Finite Group Databasebeta

← #8890

Subgroups — #8890

Order 256.

301 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 93 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
1D4×C4⋊C4C211yesmaximal, characteristic
2#1947C211yesmaximal, characteristic, complemented
3#1834C211yesmaximal, characteristic, complemented
4#1802C211yesmaximal, characteristic
5C2×D4⋊C8C211yesmaximal, characteristic
6#1296C211yesmaximal, characteristic, complemented
7#1279C211yesmaximal, characteristic, complemented
8C2×C4×D4—21no—
9C2×C4×D4C2211yescharacteristic
10C22×C4⋊C4—21no—
11C2×#107—21no—
12C2×C4⋊C8C2211yescharacteristic
13C2×C4⋊C8C2211yescharacteristic
14C2×#102—21no—
15C2×C4×C8C2211yescharacteristic
16#389—21no—
17#385—22no—
18#384C2211yescharacteristic
19#384C2211yescharacteristic
20#380—21no—
21C4×C4⋊C4C2211yescharacteristic
22#340—21no—
23#340—22no—
24#340—21no—
25D4⋊C8C414yescomplemented
26C22×D4D411yescharacteristic
27C23×C4—21no—
28C23×C4—22no—
29C22×C8—21no—
30C22×C8—21no—
31C22×C8—21no—
32C4×D4—21no—
33C4×D4—21no—
34C4×D4—42no—
35C4×D4—42no—
36C4×D4C2×C412yes—
37C4×D4Q812yes—
38C2×C4⋊C4—22no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—42no—
45C2×C4⋊C4—42no—
46C2×C4⋊C4D411yescharacteristic
47C2×C4⋊C4D411yescharacteristic
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D412yes—
50C2×C22⋊C4—21no—
51C2×C22⋊C4—22no—
52C2×C42—21no—
53C2×C42C2311yescharacteristic
54#107—42no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8C2×C412yescomplemented
59#102—42no—
60C4×C8—21no—
61C4×C8C2×C412yescomplemented
62#95—22no—
63#95—21no—
64C24—21no—
65C2×D4—21no—
66C2×D4—21no—
67C2×D4—21no—
68C2×D4C4⋊C412yes—
69C2×D4D812yes—
70C2×D4Q1612yes—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—44no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—44no—
83C22×C4—41no—
84C22×C4—82no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—41no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—41no—
95C2×C8—41no—
96C2×C8—41no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—42no—
102C4⋊C4—44no—
103C4⋊C4—42no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4C4⋊C412yes—
113C4⋊C4SD1612yes—
114C22⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—44no—
117C42—41no—
118C42—41no—
119C42C22×C411yescharacteristic
120C42C2×Q811yescharacteristic
121C42#4111yescharacteristic
122C42#4111yescharacteristic
123C23—22no—
124C23—22no—
125C23—22no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23#12011yescharacteristic
131D4—22no—
132D4—22no—
133D4—22no—
134D4#10714yescomplemented
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—88no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4C2×C4⋊C411yescharacteristic
158C2×C4C4×D411yescharacteristic
159C2×C4C4×D411yescharacteristic
160C2×C4C22⋊Q811yescharacteristic
161C2×C4C22⋊Q811yescharacteristic
162C2×C4#12311yescharacteristic
163C2×C4C2×D811yescharacteristic
164C2×C4C2×SD1611yescharacteristic
165C2×C4C2×Q1611yescharacteristic
166C2×C4#13511yescharacteristic
167C2×C4AGL(1,ℤ/8)11yescharacteristic
168C2×C4AGL(1,ℤ/8)11yescharacteristic
169C2×C4AGL(1,ℤ/8)11yescharacteristic
170C2×C4#13711yescharacteristic
171C8—42no—
172C8—42no—
173C8—81no—
174C22—24no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22#38511yescharacteristic
182C22#44711yescharacteristic
183C22#44911yescharacteristic
184C22#45011yescharacteristic
185C22#45111yescharacteristic
186C22#45411yescharacteristic
187C22#45511yescharacteristic
188C4—21no—
189C4—21no—
190C4—42no—
191C4—42no—
192C4—81no—
193C4—82no—
194C4—161no—
195C4C2×#10711yescharacteristic
196C4#42811yescharacteristic
197C4C4×SD1611yescharacteristic
198C4#44211yescharacteristic
199C4#47611yescharacteristic
200C4#47811yescharacteristic
201C4#48011yescharacteristic
202C4#48411yescharacteristic
203C2—44no—
204C2#162511yescharacteristic
205C2#163711yescharacteristic
206C2#163911yescharacteristic
207C2#164411yescharacteristic
208C2#185211yescharacteristic
209C2#186011yescharacteristic
210C2#186411yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1947C2semidirect
3. #1834C2semidirect
6. #1296C2semidirect
7. #1279C2semidirect
25. D4⋊C8C4semidirect
58. C4⋊C8C2×C4semidirect
61. C4×C8C2×C4semidirect
134. D4#107semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊C42
#19472
#18342
#18022
C2×D4⋊C82
#12962
#12792