Finite Group Databasebeta

← #9211

Subgroups — #9211

Order 256.

268 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 67 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#3106C211yesmaximal, characteristic
2C4×#131C211yesmaximal, characteristic, complemented
3#1525C211yesmaximal, characteristic, complemented
4#1492C211yesmaximal, characteristic, complemented
5#1489C211yesmaximal, characteristic, complemented
6#1299C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8C2×#131—21no—
9#532C2211yescharacteristic
10C4×#41C2211yescharacteristic
11#482—21no—
12#477C411yescharacteristic
13#476C411yescharacteristic
14#460C411yescharacteristic
15#459C411yescharacteristic
16#449—21no—
17C4×SD16—21no—
18C4×D8—21no—
19#431C2211yescharacteristic
20#423C2211yescharacteristic
21#405—21no—
22#405—22no—
23C4×M16—21no—
24C4×M16C2211yescharacteristic
25C4×M16C2211yescharacteristic
26C2×C4×C8—21no—
27C2×C4×C8C2211yescharacteristic
28#329—22no—
29Q8⋊C8—21no—
30D4⋊C8—21no—
31D4⋊C8—21no—
32D4⋊C8—21no—
33C2×#41—21no—
34C2×#41D411yescharacteristic
35C2×SD16—21no—
36C2×D8—21no—
37#131—24no—
38#131—24no—
39C2×M16—22no—
40C2×M16—21no—
41C2×M16—21no—
42C2×M16—21no—
43C22×C8—22no—
44C22×C8—21no—
45C4⋊Q8C2×C411yescharacteristic
46#127C2×C411yescharacteristic
47#124—21no—
48#121—21no—
49#121—21no—
50C4×Q8C2×C411yescharacteristic
51C4×Q8D411yescharacteristic
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4C2×C411yescharacteristic
57C4×D4D411yescharacteristic
58#117—21no—
59#117D411yescharacteristic
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62#107—21no—
63#106—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8C2×C411yescharacteristic
67C4⋊C8C2×C411yescharacteristic
68#103—21no—
69#102—21no—
70#102—21no—
71#102—21no—
72C22⋊C8—21no—
73#97—22no—
74#97—21no—
75#97D412yes—
76C4×C8—22no—
77C4×C8—21no—
78C4×C8—21no—
79C4×C8—21no—
80C4×C8—21no—
81C4×C8D411yescharacteristic
82C4×C8D411yescharacteristic
83#41—22no—
84#41—22no—
85#41—81no—
86#41—81no—
87C2×Q8—21no—
88C2×Q8C22⋊C411yescharacteristic
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—41no—
94C2×D4C22⋊C411yescharacteristic
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—41no—
99C22×C4C2×D411yescharacteristic
100C22×C4#4111yescharacteristic
101C22×C4#4111yescharacteristic
102SD16—42no—
103D8—42no—
104M16—28no—
105M16—22no—
106M16—22no—
107M16—22no—
108M16—41no—
109M16—81no—
110C2×C8—28no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—21no—
114C2×C8—22no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—41no—
119C2×C8—41no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4C22⋊C411yescharacteristic
125C4⋊C4C22⋊C411yescharacteristic
126C22⋊C4—21no—
127C22⋊C4—21no—
128C22⋊C4—21no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C42—21no—
132C42—21no—
133C42—21no—
134C42C22×C411yescharacteristic
135C42C2×D411yescharacteristic
136C42C2×D411yescharacteristic
137C42C2×D411yescharacteristic
138C23—21no—
139C23—41no—
140C23#12311yescharacteristic
141Q8—21no—
142Q8—81no—
143Q8#10412yes—
144D4—24no—
145D4—21no—
146D4—41no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—81no—
151D4—82no—
152D4#10412yes—
153C2×C4—24no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4C2×C22⋊C411yescharacteristic
171C2×C4C4×D411yescharacteristic
172C2×C4C4×D411yescharacteristic
173C2×C4#12011yescharacteristic
174C2×C4#12111yescharacteristic
175C2×C4#12111yescharacteristic
176C8—22no—
177C8—24no—
178C8—22no—
179C8—42no—
180C8—41no—
181C8—41no—
182C8—81no—
183C22—21no—
184C22—21no—
185C22—22no—
186C22—21no—
187C22—41no—
188C22—41no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22#38611yescharacteristic
193C4—21no—
194C4—21no—
195C4—22no—
196C4—21no—
197C4—41no—
198C4—41no—
199C4—81no—
200C4C2×#10411yescharacteristic
201C4C2×#10411yescharacteristic
202C4#44311yescharacteristic
203C4#44411yescharacteristic
204C4#47111yescharacteristic
205C4#47211yescharacteristic
206C2—41no—
207C2—42no—
208C2—161no—
209C2#187611yescharacteristic
210C2#189611yescharacteristic
211C2#190711yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#3106C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#131C2semidirect
3. #1525C2semidirect
4. #1492C2semidirect
5. #1489C2semidirect
6. #1299C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#31062
C4×#1312
#15252
#14922
#14892
#12992
#12782