#102D412yes—
77C22⋊C8—22no—
78#97—22no—
79C4×C8C2311yescharacteristic
80C4×C8C2311yescharacteristic
81C4×C8C2311yescharacteristic
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—42no—
85C2×Q8—44no—
86C2×Q8—82no—
87C2×Q8C2×D412yes—
88C2×D4C2×D412yes—
89C22×C4—22no—
90C22×C4—42no—
91Q16—41no—
92Q16—82no—
93SD16—44no—
94D8—41no—
95M16—82no—
96C2×C8—21no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8C2×D411yescharacteristic
100C2×C8C2×D411yescharacteristic
101C2×C8C2×D411yescharacteristic
102C2×C8C2×D411yescharacteristic
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4C2×D412yes—
111C4⋊C4C2×D412yes—
112C22⋊C4—22no—
113C22⋊C4—42no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C42—22no—
117C42—42no—
118C42C2411yescharacteristic
119C23—22no—
120Q8—22no—
121Q8—22no—
122Q8—42no—
123Q8—42no—
124Q8—42no—
125Q8—42no—
126Q8—84no—
127D4—22no—
128D4—22no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—84no—
137C2×C4C22×D411yescharacteristic
138C2×C4C22×D411yescharacteristic
139C2×C4H(5,𝔽2)11yescharacteristic
140C8—22no—
141C8—21no—
142C8—21no—
143C8—21no—
144C8—21no—
145C8—42no—
146C8—42no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22D4211yescharacteristic
151C4—21no—
152C4—42no—
153C4—42no—
154C4—42no—
155C4—82no—
156C4—82no—
157C4#57511yescharacteristic
158C4#57511yescharacteristic
159C4#57611yescharacteristic
160C4#57811yescharacteristic
161C2—82no—
162C2#327311yescharacteristic
163C2#327711yescharacteristic
164C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radicalC2×D8D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3328C2semidirect
3. #1710C2semidirect
4. #1700C2semidirect
5. #1699C2semidirect
6. #1650C2semidirect
7. #1649C2semidirect
8. #1591C2semidirect

Maximal subgroups

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

subgroupindex
#33672
#33282
#17102
#17002
#16992
#16502
#16492
#15912
#15892