Finite Group Databasebeta

← #9299

Subgroups — #9299

Order 256.

316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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#3205C211yesmaximal, characteristic
2#2328C211yesmaximal, characteristic, complemented
3#1963C211yesmaximal, characteristic, complemented
4#1545C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic, complemented
6#1286C211yesmaximal, characteristic, complemented
7#1274C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#127C2211yescharacteristic
10C2×C4×D4—21no—
11#493C412yescomplemented
12#486C412yescomplemented
13#459—22no—
14#447—22no—
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#423C2211yescharacteristic
18C2×#104—21no—
19C2×#104—21no—
20#390—21no—
21#386—22no—
22#386—22no—
23#382C2211yescharacteristic
24#376C2211yescharacteristic
25#331—22no—
26#327—22no—
27C2×#41—21no—
28C22×D4—21no—
29C22×D4—21no—
30C22×D4D412yes—
31C23×C4—22no—
32C2×D8—24no—
33C2×M16—21no—
34C4⋊Q8C2×C411yescharacteristic, complemented
35#127—21no—
36#127—21no—
37#127C2×C412yes—
38#127C2×C411yescharacteristic, complemented
39#127D412yes—
40#124—21no—
41#121—21no—
42#121—21no—
43C4×D4—21no—
44C4×D4—21no—
45C4×D4—42no—
46C4×D4—42no—
47C2×C4⋊C4—21no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—22no—
50C2×C22⋊C4—22no—
51C2×C42D411yescharacteristic
52C2×C42D411yescharacteristic
53C2×C42C2311yescharacteristic
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58#104—42no—
59#104—42no—
60#102—24no—
61C22⋊C8—21no—
62C22⋊C8—21no—
63C4×C8—21no—
64C4×C8C2×C412yescomplemented
65#95—21no—
66#95—21no—
67#95—21no—
68#95D412yes—
69C24—22no—
70#41—41no—
71#41—41no—
72#41—41no—
73#41—41no—
74C2×Q8—21no—
75C2×D4—21no—
76C2×D4—22no—
77C2×D4—21no—
78C2×D4—22no—
79C2×D4—21no—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—82no—
89C2×D4—82no—
90C2×D4C22⋊C414yes—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—44no—
95C22×C4—41no—
96C22×C4—82no—
97C22×C4—82no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101D8—84no—
102M16—41no—
103M16—41no—
104C2×C8—22no—
105C2×C8—41no—
106C2×C8—41no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C22⋊C4—44no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—82no—
115C22⋊C4—82no—
116C42—21no—
117C42—21no—
118C42—21no—
119C42D812yescomplemented
120C42C22×C411yescharacteristic
121C42C2×D411yescharacteristic
122C42#4111yescharacteristic
123C42#4111yescharacteristic
124C23—24no—
125C23—42no—
126C23—42no—
127C23—42no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23—41no—
132C23#12011yescharacteristic
133Q8—41no—
134Q8—41no—
135D4—44no—
136D4—42no—
137D4—42no—
138D4—41no—
139D4—42no—
140D4—41no—
141D4—42no—
142D4—42no—
143D4—82no—
144D4—82no—
145D4—82no—
146D4—81no—
147D4—81no—
148D4—81no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—88no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4C2×C22⋊C411yescharacteristic
165C2×C4C4×D411yescharacteristic
166C2×C4C4×D411yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12311yescharacteristic
170C2×C4C2×D811yescharacteristic
171C2×C4#13511yescharacteristic
172C2×C4AGL(1,ℤ/8)11yescharacteristic
173C2×C4AGL(1,ℤ/8)11yescharacteristic
174C8—42no—
175C8—81no—
176C8—81no—
177C22—21no—
178C22—21no—
179C22—44no—
180C22—44no—
181C22—44no—
182C22—82no—
183C22—82no—
184C22—82no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22#38611yescharacteristic
190C22#44711yescharacteristic
191C22#44911yescharacteristic
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—42no—
197C4—81no—
198C4—82no—
199C4—81no—
200C4C4×D811yescharacteristic
201C4#44211yescharacteristic
202C4#45911yescharacteristic
203C4#46311yescharacteristic
204C2—21no—
205C2—21no—
206C2—84no—
207C2—161no—
208C2#187911yescharacteristic
209C2#190411yescharacteristic
210C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3205C2
LCSG ▷ 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. #2328C2semidirect
3. #1963C2semidirect
4. #1545C2semidirect
5. #1517C2semidirect
6. #1286C2semidirect
7. #1274C2semidirect
11. #493C4semidirect
12. #486C4semidirect
34. C4⋊Q8C2×C4semidirect
38. #127C2×C4semidirect
64. C4×C8C2×C4semidirect
119. C42D8semidirect

Maximal subgroups

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

subgroupindex
#32052
#23282
#19632
#15452
#15172
#12862
#12742