Finite Group Databasebeta

← #9053

Subgroups — #9053

Order 256.

269 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 71 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#3169C211yesmaximal, characteristic
2#2935C211yesmaximal, characteristic, complemented
3#1980C211yesmaximal, characteristic, complemented
4#1502C211yesmaximal, characteristic, complemented
5#1480C211yesmaximal, characteristic, complemented
6#1279C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8C2×#131—21no—
9#532C2211yescharacteristic
10#532C2211yescharacteristic
11#500C411yescharacteristic, complemented
12#493C411yescharacteristic, complemented
13#486C411yescharacteristic, complemented
14#486C411yescharacteristic, complemented
15#466—22no—
16C4×D8—22no—
17#436—21no—
18#435—22no—
19C8×D4—21no—
20#431C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#104—21no—
23C2×#104—21no—
24#408—21no—
25C4×M16C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27C4×C4⋊C4C2211yescharacteristic
28#327—22no—
29D4⋊C8—21no—
30D4⋊C8—21no—
31C2×#41—21no—
32C2×#41—21no—
33C2×D8—22no—
34#131—42no—
35#131—42no—
36C2×M16—22no—
37C2×M16D411yescharacteristic
38C2×M16D411yescharacteristic
39C22×C8—22no—
40C22×C8—21no—
41C22×C8—21no—
42C4⋊Q8C2×C411yescharacteristic
43C4⋊Q8C2×C411yescharacteristic
44#127C2×C411yescharacteristic
45#127C2×C411yescharacteristic
46#124—21no—
47#124—21no—
48#121—21no—
49#121—21no—
50#121—21no—
51#121D412yes—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4D412yes—
56C2×C4⋊C4—21no—
57C2×C42—21no—
58C2×C42C2311yescharacteristic
59#107—22no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8D411yescharacteristic
63C4⋊C8D411yescharacteristic
64#104—42no—
65#104—42no—
66#102—22no—
67#102—22no—
68C22⋊C8—22no—
69C22⋊C8—22no—
70C22⋊C8—21no—
71#97—21no—
72C4×C8—21no—
73C4×C8—21no—
74C4×C8—21no—
75C4×C8C2×C411yescharacteristic
76C4×C8C2×C411yescharacteristic
77#95—21no—
78#41—41no—
79#41—41no—
80#41—41no—
81#41—41no—
82#41—41no—
83#41—41no—
84#41—41no—
85#41—41no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×D4—22no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—41no—
95C2×D4C22⋊C412yes—
96C22×C4—22no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4#4111yescharacteristic
102D8—44no—
103M16—21no—
104M16—21no—
105M16—21no—
106M16—44no—
107M16D812yescomplemented
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—42no—
115C2×C8—44no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—41no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—41no—
124C4⋊C4C22⋊C412yes—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C42—41no—
130C42—41no—
131C42C22×C411yescharacteristic
132C42C2×D411yescharacteristic
133C42C2×D411yescharacteristic
134C42#4111yescharacteristic
135C23—22no—
136C23—41no—
137C23#12111yescharacteristic
138Q8—41no—
139Q8—41no—
140Q8—41no—
141Q8—41no—
142D4—42no—
143D4—42no—
144D4—42no—
145D4—42no—
146D4—41no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—81no—
155D4—81no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4C2×C22⋊C411yescharacteristic
169C2×C4C4×D411yescharacteristic
170C2×C4C4×D411yescharacteristic
171C2×C4#12011yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12311yescharacteristic
174C2×C4C2×D811yescharacteristic
175C2×C4#13511yescharacteristic
176C2×C4AGL(1,ℤ/8)11yescharacteristic
177C2×C4AGL(1,ℤ/8)11yescharacteristic
178C8—22no—
179C8—22no—
180C8—41no—
181C8—41no—
182C8—41no—
183C8—41no—
184C8—81no—
185C22—21no—
186C22—21no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22#38611yescharacteristic
194C22#45911yescharacteristic
195C22#46311yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4—42no—
200C4—82no—
201C4—81no—
202C4C4×D811yescharacteristic
203C4#44211yescharacteristic
204C4#44311yescharacteristic
205C4#44411yescharacteristic
206C4#44711yescharacteristic
207C4#44911yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2—161no—
212C2#187611yescharacteristic
213C2#187911yescharacteristic
214C2#188411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#3169C2
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. #2935C2semidirect
3. #1980C2semidirect
4. #1502C2semidirect
5. #1480C2semidirect
6. #1279C2semidirect
7. #1278C2semidirect
11. #500C4semidirect
12. #493C4semidirect
13. #486C4semidirect
14. #486C4semidirect
107. M16D8semidirect

Maximal subgroups

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

subgroupindex
#31692
#29352
#19802
#15022
#14802
#12792
#12782