Finite Group Databasebeta

← #8372

Subgroups — #8372

Order 256.

394 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 79 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×#554C211yesmaximal, characteristic
2C4×#101C211yesmaximal, characteristic, complemented
3#1528C212yesmaximal, complemented
4#1490C212yesmaximal
5#1467C211yesmaximal, characteristic, complemented
6#554—24no—
7#554—24no—
8C2×C4⋊Q8C411yescharacteristic
9C2×#125C411yescharacteristic
10C2×#125C2211yescharacteristic
11C2×C22⋊Q8—21no—
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8—21no—
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C411yescharacteristic
16C2×C4×D4C411yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C22×C4⋊C4—21no—
19#432C2212yes—
20C2×#101—21no—
21C4×M16C2212yes—
22#343—21no—
23#330—24no—
24D4⋊C8—24no—
25#324—21no—
26#324—21no—
27C22×Q8—21no—
28C22×Q8D411yescharacteristic
29C22×D4C2×C411yescharacteristic
30C23×C4—21no—
31C23×C4—21no—
32C23×C4—21no—
33C2×M16—22no—
34C4⋊Q8—22no—
35C4⋊Q8—22no—
36#125—22no—
37#125—22no—
38#125—22no—
39#125D414yes—
40C22⋊Q8—28no—
41C22⋊Q8—28no—
42C22⋊Q8—28no—
43C4×Q8—22no—
44C4×Q8—22no—
45C4×D4—22no—
46C4×D4—22no—
47C4×D4—22no—
48C4×D4—22no—
49C4×D4—22no—
50C4×D4D414yes—
51C2×C4⋊C4—28no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C4⋊C4C2×C411yescharacteristic
59C2×C4⋊C4C2×C411yescharacteristic
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C42C2×C411yescharacteristic
67C2×C42C2×C411yescharacteristic
68C2×C42C2311yescharacteristic
69C4⋊C8—22no—
70#101—24no—
71C22⋊C8—22no—
72#97—22no—
73#97—22no—
74C4×C8—22no—
75C24—21no—
76C2×Q8—24no—
77C2×Q8—22no—
78C2×Q8—22no—
79C2×Q8—41no—
80C2×Q8—82no—
81C2×D4—24no—
82C2×D4—21no—
83C2×D4C22⋊C412yes—
84C22×C4—24no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—24no—
89C22×C4—24no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—82no—
93C22×C4—82no—
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22⋊C411yescharacteristic
98C22×C4C22×C411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101M16—24no—
102M16—42no—
103C2×C8—42no—
104C2×C8—42no—
105C4⋊C4—24no—
106C4⋊C4—24no—
107C4⋊C4—24no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4C22⋊C412yes—
117C4⋊C4C22⋊C412yes—
118C4⋊C4C22⋊C412yes—
119C22⋊C4—24no—
120C22⋊C4—24no—
121C22⋊C4—24no—
122C42—21no—
123C42—21no—
124C42—21no—
125C42—21no—
126C42C2×D412yes—
127C42C2×D412yes—
128C23—24no—
129C23—22no—
130C23—42no—
131C23—41no—
132C23—41no—
133C23C2×C22⋊C411yescharacteristic
134Q8—44no—
135Q8—84no—
136D4—24no—
137D4—22no—
138D4#10114yescomplemented
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—48no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—84no—
161C2×C4—84no—
162C2×C4#10412yes—
163C2×C4C2×C22⋊C411yescharacteristic
164C2×C4C2×C22⋊C411yescharacteristic
165C2×C4#12012yes—
166C2×C4#12012yes—
167C8—44no—
168C8—82no—
169C22—21no—
170C22—21no—
171C22—24no—
172C22—44no—
173C22—44no—
174C22—42no—
175C22—42no—
176C22#37911yescharacteristic
177C22C2×#10411yescharacteristic
178C22#42111yescharacteristic
179C4—22no—
180C4—21no—
181C4—22no—
182C4—22no—
183C4—42no—
184C4—42no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—42no—
189C4C2×#10112yes—
190C2—21no—
191C2—21no—
192C2—44no—
193C2#178411yescharacteristic
194C2#178911yescharacteristic
195C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#101C2semidirect
3. #1528C2semidirect
5. #1467C2semidirect
138. D4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C4×#1012
#15282
#14902
#14672