Finite Group Databasebeta

← #8217

Subgroups — #8217

Order 256.

314 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C4×C22⋊Q8C211yesmaximal, characteristic
2#1749C211yesmaximal, characteristic
3C2×#328C212yesmaximal
4#1434C211yesmaximal, characteristic
5#1294C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C2×C4×Q8C411yescharacteristic, complemented
8C22×C42C2211yescharacteristic
9C2×C22⋊C8C2212yes—
10C2×#97—21no—
11#391C411yescharacteristic, complemented
12#389C411yescharacteristic, complemented
13#387—21no—
14#385—21no—
15#380C2211yescharacteristic
16C4×C4⋊C4—21no—
17C4×C4⋊C4C411yescharacteristic, complemented
18C4×C22⋊C4—21no—
19C2×#95—21no—
20C2×#95—21no—
21C2×#95C2212yes—
22#336—21no—
23#328C418yescomplemented
24C22×Q8C2×C411yescharacteristic
25C23×C4—22no—
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C23×C4C2311yescharacteristic
29C22×C8—22no—
30C22⋊Q8—22no—
31C22⋊Q8C2×C414yes—
32C4×Q8—42no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C42—21no—
42C2×C42—21no—
43C2×C42—42no—
44C2×C42C2×C411yescharacteristic
45C2×C42C2×C411yescharacteristic
46C2×C42D411yescharacteristic
47C2×C42D411yescharacteristic
48C22⋊C8—22no—
49C22⋊C8C2×C414yes—
50#97—42no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—22no—
57#95—42no—
58#95—42no—
59#95C2×C414yes—
60C24C2×D411yescharacteristic
61C2×Q8—21no—
62C2×Q8—41no—
63C2×Q8—41no—
64C2×Q8C4212yes—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—21no—
70C22×C4—22no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—24no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22×C411yescharacteristic
90C22×C4C22×C412yes—
91C22×C4C2×D411yescharacteristic
92C22×C4#4112yes—
93C2×C8—24no—
94C2×C8—42no—
95C2×C8—42no—
96C4⋊C4—21no—
97C4⋊C4—24no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C4⋊C4C4212yes—
103C22⋊C4—24no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C42—22no—
107C42—22no—
108C42—41no—
109C42—41no—
110C42—42no—
111C42—42no—
112C23—21no—
113C23—21no—
114C23—21no—
115C23—21no—
116C23C2×C22⋊C411yescharacteristic
117C23C4×D412yes—
118C23AGL(1,ℤ/8)11yescharacteristic
119C23#13711yescharacteristic
120C23#13712yes—
121Q8—42no—
122Q8—42no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—44no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—44no—
150C2×C4—44no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—44no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4#9912yes—
163C2×C4C2×C4211yescharacteristic
164C2×C4#11711yescharacteristic
165C2×C4C4×D412yes—
166C8—44no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—22no—
175C22C4×C22⋊C411yescharacteristic
176C22C2×#9911yescharacteristic
177C22#41011yescharacteristic
178C22#41711yescharacteristic
179C22#41911yescharacteristic
180C22#42111yescharacteristic
181C22#42111yescharacteristic
182C22#44012yes—
183C22#44112yes—
184C4—22no—
185C4—21no—
186C4—21no—
187C4—22no—
188C4—42no—
189C4—42no—
190C4—41no—
191C4—41no—
192C4—41no—
193C4—84no—
194C4—81no—
195C4—81no—
196C2—22no—
197C2—22no—
198C2#151111yescharacteristic
199C2#151211yescharacteristic
200C2#151311yescharacteristic
201C2#151511yescharacteristic
202C2C4×#9911yescharacteristic
203C2#175911yescharacteristic
204C2#176311yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C2×C42
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q8C4semidirect
11. #391C4semidirect
12. #389C4semidirect
17. C4×C4⋊C4C4semidirect
23. #328C4semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
#17492
C2×#3282
#14342
#12942