Finite Group Databasebeta

← #10496

Subgroups — #10496

Order 256.

201 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 45 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#3213C211yesmaximal, characteristic
2#1967C211yesmaximal, characteristic, complemented
3#1569C211yesmaximal, characteristic, complemented
4#1557C211yesmaximal, characteristic, complemented
5#1527C211yesmaximal, characteristic, complemented
6#1525C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#124C2211yescharacteristic
10#496—21no—
11#489—21no—
12#463—21no—
13#463—21no—
14#448—21no—
15#447—21no—
16#432C2211yescharacteristic
17#432C2211yescharacteristic
18#423C2211yescharacteristic
19C2×#101—21no—
20C2×#100—21no—
21C4×M16C2211yescharacteristic
22C4×M16C2211yescharacteristic
23#335—21no—
24#334—21no—
25#332—21no—
26#331—21no—
27#329—21no—
28#329—21no—
29#329—21no—
30#329—21no—
31C2×#41—21no—
32C22×Q8D411yescharacteristic
33C22×D4D411yescharacteristic
34C2×SD16—21no—
35C2×SD16—21no—
36C2×D8—21no—
37C2×D8—21no—
38C2×M16—21no—
39C2×M16—21no—
40C2×M16—21no—
41C4⋊Q8D411yescharacteristic
42#127D411yescharacteristic
43#124—21no—
44#124—21no—
45#124—21no—
46#124—21no—
47#124—21no—
48#121—21no—
49#121—21no—
50C4×D4—21no—
51C4×D4—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C42C2311yescharacteristic
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8D411yescharacteristic
59C4⋊C8D411yescharacteristic
60#103—21no—
61#103—21no—
62#102—21no—
63#102—21no—
64#101—42no—
65#100—42no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68#97—21no—
69#97—21no—
70#97—21no—
71#97D411yescharacteristic
72#97D411yescharacteristic
73C4×C8—21no—
74C4×C8—21no—
75C24—21no—
76#41—82no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—41no—
81C2×Q8—41no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C22×C4—21no—
91C22×C4—41no—
92C22×C4C2×D411yescharacteristic
93C22×C4#4111yescharacteristic
94C22×C4#4111yescharacteristic
95SD16—81no—
96SD16—81no—
97D8—81no—
98D8—81no—
99M16—22no—
100M16—42no—
101M16—41no—
102M16—41no—
103M16—41no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C42C2×D411yescharacteristic
120C42C2×D411yescharacteristic
121C42C2×D411yescharacteristic
122C42#4111yescharacteristic
123C23—42no—
124C23—41no—
125C23—41no—
126C23—81no—
127C23—81no—
128C23#12311yescharacteristic
129Q8—41no—
130Q8—41no—
131Q8—81no—
132Q8—81no—
133D4—41no—
134D4—41no—
135D4—41no—
136D4—41no—
137D4—81no—
138D4—81no—
139D4—81no—
140D4—82no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—82no—
149C2×C4#12011yescharacteristic
150C2×C4#12111yescharacteristic
151C2×C4#12111yescharacteristic
152C2×C4#12111yescharacteristic
153C2×C4#12311yescharacteristic
154C2×C4#12411yescharacteristic
155C8—22no—
156C8—41no—
157C8—42no—
158C8—41no—
159C8—41no—
160C8—81no—
161C8—81no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—162no—
172C22#39411yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—81no—
179C4—81no—
180C4#45311yescharacteristic
181C4#45511yescharacteristic
182C2—41no—
183C2—161no—
184C2—161no—
185C2#203711yescharacteristic
186C2#204811yescharacteristic
187C2#205011yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radical#3213C2
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. #1967C2semidirect
3. #1569C2semidirect
4. #1557C2semidirect
5. #1527C2semidirect
6. #1525C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#32132
#19672
#15692
#15572
#15272
#15252
#12912