Finite Group Databasebeta

← #10090

Subgroups — #10090

Order 256.

299 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 51 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#3152C211yesmaximal, characteristic, complemented
3#2640C211yesmaximal, characteristic, complemented
4#1558C211yesmaximal, characteristic, complemented
5#1527C211yesmaximal, characteristic, complemented
6#1505C211yesmaximal, characteristic, complemented
7#1300C211yesmaximal, characteristic, complemented
8C2×#137—22no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#124C2211yescharacteristic
12#497—22no—
13#496—21no—
14#494—21no—
15#492—21no—
16#489—21no—
17#463—22no—
18#448—21no—
19#447—21no—
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22C4×M16C2211yescharacteristic
23#397—22no—
24#393—22no—
25#391—21no—
26#390—21no—
27#376C2211yescharacteristic
28#333—22no—
29#329—22no—
30#327—22no—
31C2×#41—21no—
32C22×Q8D411yescharacteristic
33C22×Q8D412yes—
34C22×D4D411yescharacteristic
35#137—48no—
36C2×Q16—24no—
37C2×SD16—24no—
38C2×SD16—22no—
39C2×D8—22no—
40C2×M16—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8—21no—
45C4⋊Q8D411yescharacteristic
46#127D411yescharacteristic
47#124—21no—
48#124—21no—
49#124—21no—
50#124—21no—
51#124—21no—
52#121—21no—
53#121—21no—
54C4×D4—21no—
55C4×D4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—24no—
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63C4⋊C8—21no—
64C4⋊C8D412yes—
65#103—22no—
66#102—22no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69#97—21no—
70#97—21no—
71#97D412yes—
72C4×C8—21no—
73#95—24no—
74#95—21no—
75#95D412yes—
76C24—21no—
77#41—44no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—22no—
81C2×Q8—21no—
82C2×Q8—22no—
83C2×Q8—42no—
84C2×Q8—44no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—42no—
91C2×D4—41no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—41no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99Q16—48no—
100SD16—48no—
101SD16—82no—
102D8—82no—
103M16—24no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—41no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—84no—
120C42—42no—
121C42C2×D411yescharacteristic
122C42C2×D411yescharacteristic
123C42C2×D411yescharacteristic
124C42#4111yescharacteristic
125C23—42no—
126C23—42no—
127C23—41no—
128C23—41no—
129C23—42no—
130C23#12011yescharacteristic
131Q8—44no—
132Q8—42no—
133Q8—42no—
134Q8—82no—
135Q8—81no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—81no—
140D4—81no—
141D4—82no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—82no—
154C2×C4#12011yescharacteristic
155C2×C4#12011yescharacteristic
156C2×C4#12111yescharacteristic
157C2×C4#12111yescharacteristic
158C2×C4#12111yescharacteristic
159C2×C4#12711yescharacteristic
160C8—24no—
161C8—42no—
162C8—81no—
163C22—21no—
164C22—21no—
165C22—84no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22#39211yescharacteristic
173C22#45312yes—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—82no—
180C4—82no—
181C4—81no—
182C4—81no—
183C4#45512yes—
184C2—22no—
185C2—161no—
186C2—161no—
187C2#200211yescharacteristic
188C2#200411yescharacteristic
189C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
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. #3152C2semidirect
3. #2640C2semidirect
4. #1558C2semidirect
5. #1527C2semidirect
6. #1505C2semidirect
7. #1300C2semidirect

Maximal subgroups

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

subgroupindex
#32132
#31522
#26402
#15582
#15272
#15052
#13002