Finite Group Databasebeta

← #8368

Subgroups — #8368

Order 256.

372 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 75 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×#556C211yesmaximal, characteristic
2C4×#101C211yesmaximal, characteristic, complemented
3#1527C212yesmaximal
4#1491C212yesmaximal, complemented
5#1467C211yesmaximal, characteristic, complemented
6#556—24no—
7#556—24no—
8C2×C4⋊Q8C411yescharacteristic
9C2×#126—21no—
10C2×#126—21no—
11C2×#124C411yescharacteristic
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8—21no—
14C2×C4×Q8C411yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C411yescharacteristic
18C2×#117—21no—
19#432C2212yes—
20C2×#101—21no—
21C4×M16C2212yes—
22#343—21no—
23#329—24no—
24Q8⋊C8—24no—
25#324—21no—
26#324—21no—
27C22×Q8C2×C411yescharacteristic
28C22×Q8C2×C411yescharacteristic
29C22×Q8D411yescharacteristic
30C22×D4C2×C411yescharacteristic
31C23×C4—21no—
32C2×M16—22no—
33C4⋊Q8—22no—
34C4⋊Q8—22no—
35#126—28no—
36#126—28no—
37#124—22no—
38#124—22no—
39#124—22no—
40#124D414yes—
41C22⋊Q8—28no—
42C4×Q8—22no—
43C4×Q8—22no—
44C4×Q8—22no—
45C4×Q8—22no—
46C4×Q8—22no—
47C4×Q8D414yes—
48C4×D4—22no—
49C4×D4—22no—
50#117—28no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4C2×C411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C42—21no—
65C2×C42—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—22no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—22no—
80C2×Q8—41no—
81C2×Q8—42no—
82C2×Q8—42no—
83C2×Q8C22⋊C412yes—
84C2×Q8C22⋊C412yes—
85C2×D4—21no—
86C2×D4—42no—
87C2×D4C22⋊C412yes—
88C22×C4—24no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—42no—
93C22×C4—42no—
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—24no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4C22⋊C412yes—
114C22⋊C4—24no—
115C22⋊C4—24no—
116C22⋊C4—24no—
117C22⋊C4—24no—
118C22⋊C4—24no—
119C42—24no—
120C42—21no—
121C42—21no—
122C42—21no—
123C42—21no—
124C42—24no—
125C42C2×D412yes—
126C42C2×D412yes—
127C23—22no—
128C23—42no—
129C23—42no—
130C23—41no—
131C23—41no—
132C23C2×C22⋊C411yescharacteristic
133Q8—24no—
134Q8—44no—
135Q8—42no—
136Q8—42no—
137Q8—42no—
138D4—42no—
139D4—42no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—48no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4#10412yes—
162C2×C4C2×C22⋊C411yescharacteristic
163C2×C4C2×C22⋊C411yescharacteristic
164C2×C4#12012yes—
165C2×C4#12012yes—
166C8—44no—
167C8—82no—
168C22—21no—
169C22—21no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—82no—
174C22—82no—
175C22#37911yescharacteristic
176C22C2×#10411yescharacteristic
177C22#42111yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—22no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—42no—
186C4—41no—
187C4—42no—
188C4#41312yes—
189C2—21no—
190C2—21no—
191C2—82no—
192C2#178511yescharacteristic
193C2#178911yescharacteristic
194C2#179611yescharacteristic
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
4. #1491C2semidirect
5. #1467C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C4×#1012
#15272
#14912
#14672