Finite Group Databasebeta

← #8219

Subgroups — #8219

Order 256.

360 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 101 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
4C4×#95C211yesmaximal, characteristic
5#1294C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C2×C4×Q8C411yescharacteristic, complemented
8C22×C42—21no—
9C22×C42C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11C2×#97—21no—
12#391C411yescharacteristic, complemented
13#389C411yescharacteristic, complemented
14#387—21no—
15#385—21no—
16#380C2211yescharacteristic
17C4×C4⋊C4—21no—
18C4×C4⋊C4C411yescharacteristic, complemented
19C4×C22⋊C4—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—24no—
42C2×C42—21no—
43C2×C42—21no—
44C2×C42—42no—
45C2×C42—42no—
46C2×C42—42no—
47C2×C42C2×C411yescharacteristic
48C2×C42C2×C411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42D411yescharacteristic
51C22⋊C8—22no—
52C22⋊C8C2×C414yes—
53#97—42no—
54#95—24no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—22no—
61#95C2×C414yes—
62C24C2×D411yescharacteristic
63C2×Q8—21no—
64C2×Q8—41no—
65C2×Q8—41no—
66C2×Q8C4212yescomplemented
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—24no—
72C22×C4—24no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—24no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22×C411yescharacteristic
92C22×C4C22×C412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4#4112yes—
95C2×C8—24no—
96C2×C8—42no—
97C2×C8—42no—
98C4⋊C4—21no—
99C4⋊C4—24no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—42no—
103C4⋊C4—42no—
104C4⋊C4C4212yescomplemented
105C22⋊C4—24no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C42—22no—
109C42—22no—
110C42—41no—
111C42—41no—
112C42—48no—
113C42—42no—
114C42—42no—
115C23—21no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23C2×C22⋊C411yescharacteristic
120C23C4×D412yes—
121C23AGL(1,ℤ/8)11yescharacteristic
122C23#13711yescharacteristic
123C23#13712yes—
124Q8—42no—
125Q8—42no—
126C2×C4—24no—
127C2×C4—24no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—28no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—44no—
147C2×C4—44no—
148C2×C4—44no—
149C2×C4—44no—
150C2×C4—44no—
151C2×C4—44no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—44no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4#9912yes—
164C2×C4#10412yes—
165C2×C4#10412yes—
166C2×C4C2×C4211yescharacteristic
167C2×C4#11711yescharacteristic
168C2×C4C4×D412yes—
169C8—44no—
170C22—22no—
171C22—22no—
172C22—21no—
173C22—21no—
174C22—22no—
175C22—22no—
176C22—21no—
177C22—21no—
178C22C4×C22⋊C411yescharacteristic
179C22C2×#9911yescharacteristic
180C22#41011yescharacteristic
181C22#41711yescharacteristic
182C22#41911yescharacteristic
183C22C2×#10411yescharacteristic
184C22C2×#10411yescharacteristic
185C22#44012yes—
186C22#44112yes—
187C4—22no—
188C4—21no—
189C4—21no—
190C4—22no—
191C4—22no—
192C4—22no—
193C4—48no—
194C4—42no—
195C4—42no—
196C4—41no—
197C4—81no—
198C4—81no—
199C2—22no—
200C2—22no—
201C2#150411yescharacteristic
202C2#150611yescharacteristic
203C2#150811yescharacteristic
204C2#150911yescharacteristic
205C2C4×#9911yescharacteristic
206C2C4×#10411yescharacteristic
207C2#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
12. #391C4semidirect
13. #389C4semidirect
18. C4×C4⋊C4C4semidirect
23. #328C4semidirect
66. C2×Q8C42semidirect
104. C4⋊C4C42semidirect

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
C4×#952
#12942