Finite Group Databasebeta

← #9057

Subgroups — #9057

Order 256.

345 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#486C211yesmaximal, characteristic
2#2366C211yesmaximal, characteristic
3#1988C211yesmaximal, characteristic, complemented
4#1981C211yesmaximal, characteristic, complemented
5C2×#331C211yesmaximal, characteristic
6#1279C211yesmaximal, characteristic, complemented
7#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8C2211yescharacteristic
9C2×#127C2211yescharacteristic
10C2×C4×D4—21no—
11C2×C4×D4—21no—
12#486—22no—
13#486C414yescomplemented
14C2×C4⋊C8C2211yescharacteristic
15C2×C4⋊C8C2211yescharacteristic
16C2×#103—21no—
17C2×#103—21no—
18C2×#102—22no—
19C2×C4×C8C2211yescharacteristic
20#390—21no—
21#390—21no—
22#386—22no—
23C4×C4⋊C4C2211yescharacteristic
24C4×C4⋊C4C2211yescharacteristic
25#340—21no—
26#340—21no—
27#331—22no—
28C22×Q8—21no—
29C22×D4—21no—
30C22×D4—21no—
31C22×D4D412yes—
32C23×C4—22no—
33C22×C8—21no—
34C22×C8—21no—
35C22×C8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8C2×C412yescomplemented
40#127—21no—
41#127—21no—
42#127C2×C412yes—
43#127D412yes—
44C4×D4—42no—
45C4×D4—42no—
46C4×D4—42no—
47C4×D4—42no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C4⋊C4D411yescharacteristic
53C2×C4⋊C4D411yescharacteristic
54C2×C22⋊C4—22no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—22no—
57C2×C42—21no—
58C2×C42—21no—
59C2×C42C2311yescharacteristic
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64#103—42no—
65#103—42no—
66#102—28no—
67C4×C8—21no—
68C4×C8C2×C412yescomplemented
69#95—21no—
70#95—21no—
71C24—22no—
72C2×Q8—22no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—42no—
84C2×D4—42no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4C22⋊C414yes—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—44no—
99C22×C4—44no—
100C22×C4—41no—
101C22×C4—82no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C2×C8—22no—
106C2×C8—41no—
107C2×C8—41no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4D812yes—
125C4⋊C4SD1612yes—
126C22⋊C4—44no—
127C22⋊C4—44no—
128C22⋊C4—82no—
129C42—41no—
130C42—41no—
131C42—41no—
132C42—41no—
133C42C22×C411yescharacteristic
134C42C2×D411yescharacteristic
135C42#4111yescharacteristic
136C42#4111yescharacteristic
137C23—24no—
138C23—42no—
139C23—42no—
140C23—42no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23#12011yescharacteristic
145Q8—42no—
146Q8—42no—
147Q8—81no—
148Q8—81no—
149D4—44no—
150D4—42no—
151D4—42no—
152D4—42no—
153D4—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—82no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—88no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4C2×C22⋊C411yescharacteristic
178C2×C4C4×D411yescharacteristic
179C2×C4C4×D411yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4#12111yescharacteristic
182C2×C4#12311yescharacteristic
183C2×C4C2×D811yescharacteristic
184C2×C4C2×SD1611yescharacteristic
185C2×C4#13511yescharacteristic
186C2×C4#13511yescharacteristic
187C2×C4AGL(1,ℤ/8)11yescharacteristic
188C2×C4AGL(1,ℤ/8)11yescharacteristic
189C2×C4AGL(1,ℤ/8)11yescharacteristic
190C2×C4#13711yescharacteristic
191C8—42no—
192C8—81no—
193C8—81no—
194C22—44no—
195C22—44no—
196C22—44no—
197C22—82no—
198C22—82no—
199C22—82no—
200C22—82no—
201C22#38611yescharacteristic
202C22#44711yescharacteristic
203C22#44911yescharacteristic
204C22#45011yescharacteristic
205C22#45211yescharacteristic
206C22#45511yescharacteristic
207C22#45511yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—82no—
213C4—81no—
214C4—81no—
215C4C4×D811yescharacteristic
216C4C4×SD1611yescharacteristic
217C4#44011yescharacteristic
218C4#44211yescharacteristic
219C4#45911yescharacteristic
220C4#46111yescharacteristic
221C4#46311yescharacteristic
222C4#46311yescharacteristic
223C2—84no—
224C2#163911yescharacteristic
225C2#164011yescharacteristic
226C2#164111yescharacteristic
227C2#164211yescharacteristic
228C2#187711yescharacteristic
229C2#187911yescharacteristic
230C2#188511yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
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
3. #1988C2semidirect
4. #1981C2semidirect
6. #1279C2semidirect
7. #1279C2semidirect
13. #486C4semidirect
39. C4⋊Q8C2×C4semidirect
68. C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4862
#23662
#19882
#19812
C2×#3312
#12792