Finite Group Databasebeta

← #8523

Subgroups — #8523

Order 256.

360 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 71 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×#542C211yesmaximal, characteristic
2#1556C212yesmaximal, complemented
3#1533C212yesmaximal, complemented
4#1464C211yesmaximal, characteristic, complemented
5#1464C211yesmaximal, characteristic, complemented
6#542—24no—
7#542—24no—
8C2×C4⋊Q8C2211yescharacteristic
9C2×#126—21no—
10C2×#124—21no—
11C2×#124C2211yescharacteristic
12C2×#123—21no—
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C411yescharacteristic
15C2×#121C411yescharacteristic
16C2×C4×Q8C411yescharacteristic
17C2×C4×D4C411yescharacteristic
18C2×#117C2211yescharacteristic
19#432C2212yes—
20#432C2212yes—
21#332—24no—
22#329—24no—
23#324—21no—
24#324—21no—
25#323—21no—
26#323—21no—
27C22×Q8—21no—
28C22×Q8C2×C411yescharacteristic
29C22×D4—21no—
30C22×D4C2×C411yescharacteristic
31C23×C4—21no—
32C23×C4C2×C411yescharacteristic
33C4⋊Q8—22no—
34C4⋊Q8D414yes—
35#126—28no—
36#124—28no—
37#124—22no—
38#124D414yes—
39#123—28no—
40C22⋊Q8—28no—
41C22⋊Q8—22no—
42C22⋊Q8—22no—
43#121—22no—
44#121—22no—
45C4×Q8—22no—
46C4×Q8—22no—
47C4×D4—22no—
48C4×D4—22no—
49#117—22no—
50#117—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4C2×C411yescharacteristic
55C2×C4⋊C4C2×C411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4D411yescharacteristic
64C2×C22⋊C4D411yescharacteristic
65C2×C42—21no—
66C2×C42C2×C411yescharacteristic
67C2×C42C2311yescharacteristic
68C4⋊C8—22no—
69C4⋊C8—22no—
70C22⋊C8—22no—
71C22⋊C8—22no—
72#97—22no—
73#97—22no—
74C24—21no—
75C24C22⋊C411yescharacteristic
76C2×Q8—21no—
77C2×Q8—24no—
78C2×Q8—42no—
79C2×Q8—82no—
80C2×Q8C22⋊C412yes—
81C2×D4—24no—
82C2×D4—21no—
83C2×D4—42no—
84C2×D4—82no—
85C2×D4C22⋊C412yes—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—24no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—42no—
93C22×C4—82no—
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22×C411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C2×C8—42no—
101C2×C8—42no—
102C4⋊C4—24no—
103C4⋊C4—24no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—24no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4C22⋊C412yes—
110C4⋊C4C22⋊C412yes—
111C22⋊C4—24no—
112C22⋊C4—24no—
113C22⋊C4—24no—
114C22⋊C4—24no—
115C22⋊C4—24no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C42—24no—
119C42—21no—
120C42—21no—
121C42C2×D412yes—
122C42C2×D412yes—
123C23—21no—
124C23—22no—
125C23—22no—
126C23—42no—
127C23—42no—
128C23—41no—
129C23—41no—
130C23—42no—
131C23C2×C22⋊C411yescharacteristic
132Q8—42no—
133Q8—42no—
134Q8—84no—
135D4—42no—
136D4—42no—
137D4—84no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—84no—
156C2×C4C2×C22⋊C411yescharacteristic
157C2×C4C2×C22⋊C411yescharacteristic
158C2×C4#12012yes—
159C2×C4#12012yes—
160C8—82no—
161C8—82no—
162C22—21no—
163C22—21no—
164C22—42no—
165C22—41no—
166C22—41no—
167C22—42no—
168C22—41no—
169C22—42no—
170C22—82no—
171C22—84no—
172C22#37911yescharacteristic
173C22#41312yes—
174C4—22no—
175C4—22no—
176C4—21no—
177C4—21no—
178C4—42no—
179C4—42no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—42no—
184C2—22no—
185C2—81no—
186C2—82no—
187C2#178511yescharacteristic
188C2#179511yescharacteristic
189C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#121C4
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. #1556C2semidirect
3. #1533C2semidirect
4. #1464C2semidirect
5. #1464C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
#15562
#15332
#14642