Finite Group Databasebeta

← #9074

Subgroups — #9074

Order 256.

347 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 99 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×#431C211yesmaximal, characteristic, complemented
2C4×#121C211yesmaximal, characteristic
3#1797C211yesmaximal, characteristic, complemented
4C2×D4⋊C8C212yesmaximal, complemented
5#1279C212yesmaximal, complemented
6C2×#121C411yescharacteristic
7C2×C4×D4—21no—
8C2×C4×D4C2211yescharacteristic
9C22×C42C2211yescharacteristic
10#431—24no—
11C2×C4⋊C8—21no—
12C2×C4⋊C8C2212yes—
13C2×C22⋊C8—21no—
14C2×C22⋊C8—21no—
15C2×C4×C8C2212yes—
16#390C411yescharacteristic
17#389C411yescharacteristic
18#388—21no—
19#386—21no—
20#380C411yescharacteristic
21C4×C4⋊C4C2211yescharacteristic
22C4×C22⋊C4—21no—
23#336—21no—
24D4⋊C8—24no—
25C22×D4—21no—
26C22×D4C2×C411yescharacteristic
27C23×C4—21no—
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4D411yescharacteristic
31C22×C8—22no—
32C22×C8—22no—
33#121—22no—
34#121C814yescomplemented
35C4×D4—22no—
36C4×D4—22no—
37C4×D4—44no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C42—22no—
48C2×C42—22no—
49C2×C42—21no—
50C2×C42C2×C411yescharacteristic
51C2×C42D411yescharacteristic
52C2×C42D411yescharacteristic
53C2×C42C2311yescharacteristic
54C4⋊C8—24no—
55C4⋊C8—22no—
56C4⋊C8—22no—
57C22⋊C8—24no—
58C22⋊C8—42no—
59C4×C8—22no—
60C4×C8D414yescomplemented
61#95—21no—
62#95—21no—
63#95—21no—
64C24—21no—
65C24C22⋊C411yescharacteristic
66C2×D4—24no—
67C2×D4—22no—
68C2×D4—21no—
69C2×D4—22no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4C2×C812yes—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4C22⋊C411yescharacteristic
93C22×C4C2×C812yes—
94C22×C4M1612yes—
95C22×C4C22×C411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C2×C8—24no—
99C2×C8—42no—
100C2×C8—42no—
101C2×C8—42no—
102C2×C8—42no—
103C2×C8—42no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—42no—
108C4⋊C4C2×C812yes—
109C22⋊C4—24no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C42—22no—
116C42—21no—
117C42—22no—
118C42—21no—
119C42—42no—
120C42C2×D412yes—
121C42#4112yes—
122C23—21no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—22no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23C22⋊C812yes—
136C23C2×C22⋊C411yescharacteristic
137D4—24no—
138D4—48no—
139D4—42no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4C22⋊C812yes—
168C2×C4C4×D412yes—
169C2×C4#12011yescharacteristic
170C2×C4#12112yes—
171C2×C4#12311yescharacteristic
172C2×C4C22×C811yescharacteristic
173C2×C4C2×M1611yescharacteristic
174C2×C4#13112yes—
175C2×C4AGL(1,ℤ/8)12yes—
176C2×C4AGL(1,ℤ/8)11yescharacteristic
177C2×C4#13711yescharacteristic
178C8—44no—
179C8—82no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22#38611yescharacteristic
195C22C2×C22⋊C811yescharacteristic
196C22#40811yescharacteristic
197C22#41711yescharacteristic
198C22#41811yescharacteristic
199C22#42111yescharacteristic
200C22#42111yescharacteristic
201C4—22no—
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—42no—
207C4—81no—
208C4—81no—
209C4C8×D412yes—
210C4#43512yes—
211C4#46812yes—
212C4#46912yes—
213C2—42no—
214C2—82no—
215C2#147711yescharacteristic
216C2#147811yescharacteristic
217C2#149511yescharacteristic
218C2#149611yescharacteristic
219C2#186911yescharacteristic
220C2#189511yescharacteristic
221C2#189711yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
socleC23C2×C22⋊C4
prime radicalC2×#121C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#431C2semidirect
3. #1797C2semidirect
4. C2×D4⋊C8C2semidirect
5. #1279C2semidirect
34. #121C8semidirect
60. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4312
C4×#1212
#17972
C2×D4⋊C82
#12792