Finite Group Databasebeta

← #8333

Subgroups — #8333

Order 256.

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

normal subgroups only

This group has 438 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
1C2×#533C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic
3C2×#328C214yesmaximal
4#1434C211yesmaximal, characteristic
5#533—28no—
6C2×C4⋊Q8C411yescharacteristic, complemented
7C2×C4⋊Q8C411yescharacteristic, complemented
8C2×#125—21no—
9C2×C22⋊Q8—21no—
10C2×C22⋊Q8C2212yes—
11C2×C4×Q8—21no—
12C2×C4×Q8C412yescomplemented
13C2×#117—21no—
14C22×C42C2211yescharacteristic
15C2×C22⋊C8C2212yes—
16C2×#95—21no—
17C2×#95—21no—
18C2×#95C2212yes—
19#336—21no—
20#336—21no—
21#328—28no—
22C22×Q8—21no—
23C22×Q8C2×C412yes—
24C23×C4—22no—
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C23×C4C2311yescharacteristic
28C22×C8—22no—
29C4⋊Q8—24no—
30C4⋊Q8—24no—
31#125—28no—
32C22⋊Q8—28no—
33C22⋊Q8—24no—
34C22⋊Q8D418yes—
35C4×Q8—28no—
36C4×Q8—24no—
37C4×Q8—24no—
38#117—28no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4C2×C412yes—
45C2×C22⋊C4—22no—
46C2×C42—22no—
47C2×C42—24no—
48C2×C42C2×C411yescharacteristic
49C2×C42C2×C411yescharacteristic
50C2×C42D411yescharacteristic
51C2×C42D411yescharacteristic
52C22⋊C8—24no—
53#95—24no—
54#95—24no—
55#95—42no—
56C24C2×D411yescharacteristic
57C2×Q8—24no—
58C2×Q8—22no—
59C2×Q8—42no—
60C2×Q8—42no—
61C2×Q8—42no—
62C2×Q8—42no—
63C2×Q8C22⋊C414yes—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—24no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C411yescharacteristic
84C22×C4SD1612yes—
85C22×C4Q1612yes—
86C22×C4C22×C411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D414yes—
89C2×C8—42no—
90C2×C8—42no—
91C2×C8—42no—
92C4⋊C4—24no—
93C4⋊C4—24no—
94C4⋊C4—28no—
95C4⋊C4—24no—
96C4⋊C4—22no—
97C4⋊C4—28no—
98C4⋊C4C22⋊C414yes—
99C22⋊C4—28no—
100C42—28no—
101C42—22no—
102C42—22no—
103C42—22no—
104C42—22no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23C2×C22⋊C411yescharacteristic
110C23#12012yes—
111C23C2×SD1611yescharacteristic
112C23C2×Q1611yescharacteristic
113C23#13712yes—
114Q8—44no—
115Q8—44no—
116Q8—48no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—24no—
128C2×C4—24no—
129C2×C4—21no—
130C2×C4—24no—
131C2×C4—24no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—44no—
135C2×C4—44no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—44no—
141C2×C4—44no—
142C2×C4—44no—
143C2×C4—82no—
144C2×C4—82no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4#10312yes—
148C2×C4#10312yes—
149C2×C4#10412yes—
150C2×C4C2×C22⋊C412yes—
151C2×C4#12012yes—
152C8—82no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—22no—
157C22—21no—
158C22—21no—
159C22—22no—
160C22—22no—
161C22#37911yescharacteristic
162C22#41011yescharacteristic
163C22#41011yescharacteristic
164C22C2×#10311yescharacteristic
165C22#41911yescharacteristic
166C22C2×#10411yescharacteristic
167C22#42111yescharacteristic
168C22#44812yes—
169C22#45112yes—
170C4—22no—
171C4—22no—
172C4—22no—
173C4—22no—
174C4—22no—
175C4—44no—
176C4—41no—
177C4—44no—
178C4—84no—
179C2—22no—
180C2—22no—
181C2#150311yescharacteristic
182C2#150611yescharacteristic
183C2#151511yescharacteristic
184C2#151811yescharacteristic
185C2#178211yescharacteristic
186C2#178711yescharacteristic
187C2#178911yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×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
6. C2×C4⋊Q8C4semidirect
7. C2×C4⋊Q8C4semidirect
12. C2×C4×Q8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
#18172
C2×#3282
#14342