Finite Group Databasebeta

← #9073

Subgroups — #9073

Order 256.

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

normal subgroups only

This group has 305 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
2C4×C22⋊Q8C211yesmaximal, characteristic
3#1797C211yesmaximal, characteristic
4C2×Q8⋊C8C212yesmaximal, complemented
5#1279C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8C22×C42C2211yescharacteristic
9#431—24no—
10C2×C4⋊C8—21no—
11C2×C4⋊C8C2212yes—
12C2×C22⋊C8—21no—
13C2×C22⋊C8—21no—
14C2×C4×C8C2212yes—
15#391C411yescharacteristic
16#389C411yescharacteristic
17#387—21no—
18#385—21no—
19#380C411yescharacteristic
20C4×C4⋊C4—21no—
21C4×C4⋊C4C2211yescharacteristic
22C4×C22⋊C4—21no—
23#336—21no—
24Q8⋊C8—24no—
25C22×Q8C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4D411yescharacteristic
28C23×C4D411yescharacteristic
29C22×C8—22no—
30C22×C8—22no—
31C22⋊Q8—22no—
32C22⋊Q8C814yescomplemented
33C4×Q8—22no—
34C4×Q8—22no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C42—22no—
44C2×C42—22no—
45C2×C42—21no—
46C2×C42—21no—
47C2×C42C2×C411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42C2311yescharacteristic
51C4⋊C8—24no—
52C4⋊C8—22no—
53C4⋊C8—22no—
54C22⋊C8—24no—
55C22⋊C8—42no—
56C4×C8—22no—
57C4×C8D414yes—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63C24C22⋊C411yescharacteristic
64C2×Q8—21no—
65C2×Q8—22no—
66C2×Q8—22no—
67C2×Q8C2×C812yes—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C2×C812yes—
86C22×C4M1612yes—
87C22×C4C22×C411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C2×C8—24no—
91C2×C8—42no—
92C2×C8—42no—
93C2×C8—42no—
94C2×C8—42no—
95C2×C8—42no—
96C4⋊C4—24no—
97C4⋊C4—21no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—42no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4C2×C812yes—
104C22⋊C4—24no—
105C22⋊C4—42no—
106C42—22no—
107C42—21no—
108C42—22no—
109C42—21no—
110C42—42no—
111C42—41no—
112C42—41no—
113C42C2×D412yes—
114C42#4112yes—
115C23—21no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23C22⋊C812yes—
122C23C2×C22⋊C411yescharacteristic
123Q8—24no—
124Q8—42no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—24no—
130C2×C4—24no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4C22⋊C812yes—
154C2×C4C4×D412yes—
155C2×C4#12011yescharacteristic
156C2×C4#12112yes—
157C2×C4#12311yescharacteristic
158C2×C4C22×C811yescharacteristic
159C2×C4C2×M1611yescharacteristic
160C2×C4#13112yes—
161C2×C4AGL(1,ℤ/8)11yescharacteristic
162C2×C4#13712yes—
163C2×C4#13711yescharacteristic
164C8—44no—
165C8—82no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22#38611yescharacteristic
174C22C2×C22⋊C811yescharacteristic
175C22#40811yescharacteristic
176C22#41711yescharacteristic
177C22#41911yescharacteristic
178C22#42111yescharacteristic
179C22#42111yescharacteristic
180C4—22no—
181C4—21no—
182C4—21no—
183C4—42no—
184C4—41no—
185C4—41no—
186C4—42no—
187C4—81no—
188C4—81no—
189C4C8×D412yes—
190C4#43512yes—
191C4#46812yes—
192C4#47012yes—
193C2—42no—
194C2#147711yescharacteristic
195C2#147911yescharacteristic
196C2#149511yescharacteristic
197C2#149711yescharacteristic
198C2#186911yescharacteristic
199C2#189411yescharacteristic
200C2#189711yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
socleC23C2×C22⋊C4
prime radicalC24C22⋊C4
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
4. C2×Q8⋊C8C2semidirect
5. #1279C2semidirect
32. C22⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#4312
C4×C22⋊Q82
#17972
C2×Q8⋊C82
#12792