Finite Group Databasebeta

← #10459

Subgroups — #10459

Order 256.

214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 57 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#3037C211yesmaximal, characteristic
2#1801C211yesmaximal, characteristic, complemented
3#1566C211yesmaximal, characteristic, complemented
4#1547C211yesmaximal, characteristic
5#1492C211yesmaximal, characteristic
6C2×Q8⋊C8C211yesmaximal, characteristic, complemented
7#1292C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#464—22no—
12#462—21no—
13#460—21no—
14#423C2211yescharacteristic
15#423C2211yescharacteristic
16#423C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18#417—21no—
19C2×#101—21no—
20C2×C4×C8C2211yescharacteristic
21#343—21no—
22#334—22no—
23#332—21no—
24#331—21no—
25Q8⋊C8—21no—
26Q8⋊C8—21no—
27D4⋊C8—21no—
28D4⋊C8—21no—
29C2×#41—21no—
30C22×Q8D411yescharacteristic
31#137—42no—
32#137—42no—
33C2×Q16—22no—
34C2×SD16—22no—
35C2×M16—21no—
36C2×M16—21no—
37C2×M16—21no—
38C22×C8—21no—
39C22×C8—21no—
40C4⋊Q8D411yescharacteristic
41#127D411yescharacteristic
42#124—21no—
43#121—21no—
44#121—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×D4—21no—
50C4×D4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4D411yescharacteristic
53C2×C42—21no—
54C2×C42C2311yescharacteristic
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8D412yes—
63#103—22no—
64#102—22no—
65#101—42no—
66C4×C8—21no—
67C4×C8D411yescharacteristic
68C4×C8D411yescharacteristic
69#41—41no—
70#41—41no—
71#41—41no—
72#41—41no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8SD1612yes—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—41no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4C2×D411yescharacteristic
88C22×C4#4111yescharacteristic
89C22×C4#4111yescharacteristic
90Q16—44no—
91SD16—44no—
92M16—41no—
93M16—41no—
94M16—41no—
95M16—41no—
96M16—41no—
97M16—41no—
98C2×C8—21no—
99C2×C8—22no—
100C2×C8—21no—
101C2×C8—42no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—81no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C42—41no—
115C42—41no—
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42#4111yescharacteristic
120C23—41no—
121C23#12311yescharacteristic
122Q8—22no—
123Q8—22no—
124Q8—41no—
125Q8—41no—
126Q8—41no—
127Q8—41no—
128D4—41no—
129D4—41no—
130D4—41no—
131D4—41no—
132D4—81no—
133D4—81no—
134D4—81no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4#12011yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12111yescharacteristic
150C2×C4#12311yescharacteristic
151C2×C4#12411yescharacteristic
152C2×C4C2×SD1611yescharacteristic
153C2×C4#13511yescharacteristic
154C2×C4AGL(1,ℤ/8)11yescharacteristic
155C2×C4AGL(1,ℤ/8)11yescharacteristic
156C8—22no—
157C8—41no—
158C8—42no—
159C8—81no—
160C8—81no—
161C8—81no—
162C22—21no—
163C22—21no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22#39411yescharacteristic
168C22#48111yescharacteristic
169C22#48211yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—42no—
174C4—81no—
175C4—81no—
176C4#44911yescharacteristic
177C4#45011yescharacteristic
178C4#46511yescharacteristic
179C4#46911yescharacteristic
180C4#47111yescharacteristic
181C4#47311yescharacteristic
182C2—21no—
183C2—21no—
184C2—161no—
185C2#203411yescharacteristic
186C2#205011yescharacteristic
187C2#205211yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1801C2semidirect
3. #1566C2semidirect
6. C2×Q8⋊C8C2semidirect
7. #1292C2semidirect

Maximal subgroups

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

subgroupindex
#30372
#18012
#15662
#15472
#14922
C2×Q8⋊C82
#12922